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+{"text":"\\section{Introduction}\nIdentifying line outages is crucial to understanding current system conditions and preventing cascading outages.\nHistorically, line outages have been identified based on breaker status indications; however, sometimes the reported line statuses are incorrect. \nFailure to update the system models can cause inaccurate state estimation and threaten system reliability~\\cite{NERC_EMSoutage_2017}, especially for systems that are under stress. \nFor example, in the 2011 San Diego blackout, operators could not detect overloaded lines because of an incorrect network model~\\cite{OutageReport2011,Chen2015Quickest}.\n\nPhasor measurement units (PMUs), because of their unique characteristics (namely, global synchronization and high reporting rate), have been installed on many modern power grids to provide better visibility of grid behavior.\nIn particular, there have been many reported efforts to use PMUs to improve topology models (in particular, to detect one or more line outages) using various approaches (\\cite{Tate_line_2008,Chen2015Quickest,Tate_double_2009,Zhu_sparse_2012,Emami_external_2013}).\nMethods in~\\cite{Tate_line_2008} and~\\cite{Tate_double_2009} are based on hypothesis testing for single-line and double-line outage cases, while reference~\\cite{Zhu_sparse_2012} proposes an overcomplete representation and formulates the problem in terms of sparse vector estimation.\nAlternative approaches include integer programming (\\cite{Emami_external_2013}) and identification based on the small-signal linearized power grid model (\\cite{Chen2015Quickest}).\nMost of the existing works use PMU voltage angle measurements rather than both magnitude and angle data since they rely on dc power flow models, which only considers angles.\nPMU-based disturbance detection methods have been developed and used by utilities and operators.\nLine switching events are identified as one of the applications ~\\cite{NASPI_using}.\nIn fact, several applications are being developed and tested (e.g., by ISO New England for external system transmission element tripping and PJM for detection and triangulation of large disturbances).\nMeanwhile, Operador Nacional do Sistema El\u00e9trico (ONS) in Brazil deployed a WAMS system which identifies transmission line tripping based on angle disturbance of nearby PMUs.\nTo the best of our knowledge, details of these applications are not publicly available for comparison.\nIn this research, we extended the single line outage identification algorithm in~\\cite{Tate_line_2008} by utilizing voltage phasor measurements from PMUs.\nFirst, the algorithm computes expected voltage phasors for all possible outage scenarios by solving ac power flows.\nNext the pre- and post-outage voltage phasor difference can be calculated (hereinafter called expected values).\nBy comparing the expected and the observed values, the hypothetical outage scenario that is closest to the observation will be identified. \nUnlike prior approaches, which have relied on the relatively inaccurate dc power flow model, the proposed method uses the full ac power flow model to identify outages.\nIdeally, system responses after different outages are distinct enough to be correctly identified.\nHowever, due to measurement uncertainty, if two outages lead to similar (but not identical) responses, they may be confused and thus misidentified.\nIn some cases, misidentification may lead to wrong operation that aggravates the situation especially when the system is under stressful conditions.\nFor example, misoperation (tripping heavily loaded but not faulted lines) played a significant role in the 2003 US-Canada blackout and the 2015 Turkish blackout~\\cite{abdullah_distance_2018}.\nIn such cases, a more conservative result (inconclusive) sometimes is better than an incorrect result (i.e., no identification is better than an erroneous identification).\nThis practical problem has not been addressed in previous research.\nIn order to acknowledge measurement uncertainty and improve identification accuracy, a rejection filtering technique is introduced.\nWith the rejection filter in place, instead of definitive identification results, the events are now labeled as conclusive or inconclusive.\nIn summary, a two-stage framework for single line outage identification is proposed.\n\nExtensive tests have been conducted to show the relationships between identification results and different conditions (including filtering methods, threshold values, and PMU placements) on the IEEE 30-bus system.\nAdditionally, results using the Ontario network are presented.\nSection~\\ref{sec: Algorithm} gives the first stage of the algorithm for single line outage identification.\nSection~\\ref{sec:filtering} discusses the rejection filtering algorithm (the second stage). \nCase studies are presented in Section~\\ref{sec:case study}.\nLastly, concluding remarks and future work are presented in Section~\\ref{sec: conclusions and future work}.\n\n\\section{Stage 1: Main Identification Algorithm}\n\\label{sec: Algorithm}\nIn this section, the main identification algorithm will be presented. \nThe assumptions we make include: \\text{(1)} all measurements are taken from a system in quasi-steady state, and \\text{(2)} the ac power flow model is available, since steady-state models are available in modern energy management systems for power flow calculations, such as contingency analysis~\\cite{wu_power_2005}.\n\nInspired by~\\cite{Tate_line_2008}, the algorithm is based on hypothesis testing using voltage phasor measurements.\nBy comparing the simulated voltage changes due to each hypothetical line outage to the observed, the case that is closest to the observations is identified as the outage source.\nOutages are assumed to be equally likely in all lines (that would not lead to islands).\n\\Cref{Eqn: criterion} to \\Cref{Eqn: V obs} describe the identification algorithm. \n\\begin{align}\n\\label{Eqn: criterion}\nl^{\\ast}&=\\text{arg} \\underset{l \\in \\mathcal{L}} {\\: \\min} \\:E(l)\\\\\n\\label{Eqn: E def}\nE(l)&=\\|\\Delta\\boldsymbol{\\bar{V}_{exp,l}}-\\Delta\\boldsymbol{\\bar{V}_{obs}}\\|\\\\\n\\label{Eqn: V exp}\n\\Delta\\boldsymbol{\\bar{V}_{exp,l}}&=(\\boldsymbol{\\bar{V}_{exp,l}}-\\boldsymbol{\\bar{V}^{pre}})\\\\\n\\label{Eqn: V obs}\n\\Delta\\boldsymbol{\\bar{V}_{obs}}&=\\boldsymbol{\\bar{V}_{obs}^{post}}-\\boldsymbol{\\bar{V}_{obs}^{pre}}\n\\end{align}\nAn error measure $E(l)$ given a hypothetical outage in line $l$ (where ${l \\in \\mathcal{L}}$) is defined as (\\ref{Eqn: E def}) to quantify the difference between the expected voltage phasor change and its observed counterpart.\nSimilar to the list commonly used in online contingency analysis~\\cite{savulescu2014real}, the set $\\mathcal{L}$ represents all lines to be checked for outage occurrence, the cardinality of which (denoted by ${L}$) is the number of lines to be checked.\nThe most likely line ${l^{\\ast}}$ that leads to the smallest ${{E}(l)}$ (\\ref{Eqn: criterion}) is identified as the cause of outage.\nThe observed voltage change after an event ($\\Delta\\boldsymbol{\\bar{V}_{obs}}$) is defined as the difference between the observed pre-event voltage ($\\boldsymbol{\\bar{V}_{obs}^{pre}}$) and the post-event voltage ($\\boldsymbol{\\bar{V}_{obs}^{post}}$).\nAnalogously, the expected voltage change (${\\Delta \\boldsymbol{\\bar{V}_{exp,l}}}$) is defined based on the pre-event voltage ($\\boldsymbol{\\bar{V}^{pre}}$, computed from state estimator) and the expected post-event voltage (${\\boldsymbol{\\bar{V}_{exp,l}}}$, computed for all potential line outages ${l \\in \\mathcal{L}}$ through power flow calculations).\n\nNote that $\\boldsymbol{\\bar{V}^{pre}}$ is used instead of the pre-outage voltage measured by PMUs because (\\ref{Eqn: V exp}) can then be updated constantly and does not require detection of an outage.\nIn contrast, (\\ref{Eqn: criterion}), (\\ref{Eqn: E def}), (\\ref{Eqn: V obs}) and (\\ref{Eqn: rank}) are only updated when an outage is detected.\nIn our study, the ac power flow method, in particular, the Newton method, is used to solve for ${\\boldsymbol{\\bar{V}_{exp,l}}}$.\nThis is the most computationally expensive step.\nFor example, one successful ac power flow solution takes around 0.1 s for the Ontario system (with 3488 buses, introduced in Section \\ref{sec: line out Ontario}) using MATPOWER~\\cite{Zimmerman_matpower_2011}.\nBy comparison, the rest of the outage identification only takes $0.0075$ s due to highly efficient sorting algorithms.  \nAll identification algorithms were implemented in MATLAB on a computer with an Intel Xeon E5-1607 processor and 8 GB RAM.  \nThe results can likely be improved using better computation resources and simple parallelism \\cite{Jun_1995,roberge_parallel_2017}.\nHowever, simulation of system responses due to hypothetical outages is commonly used in contingency analysis, which reduces the additional computation burden associated with the proposed algorithm.\nNote that unlike phase angles that are typically used in previous works, all voltage vectors in our algorithm are phasors (with both magnitude and angle) with a length of $P$, where $P$ is the number of PMUs in a system.\nEmpirically, the algorithm is based on the following ranking of the error measure ${{E}(l)}$.\n\\setlength{\\belowdisplayskip}{3pt}\n\\setlength{\\abovedisplayskip}{3pt}\n\\setlength{\\belowdisplayshortskip}{3pt}\n\\setlength{\\abovedisplayshortskip}{3pt}\n\\begin{align}\n\\label{Eqn: rank}\n0\\leq E(l^{\\ast}=l_{r_1})\\leq E(l_{r_2}) \\leq \\ldots \\leq E(l_{r_{L}})\n\\end{align}\nThe subscript of $l_{r_i}$ means this line occupies the $i^{\\text{th}}$ place in the ranking among $L$ potential candidates.\nThe line outage ($l^{\\ast}$) that leads to the smallest error (i.e., $1^{\\text{st}}$ in the ranking) with respect to the observation is identified as the cause. \nGiven PMU measurements due to the actual outage of line $l_a$, if  ${l^{\\ast} = l_a}$, then the outage is successfully identified.\n\n\\section{Stage 2: Rejection Filtering Algorithms}\n\\label{sec:filtering}\nWhen measurements are ideal without uncertainty, almost all outage cases should be identified correctly even with a small number of PMUs installed in the system. \nTwo exceptions would be series lines with zero injection at shared buses and identical parallel lines. \nGiven measurements with uncertainties, more outage cases (not the aforementioned situations) are likely to be misidentified.\nMeasurement uncertainty may stem from measurement errors (e.g., instrument transformers, the A\/D converters, and the communication cables~\\cite{zhu_enhanced_2006}) or random errors (e.g. unknown fluctuations that can not be captured deterministically in principle).\nIn our research, we use independent Gaussian distributions for voltage phasor magnitude and angle to model measurement uncertainty.\n\nWhether a case will be misidentified is determined by the system as well as the measurement uncertainty model.\nA simple example for a 4-bus system (\\cite{Zimmerman_matpower_2011,grainger_power_1994}) is used to demonstrate the impact of uncertainty on identification results.\nThe system consists of 4 buses and 4 branches with a PMU installed on bus 2 and 1 (the reference bus).\nAssuming there is no measurement error on the reference bus, the ranking in (\\ref{Eqn: rank}) solely depends on measurements at bus 2.\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{delta_phasor_misranks.pdf}\n\t\\caption[Impact of measurement uncertainty on line outage identification in a 4-bus system]{Demonstration of measurement uncertainty impact on identification, ${\\Delta\\boldsymbol{\\bar{V}_{exp,l}},l\\in \\{1,2\\}}$ and ${\\Delta\\boldsymbol{\\bar{V}_{obs,1}}}$ in 4-bus system, ${\\sigma_{v}=0.002\/\\sqrt{3}}$ pu and ${\\sigma_{\\theta}=0.01\/\\sqrt{3}}$ degrees}\n\t\\label{fig_4gs}\n\\end{figure}\nFig.~\\ref{fig_4gs} shows simulated responses after the outage of line 1 (pink crosses) considering uncertainty, along with the ideal response due to the outage of line 1 (blue cross) and line 2 (blue star).\nThe pink crosses represents 1000 random realizations corresponding to possible measurements after the line 1 outage, generated based on independent distributions.\nSpecifically, we assume the magnitude and the angle uncertainty follow a zero-mean Gaussian distribution with standard deviation ${\\sigma_{v}}$ and ${\\sigma_{\\theta}}$ respectively.\nThe main identification algorithm is represented by the solid line that partitions the graph into two regions.\nAny point in the left region is closer to ${\\Delta\\boldsymbol{\\bar{V}_{exp,1}}}$ (represented by the blue cross), which means if an observation falls in this region, line 1 will be correctly identified.\nHowever, due to measurement uncertainty, the observations may fall to the right region and be closer to ${\\Delta\\boldsymbol{\\bar{V}_{exp,2}}}$ (represented by the blue star). \nIn such cases, the event will be misidentified.\n\nIf we set up a rejection filter (represented by the band delineated by two dashed lines on each side of the original identification solid line) so that all cases in the right region are labeled as inconclusive, then they will not be misidentified.\nThey are in fact misidentified-filtered cases (those would have been misidentified but filtered out to be inconclusive).\nHowever, depending on how we define the filter threshold (band position and width), it is likely that some cases in the left region also fall within the band and become inconclusive (i.e., correct-filtered cases).\n\nThere may be different ways to set up the filter threshold. \nIn this study, the threshold is determined by $\\Delta E$, where ${\\Delta E = E(l_{r_2}) - E (l_{r_1})}$. \nNamely the difference between the first two most promising candidates is used to set up the threshold.\n\\begin{equation}\n\\Delta E^{(r)}_{l_a} = E^{(r)}_{l_a}(l_{r_2}) - E^{(r)}_{l_a} (l_{r_1}) < \\epsilon,\n\\end{equation}\nwhere ${E^{(r)}_{l_a}(l)}$ denote the error measure in (\\ref{Eqn: E def}) but for the actual outage ${l_a}$ specifically.\nDue to randomness, multiple runs are conducted based on multiple, independent uncertainty distributions.\nThe superscript $r$ indicates the error measure is computed based on the $r^{th}$ random realization.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{four_types_graffle_random.pdf}\n\t\\caption{Categories of line identification results (30-bus system with 20 PMUs)}\n\t\\label{fig_results_illustrate_1}\n\\end{figure}\nA visual representation of this process is in \\cref{fig_results_illustrate_1}.\nThe area of each circle represents the number of cases in each category.\nThe red inner circle represents misidentified cases which correspond to the cases in the right region in Fig.~\\ref{fig_4gs} (i.e., misidentified).\nIf the rejection filter is defined as an ellipse which will not be coincident with the inner circle, then by covering the inner circle (thus eliminating misidentified), some correct cases will become inconclusive (correct filtered) as well.  \nThe filter threshold defines the sizes of ellipses.\nAs will be shown in the case studies, as the threshold is increased, the misidentified cases will decrease to zero while the inconclusive cases will go up as a result of increased number of correct-filtered cases.\nThis corresponds the bottom right circle in Fig.~\\ref{fig_results_illustrate_1}. \nIn the course of minimizing the misidentified cases, we probably sacrifice some cases which would have been identified correctly. \nThe goal is to minimize misidentified cases without introducing a significant number of correct-filtered cases.\n\n\n\\section{Case Studies}\n\\label{sec:case study}\nA comprehensive comparison of different techniques and the choices of filter thresholds are presented in this section. \nThe first experiment is the comparison of the ac and the dc approach using the IEEE 30-bus test system. \nThen the results using rejection filtering are presented.\nLastly, the impact of the rejection filter threshold is discussed. \nThe identification algorithm is further tested using a 3488-bus model of the Ontario power system.\nIn all tests, we assume the measurement uncertainty in the magnitude and angle observed follows a zero mean Gaussian distribution~\\cite{chakhchoukh_pmu_2014} with standard deviation ${0.002\/\\sqrt{3}}$ pu and ${0.01\/\\sqrt{3}}$ degrees respectively.\n1000 different placements are generated randomly for each possible ${P\\in\\{2,\\ldots,B\\}}$ where ${B}$ is the total number of buses in a system.\nIf the total number of placements is less then 1000, all placements are considered.\nFor comparison, 100 randomly generated measurements for each PMU are shared among all placements.\n\n\\subsection{IEEE 30-Bus System}\n\\begin{figure}[hbtp]\n\n\t\\centering\n\t\\begin{subfigure}[b]{0.48\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim= 15 0 15 0,width=\\linewidth]{ac_dc_pcorrect.pdf}\n\t\t\\caption{correctly identified, no filter}\n\t\t\\label{fig_ac_dc_pcorrect}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.48\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim= 15 0 15 0,width=\\linewidth]{ac2_set2.pdf}\n\t\t\\caption{ac, $\\Delta E$}\n\t\t\\label{fig_ac2_set2}\n\t\\end{subfigure}\n\t\\caption[Comparison of dc and ac approaches (${E_{r_2}}$ and $\\Delta E$) for line outage identification]{Comparison of dc and ac approaches (with or without $\\Delta E$) for line outage identification (The threshold $\\epsilon$ is chosen so that the mean misidentification rate is driven under $0.00015\\%$)}\n\t\\label{fig: Comparison of DC and 2 AC}\n\\end{figure}\nFig.~\\ref{fig_ac_dc_pcorrect} shows the correctly identified rate of ac and dc approaches without filtering techniques for the IEEE 30-bus system.\nAs expected, this figure shows that the ac approach is needed to achieve high identification accuracy.\nFor example, even in the best-case scenario of complete PMU coverage, the dc accuracy is only 70.15\\%, whereas the ac accuracy is 97.4\\%.  \nFig.~\\ref{fig_ac2_set2} shows\nperformance of the rejection filter based on four types of identification results: correct, misidentified, correct-filtered and misidentified-filtered.\nThe results in Fig. \\ref{fig: Comparison of DC and 2 AC} are generated using thresholds so that the misidentification rate is driven to nearly zero (specifically the mean misidentification rate is driven under $0.00015\\%$ over all random realizations, all placements and all outages).\nIf we compare Fig.~\\ref{fig_ac_dc_pcorrect} to \\ref{fig_ac2_set2}, the drop of correctly identified rate is because of increased inconclusive cases.\nFor example, for the ac approach, with 30 PMUs the accuracy was 97.4\\%.\nIt drops to around $83\\%$ when the $\\Delta E$ filter is used.\n\n\\subsection{Results of Different Rejection Filter Thresholds}\nThis section discusses the impact of the rejection filter $\\epsilon$ on the identification results. \nTo isolate impact of the threshold, the results should be generated using different PMU placements.\nAt the beginning of the section, we mentioned 1000 different placements are generated randomly for each possible ${P\\in\\{2,\\ldots,B\\}}$ where ${B=30}$.\nAdditionally, 100 randomly generated measurements for each PMU are shared among all placements.\nThe number of lines in the 30-bus system to be checked for outages is 38.\nIf we consider all these possible combinations, $1.102\\times10^8$ cases are needed for just one threshold level.\nSince enumeration over all possible scenarios takes significant amount of time, we focus on the behavior with a varying range of $\\epsilon$ (from 0 to 0.0045) and four specific levels of PMU coverage (10\\%, 20\\%, 50\\% and 100\\%).\nThe measurement uncertainty levels remain the same as in the previous section.\n\n\n\\begin{figure}[!ht]\n\n\t\\centering\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{30bus10per.pdf}\n\t\t\\caption{10\\% coverage}\n\t\n\t\t\\label{fig_diff_threshold_30bus_10per}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{30bus20per.pdf}\n\t\t\\caption{20\\% coverage}\n\t\t\\label{fig_diff_threshold_30bus_20per}\n\t\\end{subfigure}\n\n\t\\centering\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{30bus50per.pdf}\n\t\t\\caption{50\\% coverage}\n\t\t\\label{fig_diff_threshold_30bus_50per}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.47\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim= 15 0 0 0, width=0.83\\linewidth]{30bus100per.pdf}\n\t\t\\caption{100\\% coverage}\n\t\t\\label{fig_diff_threshold_30bus_100per}\n\t\\end{subfigure}\n\t\\caption[Identification results vs $\\epsilon$ (30-bus system, $\\Delta E$ filter, four coverage levels)]{Identification results vs $\\epsilon$ (30-bus system, $\\Delta E$ filter with different coverage (legends are the same for 4 figures))}\n\t\\label{fig_ac_thresholds_complete}\n\\end{figure}\nThe trade-off between correct and misidentified cases as a result of different $\\epsilon$ is illustrated by Fig.~\\ref{fig_ac_thresholds_complete}.\nThe horizontal lines in each figure represents the correctly identified\/misidentified level without filtering.\nFor example, without any rejection filter, with $10\\%$ coverage, the misidentified rate is about $45\\%$ while the correctly identified rate is $55\\%$.\nThis corresponds to a red inner circle with an area of $45\\%$ of the entire circle in Fig.~\\ref{fig_results_illustrate_1}.\nBy increasing the threshold, more misidentified cases (in red) will be filtered out and become misidentified-filtered (in pink), whilst correctly identified cases (in green) become correct-filtered (in yellow).\n\nFig.~\\ref{fig_ac_thresholds_complete} indicates the misidentified percentage (in red) is very sensitive to the threshold when $\\epsilon < 1\\times10^{-3} $ (particularly when the coverage is low).\nIn such cases, the misidentified rate drops rapidly as $\\epsilon$ increases from zero.\nIt also decays faster than the correctly identified rate (by comparing red to green).   \nFor example, with 10\\% coverage (Fig.~\\ref{fig_diff_threshold_30bus_10per}), by increasing the threshold from 0 to $1\\times10^{-3}$, the misidentified rate drops more than 40\\% while the correctly identified rate only drops about 30\\%.\nHowever, when the misidentified rate is low (around zero), further elimination of misidentified cases leads to significant loss of correctly identified cases. \nFor example, by increasing the threshold from $1\\times10^{-3}$ to $2\\times10^{-3}$ in Fig.~\\ref{fig_diff_threshold_30bus_10per}, the green area decreases significantly (about $10\\%$) compared to the trivial gain in eliminating red cases. \nThe results also indicate that, ultimately, higher coverage is needed to achieve better identification accuracy.\nAs the number of PMUs goes up, the ratio between correctly identified and correct-filtered cases also increases (i.e., green versus yellow).\nThis means fewer correctly identified cases will be sacrificed when the number of PMUs is high. \n\nDepending on the acceptable level of the misidentification rate or the correct-filtered rate, operators can decide on an empirical value for the threshold.\nIf misidentification is considered to be worse than a lower correct rate (e.g. compared to false alarms, the operator would rather accept more inconclusive cases), then the threshold can be set accordingly to eliminate misidentified cases.\nOtherwise, a very small non-zero level (e.g. $0.0015\\%$) can be chosen instead, with minimal impact on the correctly identified cases.\n\n\\subsection{Results of the Ontario Power System}\n\\label{sec: line out Ontario}\nTo evaluate performance on a more realistic system, results were also obtained using a model of the Ontario power system, consisting of 3488 buses, 864 generators, 1290 loads, 2242 branches and 1697 transformers.\nAmong the original branches, 1762 single-line outage cases are to be checked for occurrence without introducing islands.\nPMUs are assumed to be placed on the actual PMU locations and, alternatively, a set of high voltage buses in Ontario.\n\nFirst, tests were conducted using 26 bus voltages based on the current PMU locations in the Ontario network \\cite{curtis_north_2011}.\nTo evaluate the performance of the line outage detection algorithms for realistic, future PMU deployments, we also considered cases where all buses above a certain voltage level are monitored by PMUs.\nIn particular, we consider two cases: monitoring all buses with nominal voltages greater than or equal to 230 kV (an additional 53 buses) or 220 kV (an additional 842 buses).\n\\begin{figure}[hbtp]\n\t\\centering\n\t\\begin{subfigure}[b]{0.5\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim=0 0 20 20, height = 0.15\\textheight,width=\\linewidth]{threshold_hydro_mis.pdf}\n\t\t\\caption{26 PMUs}\n\t\t\\label{fig_ieso_26PMU}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.5\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[trim=0 0 20 0, height = 0.15\\textheight,width=\\linewidth]{threshold_hydro_79_mis.pdf}\n\t\t\\caption{79 PMUs (230 kV+)}\n\t\t\\label{fig_ieso_79PMU}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.48\\linewidth}\n\t\t\\includegraphics[trim=0 0 20 0, height = 0.15\\textheight,width=\\linewidth]{threshold_hydro_868_mis.pdf}\n\t\t\\caption{868 PMUs (220 kV+)}\n\t\t\\label{fig_ieso_868PMU}\n\t\\end{subfigure}\n\t\\caption{Identification results vs $\\epsilon$ (the Ontario system, $\\Delta E$ filter, three coverage levels)}\n\t\\label{fig_ieso_complete}\n\\end{figure}\nFig.~\\ref{fig_ieso_complete} shows the identification results using different thresholds.\nConsistent with the previous results, the observations about impact of filter threshold and PMU coverage are still valid for the Ontario system.\nWhen the coverage goes up, the threshold required to eliminate misidentified cases also decreases, which results in not only a higher correctly identified rate but also higher ratio of correct versus correct-filtered.\nThe horizontal line in each figure represents the separation of correctly identified (above the line) and misidentified (below the line) without rejection filter.\nThe accuracy given by the initial 26 PMUs is low (22.7\\% correct versus 77.3\\% misidentified) due to the very limited number of PMUs and the large number of outage scenarios.\nHowever, as the coverage increases by adding PMUs at high voltage buses, the results have been improved significantly.\nWhen 220 kV+ buses are monitored (868 PMUs), the correctly identified rate without filtering rises up to more than 70\\%.\nThis indicates monitoring high voltage buses is very beneficial and should be considered for future PMU deployments.\n\n\\section{Conclusions and Future Work}\n\\label{sec: conclusions and future work}\nThe current proposed methods have been implemented and tested on several systems of different scales.\nThe results show that the proposed identification algorithm using the ac power flow model achieves better identification accuracy compared to the dc approach. \nAdditionally, relatively high identification accuracy is achieved with a small number of PMUs.\nBy using rejection filtering techniques, the misidentified rate can be further reduced, which is crucial for online utilization of the event detection.\nThe results also demonstrate there are significant benefits of having a higher PMU coverage.\nLow number of PMUs makes it difficult to identify the outage on the Ontario system, but future PMU deployments (e.g., with all high voltage buses monitored) exhibit good performance. \n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMixed action approaches have been studied by several groups,\nsuch as Domain-wall fermion (DWF) valence on Asqtad fermion sea~\\cite{Edwards:2005ym}, overlap valence on DWF sea~\\cite{Allton:2006nu}, overlap valence on \nclover sea~\\cite{Durr:2007ez}, and overlap valence\non twisted-mass fermion sea~\\cite{Cichy:2009dy}. In view of the fact that it is numerically intensive to simulate chiral fermions\n(DWF or overlap), it is deemed practical to use the cheaper fermion formulation for generating gauge configurations and\nthe more expensive fermion discretization for the valence as an expedient approach toward dynamical QCD simulations with chiral fermions.\nMany current algebra relations depend only on the chiral properties of the\nvalence sector. The mixed action theory with different fermions for the valence and the sea is a generalization of the\npartially quenched theory with different sea and valence quark masses.\nThe mixed action partially quenched chiral perturbation theory (MAPQ$\\chi$PT) has been developed for\nGinsparg-Wilson fermions on Wilson sea~\\cite{Bar:2003mh} and staggered sea~\\cite{Bar:2005tu}, and has been worked out for\nmany hadronic quantities to next-to-leading order (NLO), such as pseudoscalar masses and decay \nconstants~\\cite{Bar:2003mh,Bar:2005tu,Chen:2007ug},\nisovector scalar $a_0$ correlator~\\cite{Golterman:2005xa,Prelovsek:2005rf,Aubin:2008wk,WalkerLoud:2008bp},\nheavy-light decay constants~\\cite{Aubin:2005aq}, and baryon masses~\\cite{Tiburzi:2005is, WalkerLoud:2008bp}.\n\nIn the mixed action chiral perturbation theory with chiral valence fermions, it is shown~\\cite{Bar:2003mh} that to\nNLO there is no $\\mathcal{O}(a^2)$ correction to the valence-valence meson mass due to\nthe chiral symmetry of the valence fermion. Furthermore, both the chiral Lagrangian and the chiral extrapolation\nformulas for hadron properties to the one-loop level (except $\\theta$-dependent quantities) are independent of the\nsea fermion formulation~\\cite{Chen:2006wf}. The LO mixed-action chiral Lagrangian involves only one more term with\n$\\mathcal{O}(a^2)$ discretization dependence which is characterized by a low energy constant $\\Delta_{mix}$.\nThe LO pseudoscalar meson masses for overlap valence and DWF sea are given as\n\\begin{eqnarray}   \\label{eq:dd}\nm_{vv'}^2 &=& B_{ov}(m_v +m_{v'}), \\nonumber \\\\\nm_{vs}^2 &=& B_{ov}m_v + B_{dw}(m_s + m_{res}) + a^2\\Delta_{mix}, \\nonumber \\\\\nm_{ss'}^2 &=& B_{dw}(m_s + m_{s'} + 2 m_{res}),\n\\end{eqnarray}\nwhere $m_{vv'}\/m_{ss'}$ is the mass of the pseudoscalar meson made up of valence\/sea quark and antiquark.\n$m_{vs}$ is the mass of the mixed valence and sea pseudoscalar meson. Up to numerical accuracy, there is no\nresidual mass for the valence overlap fermion. The DWF sea has a residual mass $m_{res}$ which vanishes as\n$L_S \\rightarrow \\infty$. $\\Delta_{mix}$ enters in the mixed meson mass $m_{vs}$ as an\n$\\mathcal{O}(a^2)$ error which vanishes in the continuum limit. We should note that, unlike the partially\nquenched case, even when the quark masses in the valence and sea match, the unitarity is still violated due\nto the use of mixed actions. The degree of unitarity violation at finite lattice spacing depends on the size of $\\Delta_{mix}$.\n\n$\\Delta_{mix}$ has been calculated for DWF\nvalence and Asqtad fermion sea which gives\n$\\Delta_{mix} = 0.249(6)\\,{\\rm GeV}^4$ \\cite{Orginos:2007tw} at $a=0.125\\,{\\rm fm}$, $0.211(16)\\,{\\rm GeV}^4$ at $a=0.12\\,{\\rm fm}$ \\cite{Aubin:2008wk}, and $0.173(36)\\,{\\rm GeV}^4$ at $a=0.09\\,{\\rm fm}$ \\cite{Aubin:2008wk}. It is also calculated for overlap valence and clover sea which yields $\\Delta_{mix} = 0.35(14)\/0.55(23)\\,{\\rm GeV}^4$ at $m_{\\pi} = 190\/300\\,{\\rm MeV}$ and $a=0.09\\,{\\rm fm}$~\\cite{Durr:2007ef}.~This means that for a valence pion of $300\\,{\\rm MeV}$, $\\Delta_{mix}$ produces, at $a=0.12\\,{\\rm fm}$, a shift of $\\sim$ $110-240\\,{\\rm MeV}$ and,\nat $a =0.09\\,{\\rm fm}$, a shift of $\\sim$ $55-153\\,{\\rm MeV}$ for these cases, which are substantial portions of\nthe valence pion mass. \n\nWe have used valence overlap fermions on the 2+1-flavor DWF sea to study hadron \nspectroscopy~\\cite{Li:2010pw,Dong:2009wk,Mathur:2010ed,Gong:2011nr}.  In this work, we calculate \n$\\Delta_{mix}$ for such a mixed action approach which is needed for chiral extrapolation\nin MAPQ$\\chi$PT.\n\n\n\\section{Calculation Details}\nThe $\\Delta_{mix}$ parameter is calculated on four ensembles of the $2+1$-flavor domain wall fermion gauge configurations \\cite{Allton:2008pn, Mawhinney:2009jy}. Two different lattice spacings were used to study the dependence on the cutoff. In addition, we also used multiple sea masses, for the $32^3 \\times 64$ lattices, to study the sea quark mass dependence of $\\Delta_{mix}$.   Details of the ensembles are listed in Table \\ref{tab:ensembles}.\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nLattice Size& $a^{-1}(\\,{\\rm GeV})$& $a m_l$& $ m_{ss}(\\,{\\rm MeV})$\\\\ \n\\hline\n\\hline\n$24^3 \\times 64$ & 1.73(3) & 0.005 & 329(1)\\\\\n$32^3 \\times 64$ & 2.32(3) & 0.004 & 298(1)\\\\\n$32^3 \\times 64$ & 2.32(3) & 0.006 & 350(2)\\\\\n$32^3 \\times 64$ & 2.32(3) & 0.008 & 399(1)\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Details of the DWF ensembles used in this work.}\n\\label{tab:ensembles}\n\\end{table}\n\nThere are different parameterization schemes \\cite{Orginos:2007tw, Aubin:2008wk} one can use to relate $\\Delta_{mix}$ to the quark and pseudoscalar masses from Eq.~(\\ref{eq:dd}). In this paper we choose to parameterize $\\Delta_{mix}$ as \n \\begin{equation}\n\\label{eq:dmix}\nm_{vs}^2 -\\frac{1}{2}m_{ss}^2 = B_{ov} m_v + a^2\\Delta_{mix}.\n\\end{equation}\nThis is similar to the parameterization used in \\cite{Aubin:2008wk}. The quantity $\\delta m^2 \\equiv m_{vs}^2 -\\frac{1}{2}m_{ss}^2$, has a linear behavior in $m_v$ and can be calculated directly by computing only the pseudoscalar masses $m_{vs}$, and $m_{ss}$.  We see that in the regime where Eq.~(\\ref{eq:dmix}) is valid, $\\Delta_{mix}$ is equivalent to $\\delta m^2$ in the limit $ m_v \\rightarrow 0$.\n\nIn this work we used an overlap operator with a HYP-smeared kernel. This was shown to have better numerical properties~\\cite{Li:2010pw}. We calculated the masses $m_{vs}$ and $m_{ss}$ using 50 configurations for each ensemble.  Recall from Section I that $m_{ss}$ requires the propagators for the sea quark mass which were computed with DWF. $m_{vs}$ needs the propagators for both overlap and DWF. The DWF propagators were made available by LHPC.  The overlap propagators were computed using a polynomial approximation \\cite{Alexandru:2011sc} to the matrix sign function. They  were used to compute $14-16$ values of $m_{vs}$. A multi-shifted version of the conjugate gradient algorithm \\cite{Jegerlehner:1996pm}  was implemented for the overlap propagator calculation to compute all masses at once. To accelerate the inversions for the overlap propagators we employed a deflation technique which has been seen to speed up the calculation significantly~\\cite{Li:2010pw}.  \n\nWe fitted correlators using single state exponential fits to extract the pion masses. The fitting windows were adjusted to get a reasonable $\\chi^2\/dof$ and are the same for all the masses in each ensemble. For the $24^3 \\times 64$ ensemble the details are presented in Table \\ref{tab:pfit1}.  For one of the sea quark mass in the  $32^3 \\times 64$ ensemble, the results are presented in Table \\ref{tab:pfit2}. \n\\begin{table}[htdp]\n\\begin{center}\n\\caption{Pion mass fitting details for the $24^3 \\times 64$ lattice. $m_{ss}  \\simeq 329\\,{\\rm MeV}$.}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$a m_{v}$& $m_{vs}(\\,{\\rm MeV})$  & $\\chi^2_{vs}\/dof$ & $m_{vv}(\\,{\\rm MeV})$  &$\\chi^2_{vv}\/dof$ \\\\ \n\\hline\n\\hline\n0.0014& 274(3) & 1.0 &122(4) & 1.8 \\\\ \n0.0027& 278(2) & 1.1 &154(2) & 1.4 \\\\ \n0.0046& 287(2) & 1.2 &188(2) & 1.6 \\\\ \n0.0081& 305(2) & 1.2 &242(2) & 1.6 \\\\ \n0.0102& 315(2) & 1.2 &270(2) & 1.4 \\\\ \n0.0135& 331(2) & 1.1 &308(2) & 1.3 \\\\ \n0.0153& 339(1) & 1.1 &327(2) & 1.3 \\\\ \n0.0160& 342(1) & 1.1 &334(2) & 1.3 \\\\ \n0.0172& 347(1) & 1.1 &346(2) & 1.3 \\\\ \n0.0243& 378(1) & 1.2 &409(1) & 1.6 \\\\ \n0.0290& 397(1) & 1.2 &445(1) & 1.7 \\\\ \n0.0365& 426(1) & 1.2 &498(1) & 1.5 \\\\ \n0.0434& 451(1) & 1.2 &542(1) & 1.3 \\\\ \n0.0489& 471(1) & 1.2 &576(1) & 1.2 \\\\ \n0.0670& 531(1) & 1.0 &677(1) & 1.3 \\\\ \n0.0710& 543(1) & 1.0 &698(1) & 1.4 \\\\ \n\\hline\n\\hline\n\\end{tabular}\n\\label{tab:pfit1}\n\\end{center}\n\\end{table}\n\\begin{table}[htdp]\n\\begin{center}\n\\caption{Pion mass fitting details for the $32^3 \\times 64$ lattice for $a m_l = 0.004$.  $m_{ss}  \\simeq 298\\,{\\rm MeV}$.}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$a m_{v}$& $m_{vs}(\\,{\\rm MeV})$  & $ \\chi^2_{vs}\/dof$ & $m_{vv}(\\,{\\rm MeV})$  &$\\chi^2_{vv}\/dof$ \\\\ \n\\hline\n\\hline\n0.0007& 237(3)&1.0& 126(10)&1.5\\\\\n0.0015& 238(2)&1.0& 136(7)& 1.2\\\\\n0.0025& 246(2)&1.0& 160(4)& 1.2\\\\\n0.0035& 254(2)&1.0& 184(3)& 1.2\\\\\n0.0046& 264(2)&1.0& 209(3)& 1.2\\\\\n0.0059& 274(2)&1.0& 235(3)& 1.2\\\\\n0.0068& 282(2)&1.0& 253(3)& 1.3\\\\\n0.0076& 289(2)&1.0& 268(3)& 1.4\\\\\n0.0089& 299(2)&1.0& 288(3)& 1.6\\\\\n0.0112& 317(2)&1.1& 324(4)& 1.0\\\\\n0.0129& 329(2)&1.1& 337(5)& 1.0\\\\\n0.0152& 344(2)&1.2& 365(5)& 1.3\\\\\n0.0180& 362(2)&1.3& 396(6)& 1.5\\\\\n0.0240& 399(3)&1.6& 454(9)& 1.9\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\label{tab:pfit2}\n\\end{center}\n\\end{table}%\n\n\n\\section{Fitting strategies and Results}\n\n\\begin{figure*}[t]\n\\includegraphics[width= 3.4in]{mvvsq24_2.pdf}\n\\includegraphics[width= 3.4in]{mvvsq32_2.pdf}\n\\caption{$(a m_{vv})^2\/2(a m_v)$ as a function of $a m_v$ for\nthe $24^3 \\times 64$ ensemble (left) and \nthe three $32^3 \\times 64$ ensembles (right).}\n\\label{fitregion}\n\\end{figure*}\n\nExtracting $\\Delta_{mix}$ requires fits to squares of multiple pion propagators with different valence quark masses; the resulting \nmasses will be correlated since they all come from a single ensemble. \nTo compute the covariance matrix we would need to perform an augmented $\\chi^2$-fit involving the pion propagators for all valence quark masses simultaneously.  This is not numerically stable due to the large number of parameters in the model.  We thus have to use an alternative procedure to take into account these correlations. To gauge the systematic errors introduced by our choice of fitting method we will use multiple fitting procedures. In this section we will describe three different fitting strategies. These methods differ in the way we account for correlations among the different valence masses.\n \nIn method I we follow the standard jackknife philosophy by defining a $\\Delta_{mix}$ estimator directly in terms of the raw pion propagators; the error is then determined by the variance over the jackknife ensemble. In method II we use the jackknife procedure to estimate the $\\delta m^2$ covariance matrix and then compute $\\Delta_{mix}$ using a standard correlated fit~\\cite{Luscher:2010ae}. These two procedures are used for both the $24^3 \\times 64$ and $32^3 \\times 64$ ensembles. However, since there are three sea quark masses for the $32^3 \\times 64$ volume, we can perform an {\\em uncorrelated} fit using $\\delta m^2$ computed on the three independent ensembles---this is our method III. From the different methods we can obtain a systematic uncertainty for our results.   \n\nIn each method we performed a binned jackknife analysis of $\\delta m^2 =m_{vs}^2 -\\frac{1}{2}m_{ss}^2$. Because we performed four inversions with four different point sources per configuration we binned in units of four and eight. This allowed us to drop one or two whole configurations in each jackknife subsample. We find the uncertainties in both cases to be of comparable size, indicating that autocorrelation is negligible.  \n\n\nThe first step in our fitting is to determine a range of quark masses where the tree-level relation between $m^2_{vv}$ and $m_v$ holds so that we can use Eq.~(\\ref{eq:dd}). Below this range, one expects to see chiral logs including partially quenched logs and other non-linear $m_v$ dependence from the NLO in $\\chi$PT. Above this range, tree-level $\\chi$PT is not expected to be valid.  We do this by plotting $(a m_{vv})^2 \/2(a m_v)$ as a function of $a m_v$ as shown in Fig. \\ref{fitregion}. We choose the fitting range in the region where the ratio $(a m_{vv})^2\/2 (a m_v)$ is fairly flat; these ranges are tabulated in Table \\ref{dmix:fitrange} and are used in all three fitting methods.  We note that in the range we are fitting, $m_{\\pi}L > 4$ for both $m_{vs}$ and $m_{vv}$ so that the volume dependence is expected to be small.\n\n\\begin{table}[b]\n\\begin{center}\n \\begin{tabular}{|c|c|c|}\n  \\hline\n   Lattice&$a m_l$& $a m_v $ fit range\\\\\n   \\hline\n   \\hline\n   $24^364$&0.005&0.0243 - 0.0489 \\\\\n   $32^364$&0.004&0.0112 - 0.0240\\\\\n   $32^364$&0.006&0.0112 - 0.0240 \\\\\n   $32^364$&0.008&0.0112 - 0.0240\\\\\n   \\hline\n   \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Range of quark masses, $a m_v$, used in the fitting procedures to extract $\\Delta_{mix}$ via Eq.~(\\ref{eq:dd}).}\n\\label{dmix:fitrange}\n\\end{table}\n\n\\begin{figure*}[t]\n\\includegraphics[width= 3.4in]{dmix_24.pdf}\n\\includegraphics[width= 3.4in]{all3.pdf}\n\\caption{Extracting $\\Delta_{mix}$ from a linear extrapolation of $\\delta m^2$ for the $24^3 \\times 64$ ensemble (left) \nand the three $32^3 \\times 64$ ensembles (right). For the $32^3 \\times 64$ plot the inset shows the intercept of the \nfit and the error bars of the extracted values of $\\Delta_{mix}$.}\n\\label{dmixfit}\n\\end{figure*}\n\n\\subsection{Method I:  weighted averaging}\nIn this method we used a weighted linear fit to extract the value of $\\Delta_{mix}$ for each bin.  The weights, $\\sigma_{\\delta m^2}^2$, are given by \n\\begin{equation}\n\\sigma_{\\delta m^2}^2 = 4 m_{vs}^2\\sigma_{ m_{vs}}^2 + m_{ss}^2 \\sigma_{m_{ss}}^2,\n\\label{dmix:err1}\n\\end{equation}\nwhere $\\sigma_{m_{vs}}$, and  $\\sigma_{m_{ss}}$ are the uncertainties of the mixed and DWF pion masses respectively. Eq.~(\\ref{dmix:err1}) was derived using the standard error propagation formula neglecting correlation between $m_{vs}$ and $m_{ss}$. The cross-correlations will be accounted for by the external jackknife procedure. By using weighted fitting, less importance is given to data points with larger uncertainties.  After fitting each bin we then have a jackknife ensemble \\{$\\Delta_{mix}$\\}. We use square brackets, [~], to indicate a particular jackknife sample and angle brackets, $\\langle ~\\rangle$, to denote jackknife averages.  For the case of $\\Delta_{mix}$, its jackknife average and the corresponding uncertainty is given by\n\\begin{eqnarray}\n\\langle \\Delta_{mix} \\rangle &=& \\frac{1}{N}\\sum_{k=1}^{N}\\left[ \\Delta_{mix}\\right]_k \\nonumber \\\\\n\\sigma_{\\Delta_{mix}} &=& \\sqrt{(N-1)\\left(\\langle \\Delta_{mix}^2\\rangle-\\langle \\Delta_{mix}\\rangle^2 \\right)}\\nonumber. \\\\\n\\end{eqnarray}\n\nThe extracted values of $\\Delta_{mix}$, using this method, are presented in Table \\ref{dmix:method1}. We also list the corresponding values of $B_{ov}$.  Figure \\ref{dmixfit} shows $a^2 \\delta m^2$ as a function of $a m_{v}$ and the corresponding linear fit.\n\n\\begin{table}[b]\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|c|}\n \\hline\n   \\multirow{2}{*}{Lattice}&\\multirow{2}{*}{$a m_l$}& \\multicolumn{2}{|c|}{$\\Delta_{mix}(\\,{\\rm GeV}^4)$}& \\multicolumn{2}{|c|}{$B_{ov}$(\\,{\\rm GeV})}\\\\\n   &&\\multicolumn{1}{c}{I}&II&\\multicolumn{1}{c}{I}&II\\\\\n   \\hline\n   \\hline\n   $24^364$&0.005&0.032(6)  & 0.028(5)  &1.85(2)  &1.88(2) \\\\\n   $32^364$&0.004&0.040(14)& 0.025(9)  &1.88(10)&1.95(6)\\\\\n   $32^364$&0.006&0.054(9)  & 0.050(8)  &1.81(5)  &1.81(5)\\\\\n   $32^364$&0.008&0.059(13)& 0.063(13)&1.74(6)  &1.73(6) \\\\\n   \\hline\n   \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Extracted values of $\\Delta_{mix}$ and $B_{ov}$ using fitting methods I and II.}\n\\label{dmix:method1}\n\\end{table}\n\n\n\n \n  \n  \n  \n  \n  \n  \n  \n  \n \n  \n  \n\n\\subsection{Method II: covariance matrix}\nIn our second method we perform a correlated valence-quark mass fit to the function $\\delta m^2  = B_{ov} m_v + a^2 \\Delta_{mix}$. The central values are taken to be the jackknife averages,\n\\begin{equation}\n\\langle(\\delta m^2)_i \\rangle = \\frac{1}{N}\\sum_{k=1}^N\\left [(\\delta m^2)_i\\right]_k \\,.\n\\end{equation}\nWe minimize the correlated $\\chi^2$-function,\n\\begin{eqnarray*}\n\\chi^2& =& \\sum_{i,j} \\left(\\langle(\\delta m^2)_i \\rangle - (\\delta m^2)_i\\right)C_{i,j}^{-1} \\nonumber\\\\\n&\\times&\\left(\\ \\langle(\\delta m^2)_j \\rangle -(\\delta m^2)_j\\right) \\,, \\nonumber \\\\\n\\end{eqnarray*}\nwith the covariance matrix, $C_{i,j}$, given by the jackknife estimate\n\\begin{eqnarray}\nC_{ij} &=& \\frac{(N-1) } {N} \\sum_{k=1}^N \\left(\\left[(\\delta m^2)_i\\right]_k - \\langle(\\delta m^2)_i \\rangle \\right) \\nonumber \\\\\n&\\times& \\left(\\left[(\\delta m^2)_j\\right]_k - \\langle (\\delta m^2)_j \\rangle\\right) \\,. \\nonumber \\\\\n\\label{covmat}\n\\end{eqnarray}\nThe subscripts $i$ and $j$ index the valence-quark mass. The factor $(N-1)\/N$ differs from the usual definition of the covariance matrix to account for the correlation of the jackknife samples~\\cite{Luscher:2010ae}. The fit uncertainties are obtained by constructing the standard Hessian matrix. Results of this method are tabulated in Table \\ref{dmix:method1}.\n  \n   \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\\subsection{Method III: uncorrelated fitting}\nThe methods described in the two previous sections closely parallel the method performed by \\cite{Aubin:2008wk} in which for each lattice ensemble a string of partially quenched meson masses are calculated; from them we extract $\\Delta_{mix}$.  In this section we perform a fit on the three $32^3 \\times 64$ ensembles based on the value of $\\delta m^2$ measured at the point where the valence and sea quark masses match \\footnote{As mentioned in \\cite{Aubin:2008wk} one can never get rid of partial quenching effects even in the case where the pseudoscalar mesons of both the DWF and overlap actions match. This is because discretization errors are different for the two actions.}.  There are no cross-correlations among the masses since the ensembles are independent. \n\nThe value of $\\delta m^2$ is computed for the pion mass that most closely satisfies the condition $m_{ss} \\approx m_{vv}$. Because it is not  easy to match a priori the pseudoscalar masses and it is expensive to regenerate overlap propagators with different masses, we performed interpolation among the existing data points to obtain a better approximation of where $m_{ss}$ and $m_{vv}$ match.\n\nWe do an uncorrelated $\\chi^2$-fit with the error bars obtained by a jackknife procedure. Figure \\ref{dmix:32method3} shows the data points and the fit results. For this case we find $\\Delta_{mix} = 0.042(24)\\,{\\rm GeV}^4$, and $B_{ov} = 1.82(16)\\,{\\rm GeV}$. \n\n\\begin{figure}[t]\n\\includegraphics[width= 3.4in]{3264_full3.pdf} \n\\caption{Determining $\\Delta_{mix}$ by performing the linear extrapolation as in  Fig.~\\ref{dmixfit}.  Each point corresponds to one of the $32^3 \\times 64$ lattices where $m_{ss} \\approx m_{vv}$. A magnified view of the extrapolation in the neighborhood of the chiral limit, along with the extrapolated value, is shown in the inset.}\n\\label{dmix:32method3}\n\\end{figure}\n\n\\section{Discussion}\n\n\nAs a first step, we look at the lattice spacing dependence. We compare the two ensembles with $m_{ss}$ close to $300\\,{\\rm MeV}$.  \nThese correspond to $a m_l = 0.005$ and $a m_l = 0.004$ with $a = 0.114\\,{\\rm fm}$ and $a=0.085\\,{\\rm fm}$ respectively.\nWe average the central values and the errors from the two fitting methods, separately for each lattice spacing. \nWe find at $a = 0.114\\,{\\rm fm}$ $\\Delta_{mix} = 0.030(6)\\,{\\rm GeV}^4$ and for $a = 0.085\\,{\\rm fm}$ $\\Delta_{mix} = 0.033(12)\\,{\\rm GeV}^4$. \nWe see that the lattice spacing dependence is smaller than our errors; this indicates that $\\Delta_{mix}$ is capturing \nthe dominant lattice artifact for the parameters used in this study.  \n\nTo discuss the sea quark dependence of $\\Delta_{mix}$ we plot in Fig.~\\ref{dmix:32method4} the results of \nmethods~I, II, and III for the ensembles with $a = 0.085\\,{\\rm fm}$.  We note that the results are consistent within \ntwo sigma. This is consistent with LO MAPQ$\\chi$PT in Eq.~(\\ref{eq:dd}) where $\\Delta_{mix}$ is a low energy \nconstant, independent of the valence and sea masses.\n\nWe now combine the $\\Delta_{mix}$ values extracted from the two lightest sea mass ensembles to produce our final result.\nWe use these ensembles because the LO MAPQ$\\chi$PT is expected to describe the data better at lower sea quark masses.\nFor each of the fitting methods we combine the results of the $a = 0.114\\,{\\rm fm}$ and $a=0.085\\,{\\rm fm}$ ensembles.\nSince the two ensembles are statistically independent, it is straightforward to combine these results: for method~I we get \n$\\Delta_{mix} = 0.033(6)\\,{\\rm GeV}^4$ and for method~II we get $\\Delta_{mix} = 0.027(5)\\,{\\rm GeV}^4$. We can now use the values\ndetermined using these two methods to estimate the systematic fitting errors. \nWe quote the final result with two uncertainties, the first statistical and the second associated with fitting systematics.\nThe central value and the statistical error are taken to be the average of the results from methods~I and II. \nThe systematic error is the standard deviation of the results from these two methods. \nWe get $\\Delta_{mix}=0.030(6)(5)\\,{\\rm GeV}^4$.\n\n\n\\begin{figure}[t]\n\\includegraphics[width= 3.4in]{3264_vs_chiral.pdf} \n\\caption{$\\Delta_{mix}$ for $ a= 0.085\\,{\\rm fm}$. The empty symbols indicate the results of method I and the full symbols are from method II. The continuous line and the shaded region are the results of the chiral extrapolation in method III.}\n\\label{dmix:32method4}\n\\end{figure}\n\n\n\n\n\nTable \\ref{dmix:others} lists calculated values of $\\Delta_{mix}$ for\npion masses close to $300\\,{\\rm MeV}$ using different mixed actions.  We\nnotice that our values of $\\Delta_{mix}$ are  significantly smaller\nthan overlap on clover or DWF on Asqtad for comparable lattice spacings and pion masses. \nFor both $m_{vv}$ and $m_{ss}$ at $300\\,{\\rm MeV}$, our calculated $\\Delta_{mix}$ will\nshift the pion mass up by $10$ and $16\\,{\\rm MeV}$ for the $32^3 \\times 64$ lattice at\n$a = 0.085\\,{\\rm fm}$ and $24^3 \\times 64$ lattice at $a = 0.114\\,{\\rm fm}$, respectively.\nThese are substantially smaller than the corresponding $55-153\\,{\\rm MeV}$ and\n$110-240\\,{\\rm MeV}$ shifts that we mentioned in Sec. I.\nThe value of $\\Delta_{mix}$ is significantly smaller in our case probably because \nthe sea and valence fermion actions are similar.  Both of them are approximations of \nthe matrix sign function, but with different kernels.\n\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n~~~~Mixed Action& Ref. & $a\\,{\\rm fm}$&$\\Delta_{mix}(\\,{\\rm GeV}^4)$\\\\\n\\hline\n\\hline\nDWF on staggered& \\cite{Orginos:2007tw}&0.125& 0.249(6)\\\\\nDWF on staggered&\\cite{Aubin:2008wk}& 0.12& 0.211(16)\\\\\nDWF on staggered& \\cite{Aubin:2008wk}& 0.09& 0.173(36)\\\\\noverlap on clover& \\cite{Durr:2007ef}&0.09&0.55(23)\\\\\noverlap on DWF    & this work & 0.114 & 0.030(6)\\\\\noverlap on DWF    & this work & 0.085& 0.033(12)\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{$\\Delta_{mix}$ values for DWF valence quarks on staggered sea quarks for pion mass at $300\\,{\\rm MeV}$.}\n\\label{dmix:others}\n\\end{table}\n\nThe previous calculation \\cite{Li:2010pw} of $\\Delta_{mix}$, for\noverlap on the DWF sea, has roughly the same value but the sign is\nnegative. That calculation measured $\\Delta_{mix}$ by examining the\nstates that wrap around the time boundary.  This indirect \nmethod is less reliable and the errors are large. \n\n\\section{Conclusion}\nWe calculated the additive mixed action pseudoscalar meson mass parameter,\n$\\Delta_{mix}$, for the case of valence overlap fermions on a DWF sea.\n$\\Delta_{mix}$ is significant because it enters\ninto mixed action partially quenched chiral perturbation theory (MAPQ$\\chi$PT) for\nchiral extrapolation of many low energy observables.\n\nTwo different lattice spacings were used to examine the cut-off behavior.\nFor a pion mass close to $300\\,{\\rm MeV}$ we find $\\Delta_{mix}=0.030(6)\\,{\\rm GeV}^4$ at $a =0.114\\,{\\rm fm}$ and \n$\\Delta_{mix}=0.033(12)\\,{\\rm GeV}^4$ at $a =0.085\\,{\\rm fm}$.  They are the same within errors. \nOur calculated $\\Delta_{mix}$ will\nshift the pion mass up by $10$ and $16\\,{\\rm MeV}$ for the $32^3 \\times 64$ lattice at\n$a = 0.085\\,{\\rm fm}$ and $24^3 \\times 64$ lattice at $a = 0.114\\,{\\rm fm}$, respectively.\nWe studied the sea quark mass dependence of $\\Delta_{mix}$ at $a = 0.085\\,{\\rm fm}$ and\nwe find that they agree within two sigma. Combining the results extracted from\nthe ensembles with the lightest sea quarks, we get $\\Delta_{mix}=0.030(6)(5)\\,{\\rm GeV}^4$,\nwhere the first error is statistical and the second is the systematic error\nassociated with the fitting method.\n \nWhen compared to previous mixed action studies, DWF on staggered or overlap on clover, the values of $\\Delta_{mix}$ of overlap on  DWF\nare almost an order of magnitude smaller. This is most likely due to the fact that the sea and valence fermions\nare similar. \n\n\\begin{acknowledgements}\nWe would like to thank J. Negele, A. Pochinsky, M. Engelhardt, and\nLHPC for generously making available the DWF propagators for all the\nensembles used in this paper. This work is supported in part by\nU.S. Department of Energy grants DE-FG05-84ER40154,\nDE-FG02-95ER40907, GW IMPACT collaboration, DE-FG02-05ER41368, and DST-SR\/S2\/RJN-19\/2007, India.\n\\end{acknowledgements}\n\n\\newpage\n\\bibliographystyle{jhep-3}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nThe compensation of quantum numbers plays a key role in our understanding of the\nfragmentation process whereby partons transform into observable hadrons. \nConsequently, baryon production in hadronic \\ensuremath{{\\rm e}^+{\\rm e}^-}\\ annihilation final states\nprovides data very well suited to test \nphenomenological fragmentation models. In particular, the study of di-lambda \npairs allows a subtle testing of model predictions because of the relatively \nlarge rates and the necessity to compensate two quantum numbers:  \nbaryon number and strangeness.  \n\nFragmentation models such as \\jt \\cite{jt} and \\hw \\cite{hw} are based on\na chainlike production of hadrons with local compensation of quantum numbers.\nIn \\jt, particle production is implemented via string fragmentation. Baryons (B)\nare formed when a diquark pair is contained in the string \n(see diagram a below), thus resulting in a strong baryon-antibaryon \ncorrelation. This correlation can be softened by the ``popcorn effect'' when \nan additional meson (M) is produced between the baryon pair as shown in \nthe diagrams b and c below.\nIn contrast, \\hw\\ describes fragmentation via the formation of clusters and \ntheir subsequent decay. Baryons are produced by the isotropic cluster decay \ninto a baryon pair, which can result in stronger correlations than those \npredicted by \\jt .\n\n\\vspace*{-2.5cm}\n\\begin{center}\n  \\resizebox{\\textwidth}{!}{\n    \\includegraphics{pr251_diagr.eps}\n    }\n\\end{center}\n\\vspace{-2.5cm}\n\nDi-lambda production in multihadronic \\ensuremath{{\\rm Z}^0}\\ decays has been studied over the \npast years by experiments at {\\sc Petra}, {\\sc Pep} and \\lep \n\\cite{petra,pep,aleph_corr,delphi_corr,opal_corr}. These experiments \nreport short-range correlations as observed in the distributions of the \nrapidities $y$ or rapidity differences \\ensuremath{|\\Delta y|}\\ of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs.\nThe rapidity of a particle is defined as\n$y=\\frac{1}{2}\\ln\\left(\\frac{E+p_{\\scriptscriptstyle\\parallel}}{E-\np_{\\scriptscriptstyle\\parallel}}\\right)$, \nwhere $E$ is the energy of the particle and $p_{\\scriptscriptstyle\\parallel}$ \nthe longitudinal momentum with respect to the thrust axis. Rapidity differences \nare Lorentz-invariant under boosts along the event axis.\nThese correlations are compared to predictions of \\jt\\ and \\hw . \nSatisfactory agreement is found with the predictions of \\jt\\ when $\\rho$, the \n``popcorn parameter''\\footnote{The value of $\\rho$ can be set in \\jt\\ with the \nparameter PARJ(5): $\\rho=\\frac{BM\\bar{B}\\rule{0cm}{2.5ex}}{B\\bar{B}+BM\\bar{B}} = \n\\frac{\\rm PARJ(5)}{0.5+{\\rm PARJ(5)}}$.},\nis set to the default value, $\\rho=0.5$. However, the tune of other parameters \nmodeling baryon production significantly influences the predictions \n\\cite{james_early}. \n\\hw\\ on the other hand predicts correlations much larger than those \nexperimentally observed.\n\nThe full data sample of 4.3~million hadronic \\ensuremath{{\\rm Z}^0}\\ decays collected with \nthe {\\sc Opal}\\ detector at \\lep\\ in the region of the \\ensuremath{{\\rm Z}^0}\\ peak is used in this \ninvestigation. It supplements the earlier {\\sc Opal}\\ \nwork\\cite{opal_corr} by increased statistics and a more robust technique  \nto remove the background contributions from the \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ \nsamples in order to obtain a correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ sample which is as clean as \npossible. \nThe \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations are investigated mainly via rapidity differences. \nThey are compared to the earlier \\lep\\ results and to the predictions of  \n\\jt\\ and \\hw. The predictions of the recent \\jt\\ modification \n{\\sc Mops}\\ (MOdified Popcorn Scenarium)~\\cite{mops} are also considered. \nCorrelated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs are also studied in 2-jet events in which models can\nbe tested with improved sensitivity (compared to the full data sample)\nwhen rapidity differences are investigated. \nFinally, we study correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs within the same and within different\njets. \n\nSection~\\ref{sec:procedure} gives a short description of the {\\sc Opal}\\ detector \nand\npresents the selection of the \\l\\ events\\footnote \n{For simplicity \\l\\ refers to both \\l\\ and \\ensuremath{\\bar{\\Lambda}}.}\nin the total sample and also in 2-jet and 3-jet events, for both experimental \nand simulated data.\nIn section~\\ref{sec:method} the separation of the  \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and  \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ samples \nfrom the background and the determination of the rates of correlated \n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs as a function of the rapidity differences \\ensuremath{|\\Delta y|}\\ are discussed. \nSection~\\ref{sec:results} contains the measured rates with their errors and a \ncomparison to earlier results as well as the presentation of the differential \ndistributions as a function of \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*}, where \\ensuremath{\\theta^*}\\  is the angle \nbetween the thrust axis and the \\l\\ momentum calculated in the rest frame of \nthe di-lambda pair.\nIn section~\\ref{sec:comparison} the models are tested using the production \nrates of \\l\\ pairs as well as the \\ensuremath{\\cos \\theta^*}\\ and \\ensuremath{|\\Delta y|}\\ spectra of correlated \n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs. \nThe range of di-lambda correlations is investigated in section~\\ref{sec:jets}\nby the assignment of the \\l's to the jets. \nConclusions are drawn in section~\\ref{sec:summary}.\n\\section{Experimental Procedure}\n\\label{sec:procedure}\n\\subsection{The OPAL Detector}\n\\label{subs:detec}\nA detailed description of the {\\sc Opal}\\ detector can be found in \nRef.~\\cite{detector}. \nOf most relevance for the present analysis is the tracking system and \nthe electromagnetic calorimeter. \nThe tracking system consists of a silicon microvertex detector, an inner\nvertex gas chamber, a large-volume jet chamber and specialized chambers at\nthe outer radius of the jet chamber which improve the measurements in\nthe $z$ direction ($z$-chambers)\\footnote\n{The coordinate system is defined so that $z$~is the coordinate parallel \nto the e$^-$ beam axis, $r$~is the coordinate normal to the beam axis, \n$\\phi$~is the azimuthal angle around the beam axis, and $\\theta$~is the \npolar angle \\mbox{with respect to~$z$.}}. The tracking system covers the region\n$|\\cos\\theta|<0.95$ and is located within a solenoidal magnet coil with\nan axial field of~0.435~T. \nThe tracking detectors provide momentum measurements of charged\nparticles, and particle identification from measurements of the\nionization energy loss,  d$E$\/d$x$.\nElectromagnetic energy is measured by a lead-glass calorimeter \nlocated outside the magnet coil, which covers $|\\cos\\theta|<0.98$.\n\\subsection{Data Samples}\n\\label{subs:data}\nThe analysis is based on hadronic \\ensuremath{{\\rm Z}^0}\\ decays collected around the \n\\ensuremath{{\\rm Z}^0}\\ peak from 1990 to 1995 (total \\lep\\ 1 statistics). The hadronic events \nwere selected with the standard {\\sc Opal}\\ procedure~\\cite{tkmh} based on the \nnumber and quality of the measured tracks and the electromagnetic clusters and \non the amount of visible energy in the event. In addition, events with the \nthrust axis close to the beam direction were rejected by requiring \n$|\\cos \\theta_{\\rm {thrust}}|< 0.9$, where $\\theta_{\\rm thrust}$ is the polar \nangle of the thrust axis. With the additional requirement that the jet chamber\nand the z-chambers were fully operational, a total of 3.895 million hadronic \nevents remained for further analysis, with an efficiency of ($98.4 \\pm 0.4$)\\%. \nThe remaining background processes, \nsuch as $\\ensuremath{{\\rm e}^+{\\rm e}^-} \\rightarrow \\tau^+ \\tau^-$ and two photon events, were estimated \nto be at negligible level (0.1\\% or less).\n\nAfter the \\l-selection which will be described below, the selection of 2- and \n3-jet events was performed. Charged tracks and electromagnetic \nclusters not associated with any track were grouped into jets using the \n\\ensuremath{{\\rm k}_{\\perp}}\\ recombination algorithm \\cite{durham} with a cut value\n$y_{\\rm cut}=0.005$. In addition to the standard selection criteria, the\nenergy of the clusters and the momenta of the charged tracks had to be less \nthan 60~GeV\/$c$.\nTo improve the quality of the jets it was finally required that there be at \nleast two charged particles per jet (in addition to the possible tracks from \n\\l\\ decays) and that the minimum energy per jet was $5$~GeV.\nThe cuts on the quality of jets were chosen to be this loose to keep the \nkinematic range as large as possible for comparison with fragmentation models. \nIn total, samples of 1.7 million 2-jet events and 1.4 million 3-jet events \nwere available for further analysis corresponding to $45\\%$ and $36\\%$,\nrespectively, of the entire data set.\n\\subsection{Monte Carlo Event Samples} \n\\label{subs:mc} \nMonte Carlo hadronic events with a full simulation of the OPAL detector \n\\cite{gopal} and including initial-state photon radiation were used \n(a) for evaluation of detector acceptance and resolution \nand (b) for studying the efficiency of the di-lambda reconstruction as a \nfunction of the rapidity differences.  \nIn total, seven million simulated events were available, of which four million\nwere generated by \\jt~7.4 with fragmentation parameters  \ndescribed in~\\cite{jt7.4}, and three million were generated by \\jt~7.3  with  \nfragmentation parameters described in~\\cite{jt7.3}.\nThe two \\jt\\ versions differ in the particle decay tables and heavy meson  \nresonances. \nThere are also some differences in the simulation of baryon production between \nthe two samples.\nTheir small influence on the efficiency correction to the experimental \ndata is accounted for in the systematic errors (see \nsection~\\ref{subs:syserr}). \n\nFor comparison with the experimental results, the Monte Carlo models \n\\jt~7.4 and \\hw~5.9\\cite{hw5.8}\\footnote \n{The fragmentation parameters of \\hw~5.9 were identical to those used in  \n our tuned version of \\hw~5.8\\cite{hw5.8} with the exception of the maximum\n cluster mass ({\\tt CLMAX}) which was set to 3.75 GeV in order to improve the  \n description of the mean charged particle multiplicity in inclusive  \n hadronic \\ensuremath{{\\rm Z}^0}\\ decays.} \nwere used. Both models give a good description of global  \nevent shapes and many inclusive particle production rates, but differ in  \ntheir description of the perturbative phase and their implementation of the  \nhadronization mechanism. \n\nTracks and clusters are selected in the Monte Carlo events, which \ninclude detector simulation, in the same way as for the data, and the \nresulting four-vectors of particles are referred to as being at the \n`detector level'.  Alternatively, for testing the model predictions, \nMonte Carlo samples without \ninitial-state photon radiation nor detector simulation are used, with \nall charged and neutral particles with mean lifetimes greater than \n$3\\times10^{-10}$~s treated as stable.  The four-vectors of the \nresulting particles are referred to as being at `generator level'. \n\\subsection{\\l\\ Reconstruction } \n\\label{subs:lambda} \nNeutral strange \\l\\ baryons were reconstructed in their decay  \nchannel \\l ~$\\rightarrow \\pi^-$p as described in \\cite{opal_sp}. \nBriefly, tracks of opposite charge were paired and regarded as a secondary  \nvertex candidate if the track pair intersection in the plane  \nperpendicular to the beam axis satisfied the criteria of a neutral two-body  \ndecay with a decay length of at least 1 cm.\n \nEach candidate track pair was refitted with the \nconstraint that the tracks originated from a common vertex, and \nbackground from photon conversions was suppressed. Information from  \nd$E$\/d$x$ measurements was used as in \\cite{opal_sp} to help identify the \n$\\pi$ and p for further background suppression, primarily due to \n\\ensuremath{{\\rm K}^{0}_{\\rm S}} $\\rightarrow \\pi^+ \\pi^-$. Two sets of cuts, called `method 1' and \n`method 2' are described in \\cite{opal_sp} for \\l\\ identification.   \nFor the present analysis, \\l\\ candidates were reconstructed using method~1,  \nwhich is optimized to have good mass and momentum resolution. \n\n\\begin{sloppypar} \nBy these means a narrow \\l\\ mass peak above a small background has been \nobtained. The selection of di-lambda candidates with both invariant masses in  \nthe range \\mbox{1.1057~GeV\/$c^2<$ $m_{\\pi p}<$ 1.1257~GeV\/$c^2$} (region A in \nfigure~\\ref{fig:m2dim}) retains most of the \\l\\ signal for further analysis. \n\\end{sloppypar}\n\\section{Selection of Correlated \\l-pairs}\n\\label{sec:method}\n\\subsection{Method}\n\\label{subs:mgeneral}\nEvents with more than one \\l\\ candidate that had passed the above selection \ncriteria were considered and all possible pair combinations of the \\l\\ and \n\\ensuremath{\\bar{\\Lambda}}\\ baryons within an event were formed. This resulted in pairs of \\ensuremath{\\Lambda\\bar{\\Lambda}}, \n\\l\\l\\ and \\ensuremath{\\bar{\\Lambda}}\\lb. Combinations were rejected if the pair had a track in \ncommon. The remaining pairs are henceforth referred to as \\l-pair candidates. \n\nThe three types of baryon pairs can be grouped into two classes: pairs with \ndifferent baryon numbers \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and pairs with equal baryon numbers \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}. \nOnly in \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs can the baryon and flavor quantum numbers be compensated \nby correlated production. \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs can never be produced in correlation\nand hence they will occur only in events with more than one baryon-antibaryon \npair (B$\\bar{\\rm B}$). In such events uncorrelated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs from \ndifferent (B$\\bar{\\rm B}$) pairs are also possible. The number of uncorrelated \n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs corresponds to the number of pairs with same baryon number. \nHence, the number of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs can be derived via \n\\begin{equation}\nN_{\\ensuremath{\\Lambda\\bar{\\Lambda}}}^{\\rm corr.} = N_{\\ensuremath{\\Lambda\\bar{\\Lambda}}} - (N_{\\ll} + N_{\\ensuremath{\\bar{\\Lambda}\\bar{\\Lambda}}})\\: .\n\\end{equation}\nAt this stage 9479 \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and 4217 (\\l\\l+\\ensuremath{\\bar{\\Lambda}}\\lb) pair candidates are \nselected.\n\\subsection{Background Subtraction and Efficiency Correction}\n\\label{subs:backgr} \nDue to the small statistical errors it is necessary to keep systematic \nuncertainties as low as possible in this analysis. \nThe correct subtraction of {\\mbox{non-\\l\\ }} background from the pairs is \ntherefore of  particular importance. This background consists mainly of other \nlong-lived particles with similar decay topologies (namely \\ensuremath{{\\rm K}^{0}_{\\rm S}} $\\rightarrow\n\\pi^+ \\pi^-$) and random \ntrack  combinations. An important contribution to the contamination is the \nso-called correlated background from \\l candidates that have been reconstructed \nwith one false decay track.   \nThey are more numerous in pairs with opposite baryon number because the \nnumber of \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs is far higher than the number of \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs.  \nFor this reason the background has to be estimated in the two samples  \nseparately. \nBackground pairs occur when either one or both \\l-candidates are fake.\nIn the two-dimensional mass plane in figure~\\ref{fig:m2dim}, pairs with one \nfake \\l\\ form horizontal and vertical bands of background, while pairs with \ntwo fake candidates are uniformly distributed in the region above the lower \nmass bounds. \n\nThe background was subtracted using a two-dimensional sideband method. The  \nbackground in the signal region A was measured from two mass windows \n(sidebands) of the same size (regions B$_1$ and B$_2$) placed \nin the two bands of background.  \nIn this way the background with two fake candidates is counted twice. The latter\nwas determined from region C outside the bands.\nHence, the signal is obtained with the subtraction:  \n\\begin{center} \n   Signal = $N_{\\rm A} - (N_{\\rm B_1} + N_{\\rm B_2} - N_{\\rm C})$ .   \n\\end{center}\nWe optimized the position of the sidebands with a MC test investigating the\ndeviations between the background-corrected sample and the true-\\l\\ sample.  \nThe stability of this method was tested in the experimental data by shifting \nthe position of the sidebands by one half of the band size from the optimized \nposition. The fluctuations were of the same size as the deviations found in \nthe MC. \n\nFinally the background-corrected \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ signal distributions were \ncorrected for detector acceptance and reconstruction efficiency as functions \nof \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*} . The average efficiency in the total hadronic sample \nwas found to be $\\approx 2\\%$, varying between 1.3\\% and 2.5\\% over the \n\\ensuremath{|\\Delta y|}\/\\ensuremath{\\cos \\theta^*}\\ range.\n\\section{Experimental Results}\n\\label{sec:results}\nIn an earlier OPAL paper~\\cite{opal_corr} based on the 1990 and 1991 data \nsamples we already investigated the production dynamics of baryon-antibaryon \npairs. \nIn this section, we present the rates and differential distributions of \n\\l~pairs using the full 1990 to 1995 LEP~1 data in three samples: \nthe entire set of multihadronic events, the 2-jet and the 3-jet events. \n\\subsection{Pair Production Rates}\n\\label{subs:rates}\nThe resulting rates for \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs in all hadronic events, \ndetermined as sum over all corrected \\ensuremath{|\\Delta y|}\\ bins, are \ngiven in table~\\ref{tab:rates}. The rates for the correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs \nare derived according to equation (1) from the difference of the opposite \nand same baryon number pairs. Compared to the results from other \\lep\\ \nexperiments and to the previous {\\sc Opal}\\ publication, good agreement is found. \n\nThe di-lambda rates in 2- and 3-jet events are listed in  \ntable~\\ref{tab:2_3jrates}.\nIn 3-jet events, due to the higher color charge of the gluons, the average \npair multiplicity is higher. \n\\subsection{Differential Distributions}\n\\label{subs:distributions}\nWe studied the correlations in the differential \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ spectra using the \nobservables \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*}\\ as they are particularly sensitive for  \ncomparison with Monte Carlo models.\nThe differential distributions are shown in figure~\\ref{fig:distrib}.\nThe short-range correlations show up as a peak in the region \\ensuremath{|\\Delta y|}\\ $\\le$ 2.0.\n\nWhen investigating \\ensuremath{|\\Delta y|}\\ distributions, we will restrict ourselves to \n2-jet events.\nThis is due to the fact that in 3-jet events many particle momenta have large \nangles to the thrust axis, resulting in smaller longitudinal momenta and \nsmaller rapidity differences, independent of correlations. As a result the \\ensuremath{|\\Delta y|}\\ \ndistribution is broader and less steep in 2-jet events than in the 3-jet or the \ntotal sample (see figure~\\ref{fig:distrib}a). Consequently, also the range of \nvariations is larger in 2-jet events and yields a higher sensitivity in the \ncomparison with model predictions.\n\\subsection{Systematic Errors}\n\\label{subs:syserr}\nThe systematic error is found to be largely independent of \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*} ,  \nand in the subsequent discussion of the differential distributions of \nthe correlated pairs, only normalized distributions are considered. These are \nlargely insensitive to effects of systematic uncertainties. Consequently, the \nsystmatic errors discussed below are mainly relevant for the total rates.\n\nFor the determination of the experimental uncertainties we considered the \nfollowing sources of systematic effects:\n\\begin{itemize}\n\\item Uncertainties due to the subtraction of background via the \n  sidebands. These were estimated using simulated events by applying the \n  analysis to the fully detector simulated MC and comparing the rate from the \n  background corrected sample to the true number.\n\\item Efficiency uncertainties. These were estimated from the difference of \n  the results when the efficiency\n  correction was done using both \\jt\\ versions 7.3 \n  and 7.4 samples in combination and using them separately.\n\\item The statistical error of the efficiency due to the limited sample size\n  of the simulated events at detector level.\n\\item Uncertainties in the modelling of the cut variables used for the \\l\\\n  selection. This error is taken from a former analysis \n  \\cite{opal_sp} where it was determined very precisely for single \\l 's. \n  The error given there is doubled for the \\l\\ pairs in the present \n  analysis.\n\\end{itemize}\n\nThese effects contribute to the total systematic error as shown in\ntable~\\ref{tab:syserr}, where the relative systematic errors from the \ndifferent sources are compared to the total systematic as well as to the \nstatistical error. Statistical and total systematic errors contribute about \nequally.\n\\section{Comparison with Fragmentation Models}\n\\label{sec:comparison}\n\nWe start the discussion with the numbers and distributions of the models with\nOPAL default tunes that optimize the general performance of the models and the \nagreement with the measured single particle rates. \n\n\\subsection{Pair Production Rates}\n\\label{subs:comp_rates}\n\nWe investigate the di-lambda rates first in the total hadronic data sample\ncomparing the measured rates to the predictions of the models \\jt~7.4, {\\sc Mops}\\ \nand \\hw~5.9 (see table~\\ref{tab:rates}). None of the models gives a perfect \ndescription of the data but \\hw\\ clearly exhibits the largest disagreement. \n\nThe comparison of the di-lambda rates in 2- and 3-jet events is given in \ntable~\\ref{tab:2_3jrates}. The higher multiplicity in 3-jet events compared to \n2-jet events is qualitatively well described by all three models. \nHowever, only \\jt\\ yields a prediction compatible with the measured numbers.\nIn the 2-jet event sample the agreement is excellent.\nIn the 3-jet sample all the measured rates exceed the \\jt\\ predictions. \nThis can be compared to the observation that \\l\\ rates in gluon jets are too low\nin \\jt~\\cite{opal_jets} .\n\n\\subsection{Differential Distributions}\n\\label{subs:comp_distributions}\n\nTo further investigate the nature of the \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations we \ncompare the differential distributions of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ pairs with the \npredictions of the various models. We use the variables \\ensuremath{\\cos \\theta^*}\\ and \\ensuremath{|\\Delta y|}\\ \nand test their sensitivity to distinguish between the different fragmentation \nmodels and baryon production mechanisms. All distributions are of the type \n$ \\frac{1}{N} \\frac{dN}{d(\\ensuremath{|\\Delta y|})}$, $N$ being the total number of entries. \nThis has the advantage that they are independent of the total rates and \nthat the systematic errors mostly cancel out, since they are nearly independent \nof both \\ensuremath{|\\Delta y|}\\ and \\ensuremath{\\cos \\theta^*} .\n\nThe angle \\ensuremath{\\theta^*}\\ is particularly suited to distinguish between string and \ncluster fragmentation.  The mostly isotropic cluster decay (\\hw) results in a \nrelatively flat \\ensuremath{\\cos \\theta^*}\\ distribution  whereas string fragmentation produces \nthe correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ system predominantly close to the thrust axis, i.e., with \n$\\ensuremath{\\cos \\theta^*} \\approx 1$. These predictions are compared to the measurement \nin figure~\\ref{fig:comp_theta}. The data show a distribution that is strongly \npeaked towards \\ensuremath{\\cos \\theta^*} = 1 and therefore clearly rule out the \\hw\\ cluster \nmodel. \nThe predictions of {\\sc Mops}\\ agree somewhat better with the experimental \ndistribution but they also fail to model the forward peak correctly. \nOnly \\jt\\ yields a good description of the data.\n\nOn the other hand, especially in 2-jet events, the rapidity difference \\ensuremath{|\\Delta y|}\\ is \nmore sensitive to show differences in the strength of the correlations.\nThe experimental data and model predictions are compared in \nfigure~\\ref{fig:compall_2j}.\nAgain \\jt\\ gives the best, albeit not completely satisfactory, description of \nthe measured distribution. \\hw\\ generates correlations which are far too \nstrong .\nThe {\\sc Mops}\\ model with its built-in facility to allow for several ``popcorn \nmesons'' should yield weaker correlations than \\jt ; however, in contrast to \nthis naive expectation it produces a narrower \\ensuremath{|\\Delta y|}\\ distribution, i.e. \nstronger correlations. \nWe see the following possible reasons for this: first of all, and different \nfrom \\jt , a new kinematic property is built into {\\sc Mops}: the \nlow-$\\Gamma$-suppression\\cite{mops}. This suppresses popcorn fluctuations \nat early times in the color field, resulting in very strong correlations. \nSecondly, it appears that the strength of the correlations is influenced more\nby the rate of baryon production via the popcorn mechanism than by the actual \nnumber of intermediate mesons produced.  \nAs \\jt\\ and {\\sc Mops}\\ are tuned to show the same mean number of popcorn mesons \ninstead of popcorn systems, {\\sc Mops}\\ has fewer popcorn systems and therefore \nstronger correlations.\n\n\\subsection{Tuning of Models}\n\\label{subs:tuning}\n\nIn an earlier {\\sc Opal}\\ analysis of strange baryons\\cite{james_early} it was \nobserved that the agreement between experimental data and \\jt\\ predictions\ncan be improved by adjusting some of the diquark parameters: \nimproving the predicted shape of the \\ensuremath{|\\Delta y|}\\ distribution was possible by varying \nthe popcorn parameter, $\\rho$=PARJ(5), that influences the frequency of \npopcorn production and hence the correlation strength. It was found that \n$\\rho$ acts on both the shape of the rapidity spectrum and the production \nrates.\nTwo other parameters were used to correct for this change of predicted \nmultiplicities:\nthe ratio of the strange to non-strange diquarks over strange to \nnon-strange quarks, (us:ud\/s:d) = PARJ(3), and the ratio of spin-1 to spin-0 \ndiquarks, $(1\/3 \\cdot [{\\rm qq}]_1\/[{\\rm qq}]_0)$=PARJ(4). \nThese last parameters affect mainly the rates and leave the spectra nearly\nunmodified.\nWhen attempting to improve the predictions of the {\\sc Mops}\\ model in the same \nmanner, the most direct correspondence to $\\rho$ in \\jt\\ is the \n{\\sc Mops}\\ parameter PARJ(8)=$\\beta({\\rm u})$, the transverse mass of an \nintermediate u-quark. The higher the transverse mass of the intermediate \nsystem (with several quark pairs possible), the lower the probability to \nproduce this popcorn system and the stronger the correlations. \nTherefore, in both \\jt\\ and {\\sc Mops}\\ we tried first to improve the \nagreement with the data distributions by tuning \nthe parameters that influence \nthe correlation strength (figure~\\ref{fig:compjt_tunes} for \\jt .) \nIn \\jt , the popcorn probability $\\rho$ was varied from 0\\%-90\\%, while in \n{\\sc Mops}\\ the transverse mass of a u-quark, $\\beta({\\rm u})$=PARJ(8), was altered \nbetween 0.2 and 1.0~GeV$^{-1}$. \nAll other parameters remained at the {\\sc Opal}\\ default values.\nAs can be seen from table~\\ref{tab:jtrates_tune}, for the $\\rho$ parameter,\nthese variations affect not only the shape of the \\ensuremath{|\\Delta y|}\\ distribution but \nalso the di-lambda rates, as expected.\n\nThe predictions with the different popcorn parameter values in \\jt\\ are \ncompared to the data in figure~\\ref{fig:compjt_tunes}a.\nOnly the results from parameter settings above the default value \nof $\\rho = 0.5$ are shown, since lower values give a poorer agreement with the \ndata. \nThe best agreement \nis found in the range $0.6<\\rho<0.8$.\nPopcorn values within this range also yield good agreement between data and \npredictions for the \\ensuremath{\\cos \\theta^*}\\ distribution.\nHowever, when the influence of the popcorn parameter on the predicted\ndi-lambda rates is also considered (table~\\ref{tab:jtrates_tune})  \nuse has to be  made of the other two \\jt\\ parameters that affect the strange \nbaryon production in order to tune the rates back to values corresponding to\nthe measurement. It can be seen from table~\\ref{tab:jtrates_tune} and \nfigure~\\ref{fig:compjt_tunes}b that such\na tune clearly produces a better agreement with the rates (also for single \nparticle production) while it does not change the spectra of rapidity \ndifferences significantly. Using the results of other {\\sc Opal}\\ analyses,\nit can also be seen that the tune does not change the strange meson (\\ensuremath{{\\rm K}^{0}_{\\rm S}})\nrate, nor does it affect the non-strange baryon (p) rate significantly.\nThe known problems~\\cite{james_early} in modeling the decuplet baryon rates \nare also seen here.\nNo further attempt has been made to globally optimize the parameter set, \nhowever. \n\nThe tune of parameter PARJ(8) in {\\sc Mops}\\ did not result in an improvement. \nAlthough the value of the parameter was varied in a comparatively wide \nrange, the effect on the \\ensuremath{|\\Delta y|}\\ distribution was almost imperceptible. \nPARJ(8) clearly is not suited to adjust the {\\sc Mops}\\ model to the data. \nTherefore, we tested another parameter of the model using the relative \ndifference between the fragmentation function $f(z)$ for baryons and mesons, \nthe parameter PARJ(45). Again, the variation did not notably change the shape \nof the distribution. \nThis relatively poor performance of the {\\sc Mops}\\ Monte Carlo in describing \nthe \\ensuremath{|\\Delta y|}\\ dependent \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations seems to be connected to the known \nshortcomings of the model in describing  $p_{\\perp}$-related \ndistributions~\\cite{mops}.\n\\section{Di-lambdas in Jets}\n\\label{sec:jets}\nAfter studying the strength of \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations in the \\ensuremath{|\\Delta y|}\\ spectra, we will \nnow present results on the range of the correlations by assigning \nboth partners from a correlated pair to the reconstructed jets in an event. \nFor short-range correlations both partners are expected \nwithin the same jet whereas long-range correlations (which can be obtained by \nthe production of baryons from the primary quarks) should result in an \nassignment to different jets. \nWe use the following two classifications for the assignment study: \nboth partners within the same jet, and each partner in a different jet. \n\nDue to the fact that it is impossible to map 2-jet events at detector level \nto 2-jet events at generator level, we do not attempt to apply efficiency \ncorrections but compare our uncorrected results with the \\jt\\ predictions at \ndetector level.\nWe count the number of \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ and \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ pairs in \neach sample and obtain the number of correlated pairs again from the \nrelation \n\\mbox{$N_{\\ensuremath{\\Lambda\\bar{\\Lambda}}}^{\\rm correlated} = N_{\\ensuremath{\\Lambda\\bar{\\Lambda}}} - (N_{\\ll} + N_{\\ensuremath{\\bar{\\Lambda}\\bar{\\Lambda}}})$}.\nThe amount of background in like- and unlike-sign pairs approximately cancels \nout in this subtraction as long as the contribution from the correlated \nbackground (see section 3.2) can be neglected. \nThe numbers of pairs obtained from the same jet and from different jets are \nlisted in table~\\ref{tab:injets} \nfor both 2- and 3-jet events.\nThe major part of the correlated pairs is reconstructed \nwithin the \nsame jet (about 96\\% in 2-jet events, 81\\% in 3-jet events) whereas only a \nvery small fraction is found in different jets. These experimental numbers are \nin excellent agreement with the \\jt\\ predictions at detector level and support\nthe assumption of short-range compensation of baryon number \nand strangeness in the fragmentation process.\n\\section{Summary}\n\\label{sec:summary}\n\\ensuremath{\\Lambda\\bar{\\Lambda}}\\ correlations have been studied in 4.3 million multihadronic \\ensuremath{{\\rm Z}^0}\\\ndecays, with the correlated sample obtained from the difference: \n\\ensuremath{\\Lambda\\bar{\\Lambda}_{\\rm corr}} =\\ensuremath{\\Lambda\\bar{\\Lambda}}--(\\ll+\\ensuremath{\\bar{\\Lambda}\\bar{\\Lambda}}).\nThe analysis has been performed in terms of \\ensuremath{\\cos \\theta^*}\\ and rapidity differences \n\\ensuremath{|\\Delta y|} . As the rapidity is defined with respect to the event (thrust) \naxis, the sensitivity of the analysis is seen to be higher in 2-jet events.\nTherefore three data samples have been analyzed: the \nentire hadronic event sample, 2-jet events ($45\\%$), and 3-jet events ($36\\%$).\nThe experimental findings have been used to study the baryon production\nmechanism implemented in various phenomenological fragmentation models.\n\nThe following results have been obtained:\n\\begin{itemize}\n\\item  In the full data set, the measured production rates of \\ensuremath{\\Lambda\\bar{\\Lambda}}, \\ensuremath{\\Lambda\\Lambda(\\bar{\\Lambda}\\bar{\\Lambda})}\\ \n  and, consequently, \\ensuremath{\\Lambda\\bar{\\Lambda}_{\\rm corr}}\\ are in good agreement with a previous {\\sc Opal}\\ \n  measurement and results from \\aleph\\ and {\\sc Delphi} , but show significantly \n  smaller errors. \n\\item The \\ensuremath{\\cos \\theta^*}\\ distribution of correlated \\l-pairs is well suited to \n  distinguish between isotropic cluster and non-isotropic string decay, and \n  clearly favours the latter, implemented in \\jt . The predictions of the \n  isotropic cluster model \\hw\\ are ruled out by the data: they do not describe \n  the features of correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ production. \n\\item  The rapidity difference \\ensuremath{|\\Delta y|}\\ is used to study the strength of correlated \n  di-lambda production. The measured distribution \n  exhibits strong local correlations.\n\\item  Satisfactory reproduction of the experimental results is obtained\n  with the predictions of the string fragmentation model \\jt . Improved\n  agreement can be found by tuning some of the default parameters used \n  by {\\sc Opal} . After adjusting the popcorn parameter, to improve the description \n  of the \\ensuremath{|\\Delta y|}\\ spectrum, other parameters, fixing the fraction of diquarks with \n  strangeness and spin1, have to be modified to readjust the predicted rates \n  to the experimental ones. This procedure does not affect the previously \n  optimized \\ensuremath{|\\Delta y|}\\ distribution. \n  The \\hw\\ model cannot describe the measured \\ensuremath{|\\Delta y|}\\ spectra, and the predictions \n  of {\\sc Mops} , a recently published modification of the \\jt\\ model, also fail to \n  reproduce the experimental data, even after some parameter tuning. \n\\item  In the 2-jet and 3-jet event samples it is found that correlated \\ensuremath{\\Lambda\\bar{\\Lambda}}\\ \n  pairs are produced predominantly within the same jet, supporting the \n  assumption of a short-range compensation of quantum numbers.\n  Again, the \\jt\\ predictions are in good agreement with the experimental \n  results. \n\\end{itemize} \nIn conclusion, the analysis of correlated di-lambda pairs proves to be a\nvery effective tool to test fragmentation models. \\jt\\ is the only candidate \nmodel studied, describing the data successfully. \n\n\\bigskip \\bigskip \\bigskip \\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{INTRODUCTION}\n\nThe connection between endomorphisms of factors and families of isometric operators has been explored most notably by Arveson \\cite{arveson}, among others \\cite{brenken1,brenken2,laca,longo}. In \\cite{laca}, Laca determines that given a normal $*$-endomorphism $\\alpha$ of $B(H)$ there exists an $n\\in\\mathbb{N}\\cup\\{\\infty\\}$ and $*$-representation $\\pi:\\mathcal{E}_n\\to B(H)$, where $\\mathcal{E}_n$ denotes the Toeplitz algebra for $n$ orthogonal isometries $v_1,...,v_n$, such that\n\\begin{equation*}\\alpha(T)=\\sum_{i=1}^n\\pi(v_i)T\\pi(v_i)^*\\end{equation*}\nfor each $T\\in B(H)$. The $n$ value is unique but the representation $\\pi$ may differ by automorphisms of $\\mathcal{E}_n$ which arise specifically from unitary transformations of the Hilbert space $\\ell^2(\\{v_1,...,v_n\\})\\subseteq\\mathcal{E}_n$ \\cite[Proposition 2.2]{laca}. \n\nIn \\cite{brenken1} and \\cite{brenken2} Brenken extends the connection by determining that, for a von Neumann algebra which can be decomposed into a direct sum of Type I factors, a certain class of $*$-endomorphisms correspond to representations of certain $C^*$-algebras associated with (possibly infinite) matrices which arise as the adjacency matrices for directed graphs. The $*$-endomorphisms studied by Brenken are, however, required to either be unital \\cite{brenken1} or satisfy a number of restrictive conditions \\cite[pp 25]{brenken2}.\n\nOur results extend the work of Brenken by eliminating the conditions imposed on the $*$-endomorphisms. The fundamental difference in our approach is that we will investigate representations of the Toeplitz algebra of a $C^*$-correspondence, whereas Brenken concerned himself with the so-called relative Cuntz-Pimnser algebras. We recover Brenken's results as a special case when the endomorphisms are assumed unital. Our primary result in this line is Theorem \\ref{theorem3}.\n\nOur results also continue the spirit of Laca's investigations which allow for equivalencies between $*$-endomorphisms to be encoded as transformations of an underlying linear object: a Hilbert space in the case of Laca and a $C^*$-correspondence in the present work. Our primary results along these lines are Theorems \\ref{theorem1} and \\ref{theorem2}.\n\n\\section{PRELIMINARIES}\nFirst we will establish our terminology and notation.\n\n\\begin{defi} A \\emph{graph} is a tuple $E=(E^0,E^1,r,s)$ consisting of a \\emph{vertex set} $E^0$, an \\emph{edge set} $E^1$, and \\emph{range} and \\emph{source} maps $r,s:E^1\\to E^0$.\n\\end{defi}\n\nWe will only consider graphs where $E^0$ and $E^1$ are at most countable.\n\n\\begin{defi} Let $A$ be a $C^*$-algebra. A set $X$ is a \\emph{$C^*$-correspondence over $A$} provided that it is a right Hilbert $A$-module and there is a $*$-homomorphism $\\phi:A\\to L(X)$, where $L(X)$ denotes the space of adjointable $A$-module homomorphisms from $X$ to itself. \n\\end{defi} \n\nGiven a $C^*$-correspondence $X$ over $A$, we will denote the $A$-valued inner product of $x,y\\in X$ by ``$\\langle x,y\\rangle_A$\" (perhaps omitting the $A$) and the right action of $a\\in A$ on $x\\in X$ will be written as ``$x\\cdot a$\". The map $\\phi:A\\to L(X)$ may sometimes be written as $\\phi_X$ for clarity.\n\nOur primary objects of study will be certain $C^*$-correspondences which arise from graphs. The following constructions are due originally to Fowler and Raeburn \\cite[Example 1.2]{fowlerraeburn}, although we will adopt the modern convention for the roles of $r$ and $s$. \n\n\\begin{defi} Given a graph $E$, the \\emph{graph correspondence} $X(E)$ is the set of all functions $x:E^1\\to\\mathbb{C}$ for which $\\hat{x}(v):=\\sum_{e\\in s^{-1}(v)}|x(e)|^2$ extends to a function $\\hat{x}\\in C_0(E^0)$. We give $X(E)$ the structure of a $C^*$-correspondence over $C_0(E^0)$ as follows:\n\\begin{align*}x\\cdot a&:e\\mapsto x(e)a(s(e)),\\\\\n\\phi(a)x&:e\\mapsto a(r(e))x(e),\\\\\n\\langle x,y\\rangle&:v\\mapsto\\sum_{e\\in s^{-1}(v)}\\overline{x(e)}y(e).\\end{align*}\nwhich is to say that $a\\in C_0(E^0)$ acts on the right of $X(E)$ as multiplication by $a\\circ s$ and acts on the left as multiplication by $a\\circ r$.\n\\end{defi}\n\nThe sets $\\{\\delta_e:e\\in E^1\\}$ and $\\{\\delta_v:v\\in E^0\\}$ are dense in $X(E)$ and $C_0(E^0)$, respectively, in the appropriate senses. For $e\\in E^1$ and $v\\in E^0$ we have the following useful relations: $\\langle \\delta_e,\\delta_e\\rangle=\\delta_{s(e)}$, $\\delta_e\\cdot\\delta_v=\\delta_e$ if $v=s(e)$ and is $0$ otherwise, and $\\phi(\\delta_v)\\delta_e=\\delta_e$ if $v=r(e)$ and is $0$ otherwise.\n\n\n\\begin{defi}\\label{toeplitzrepdef} Given a $C^*$-correspondence $X$ over $A$ and given another $C^*$-algebra $B$, a \\emph{Toeplitz representation} of $X$ in $B$ is a pair $(\\sigma,\\pi)$ consisting of a linear map $\\sigma:X\\to B$ and a $*$-homomorphism $\\pi:A\\to B$ such that for all $x,y\\in X$ and $a\\in A$\n\t\\begin{enumerate}\n\t\\item $\\sigma(x\\cdot a)=\\sigma(x)\\pi(a)$,\n\t\\item $\\sigma(\\phi(a)x)=\\pi(a)\\sigma(x)$, and\n\t\\item $\\pi(\\langle x,y\\rangle)=\\sigma(x)^*\\sigma(y).$\n\t\\end{enumerate}\n\\end{defi}\n\nFor a graph correspondence $X(E)$, a Toeplitz representation $(\\sigma,\\pi)$ is determined entirely by the values $\\{\\sigma(\\delta_e):e\\in E^1\\}$ and $\\{\\pi(\\delta_v):v\\in E^0\\}$. Property (iii) of a Toeplitz representation guarantees that $\\sigma(\\delta_e)$ is a partial isometry with source projection $\\pi(\\delta_{s(e)})$.\n\n\\begin{defi}\\cite[Proposition 1.3]{fowlerraeburn} Given a $C^*$-correspondence $X$ over $A$, the \\emph{Toeplitz algebra of $X$}, denoted $\\mathcal{T}_X$, is the $C^*$-algebra which is universal in the following sense: there exists a Toeplitz representation $(\\sigma_u,\\pi_u)$ of $X$ in $\\mathcal{T}_X$ such that if $(\\sigma,\\pi)$ is another Toeplitz representation of $X$ in a $C^*$-algebra $B$ then there exists a unique $*$-homomorphism $\\rho_{\\sigma,\\pi}:\\mathcal{T}_X\\to B$ such that $\\sigma=\\rho_{\\sigma,\\pi}\\circ\\sigma_u$ and $\\pi=\\rho_{\\sigma,\\pi}\\circ\\pi_u$.\n\\end{defi}\n\nThat $\\mathcal{T}_X$ exists was proven by Pimnser in \\cite{pimsner}.\n\nGiven a graph $E$ we may consider the Toeplitz algebra of its graph correspondence, cumbersomely denoted $\\mathcal{T}_{X(E)}$. Unless there is danger of confusion, we will abuse notation and make no distinction between elements of $X(E)$ and $C_0(E^0)$ and their images in $\\mathcal{T}_{X(E)}$ under the universal maps $\\sigma_u$ and $\\pi_u$. \n\nIf $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$ is a $*$-representation then, for each $e\\in E^1$, $\\tau(\\delta_e)$ is a partial isometry with source projection $\\tau(\\delta_{s(e)})$ and range projection contained in $\\tau(\\delta_{r(e)})$.\n\nIf $E$ is the graph with but a single vertex and $n$ edges then $X(E)$ is a Hilbert space of dimension $n$ and $\\mathcal{T}_{X(E)}$ is isomorphic to the classical Toeplitz algebra $\\mathcal{E}_n$. In this case the elements $\\{\\delta_e:e\\in E^1\\}$ are precisely the generating isometries of $\\mathcal{E}_n$. The space $X(E)$ plays a significant role in the analysis of endomorphisms of $B(H)$ in \\cite{laca}, and it is for this reason that we are considering the generalized Toeplitz algebras $\\mathcal{T}_{X(E)}$ in our investigations.\n\n\\section{COHERENT UNITARY EQUIVALENCE}\n\nTwo graphs $E$ and $F$ are isomorphic if there are two bijections $\\psi^0:E^0\\to F^0$ and $\\psi^1:E^1\\to F^1$ for which $r_F\\circ\\psi^1=\\psi^0\\circ r_E$ and $s_F\\circ\\psi^1=\\psi^0\\circ s_E$. In order to encode such an isomorphism at the level of the graph correspondences $X(E)$ and $X(F)$, we offer the following novel definition.\n\n\\begin{defi} Let $X$ and $Y$ be $C^*$-correspondences over $A$ and $B$, respectively. A \\emph{coherent unitary equivalence} between $X$ and $Y$ is a pair $(U,\\alpha)$ consisting of a bijective linear map $U:X\\to Y$ and a $*$-isomorphism $\\alpha:A\\to B$ for which \n\t\\begin{enumerate}\n\t\\item $U(x\\cdot a)=(Ux)\\cdot \\alpha(a)$ for all $x\\in X$ and $a\\in A$,\n\t\\item $U(\\phi_X(a)x)=\\phi_Y(\\alpha(a))Ux$ for all $x\\in X$ and $a\\in A$, and\n\t\\item $\\langle Ux,y\\rangle_Y=\\alpha(\\langle x,U^{-1}y\\rangle_X)$ for all $x\\in X$ and $y\\in Y$.\n\t\\end{enumerate}\nRoutine calculations will verify that coherent unitary equivalence is an equivalence relation.\n\\end{defi}\n\n\\begin{proposition} If $E$ and $F$ are isomorphic graphs then $X(E)$ and $X(F)$ are coherently unitarily equivalent.\n\\end{proposition} \n\\begin{proof} We'll assume $(\\psi^0,\\psi^1)$ to be an isomorphism from $F$ to $E$.\n\nFor $a\\in C_0(E^0)$, $\\alpha(a):=a\\circ\\psi^0$ clearly defines a $*$-isomorphism $\\alpha:C_0(E^0)\\to C_0(F^0)$. For $x\\in X(E)$ define $Ux:=x\\circ\\psi^1$.\nFor $v\\in F^1$ we have\n\\begin{equation*}\\sum_{e\\in s_F^{-1}(v)}|Ux(e)|^2=\\sum_{e\\in s_F^{-1}(v)}|x(\\psi^1(e))|^2=\\sum_{f\\in s_E^{-1}(\\psi^0(v))}|x(f)|^2\\end{equation*}\n(using the fact that if $s_E(e)=v$ then $s_F(\\psi^1(e))=\\psi^0(v)$) and so $\\widehat{Ux}(v)=\\hat{x}(\\psi^1(v))$. As $\\hat{x}\\in C_0(E^0)$ it follows immediately that $\\widehat{Ux}\\in C_0(F^0)$, i.e. $Ux\\in X(F)$. Identical arguments show that $U^{-1}y:=y\\circ(\\psi^1)^{-1}$ is a map from $X(F)$ to $X(E)$ which is a two-sided inverse for $U$. Hence $U:X(E)\\to X(F)$ is a bijection which is naturally linear.\n\nGiven $x\\in X(E)$, $a\\in C_0(E^0)$, and $e\\in E^1$ we have\n\\begin{align*}\nU(x\\cdot a)&=(x(a\\circ s_E))\\circ\\psi^1=(x\\circ\\psi^1)(a\\circ s_E\\circ\\psi^1)=(Ux)(a\\circ\\psi^0\\circ s_F)=Ux\\cdot\\alpha(a)\\\\\nU(\\phi_E(a)x)&=((a\\circ r_E)x)\\circ\\psi^1=(a\\circ r_E\\circ\\psi^1)(x\\circ\\psi^1)=(a\\circ \\psi^0\\circ r_F)(Ux)=\\phi_F(\\alpha(a))Ux\n\\end{align*}\nand, given $v\\in F^0$,\n\\begin{align*}\n\\langle Ux,y\\rangle(v)&=\\sum_{e\\in s_F^{-1}(v)}\\overline{Ux(e)}y(e)=\\sum_{e\\in s_F^{-1}(v)}\\!\\!\\overline{x(\\psi^1(e))}y(e)=\\!\\!\\!\\sum_{f\\in s_E^{-1}(\\psi^0(v))}\\hspace{-.5cm}\\overline{x(f)}y((\\psi^1)^{-1}(f))\\\\\n&=\\sum_{f\\in s_E^{-1}(\\psi^0(v))}\\overline{x(f)}U^{-1}y(f)=\\langle x,U^{-1}y\\rangle(\\psi^0(v))= \\alpha(\\langle x,U^{-1}y\\rangle)(v)\n\\end{align*}\n(the first inner product is that of $X(F)$ and the later two are that of $X(E)$). Thus the pair of $U$ and $\\alpha$ satisfies the definition of a coherent unitary equivalence.\n\\end{proof}\n\nNot every coherent unitary equivalence comes from a graph isomorphism in the sense of the preceding Proposition. As a simple example, consider the graph $E$ with but a single vertex and two edges. In this case $C_0(E^0)=\\mathbb{C}$ and $X(E)=\\mathbb{C}^2$. Hence any unitary $U\\in M_2(\\mathbb{C})$ forms (with the identify on $C_0(E^0)$) a coherent unitary equivalence. However, the only such equivalences arising from graph isomorphisms would be those of the two permutation matrices.\n\n\\begin{proposition}\\label{isotoeplitz} If there is a coherent unitary equivalence between $X$ and $Y$ then $\\mathcal{T}_X$ and $\\mathcal{T}_Y$ are $*$-isomorphic. \n\\end{proposition}\n\\begin{proof} Let $A$ and $B$ be the coefficient $C^*$-algebras for $X$ and $Y$, respectively.\nSuppose that $(U,\\alpha)$ is a coherent unitary equivalence between $X$ and $Y$ and let $(\\sigma,\\pi)$ be a Toeplitz representation of $Y$. For $x\\in X$ and $a\\in A$\n\\begin{align*}\n\\sigma(U(x\\cdot a))&=\\sigma(Ux\\alpha(a))=\\sigma(Ux)\\pi(\\alpha(a))\\\\\n\\sigma(U(\\phi_X(a)x))&=\\sigma(\\alpha(a)Ux)=\\pi(\\alpha(a))\\sigma(Ux)\n\\end{align*}\nand for $x_1,x_2\\in X$\n\\begin{equation*}\\pi\\circ\\alpha(\\langle x_1,x_2\\rangle_A)=\\pi(\\langle Ux_1,Ux_2\\rangle_B)=\\sigma(Ux_1)^*\\sigma(Ux_2).\\end{equation*}\nHence $(\\sigma\\circ U,\\pi\\circ \\alpha)$ is a Toeplitz representation of $X$.\n\nIn particular, $(\\sigma_Y\\circ U,\\pi_B\\circ\\alpha)$ is a Toeplitz representation of $X$ where $(\\sigma_Y,\\pi_B)$ is the universal Toeplitz representation of $Y$ in $\\mathcal{T}_Y$. By the universal property of $\\mathcal{T}_X$, there is a $*$-homomorphism $\\theta:\\mathcal{T}_X\\to\\mathcal{T}_Y$ such that $\\theta\\circ\\sigma_X=\\sigma_Y\\circ U$ and $\\theta\\circ\\pi_A=\\pi_B\\circ \\alpha$, where $(\\sigma_X,\\pi_A)$ is the universal representation of $X$ in $\\mathcal{T}_X$.\n\nSimilarly $(\\sigma_X\\circ U^{-1},\\pi_A\\circ \\alpha^{-1})$ is a Toeplitz representation of $Y$ and induces a $*$-homomorphism $\\theta':\\mathcal{T}_Y\\to \\mathcal{T}_X$ for which $\\theta'\\circ\\sigma_Y=\\sigma_X\\circ U^{-1}$ and $\\theta'\\circ\\pi_B=\\pi_A\\circ \\alpha^{-1}$. Thus\n$$\\sigma_Y=\\sigma_Y\\circ U\\circ U^{-1}=\\theta\\circ\\sigma_X\\circ U^{-1}=\\theta\\circ\\theta'\\circ\\sigma_Y$$\nand similarly $\\pi_B=\\theta\\circ\\theta'\\circ\\pi_B$. Since the identity $id$ on $\\mathcal{T}_Y$ also has the property that $\\pi_B=id\\circ\\pi_B$ and $\\sigma_Y=id\\circ\\sigma_Y$, it follows by the universal property of $\\mathcal{T}_Y$ that $\\theta\\circ\\theta'=id$. Identical reasoning verifies that $\\theta'\\circ\\theta$ is the identity on $\\mathcal{T}_X$. Thus $\\theta$ is our desired $*$-isomorphism.\n\\end{proof}\n\nGoing forward we will be exclusively interested in Toeplitz algebras associated to graph correspondences, and so offer the following corollary.\n\\begin{corollary}\\label{autosfromcoherent} Let $E$ and $F$ be graphs. If $(U,\\alpha)$ is a coherent unitary equivalence between $X(E)$ and $X(F)$ then there is a $*$-isomorphism $\\Gamma_{U,\\alpha}:\\mathcal{T}_{X(E)}\\to\\mathcal{T}_{X(F)}$ for which $\\Gamma_{U,\\alpha}(\\delta_e)=U\\delta_{e}$ and $\\Gamma_{U,\\alpha}(\\delta_v)=\\alpha(\\delta_v)$ for all $e\\in E^1$ and $v\\in E^0$.\n\\end{corollary}\nThis is immediately seen from the proof of the previous Proposition if we recall that we identify $X(E)$ and $X(F)$ with their images in $\\mathcal{T}_{X(E)}$ and $\\mathcal{T}_{X(F)}$, respectively, under the appropriate universal maps. This also implies that if $E$ and $F$ are isomorphic graphs then $\\mathcal{T}_{X(E)}$ and $\\mathcal{T}_{X(F)}$ are $*$-isomorphic, which is unsurprising.\n\n\\section{ENDOMORPHISMS FROM GRAPHS}\n\nThroughout this section we will let $E=(E^0,E^1,r,s)$ be a given graph whose vertex and edge sets are at most countable. All $*$-representations will be assumed non-degenerate.\n\n\\begin{proposition} Given a $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$, the assignments\n\\begin{equation*}Ad_\\tau(w):=\\sum_{e\\in E^1}\\tau(\\delta_e)w\\tau(\\delta_e)^*\\end{equation*}\n(the sum is taken as a SOT limit) define a $*$-endomorphism $Ad_\\tau$ of the von Neumann algebra $W=\\{\\tau(\\delta_v):v\\in E^0\\}'$ (this notation will denote the relative commutant in $B(H)$).\n\\end{proposition}\n\\begin{proof} First, notice that for $e\\in E^1$ and $w\\in W$ the term $\\tau(\\delta_e)w\\tau(\\delta_e)^*$ has its support projection contained in $\\tau(\\delta_e^*\\delta_e)$. Since the partial isometries $\\tau(\\delta_e)$ have mutually orthogonal ranges, it follows that for every $h\\in H$, $\\tau(\\delta_e)w\\tau(\\delta_e)^*h$ is nonzero for at most one $e\\in E^1$. Thus the sum converges in the SOT.\n\nCertainly $Ad_\\tau$ is linear and has $Ad_\\tau(w^*)=Ad_\\tau(w)^*$ for each $w\\in W$. Given $w_1,w_2\\in W$ we find that\n\\begin{align*}Ad_\\tau(w_1)Ad_\\tau(w_2)&=\\bigg(\\sum_{e\\in E^1}\\tau(\\delta_e)w_1\\tau(\\delta_e)^*\\bigg)\\bigg(\\sum_{f\\in E^1}\\tau(\\delta_f)w_2\\tau(\\delta_f)^*\\bigg)\\\\\n\t&= \\sum_{e,f\\in E^1}\\tau(\\delta_e)w_1\\tau(\\delta_e)^*\\tau(\\delta_f)w_2\\tau(\\delta_f)^*\\\\\n\t&= \\sum_{e\\in E^1}\\tau(\\delta_e)w_1\\tau(\\delta_{s(e)})w_2\\tau(\\delta_e)^*\\\\\n\t&=\\sum_{e\\in E^1}\\tau(\\delta_e)\\tau(\\delta_{s(e)})w_1w_2\\tau(\\delta_e)^*\\\\\n\t&=\\sum_{e\\in E^1}\\tau(\\delta_e)w_1w_2\\tau(\\delta_e)^*\\\\\n\t&=Ad_\\tau(w_1w_2)\n\t\\end{align*}\nand so $Ad_\\tau$ is multiplicative. Note that any potential issues with SOT-convergence of the product are circumvented by $E^1$ being at most countable. All that remains is to verify that $Ad_\\tau(w)\\in W$ for each $w\\in W$. To that end we first note that $\\delta_e^*\\delta_v=\\delta_e^*$ if $v=r(e)$ and is zero otherwise. By taking adjoints, $\\delta_v\\delta_e=\\delta_e$ if $v=r(e)$ and is zero otherwise. Thus, given $w\\in W$ and $v\\in E^0$ we find\n\\begin{equation*}Ad_\\tau(w)\\tau(\\delta_v)=\\sum_{e\\in r^{-1}(v)}\\tau(\\delta_e)w\\tau(\\delta_e)^*=\\tau(\\delta_v)Ad_\\tau(w)\\end{equation*}\nand so $Ad_\\tau(w)$ commutes with each $\\tau(\\delta_v)$.\n\\end{proof}\n\nWe note that the von Neumann algebra $W=\\{\\tau(\\delta_v):v\\in E^0\\}$, because the $\\tau(\\delta_v)$ are a family of mutually orthogonal projections, is precisely equal to $\\displaystyle{\\bigoplus_{v\\in E^0} \\tau(\\delta_v)B(H)\\tau(\\delta_v)}$ which is a sum of Type I factors. \n\nThe following is a construction which we believe to be folklore, but our use of it is motivated by observations made by Muhly and Solel \\cite{muhlysolel}. Given a $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$ let $W=\\{\\tau(\\delta_v):v\\in E^0\\}'$. The space\n\\begin{equation*}\\mathcal{I}_\\tau:=\\left\\{T\\in B(H):Ad_\\tau(w)T=Tw\\ ,\\ w\\in W \\right\\}\\end{equation*}\nis a $C^*$-correspondence over $W'$. The left and right actions of $W'$ are simply multiplication within $B(H)$ and the $W'$-valued inner product is defined by $\\langle T,S\\rangle_{W'}:=T^*S$.\n\nBecause our endomorphism is of the form $Ad_\\tau$,  we can say more: for $w\\in W$ and $e\\in E^1$\n\\begin{equation*}Ad_\\tau(w)\\tau(\\delta_e)=\\sum_{f\\in E^1}\\!\\!\\tau(\\delta_f)w\\tau(\\delta_f)^*\\tau(\\delta_e)=\\tau(\\delta_e)w\\tau(\\delta_{s(e)})=\\tau(\\delta_e)\\tau(\\delta_{s(e)})w=\\tau(\\delta_e)w\\end{equation*}\nand so $\\tau(\\delta_e)\\in\\mathcal{I}_\\tau$ for each $e\\in E^1$. As $\\tau(\\delta_v)\\in W'$ for each $v\\in E^0$ we finally have $\\tau(X(E))\\subseteq \\mathcal{I}_\\tau$.\n\n\\begin{theorem}\\label{theorem1} Suppose that $E$ and $F$ are graphs and $\\tau_1:\\mathcal{T}_{X(E)}\\to B(H)$ and $\\tau_2:\\mathcal{T}_{X(F)}\\to B(H)$ are two faithful $*$-representations. If $Ad_{\\tau_1}=Ad_{\\tau_2}$ on $W=\\{\\tau_1(\\delta_v):v\\in E^0\\}'=\\{\\tau_2(\\delta_v):v\\in F^0\\}'$ then there is a coherent unitary equivalence $(U,\\alpha)$ between $X(E)$ and $X(F)$ such that $\\tau_2=\\tau_1\\circ\\Gamma_{U,\\alpha}$. \n\\end{theorem}\nHere $\\Gamma_{U,\\alpha}$ is the $*$-isomorphism from $\\mathcal{T}_{X(E)}$ to $\\mathcal{T}_{X(F)}$ arising from $(U,\\alpha)$ as given in Corollary \\ref{autosfromcoherent}.\n\\begin{proof} \nSince $\\{\\tau_1(\\delta_v):v\\in E^0\\}$ and $\\{\\tau_2(\\delta_v):v\\in F^0\\}$ are sets of orthogonal projections with the same commutant they are in fact equal, and in particular $E^0$ and $F^0$ are of the same cardinality. To ease notation we'll denote these projections by $P_v$, $v\\in E^0$, (with no assumption that $P_v=\\tau_1(\\delta_v)$ or similar) hence\n\\begin{equation*}\\{P_v:v\\in E^0\\}=\\{\\tau_1(\\delta_v):v\\in E^0\\}=\\{\\tau_2(\\delta_v):v\\in F^0\\}.\\end{equation*}\n\nAs $Ad_{\\tau_1}=Ad_{\\tau_2}$ we have that $\\mathcal{I}_{\\tau_1}=\\mathcal{I}_{\\tau_2}$ and we'll call this module simply $\\mathcal{I}$.\n\nAs $\\tau_1(\\delta_e)\\in\\mathcal{I}$ for each $e\\in E^1$ we have\n\\begin{equation*}\\tau_1(\\delta_e)=\\tau_1(\\delta_e)I=Ad_{\\tau_2}(I)\\tau_1(\\delta_e)=\\sum_{f\\in F^1}\\tau_2(\\delta_f)\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)\\end{equation*}\nhence $\\tau_1(\\delta_e)$ is in the $W'$-submodule of $\\mathcal{I}$ generated by $\\tau_2(X(E))$. Similarly, for each $f\\in F^1$, $\\tau_2(\\delta_f)$ is in the $W'$-submodule generated by $\\tau_1(X(E))$. Thus $\\tau_1(X(E))$ and $\\tau_2(X(E))$ generate the same $W'$-submodule of $\\mathcal{I}$.\n\nGiven $e\\in E^1$ and $f\\in F^1$ we have seen that \n\\begin{equation*}\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)\\in W'=\\{P_v:v\\in E^0\\}''=\\ell^\\infty(\\{P_v:v\\in E^0\\}).\\end{equation*}\nNotice however that $\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)\\tau_1(\\delta_v)=0$ unless $v=s(e)$ and hence $\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)$ is a multiple of $\\tau_1(\\delta_{s(e)})$ only, i.e. is an element of $C_0(\\{P_v:v\\in E^0\\})$. Since before we obtained $\\tau_1(\\delta_e)=\\sum_{f\\in F^1}\\tau_2(\\delta_f)\\tau_2(\\delta_f)^*\\tau_1(\\delta_e)$ for all $e\\in E^1$, it now follows that $\\tau_1(X(E))$ and $\\tau_2(X(E))$ generate the same correspondence over $C_0(\\{P_v:v\\in E^0\\})$. It is important to note that this correspondence has three different actions of $C_0(\\{P_v:v\\in E^0\\})$: the ones inherited through $\\tau_1$ and $\\tau_2$ and simple operator multiplication in $B(H)$.\n \nFinally we have that $\\tau_1(C_0(E^0))=\\tau_2(C_0(F^0))$ and $\\tau_1(X(E))=\\tau_2(X(F))$ as sets and so, because both representations are faithful by hypothesis, $\\tau_2^{-1}\\circ\\tau_1$ is a well-defined bijection between $X(E)$ and $X(F)$ and between $C_0(E^0)$ and $C_0(F^0)$. Denote by $U$ and $\\alpha$ the restrictions of $\\tau_2^{-1}\\circ\\tau_1$ to $X(E)$ and to $C_0(E^0)$, respectively.\n\nGiven $x\\in X(E)$, $y\\in X(F)$, and $a\\in C_0(E^0)$ we have\n\\begin{align*}U(xa)&=\\tau_2^{-1}\\circ\\tau_1(xa)=\\tau_2^{-1}\\circ\\tau_1(x)\\tau_2^{-1}\\circ\\tau_1(a)=(Ux)\\alpha(a),\\\\\nU(\\phi(a)x)&=\\tau_2^{-1}\\circ\\tau_1(\\phi(a)x)=\\tau_2^{-1}\\circ\\tau_1(a)\\tau_2^{-1}\\circ\\tau_1(x)=\\alpha(a)Ux,\\\\\n\\langle Ux,y\\rangle&=[\\tau_2^{-1}\\circ\\tau_1(x)]^*y=\\tau_2^{-1}\\circ\\tau_1(x^*\\tau_1^{-1}\\circ\\tau_2(y))=\\alpha(\\langle x,\\tau_1^{-1}\\circ\\tau_2(y)\\rangle)=\\alpha(\\langle x,U^{-1}y\\rangle).\\end{align*}\nand so $(U,\\alpha)$ is a coherent unitary equivalence between $X(E)$ and itself.\n\nIt follows from Corollary \\ref{autosfromcoherent} that $(U,\\alpha)$ induces a $*$-isomorphism $\\Gamma_{U,\\alpha}:\\mathcal{T}_{X(E)}\\to\\mathcal{T}_{X(F)}$ and, by construction, $\\tau_2\\circ \\Gamma_{U,\\alpha}=\\tau_1$.\n\\end{proof}\n\nNotice we have shown that if $\\tau_1:\\mathcal{T}_{X(E)}\\to B(H)$ and $\\tau_2:\\mathcal{T}_{X(F)}\\to B(H)$ generate identical $*$-endomorphims $Ad_{\\tau_1}$ and $Ad_{\\tau_2}$ then $\\mathcal{T}_{X(E)}$ and $\\mathcal{T}_{X(F)}$ are $*$-isomorphic as $C^*$-algebras. Hence in subsequent results we will concern ourselves only with two representations $\\tau_1$, $\\tau_2$ of the same Toeplitz algebra $\\mathcal{T}_{X(E)}$.\n\n\nOur result is a generalization of Laca's \\cite[Proposition 2.2]{laca}. When $E$ is the graph with a single vertex and $n\\in\\mathbb{N}\\cup\\{ \\infty\\}$ edges we have already seen that $\\mathcal{T}_{X(E)}=\\mathcal{E}_n$. If $\\tau_1$ and $\\tau_2$ are faithful and nondegenerate then $W=B(H)$. The map $\\alpha$ is the identity on $C_0(E^0)=\\mathbb{C}$ and $U$ is a unitary operator on the Hilbert space $X(E)=\\ell^2(\\{v_1,...,v_n\\})$. Hence $\\Gamma_{U,\\alpha}$ is a $*$-automorphism of $\\mathcal{E}_n$ which fixes the Hilbert space $X(E)$ and implements the equivalence between the two representations.\n\nWe will conclude this section with a discussion of conjugacy conditions for endomorphisms of the type we've been examining. Recall that two endomorphisms $\\alpha$ and $\\beta$ are said to be \\emph{conjugate} if there is an automorphism $\\gamma$ such that $\\alpha\\circ\\gamma=\\gamma\\circ\\beta$.\n\n\\begin{lemma}\\label{spatialdiagonal} If $P_1,P_2,...\\in B(H)$ is an at most countable family of orthogonal projections and $\\gamma$ is a $*$-automorphism of $W=\\{P_1,P_2,...\\}'$ then there exists a unitary $U\\in B(H)$ such that $\\gamma(w)=UwU^*$ for all $w\\in W$.\n\\end{lemma}\n\\begin{proof} Note that for each $n$, $\\gamma$ restricts to a $*$-isomorphism $\\gamma_n$ between $P_nB(H)P_n=B(P_nH)$ and $\\gamma(P_n)B(H)\\gamma(P_n)=B(\\gamma(P_n)H)$. Such isomorphisms are always spatial and so there are unitaries $U_n:B(P_nH)\\to B(\\gamma(P_n)H)$ such that $\\gamma_n(w)=U_nwU_n^*$. It is then immediate that $U:=U_1\\oplus U_2\\oplus...$ is a unitary in $B(H)$ and $UwU^*=\\gamma(w)$ for each $w\\in W$.\n\\end{proof}\n\n\\begin{theorem}\\label{theorem2} Suppose that $\\tau_1,\\tau_2:\\mathcal{T}_{X(E)}\\to B(H)$ are two faithful $*$-representations such that $Ad_{\\tau_1}$ and $Ad_{\\tau_2}$ are conjugate $*$-endomorphisms of $W=\\{\\tau_1(\\delta_v):v\\in E^0\\}'=\\{\\tau_2(\\delta_v):v\\in E^0\\}'$. Then there is a coherent unitary equivalence $(U,\\alpha)$ between $X(E)$ and itself such that $\\tau_2$ and $\\tau_1\\circ\\Gamma_{U,\\alpha}$ are unitarily equivalent $*$-representations.\n\\end{theorem}\n\n\\begin{proof} Let $\\gamma$ be an $*$-automorphism of $W$ such that $Ad_{\\tau_1}\\circ\\gamma=\\gamma\\circ Ad_{\\tau_2}$ and let $V\\in B(H)$ be the unitary for which $\\gamma(w)=VwV^*$ according the Lemma \\ref{spatialdiagonal}. Then $Ad_{\\tau_2}(w)=V^*Ad_{\\tau_1}(VwV^*)V$ for all $w\\in W$. Define $\\kappa:\\mathcal{T}_{X(E)}\\to B(H)$ by $\\kappa(t):=V\\tau_1(t)V^*$ and note that $\\kappa$ is a $*$-representation of $\\mathcal{T}_{X(E)}$ such that \n\\begin{equation*}Ad_{\\kappa}(w)=\\sum_{e\\in E^1}\\kappa(\\delta_e)w\\kappa(\\delta_e)^*=\\sum_{e\\in E^1}V\\tau_1(\\delta_e)V^*wV\\tau_1(\\delta_e)^*V^*=VAd_{\\tau_1}(V^*wV)V^*\\end{equation*}\nand so $Ad_{\\kappa}=Ad_{\\tau_2}$ on $W$. Applying Theorem \\ref{theorem1} we obtain a coherent unitary equivalence $(U,\\alpha)$ inducing the $*$-automorphism $\\Gamma_{U,\\alpha}$ of $\\mathcal{T}_{X(E)}$ such that $\\tau_2=\\kappa\\circ\\Gamma_{U,\\alpha}$. As now $\\tau_2(t)=V[\\tau_1\\circ\\Gamma_{U,\\alpha}(t)]V^*$ for each $t\\in \\mathcal{T}_{X(E)}$, we have that $\\tau_2$ and $\\tau_1\\circ\\Gamma_{U,\\alpha}$ are unitarily equivalent, as desired.\n\\end{proof}\n\n\\section{GRAPHS FROM ENDOMORPHISMS}\n\nIn this final section we will demonstrate that all $*$-endomorphisms of von Neumann algebras which are sums of Type I factors are obtained in the natural way from representations of Toeplitz algebras for graph correspondences. Our result is a significant generalization of \\cite[Theorem 3.9]{brenken2} which places technical restrictions on the endomorphisms. In the case of unital endomorphisms our results are comparable.\n\n\\begin{theorem}\\label{theorem3} Let $W=\\bigoplus W_i\\subseteq B(H)$ be a countable sum of Type I factors. Let $P_1,P_2,...\\in B(H)$ be projections such that $W_i=P_iB(H)P_i$. If $\\alpha$ is a normal $*$-endomorphism of $W$ then there exists a graph $E$ and $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$ such that $\\alpha=Ad_\\tau$.\n\\end{theorem}\n\\begin{proof} Without loss of generality we may assume that $\\sum P_i=I$. If this were not the case we may define $P_0=(\\sum P_i)^\\perp$ and $W_0=P_0B(H)P_0$ and extend $\\alpha$ to a normal $*$-endomorphism of $W\\oplus W_0$ with $\\alpha|_{W_0}=id$. \n\nFor $i>0$ define $H_i=P_iH$. For $i,j>0$ and $x\\in W$ define $\\alpha_{ij}(x)=P_j\\alpha(P_ix)$. Then $\\alpha_{ij}$ restricts to a $*$-homomorphism between $B(H_i)=P_iB(H)P_i=W_i$ and $B(H_j)=P_jB(H)P_j=W_j$ as seen by\n\\begin{equation*}P_j\\alpha(P_ix)P_j\\alpha(P_iy)=P_j\\left(P_j\\alpha(P_ix)\\right)\\alpha(P_iy)=P_j\\alpha(P_ixP_iy)=P_j\\alpha(P_ixy).\\end{equation*}\n\nSo $\\alpha_{ij}$ is a $*$-homomorphism between two Type I factors, and thus by \\cite[Proposition 2.1]{arveson} if $\\alpha_{ij}$ is nonzero there exists $n_{ij}\\in\\mathbb{N}\\cup\\{\\infty\\}$ and isometries $V_k^{(ij)}\\in B(H_i,H_j)$, $k=1,...,n_{ij}$ such that $\\alpha_{ij}|_{B(H_i)}(T)=\\sum_{k=1}^{n_{ij}}V_k^{(ij)}TV_k^{(ij)*}$.\nWe will identify the $V_k^{(ij)}$ with their associated partial isometries in $B(H)$, so that $V_k^{(ij)*}V_k^{(ij)}=P_i$ and $V_k^{(ij)}V_k^{(ij)*}\\leq P_j$.\n\nSet $E^0:=\\{P_1,P_2...,\\}$ and $E^1:=\\bigcup_{i,j}\\{V_k^{(ij)}:k=1,...,n_{ij}\\}$. Define maps $r,s:E^1\\to E^0$ by $r(V_k^{(ij)})=P_j$ and $s(V_k^{(ij)})=P_i$. Then $E=(E^0,E^1,r,s)$ is a graph. It is trivial to see that the identity maps on $E^1$ and $E^0$ extend to a Toeplitz representation of $X(E)$ which in turn induces a $*$-representation $\\tau:\\mathcal{T}_{X(E)}\\to B(H)$.\n\nFinally, we have that for each $x\\in W$\n\\begin{equation*}\\alpha(x)=\\sum_{i,j>0} P_j\\alpha(P_ix)=\\sum_{i,j>0}\\alpha_{ij}(x)=\\sum_{i,j>0}\\sum_{k=1}^{n_{ij}} V_k^{(ij)}xV_k^{(ij)*}=\\sum_{f\\in E^1}\\tau(\\delta_f)x\\tau(\\delta_f)^*\\end{equation*}\nas desired.\n\\end{proof}\nAs a consequence, the possible $*$-endomorphisms of a given von Neumann algebra $W=\\{P_1,P_2,...\\}'$ coincide, up to conjugacy, with the coherent unitary equivalence classes of $C^*$-correspondences over $C_0(\\{P_1,P_2,...\\})$. \n\n\\subsection{Cuntz-Pimsner Algebras} To discuss the special case of unital $*$-endomorphisms we will need to briefly sketch the defining features and universal properties of the so-called Cuntz-Pimsner algebras. The relationship of these Cuntz-Pimnser algebras to our Toeplitz algebras is analogous to the relationship between the classical Toeplitz algebras $\\mathcal{E}_n$ and classical Cuntz algebras $\\O_n$. The Cuntz-Pimsner algebras were originally defined in \\cite{pimsner} though our treatment will take its cues from \\cite{fowlerraeburnmuhly} and \\cite{fowlerraeburn}.\n\nRecalling that $X(E)$ is a $C^*$-correspondence over $C_0(E^0)$, notice that $I=C_0(\\{v\\in\\ E^0:r^{-1}(v)\\text{ is finite }\\})$ is a closed, two-sided ideal in $C_0(E^0)$. As noted in \\cite[Proposition 4.4]{fowlerraeburn} (and recalling our modern reversal of $r$ and $s$ from the presentation in that work), elements of $I$ are precisely those elements of $C_0(E^0)$ whose left action on $X(E)$ is compact. Since our graphs have at most countable vertices and edges, $X(E)$ is countably generated as a $C^*$-correspondence over $C_0(E^0)$. It follows, \\cite[Remark 3.9 following Definition 3.8]{pimsner} that the \\emph{Cuntz-Pimnser algebra} of $X(E)$ is the $C^*$-algebra $\\O_{X(E)}$ which is universal for Toeplitz representations $(\\sigma,\\pi)$ (as in Definition \\ref{toeplitzrepdef}) which additionally satisfy:\n\\begin{enumerate}\n\\item[(iv)] $\\displaystyle{\\pi(a)=\\sum_{f\\in E^1}\\sigma(\\phi(a)\\delta_f)\\sigma(\\delta_f)^*}$ for all $a\\in I$.\n\\end{enumerate}\nor, equivalently  (see \\cite[Example 1.5]{fowlerraeburnmuhly}),\n\\begin{enumerate}\n\\item[(iv)$'$] $\\displaystyle{\\pi(\\delta_v)=\\sum_{f\\in r^{-1}(v)}\\sigma(\\delta_f)\\sigma(\\delta_f)^*}$ for all $v\\in E^0$ with $|r^{-1}(v)|<\\infty$.\n\\end{enumerate}\nRepresentations satisfying (i)-(iv)$'$ are often termed ``coisometric\" Toeplitz representations. With this characterization of $\\O_{X(E)}$ we may recognize that Theorem \\ref{theorem3} has more to say when the endomorphism is unital.\n\n\\begin{cor} Let $W=\\bigoplus W_i\\subseteq B(H)$ be a countable sum of Type I factors. Let $P_1,P_2,...\\in B(H)$ be projections such that $W_i=P_iB(H)P_i$. If $\\alpha$ is a normal, unital $*$-endomorphism of $W$ then there exists a graph $E$ and $*$-representation $\\tau:\\O_{X(E)}\\to B(H)$ such that $\\alpha=Ad_\\tau$.\n\\end{cor}\n\\begin{proof} We define the partial isometries $V_k^{(ij)}$ and graph $E$ in the same manner as in the proof of Theorem \\ref{theorem3}. Since $\\alpha$ is unital we have\n\n$$P_n=P_n\\alpha(I)=\\sum_{i,j>0}\\sum_{k=1}^{n_{ij}} P_nV_k^{(ij)}V_k^{(ij)*}=\\sum_{i>0}\\sum_{k=1}^{n_{in}} V_k^{(in)}V_k^{(in)*}$$\nfor every $P_n$. Recalling that the representation of $(\\sigma,\\pi)$ $E$ on $B(H)$ is the pair of identity maps, this becomes\n$$\\pi(\\delta_v)=\\sum_{f\\in E^1}\\pi(\\delta_v)\\sigma(\\delta_f)\\sigma(\\delta_f)^*=\\sum_{f\\in r^{-1}(v)}\\sigma(\\delta_f)\\sigma(\\delta_f)^*$$\nfor all $v\\in E^0$. In particular this holds for all edges $v$ with $r^{-1}(v)$ finite (i.e.\\ all $P_j$ such that $\\{V_k^{(ij)}: i>0, k=1,...,n_{ij}\\}$ is finite). Thus condition (iv)$'$ is satisfied and the identity maps on $E^0$ and $E^1$ induce a representation of $\\O_{X(E)}$. The fact that $\\alpha=Ad_\\tau$ follows as before.\n\\end{proof}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\nSequential execution of multi-robot coordinated behaviors can be employed to solve real-world complex missions. However, sequences of behaviors can be executed only if the robots meet all required communication constraints in finite time. In this paper, we described a distributed framework for the sequential composition of coordinated behaviors designed on finite-time convergence control barrier functions. The resulting composition framework is formulated in the form of a quadratic program, which is solved locally by individual robots. \\mymod{Although the focus of this paper is on coordinated motion, the application of the proposed framework is relevant to other form of autonomous collaborations where the robots need to satisfy prescribed pair-wise proximity requirements that change over time}. Finally, a large-scale multi-task scenario, denoted ``Securing a Building\" mission is proposed as an ideal environment for testing multi-robot techniques.\n\n\\appendices\n\\input{sections\/researchOpportunities}\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Behaviors Dynamic} \\label{sec:appendixB}\n\\paragraph{Rendezvous}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} x_j - x_i\n\\end{equation}\n\n\\paragraph{Scatter}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} x_i - x_j\n\\end{equation}\n\n\n\\paragraph{Formation control}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i)\n\\end{equation}\n\nwhere $\\theta_{ij}$ must be feasible.\n\n\\paragraph{Leader-follower}\n\\begin{align}\n\tu_f &= \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i) \\\\\n\tu_l &= \\sum_{j \\in \\mathcal{N}_i} \\left( (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_i - x_i) \\right) + \\gamma_g(x_g-x_i) \n\\end{align}\n\n\\paragraph{Leader formation}\n\\begin{align}\n\tu_f &= \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i) \\\\\n\tu_l &= \\sum_{j \\in \\mathcal{N}_i} \\left( (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_i - x_i) \\right) + \\gamma_g(x_g-x_i) \n\\end{align}\nwhere $\\theta_{ij}$ must be feasible.\n\n\\paragraph{Cyclic pursuit}\n\\begin{equation}\n\tu_i = \\sum_{j \\in \\mathcal{N}_i} R(\\theta)\\,(x_j - x_i) \n\\end{equation}\nwhere $\\mathcal{G}$ must be $C_N$ (cycle graph).\n\n\\paragraph{Lattice}\n\\begin{equation}\nu_i = \\sum_{j \\in \\mathcal{N}_i} (\\|x_i-x_j\\|^2-\\theta^2)(x_j - x_i)\n\\end{equation}\nwhere $\\theta \\simeq 0.8$ max sensing radius. \n\n\\paragraph{Coverage}\n\\begin{equation}\nu_i = c_i - x_i\n\\end{equation}\nwhere $c_i$ centroid of Voronoi tessellation.\n\n\\paragraph{Swarming}\n\\begin{equation}\nu_i = \\sum_{j \\in \\mathcal{N}_i} \\dots (x_j - x_i)\n\\end{equation}\n\n\\section{Securing a Building as Benchmark Scenario} \\label{sec:appendixA}\nTesting the performance of techniques and algorithms for the control of multi-robot systems in real-world scenarios is a challenging task. \\mymod{This is particularly true when addressing novel approaches, as the focus on specific aspects of the problem might obscure all-around performance assessments.} To this end, thanks to its modularity, the {\\it Securing a Building} mission is an ideal testing framework. In this section, we suggest a number of selected research topics, for which this mission could serve as a testing framework when aiming to evaluate performance of new techniques. \\mymod{This appendix is by no mean proposed as a complete list of subjects relevant to multi-robot systems but rather as a discussion to stimulate application of the {\\it Securing a Building} as a versatile, real-world testing scenario.}\n\n\\paragraph{Team Assembly} Considerable efforts have been devoted to the development of team composition techniques for heterogeneous robots~\\cite{prorok2016formalizing},~\\cite{koes2005heterogeneous}. Based on the skill set required to solve a particular task, e.g., certain actuation, sensing, locomotion, or communication capabilities, the question is to find a recruitment rule that produces a team capable of delivering the best performance. For instance, in the RESCUE phase, robots capable of opening doors may be required for the {\\it maneuvering} agents, while agility and communication capabilities might be preferred during the FIND phase.\n\n\\paragraph{Communication} In the context of autonomous networked systems, central roles are played by the flow of information between agents, and the infrastructure required for it~\\cite{gupta2016survey}. A number of questions can be posed in relation to the distribution of agents over a domain, given the constraints of communication systems, such as limited range, power requirements, and privacy of the information.\n\n\\paragraph{Unknown Environment} The amount of prior knowledge about the environment plays an important role in the definition of both low-level robot controllers and high-level mission plans. The performance of distributed solutions to the localization and mapping problems~\\cite{forster2013collaborative} can be tested on the {\\it Securing a Building}. Aspect of interest include balancing between exploitation and exploration of the environment applied, for instance, to the building exploration planning.\n\n\\paragraph{Resilience} Failure of the mission can be attributed to factors such as damaged components, sensing errors, communication dropouts, delays, control disturbances, reduction of functionalities due to adversarial attacks, etc. A number of different research thrusts focus on the problem of detecting and responding to faults and malicious attacks in multi-agent and cyber-physical systems~\\cite{pasqualetti2011consensus,pierpaoli2018fault,fawzi2014secure}.\n\n\n\\section{\\mymod{Distributed Composition of Behaviors}}\\label{sec:multAgImp}\nThe composition framework discussed in the previous section reduces to the team-wise minimum norm controller~(\\ref{eq:minQP1}), which is not directly solvable by individual robots. \\mymod{In addition to this, a centralized supervisor is needed in order to synchronize behavior transitions.} In this section, we \\mymod{formulate a decentralized solution to problem~\\ref{pr:problem} which can be implemented by the robots using only information from their neighbors. Furthermore, we also include those} additional constraints necessary for the safe operations of the robots, e.g., inter-agent collisions and obstacles avoidance~\\cite{wang2016multi}. The formulation is derived following the approach described in~\\cite{squires2019composition}, which we adapt here to our framework.\n\n\\subsection{Distributed Finite-Time Convergence Control Barrier Functions}\nThe limitation in solving problem~(\\ref{eq:minQP1}) in a distributed fashion is represented by the fact that knowledge of dynamics, input $\\hat{u}$, and state $x$ for the entire team need to be available. In addition, solution of~(\\ref{eq:minQP1}), provides the control inputs for the entire team, which are unnecessary to the individual robots.\n\nIn order to develop the correct decentralized formulation of~(\\ref{eq:minQP1}), we first define a decomposition of the dynamics~(\\ref{eq:ensembleDynamics}). We denote by $\\mathcal{D}_i\\subset$ \\mymod{$\\mathbb{R}^d$} and $U_i \\subset$ \\mymod{$\\mathbb{R}^m$} configuration space and set of feasible controls for agent $i$ respectively. In addition, by denoting with $\\bar{f},\\,\\bar{g}: \\mathcal{D}_i \\mapsto$ \\mymod{$\\mathbb{R}^d$} the node-level terms of the control affine dynamics of agent $i$, the ensemble dynamics can be written as:\n\\begin{equation} \\label{eq:decoup_dyn}\n\t\\dot{x} = \\bar{f}(x_i)\\otimes {\\bf 1}_n + (\\bar{g}(x_i) \\otimes I_n)\\,\\begin{bmatrix} u_1 \\\\ \\vdots \\\\ u_n \\end{bmatrix},\n\\end{equation}\nwhere $u_i \\in U_i$ is the $i^{\\text{th}}$ robot's control input, $\\otimes$ is  the Kronecker product, and ${\\bf 1}_n$ and $I_n$ are vector of ones and identity matrix of size $n$ respectively.\n\nLet's consider two sequential behaviors $\\mathcal{B}_{k-1}$ and $\\mathcal{B}_{k}$. Upon completion of $\\mathcal{B}_{k-1}$, for all edges $(i,j)\\in E_k$, robots' configuration should satisfy \n\\begin{equation} \\label{eq:dec_const_edge}\n\t\\dot{h}_{ij}^c(x_i,x_j) + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c(x_i,x_j)) \\geq 0.\n\\end{equation}\nFrom the $i^{\\text{th}}$ robot's point of view, the set of constraints that need to be satisfied in order to execute the new behavior are\n\\begin{equation} \\label{eq:dec_const_agenti}\n\t\\dot{h}^c_{ij}(x_i,x_j) + \\bar{\\alpha}_{\\rho,\\gamma}(h^c_{ij}(x_i,x_j)) \\geq 0 \\quad \\forall j\\in \\mathcal{N}_{k}^i,\n\\end{equation}\nwhere we recall that $\\mathcal{N}_{k}^i$ is the set of neighbors to robot $i$ required by behavior $\\mathcal{B}_k$. However, since constraint~(\\ref{eq:dec_const_agenti}) appears exactly twice across the team of robots, it can be relaxed by considering the admissible set of control inputs\n\\begin{equation} \\label{eq:admGraphControl}\nK_{k}^{c,i} = \\bigcap_{j\\in\\mathcal{N}_k^i} K_{k,ij}^{c,i}\n\\end{equation}\nwith\n\\begin{equation}\nK_{k,ij}^{c,i} = \\{u_i\\in U_i \\,| \\,L_{\\bar{f}}h_{ij}^c + L_{\\bar{g}}h_{ij}^cu_i + \\frac{\\bar{\\alpha} _{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0 \\},\n\\end{equation}\nwhere dependence from $x_i$ and $x_j$ is omitted for clarity.\n\n\\begin{theorem} \\label{thm:fcbfControl_dist}\n\tDenoting with $x_0 = [x_{0,1}^T,\\dots,x_{0,n}^T]^T$ the initial state of a multi-agent system with dynamics described as in~(\\ref{eq:decoup_dyn}), any controller $\\mathcal{U}_i:\\mathcal{D}_i^{|\\mathcal{N}_k^i|} \\mapsto U_i$ such that $\\mathcal{U}_i(x_{0}) \\in K_{k}^{c,i}$ for all $x_{0} \\in \\mathcal{D}_i^{|\\mathcal{N}_k^i|}$, will drive the ensemble state to $\\mathcal{C}^c_k $ within time\n\\begin{equation}\n\tT_k = \\max_{\\substack{(i,j) \\in E_k \\\\ \\text{s.t.} \\,\\, h^c_{ij}(x_{0,i},x_{0,j})<0}} \\left\\{ \\frac{1}{ \\gamma(1-\\rho)} | h_{ij}^c(x_{0,i},x_{0,j}) |^{1-\\rho} \\right\\}.\n\\end{equation}\n\\end{theorem}\n\n\n\n\\begin{IEEEproof}\nFrom Theorem~\\ref{thm:FCBF}, agents $i$ and $j$, with $(i,j)\\in E_k$, will satisfy $h_{ij}^c \\geq 0$ in finite time if\n\\begin{equation} \\label{eq:connect_const}\n\\dot{h}_{ij}^c + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0. \n\\end{equation}\n\nConsidering the node level dynamics in~(\\ref{eq:decoup_dyn}), the constraint~(\\ref{eq:connect_const}) reduces to\n\n\\begin{equation} \\label{eq:const2agents}\n\\begin{aligned}\n&\\frac{\\partial h_{ij}^c}{\\partial x_i} \\left( \\bar{f} + \\bar{g}u_i \\right) \\, + \\, \\frac{\\partial h_{ij}^c}{\\partial x_j}\\left(\\bar{f} + \\bar{g}u_j\\right) + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0 \\\\\n&2\\,L_{\\bar{f}}h_{ij}^c + L_{\\bar{g}}h_{ij}^c\\,u_i + L_{\\bar{g}}h_{ij}^c u_j + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0\n\\end{aligned}\n\\end{equation}\nwhich will be satisfied if both agents $i$ and $j$ satisfy the constraint \n\\begin{equation} \\label{eq:dec_const}\n\t\\dot{h}_{ij}(x_i,x_j) + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}(x_i,x_j))}{2} \\geq 0.\n\\end{equation}\nIn addition, as discussed in Theorem~\\ref{thm:fcbfControl},  constraint~(\\ref{eq:const2agents}) will still be satisfied at time\n\n\\begin{equation}\n\tT_{ij} \\leq \\frac{1}{\\gamma(1-\\rho)} | h_{ij}^c(x_{0,i},x_{0,j}) |^{1-\\rho}.\n\\end{equation}  \n\nThe same argument can be repeated for all pairs $(i,j) \\in E_k$, and condition $\\mathcal{G}_k \\subseteq \\mathcal{G}(t)$ will be satisfied within time\n\n\\begin{equation}\n\tT_k = \\max_{\\substack{(i,j) \\in E_k \\\\ \\text{s.t.} \\,\\, h^c_{ij}(x_{0,i},x_{0,j})<0}} \\left\\{ T_{ij} \\right\\}.\n\\end{equation}\t\n\\end{IEEEproof}\n\nApplying the same design principle described in Section~\\ref{sec:ftcontrolBF}, the minimally invasive control action can be computed by each robot as\n\\begin{equation}\\label{eq:minQP1_dist}\nu^*_i = \\argmin_{u_i \\in U_i}  \\| \\hat{u}_{k-1,i} - u_i \\|^2 \\\\\n\\end{equation}\nsubject to\n\\begin{equation}\\label{eq:constrTransition_dist}\nL_{\\bar{f}}\\,h_{ij}^c + L_{\\bar{g}}\\,h_{ij}^c\\,u_i + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0 , \\quad \\forall j\\in  \\mathcal{N}_{k-1}^i \\cup \\mathcal{N}_{k}^i.\n\\end{equation}\nSimilarly to constraint~(\\ref{eq:constrTransition}), once all edges in $E_k$ are available, constraint~(\\ref{eq:constrTransition_dist}) is substituted with\n\\begin{equation}\\label{eq:constrExecution_dist}\nL_{\\bar{f}}\\,h_{ij}^c + L_{\\bar{g}}\\,h_{ij}^c\\,u_i + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0 , \\quad \\forall j\\in\\mathcal{N}_{k}^i,\n\\end{equation}\nwhich remains active until $\\mathcal{B}_k$ is completed.\n\nWe note that, in order for agent $i$ to respect~(\\ref{eq:constrExecution_dist}), the only external information needed is the state of all current neighbors, i.e. $x_j$ for all $j\\in\\mathcal{N}^i_k$. On the other side, in order to respect (\\ref{eq:constrTransition_dist}), robots need to have access to the state of the future neighbors. This requirement can be satisfied through a state estimation scheme (e.g. EKF~\\cite{williams2015observability}), which in turn requires knowledge of robots' dynamics (known for homogeneous teams) or network localization techniques~\\cite{aspnes2006theory}.\n\\mymod{\n\\begin{remark}\nThe ability of each robot to have access to an estimate of their future neighbors' state does not eliminate the necessity of establishing neighborhood relationships. In fact, a certain proximity structure between robots might be required by desired controllers' performance that cannot be met through state estimations, or by collaboration tasks that require physical interaction between the robots, e.g. collaborative manipulation~\\cite{culbertson2018decentralized}, sharing of resources~\\cite{ramachandran2019resilience}.\n\\end{remark}}\n\n\\subsection{Additional Constraints}\nIn addition to the proximity constraints discussed above, additional limitations might be imposed on the robots' configuration by the mission and the environment. For illustrative purposes, we consider inter-robots collisions and obstacle avoidance. Following the approach described in~\\cite{wang2016multi}, we encode each pair-wise separation condition through the following barrier certificate\n\\begin{equation}\n\th_{ij}^a(x) = \\| x_i-x_j \\|^2 - D_a^2 \n\\end{equation}\nand the minimum separation $D_a$ between the robots is satisfied if $h_{ij}^a(x) \\geq 0$, for all physical neighbors $j\\in \\mathcal{N}^i(t)$.\n\nSimilarly, avoidance of fixed obstacles can be introduce by considering $M$ ellipsoidal regions of the domain, described by centers $o=[o_1^T,\\dots,o_M^T]^T$. For every agent-obstacle pair $(i,m)$ we define a pairwise barrier function as\n\\begin{align}\nh_{im}^o(x) &=(x_i-o_m)^T\\,P_m\\,(x_i-o_m) - 1 \\\\\nP_m &= \\begin{bmatrix}\n\ta_m & 0 \\\\ 0 & b_m \n\\end{bmatrix}\n\\quad a_m,b_m > 0.\n\\end{align}\nThe object avoidance constraints are satisfied if $h_{im}^o(x)\\geq 0$, for all $i\\in V$ and $m \\in \\{1,\\dots,M \\}=\\mathcal{I}_M$.\n\nCollecting all the constraints, we expand the problem formulation in~(\\ref{eq:minQP1}) to\n\n\\begin{equation} \\label{eq:minQP2}\n\\begin{aligned}\n&\\qquad u_i^* = \\arg \\min_{u_i \\in U_i}  \\| \\hat{u}_{k-1,i} - u_i \\|^2 &\\\\\n&L_f\\,h_{ij}^c + L_g\\,h_{ij}^c\\,u_i + \\frac{\\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c)}{2} \\geq 0, & \\forall j\\in \\mathcal{N}_{k}^i \\\\\n&L_f\\,h_{ij}^s + L_g\\,h_{ij}^s\\,u_i + \\alpha(h_{ij}^{s}) \\geq 0, &\\forall j\\in \\mathcal{N}^i(t)\\\\\n&L_f\\,h_{im}^o + L_g\\,h_{im}^o\\,u_i + \\alpha(h_{ij}^{s}) \\geq 0, &\\forall m \\in \\mathcal{I}_M\n\\end{aligned}\n\\end{equation}\nwhere $\\alpha$ is a locally Lipschitz extended class-$\\mathcal{K}$ function and the first constraint is replaced by~(\\ref{eq:constrTransition_dist}) during transitions. In conclusion, \\mymod{if there exists a set of control inputs $u = [u_1,\\dots,u_N]$ that simultaneously satisfies all constraints in~(\\ref{eq:minQP2}), for all behaviors $k=1,\\dots,M$, Problem~\\ref{pr:problem} will be solved by the robots.}\n\n\\subsection{\\mymod{Decentralized Behaviors Sequencing}}\n\\mymod{For the correct execution of the behaviors sequencing, each robot should start assembling a new graph only after all other robots have completed the current behavior. Similarly, a new behavior should start once all robots satisfy the neighbors' requirements for it. Now, we describe a decentralized strategy that allows execution of these two transitions without the need of a supervisor, nor synchronization between the robots.}\n\n\\mymod{With reference to Fig.~\\ref{fig:automata}, at any given time, each robot's mode of operation is described by a binary variable $\\alpha_i$ that describes whether robot $i$ is assembling the graph for an upcoming behavior ($\\alpha_i=1$) or executing a behavior ($\\alpha_i=0$). Without loss of generality, assume robots' initial configuration satisfies the communication requirements for the first behavior, which is then executed ($\\alpha_i = 0$). Once all robots have completed the first behavior, they start assembling the graph required by the following one ($\\alpha_i = 1$), while minimally perturbing the behavior just concluded. Once the new graph is satisfied $\\mathcal{G}_2 \\subseteq \\mathcal{G}(t)$, robots start behavior $\\mathcal{B}_2$ and exit from assembly mode ($\\alpha_i = 0$). This process repeats, until no successive behavior exists.} \n\n\\mymod{A correct execution of this process requires robot to agree on when to perform transitions $\\alpha_i = 0 \\rightarrow 1$ and $\\alpha_i = 1 \\rightarrow 0$. To this end, we take inspiration from the consensus-based algorithm described in~\\cite{wagenpfeil2009distributed} and we note that this choice is not central to the contribution of this paper. For each robot, we define a binary variable available only to robot $i$, $s_{t,i} \\in \\{0,1\\}$ that denotes whether robot $i$ itself has completed its current task $s_{t,i} = 1$ ($s_{t,i} = 0$ if robot has not completed its current task). In addition, we introduce a variable $\\sigma_i \\in \\mathbb{R}_+$, shared among neighbors, continuously updated through the following consensus-based process\n\\begin{equation} \\label{eq:sync}\n    \\sigma_i^+ = s_{t,i} \\frac{1}{|\\mathcal{N}_i(t)|+1}\\left(\n    \\sum_{j \\in \\mathcal{N}_i(t)} \\sigma_j + 1 \\right),\n\\end{equation}\nwhere $\\sigma_i^+$ represent the variable's value after the update. Owing to the diffusion of $\\sigma_1,\\dots,\\sigma_N$ throughout the network, we can interpret $\\sigma_i$'s as local measures of the team-wise completion of a task. As proved in~\\cite{wagenpfeil2009distributed}, if $s_{t,i}=1$ for all $i=1,\\dots N$ (i.e., all robots are capable to complete the current behavior), by following~(\\ref{eq:sync}), $\\lim_{t \\rightarrow \\infty} \\sigma_i = 1$, for all $i=1,\\dots N$. Therefore, robot $i$ starts assembling a new communication graph once the value of $\\sigma_i$ is close enough to $1$ (see~\\cite{wagenpfeil2009distributed} for a discussion on how to choose the switching threshold). A similar process is used for the transition $\\alpha_i=1 \\rightarrow 0$, where we replace $s_{t,i}$ and $\\sigma_i$ with $s_{a,i}$ and $\\eta_i$ respectively. The distributed sequencing procedure is summarized in Algorithm~\\ref{alg:dis_sequence}.}\n\n\n\\begin{algorithm}[h]\n\\caption{Distributed composition of behaviors.} \\label{alg:dis_sequence}\n$\\pi \\gets \\{\\mathcal{B}_1,\\dots, \\mathcal{B}_M\\}$ \\tcc*[r]{initialize behaviors}\n$k = 1$\\;\n$\\alpha_i \\gets 0$\\;\n\\While{$k < M+1$}{\n\\tcc*[h]{Aggregate data from neighbors} \\\\\n\\For{$j \\in \\mathcal{N}_i(t)$}\n{\n$\\{X_i, \\Sigma_i, H_i\\} \\gets \\{X_i, \\Sigma_i, H_i\\} \\cup \\{ x_j , \\sigma_j , \\eta_j \\}$ \\;\n}\n\\tcc*[h]{Compute nominal control} \\\\\n$\\hat{u}_i \\gets \\mathcal{U}_k(x_i,X_i)$\\; \n\\tcc*[h]{Compute team-wise completion states} \\\\\n\n\\eIf{$\\alpha_i == 0$}{\n\\lIf{{\\tt task complete}}\n{$s_e \\gets 1$}\n\\lElse{$s_e \\gets 0$} \n$\\sigma_i:=s_e\\,\\frac{1}{|\\mathcal{N}_k^i|+1}(\\sum_{j \\in \\mathcal{N}_i(t)} \\sigma_j + 1)$\\;\n\\If{$\\sigma_i > \\bar{\\sigma}$}{\n$\\alpha_i\\gets 1$\\; $k \\gets k+1$ \\;\n}\n}{\n\\lIf{{\\tt assembly complete}}\n{$s_a \\gets 1$}\n\\lElse{$s_a \\gets 0$} \n$\\eta:=s_a\\,\\frac{1}{|\\mathcal{N}_k^i|+1}(\\sum_{j \\in \\mathcal{N}_i(t)} \\eta_j + 1)$\\;\n\\If{$\\eta>\\bar{\\eta}$}{\n$\\alpha_i\\gets 0$\\; $s_e\\gets 0$ \\;}\n}\n\\tcc*[h]{Solve FCBF QP} \\\\\n$u_i \\gets QP(\\hat{u}_i,X_i,x_i)$\n}\n\\end{algorithm}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{graphics\/automata.pdf}\n    \\caption{Representation of the distributed sequencing framework and information flow. At all times, each robot's state is in either {\\tt behavior execution} ($\\alpha_i=0$) or {\\tt graph assembly} ($\\alpha_i=0$) modes. Switching between the two modes is triggered by the variables $\\sigma_i$ and $\\eta_i$ whose values is continuously) updated through~(\\ref{eq:sync}). When a switching between {\\tt graph assembly} and {\\tt behavior execution} occurs, a new behavior is started.}\n    \\label{fig:automata}\n\\end{figure}\n\n\\subsection{Applications}\n\\begin{figure*}[t]\n\\centerline{ \n\\subcaptionbox{\\label{fig2:a}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T0_task2_fixed}}~\n\\subcaptionbox{\\label{fig2:b}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T1_task2_fixed}}~\n\\subcaptionbox{\\label{fig2:c}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T2_task23_fixed}}\n} \n\\vspace{0.25cm}\n\\centerline{ \n\\subcaptionbox{\\label{fig2:d}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T3_task3_fixed}}~\n\\subcaptionbox{\\label{fig2:e}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T4_task3_fixed}}~\n\\subcaptionbox{\\label{fig2:f}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/T5_task3_fixed}}\n}\n\\caption{Overhead screen-shots from experiments on the Robotarium. Five robots execute two behaviors in sequence (cyclic-pursuit and formation). In figure, green patches represent robots that have completed their task, black rings represent robots that have all neighbors needed for the following task, and green lines represent edges that are available in the current communication graph. From (a) to (b) robots complete the first behavior. During second behavior, additional edges $(2,5)$ and $(3,5)$ are required (red dashed line represent missing edges). From (c) to (d), robots $2,3,5$ reduce their distance below the communication threshold. After the new graph is complete (d), robots initiate the second behavior (e) and complete it (f).\n\\label{fig:5RobotsExp}}\n\\end{figure*}\n\n\\mymod{We implemented the distributed sequencing framework on the Robotarium~\\cite{pickem2017robotarium}, on a team of $5$ differential drive robots. For this example, controllers are designed assuming a single integrator model, i.e. $\\bar{f}(x_i)=[0,0]^T$ and $\\bar{g}(x_i)=I_2$. In this example, robots execute a transition between two behaviors, where $\\mathcal{B}_1$ is a cyclic-pursuit behavior and $\\mathcal{B}_2$ is a formation assembly with leader. Cyclic-pursuit behavior is obtained through the following controller:\n\\begin{equation*}\n\t\\hat{u}_i = \\sum_{j \\in \\mathcal{N}_1^i} R(\\phi)\\,(x_j - x_i) \\quad \\forall \\, i=1,\\dots,5,\n\\end{equation*}\nwhere $R(\\phi)\\in SO(2)$ is the rotation matrix of angle $\\phi$, which is related to the desired cycle radius. Importantly, for this behavior to work, the communication graph $\\mathcal{G}_1$ must be a cycle graph. Considering robot $1$ as leader, the formation control behavior can be achieved with\n\\begin{align*}\n\\hat{u}_1 &= \\sum_{j \\in \\mathcal{N}_2^i} \\left( (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_i - x_i) \\right) + \\gamma_g(x_g-x_i) \\\\\n\t\\hat{u}_i &= \\sum_{j \\in \\mathcal{N}_2^i} (\\|x_i-x_j\\|^2-\\theta_{ij}^2)(x_j - x_i) \\quad i = 2,\\dots 5\n\\end{align*}\nwhere $\\theta_{ij} \\in \\mathbb{R}_+$ is the desired inter-robot distance, $x_g \\in \\mathcal{D}$ is the leader's goal, and $\\gamma_g \\in \\mathbb{R}_+$ the corresponding proportional gain. In the case of formation control, it is known that the Euclidean embedding of $\\mathcal{G}_2$ must be a rigid framework (see for instance~\\cite{mesbahi2010graph} and references therein). With reference to Fig.~\\ref{fig:5RobotsExp}, robots initially execute $\\mathcal{B}_1$ for a certain amount of time (a). Once completed (b) (green patches represent robots that have completed their current behavior), robots start assembling $\\mathcal{G}_2$ (c), after which $\\mathcal{B}_2$ is executed until $\\| \\hat{u}_i \\|$ is below a pre-defined threshold (d-f).}\n\n\\mymod{In Fig.~\\ref{fig:transition} we can observe the value of the two consensus variables $\\sigma_i$ and $\\eta_i$ for all robots during the behavior transition. Background colors represent the time intervals during which the two behaviors were executed, while the darker region in the middle corresponds to the assembly of the new graph. We observe the assembly and task variables $\\eta_i$ and $\\sigma_i$ approaching the value $1$ simultaneously for all robots, thus triggering a synchronized start of the successive phase.}\n\n\\mymod{The robustness of our technique was tested by simulating uniformly distributed delays between the robots. Results for this case are shown in Fig.~\\ref{fig:transition_delay} where we observe that although convergence of $\\eta_i$ and $\\sigma_i$ is no longer monotonic, robots still reach agreement on when to switch to the successive phase.}\n\n\\mymod{Finally, in order to show the benefits of the minimally invasive approach, we compare it with an alternative technique inspired by~\\cite{twu2010graph}, where, upon collective completion of a behavior, robots execute rendezvous until the communication graph required by the successive behavior is assembled. As shown by the simulation results for a sequence of $7$ behaviors (Fig.~\\ref{fig:energyComp}), the mean of the input's norm when considering our framework (red solid line) is always lower than the one obtained using the rendezvous as {\\it glue} behavior. Importantly, since transitions between behaviors occur faster in the minimally invasive case, the lower control effort cannot be attributed to a more {\\it relaxed} choice of controller gains.}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width = \\columnwidth]{graphics\/transition.png}\n    \\caption{Task and assembly consensus variables $\\sigma_i$ and $\\eta_i$ for $i=1,\\dots,N$ during a transition between two behaviors.}\n    \\label{fig:transition}\n\\end{figure}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width = \\columnwidth]{graphics\/transition_delay.png}\n    \\caption{Task and assembly consensus variables $\\sigma_i$ and $\\eta_i$ for $i=1,\\dots,N$ during a transition between two behaviors with communication delays.}\n    \\label{fig:transition_delay}\n\\end{figure}\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[trim ={2cm 0 0 0}, width=1.05\\columnwidth]{graphics\/energyComparison.png}\n    \\caption{Control input comparison between the minimally invasive sequencing framework proposed in this paper (red) and a sequencing based on rendezvous as {\\it gluing} behavior (blue). Solid lines represent the mean of the control input across all robots, while shaded regions represent the interval between minimum and maximum control input.}\n    \\label{fig:energyComp}\n\\end{figure}\n\\section{Problem Formulation}\n\\label{sec:problem}\nWe denote the state of a team of $n$ homogeneous mobile robots operating in a $d$-dimensional and connected domain $\\mathcal{D}$ as $x(t) = [x_1(t)^T,\\dots,x_n (t)^T]^T \\in \\mathcal{D} \\subset$ $\\mathbb{R}^{dn}$ where $x_i(t)\\in\\mathbb{R}^d$ is the position of robot $i$ at time $t$. As part of the coordinated nature of the behaviors being performed by the robots, each robot executes a control protocol which depends on the state of the subset of robots with which it interacts. We assume robots can communicate if the distance between them is less or equal to a sensing threshold $\\Delta\\in\\mathbb{R}_{>0}$. Thus, the list of possible interactions between agents are described by a time-varying, undirected, proximity graph $\\mathcal{G}(t)=(V,E(t))$, where $V=\\{1,\\dots,n \\}$ is the set of nodes representing the robots and $E(t)$ is the set of interacting pairs at time $t$, where \n\\begin{equation}\nE(t) = \\{ (i,j) \\in V \\times V\\, | \\, \\| x_i(t)-x_j(t)\\| \\leq \\Delta \\}.\t\n\\end{equation}\nFor each robot $i=1,\\dots,n$, we denote the set of available neighbors at time $t$ as $\\mathcal{N}_i(t) = \\{ j \\in V\\, | \\, (i,j) \\in E(t) \\}$, which depends on the position of the robots at time $t$.\n\nThe ensemble dynamics of the multi-agent system is described by\n\\begin{equation}\n\t\\dot{x} = f(x) + g(x)\\,u\n\t\\label{eq:ensembleDynamics}\n\\end{equation}\nwhere $f$ and $g$ are continuous locally Lipschitz continuous functions and $u = [u_1^T,\\dots,u_n^T]^T \\in U \\subset \\mathbb{R}^m$ is the vector of inputs, which depends on the particular behavior being executed. At all times, the control input $u$ in~(\\ref{eq:ensembleDynamics}) is given by a controller $\\mathcal{U}$, which can be defined as a state feedback law $\\mathcal{U}: \\mathcal{D} \\mapsto U$ or by a combination of both external parameters and state feedback law $\\mathcal{U}: \\mathcal{D} \\times \\Theta  \\mapsto U$, where $\\Theta$ is a space of parameters appropriate for the behavior. For instance, the controller corresponding to a {\\it weighted consensus} belongs to the first case. On the other side, a leader-follower protocol where followers maintain prescribed inter-agent distances is described by a controller that depends on both state feedback (followers' control) and exogenous parameters (leader's goal) (see Section~\\ref{sec:multAgImp} for examples).\n\nWe represent a {\\it mission} by an ordered sequence of $M$ coordinated behaviors\n\\begin{equation}\n\t\\pi = \\{ \\mathcal{B}_1,\\dots, \\mathcal{B}_M\\}.\n\\end{equation}\nThe $k^{\\text{th}}$ behavior in $\\pi$ is defined by the pair\n\\begin{equation}\\label{eq:tuple}\n\t\\mathcal{B}_k = \\{ \\mathcal{U}_k,\\, \\mathcal{G}_k\\},\n\\end{equation}\nwhere $\\mathcal{U}_k$ represents the coordinated controller described above and $\\mathcal{G}_k$ is the interaction graph required by behavior $\\mathcal{B}_k$ to function properly. We assume the list of behaviors $\\pi$ to be fixed and available to all robots. \\mymod{We will use the term {\\it behavior} to refer to a generalized multi-robot controller in the form~(\\ref{eq:tuple}) and to {\\it task} as the objective of the controller.}\n\nAs discussed in Section~\\ref{sec:intro}, each behavior requires a certain interaction structure between the robots (i.e., pairs of robots that need to be neighbors). With reference to~(\\ref{eq:tuple}), we describe an interaction structure via the graph $\\mathcal{G}_k=(V,E_k)$. Thus, denoting by $t_k^\\vdash$ and $t_k^\\dashv$ the starting and ending times for behavior $k$, the robots' configuration needs to satisfy $\\mathcal{G}_k \\subseteq \\mathcal{G}(t)$ for all $t \\in [t_k^\\vdash, t_k^\\dashv]$. In other words, as shown in Fig.~\\ref{fig:beh_seq}, the interaction structure required by each behavior needs to be a spanning graph of the graph induced by the state of the agents during the interval of time the behavior is executed. \\mymod{At this point, given a list of behaviors constituting the mission $\\pi$ and the corresponding multi-robot controllers, we want to design a procedure that enables robots to assemble and maintain the communication graph required by each behavior.}\n\n\n\\mymod{\n\\begin{problem} \\label{pr:problem}\nGiven an ordered sequence of coordinated behaviors $\\pi= \\{ \\mathcal{B}_1,\\dots, \\mathcal{B}_M\\}$, where each $\\mathcal{B}_k = \\{ \\mathcal{U}_k,\\, \\mathcal{G}_k\\}$ can be completed by the robots in finite-time, design a feedback control policy to compose the behaviors such that\n\\begin{equation}\n\\mathcal{G}(t) \\supseteq\n    \\begin{cases}\n    \\mathcal{G}_k \\, &t \\in [t_k^\\vdash , t_k^\\dashv] \\\\\n    \\mathcal{G}_k \\cup \\mathcal{G}_{k+1} \\, &t \\in (t_k^\\dashv,t_{k+1}^\\vdash)\n\\end{cases} \\quad \\forall \\, k=1,\\dots,M-1.\n\\end{equation}\n\\end{problem}\n}\n\\section{Finite-Time Barrier Functions} \\label{sec:fcbf}\nIn this section we review the general definition of Finite-time Convergence Control Barrier Function (FCBF) which was first introduced in~\\cite{li2018formally} and inspired by the finite-time stability analysis for autonomous system introduced in~\\cite{bhat2000finite}. Given a dynamical system operating in an open set $\\mathcal{D} \\subseteq \\mathbb{R}^n$ and a set $\\mathcal{C}\\subset\\mathcal{D}$, barrier functions~\\cite{xu2015robustness} are Lyapunov-like functions that guarantee forward invariance of $\\mathcal{C}$ with respect to the state of the system. In other words, if an appropriate barrier function exists, it can be used to show that if the state of a system is in $\\mathcal{C}$ at some time, it will be in $\\mathcal{C}$ thereafter. The concept of barrier functions was extended to Zeroing Control Barrier Functions (ZCBF) in~\\cite{xu2015robustness}, where asymptotic convergence of the state to the set $\\mathcal{C}$ was discussed. Thus, provided that an appropriate ZCBF exists, if the state of the system is not in $\\mathcal{C}$ at some initial time, it will asymptotically converge to $\\mathcal{C}$. \n\nAs discussed in the introduction, before execution of a coordinated behavior, robots need to satisfy certain spatial configurations imposed by the behavior itself. Importantly asymptotic convergence to the correct configuration is not sufficient. In fact, if we consider $\\mathcal{C}$ as the joint set of all initial configurations required for a particular behavior, the state must strictly belong to $\\mathcal{C}$ for the behavior to work properly. Following this observation, the need for a finite-time convergence extension of the previous concepts becomes clear. In particular, denoting the state of the system as $x(t)\\in\\mathcal{D}$, we are interested in verifying the following conditions:\n\\begin{itemize}\n\t\\item if $x(t_0)\\in\\mathcal{C}$, then $x(t)\\in\\mathcal{C}$ for all $t > t_0$\n\t\\item if $x(t_0) \\notin \\mathcal{C}$, then $x(t)\\in\\mathcal{C}$ for some $t_0<t<\\infty$.\n\\end{itemize}\nIn order to do this, we encode the set $\\mathcal{C} \\subset \\mathcal{D} \\subseteq \\mathbb{R}^n$, through the superzero-level set of a continuously differentiable function $h: \\mathcal{D} \\rightarrow \\mathbb{R}$, i.e.,\n\\begin{equation}\n\t\\mathcal{C} = \\{ x\\in \\mathcal{D} \\, | \\, h(x)\\geq 0 \\}.\n\t\\label{eq:sefdef}\n\\end{equation}\n\\begin{definition}\nWe introduce the following class-$\\mathcal{K}$ function\n\\begin{equation}\n\t\\bar{\\alpha}_{\\rho,\\gamma}(h(x)) = \\gamma \\cdot \\,\\text{sign}(h(x))\\, \\cdot |h(x)|^\\rho,\t\\label{eq:alpha}\n\\end{equation}\nwith $\\rho \\in [0,1)$ and $\\gamma>0$, which is continuous everywhere and locally Lipschitz everywhere except at the origin~\\cite{bhat2000finite}.\n\\end{definition}\n\n\\begin{definition}~\\cite{li2018formally} For a dynamical system \n\\begin{equation} \\label{eq:affinesystem}\n\t\\dot{x} = f(x)+g(x)u\n\\end{equation} with $x\\in\\mathcal{D}$, $u \\in U \\subset \\mathbb{R}^m$, and for a set $\\mathcal{C}$ induced by $h$, if there exists a function $\\bar{\\alpha}_ {\\rho,\\gamma}(h(x))$ of the form~(\\ref{eq:alpha}) such that\n\t\\begin{equation}\n\t\t\\sup_{u \\in U}\\bigg\\{ L_fh(x) + L_gh(x)u + \\bar{\\alpha} _{\\rho,\\gamma}(h(x)) \\bigg\\} \\geq 0 \\quad \\forall x\\in\\mathcal{D},\n\t\\end{equation}\nthen, the function $h$ is a {\\it Finite-time Convergence Barrier Function }(FCBF) defined on $\\mathcal{D}$.\n\\end{definition}\n\nFollowing from the definition above, we define the set of admissible control inputs as\n\\begin{equation} \\label{eq:admisU}\n\tK(x) = \\{ u \\in U \\, | \\, L_fh(x) + L_gh(x)u + \\bar{\\alpha} _{\\rho,\\gamma}(h(x)) \\geq 0\\}.\n\\end{equation}\n\n\\begin{theorem} \\label{thm:FCBF} \\cite{li2018formally}~Given a set $\\mathcal{C}\\subset \\mathbb{R}^n$, any Lipschitz continuous controller $\\mathcal{U}: \\mathcal{D} \\mapsto U$ such that \n\\begin{equation} \\label{eq:fcbf_controllers}\n\t\\mathcal{U}(x) \\in K(x) \\qquad \\forall x\\in\\mathcal{D},\n\\end{equation}\nrenders $\\mathcal{C}$ forward invariant for the system~(\\ref{eq:affinesystem}). Moreover, given an initial state $x_0 \\in \\mathcal{D} \\backslash \\mathcal{C}$, the same controller $\\mathcal{U}$ results in $x(T)\\in\\mathcal{C}$, where\n\t\\begin{equation}\n\t\tT \\leq \\frac{1}{\\gamma(1-\\rho)}|h(x_0)|^{1-\\rho}.\n\t\\end{equation}\n\\end{theorem}\n\n\\iffalse\n\\begin{IEEEproof}\n\tLet's consider the following Lyapunov function $V(x)=\\max\\{0,-h(x)\\}$. It can be verified that\n\\begin{align}\n\t\tV(x) &> 0 \\qquad  x\\in\\mathcal{D}\\backslash \\mathcal{C} \\\\\n\t\tV(x) &= 0 \\qquad  x\\in\\mathcal{C}\n\\end{align}\t\n\nIn addition, since\n\\begin{equation}\n\t\\frac{\\partial V(x)}{\\partial h(x)}=\n\t\\begin{cases}\n\t-1 &\\qquad x\\in\\mathcal{D}\\backslash \\mathcal{C} \\\\\n\t 0 &\\qquad  x\\in\\mathcal{C}\n\t\\end{cases}\n\\end{equation} \t\nit follows that $\\dot{V}(x(t)) \\leq -\\gamma\\,V^\\rho(x(t))$, for all $t$. \n\nConsider $x_0 = x(t_0) \\in \\mathcal{C}$. Since $V(x_0)=0$ and $\\dot{V}(t)=0$, we have $x(t)\\in\\mathcal{C}$ for all $t>t_0$, from which forward invariance of $\\mathcal{C}$ under $\\mathcal{U}$ follows. Now, consider $x_0 \\in \\mathcal{D}\\backslash\\mathcal{C}$. As shown in~\\cite{bhat2000finite}, the dynamics $\\dot{h} _1(t) = -\\gamma\\,\\text{sign}(h_1(t))\\, |h_1(t)|^\\rho$, with $\\rho \\in [0,1)$ and $\\gamma>0$ drives $h_1$ to the origin, and the minimum time $T_1$ for which $h_1$ reaches $0$ (denoted finite-settling time) is\n\\begin{equation}\n\tT_1 = \\frac{1}{\\gamma(1-\\rho)}|h_1(t_0)|^{1-\\rho}.\n\\end{equation} \nBy applying the comparison lemma~\\cite{pachpatte1997inequalities}, if $h(0) \\geq h_1(0)$ and $\\dot{h}(t) \\geq \\dot{h}_1(t)$, then $h(t) \\geq h_1(t)$ for all $t \\geq 0$ and consequently under the effect of $\\mathcal{U}$ we have\n\\begin{equation}\n\th(x(T)) = 0, \\quad T \\leq T_1.\n\\end{equation}\n\\end{IEEEproof}\n\\fi\n\nIn conclusion, by selecting a controller that verifies condition~(\\ref{eq:fcbf_controllers}), both forward invariance and finite-time convergence to the desired set are guaranteed. \n\n\n\\section{Composition of Coordinated Behaviors}\n\\label{sec:seq_framework}\n\nIn addition to the list $\\pi$, transitions between behaviors need to be synchronized, i.e., for each behavior $\\mathcal{B}_k$, $k=1,\\dots, M$, robots must 1) start assembling $\\mathcal{G}_{k+1}$ only after all robots have completed $\\mathcal{B}_k$ and 2) start executing $\\mathcal{B}_{k+1}$ only after condition $\\mathcal{G}_{k+1} \\subseteq \\mathcal{G}(t)$ is satisfied. We assume the existence of a discrete counter $\\sigma \\in [1,\\dots,M]$ which indicates the active behavior and a binary signal\n\\begin{equation}\n\\eta(\\sigma) = \\begin{cases}\n\t1 \\quad \\text{if} \\quad \\mathcal{G}_k \\subseteq \\mathcal{G}(t) \\\\\n\t0 \\quad \\text{o.w.}\n\\end{cases}\n\\end{equation} \nwhich describes whether the interaction structure required by behavior $\\mathcal{B}_\\sigma$ is available. In this section, we assume both signals to be controlled by a supervisor and made available to the robots at all times, e.g., through a dedicated static communication network. \\mymod{In the next section, we discuss the extension to a fully distributed framework.}\n\n\\begin{figure}[h!]\n\\includegraphics[width=\\columnwidth]{graphics\/figure_Sequence.pdf}\n  \\caption{Schematic representation of the behaviors sequencing framework. Behavior $\\mathcal{B}_k$ is executed during the blue portion of the timeline and $\\mathcal{B}_{k+1}$ is executed during the orange portion. Sequential execution of behaviors requires each agent to reach a spatial configuration such that the desired graph is a spanning graph of the communication graph, i.e., $\\mathcal{G}_k \\subseteq \\mathcal{G}(t_k^\\vdash)$ and $\\mathcal{G}_{k+1} \\subseteq \\mathcal{G}(t_{k+1}^\\vdash)$ respectively. \\label{fig:beh_seq}}\n\\end{figure}\n\nFollowing from the communication modality assumed for the robots, communication constraints can be expressed in terms of relative distance between the robots. In other words, behavior $\\mathcal{B}_k$ can be correctly executed if, for all $t\\in[t_k^\\vdash,t_k^\\dashv]$, all the distances between pairs in $E_k$ are below the proximity threshold $\\Delta$. To this end, a convenient pair-wise connectivity FCBF can be defined as\n\\begin{equation}\n\th_{ij}^c(x) = \\Delta^2 - \\| x_i - x_j \\|^2,\n\t\\label{eq:commBarriers}\n\\end{equation}\nand we note that if $\\|x_i-x_j\\| \\leq \\Delta$, then $h_{ij}^c(x)\\geq 0$.\nIn addition, the edge-level and ensemble-level connectivity constraint sets for behavior $\\mathcal{B}_k$ are\n\\begin{align}\n\\mathcal{C}_{ij}^c &= \\{ x \\in \\mathcal{D} \\,|\\, h_{ij}^c(x) \\geq 0 \\} \\\\\n\\mathcal{C}^c_k &= \\{ x \\in \\mathcal{D} \\,|\\, h_{ij}^c(x) \\geq 0, \\, \\forall (i,j)\\in  E_k\\}.\n\\end{align}\n\nFollowing the definition given in~(\\ref{eq:admisU}), the admissible set of control inputs that guarantees finite-time convergence to $\\mathcal{C}^c_k$ is:\t\n\\begin{multline} \\label{eq:admGraphinput}\n\tK_k^c (x) = \\{ u \\in U \\, | \\, \\dot{h}_{ij}^c(x) + \\bar{\\alpha} _{\\rho,\\gamma}(h_{ij}^c(x)) \\geq 0 , \\\\ \\forall (i,j)\\in E_k \\}\n\\end{multline}\n\n\\begin{theorem} \\label{thm:fcbfControl}\n\tDenoting with $x_0$ the initial state of the system with dynamics~(\\ref{eq:ensembleDynamics}), any controller $\\mathcal{U}:\\mathcal{D} \\mapsto U$ such that $\\mathcal{U}(x_0) \\in K_k^c(x_0)$ for all $x_o \\in \\mathcal{D}$, will drive the system to $\\mathcal{C}^c_k $ within time\n\\begin{equation} \\label{eq:fcTime}\n\tT_k = \\max_{ (i,j) \\in E_k |  h^c_{ij}(x_0)<0} \\left\\{ \\frac{1}{ \\gamma(1-\\rho)} | h_{ij}^c(x_0) |^{1-\\rho} \\right\\}.\\end{equation}\n\\end{theorem}\n\n\\begin{IEEEproof} \n\tConsider all pairs of agents $i$ and $j$, such that $(i,j) \\in E_k$. If $h_{ij}^c(x_0) \\geq 0$, i.e., agents $i$ and $j$ are within communication distance, the forward invariance property of $\\mathcal{U}$, guarantees that $i$ and $j$ will stay connected. In this case, the state will reach $\\mathcal{C}_{ij}^c$, within time $T_{ij}=0$. On the other side, consider $h_{ij}^c(x_0)<0$. Any $\\mathcal{U}(x_0) \\in K_k^c(x_0)$ satisfies the finite-time convergence barrier certificates, and because of Theorem~\\ref{thm:FCBF}, if $x_0 \\notin \\mathcal{C}_{ij}^c$, then $x(T_{ij})\\in \\mathcal{C}_{ij}^c $, with\n\\begin{equation}\nT_{ij} \\leq  \\frac{1}{ \\gamma(1-\\rho)} | h_{ij}^c(x_0) |^{1-\\rho}.\n\\end{equation}   \nSince every communication constraint $\\mathcal{C}_{ij}^c $ will be reached within time $T_{ij}$, the total time required to drive $x(t)$ to $\\mathcal{C}_k^c$ is upper bounded by\n\\begin{equation}\n\tT_k = \\max_{(i,j) \\in E_k |  h^c_{ij}(x_0)<0} T_{ij}.\n\\end{equation}\t\n\\end{IEEEproof} \nWhen selecting control inputs from set~(\\ref{eq:admGraphinput}), the system~(\\ref{eq:ensembleDynamics}) will satisfy requirements for behavior $\\mathcal{B}_k$ in finite time.\n\n\\subsection{Finite-Time Convergence Control Barrier Functions}\n\\label{sec:ftcontrolBF}\nOnce behavior $\\mathcal{B}_{k-1}$ is completed, robots are required to converge to the set $\\mathcal{C}^c_{k}$ before behavior $\\mathcal{B}_{k}$ can start. Under the lead of the external supervisor, the change of behavior is communicated to the robots through the signal $\\sigma$, which transitions from $k-1$ to $k$ once $\\mathcal{B}_{k-1}$ is completed. Now, although finite-time convergence to $\\mathcal{C}^c_{k}$ can be achieved by selecting any control input in $K_{k}^c(x)$, we seek to minimally perturb the execution of the behavior just concluded, namely $\\mathcal{B}_{k-1}$. This can be accomplished by solving a problem similar to the one proposed in~\\cite{ames2014control}, which we adapt to our framework. Denoting with $\\hat{u}_{k}=\\mathcal{U}_{k}(x)$ the nominal control input from behavior $\\mathcal{B}_{k}$, during transition between $\\mathcal{B}_{k-1}$ and $\\mathcal{B}_{k}$ the actual control input to the robots $u^*$ is defined as\n\\begin{equation}\\label{eq:minQP1}\nu^* = \\argmin_{u \\in U}  \\| \\hat{u}_{k-1} - u \\|^2 \\\\\n\\end{equation}\nsubject to\n\\begin{equation} \\label{eq:constrTransition}\n   L_f\\,h_{ij}^c + L_g\\,h_{ij}^c\\,u + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0, \n\\end{equation}\nfor all $(i,j) \\in  E_{k-1} \\cup E_{k}$. Once all required edges $E_{k}$ are established (i.e., $\\eta=1$), edges in $E_{k-1}$ are no longer necessary. At this point, under the effect of the controller $\\mathcal{U}_{k}$, the list of constraints in~(\\ref{eq:constrTransition}) is substituted with\n\\begin{equation} \\label{eq:constrExecution}\n   L_f\\,h_{ij}^c + L_g\\,h_{ij}^c\\,u + \\bar{\\alpha}_{\\rho,\\gamma}(h_{ij}^c) \\geq 0,\n\\end{equation}\nfor all $(i,j)\\in  E_{k}$. Since the cost function is convex and the inequality constraints~(\\ref{eq:constrTransition}) and~(\\ref{eq:constrExecution}) are control affine, the problem can be solved in real-time. In conclusion, because of the finite-time convergence and forward invariance properties of the above formulation, \\mymod{if $\\mathcal{B}_{k-1}$ can be completed and a solution to~(\\ref{eq:minQP1}-\\ref{eq:constrTransition}) (or (\\ref{eq:minQP1}-\\ref{eq:constrExecution})) exists}, robots will converge to the configuration required by $\\mathcal{B}_{k}$, and maintain it throughout its execution.\n\\mymod{\n\\begin{remark}\nThe solution of~(\\ref{eq:minQP1}-\\ref{eq:constrTransition}) (or (\\ref{eq:minQP1}-\\ref{eq:constrExecution})) is contingent upon the existence of a control input capable to solve all constraints. In other words, $K_k^c(x) \\cap K_{k+1}^c(x)$ (or $K_k^c(x)$) should not be empty for all times. For this, it is necessary that a robot's configuration that satisfies all constraints of the problem exists. However, this is not sufficient as the progress towards the desired configuration might be obstructed by constraints on the actuators or deadlock configurations. Although we do not address this directly, it is possible to mitigate feasibility issues by considering, for example, constraints relaxation, sum of squares barrier functions, or pre-defined back-up controllers (see~\\cite{ames2019control} and references therein). \\label{rmk:feasibility}\n\\end{remark}}\n\n\n\n\\subsection{Initial Constraints}\nIn addition to the communication constraints considered above, certain missions might require additional conditions to be met before each behavior can start. For example, during the exploration tasks it might be desirable for one robot to always stay within range of communication with a human-operator, or to maintain a minimum distance from an unsafe area. Assuming $\\mathcal{B}_k$ requires a number of distinct $s_k$ of such constraints, we encode the entire set of initial conditions through a list of barrier functions $h_\\ell^s(x)$, with $\\ell = 1,\\dots,s_k $:\n\\begin{equation} \\label{eq:initialSet}\n\\mathcal{C}^{s}_k = \\{ x \\in \\mathcal{D} \\,|\\, h_\\ell^s(x) \\geq 0, \\, \\forall \\ell = 1,\\dots,s_k \\}.\n\\end{equation}\nFollowing this definition, we define a set of admissible control inputs similar to the one in~(\\ref{eq:admGraphinput}) that will drive the state of the system to the desired set within finite time:\n\\begin{multline} \\label{eq:admInitCond}\n\tK_k^s (x) = \\{ u\\in U \\, | \\, \\dot{h}_\\ell^s(x) + \\bar{\\alpha} _{\\rho,\\gamma}(h_\\ell^s(x)) \\geq 0, \\\\\n\t \\forall \\ell = 1,\\dots,s_k \\}.\n\\end{multline}\n\nThe set of controls satisfying both communication and initial conditions constraints can thus be obtained by intersection of set~(\\ref{eq:admInitCond}) and~(\\ref{eq:admGraphinput}):\n\\begin{equation}\n\tK_k(x) = K_k^c (x) \\bigcap K_k^s (x).\n\\end{equation}\n\nWe note that the results in Theorem~\\ref{thm:fcbfControl} and the formulation of minimally invasive controller in~(\\ref{eq:minQP1}) still holds valid by considering the set $K_k(x)$ instead of $K_k^c (x)$ as the set of admissible control inputs.\n\\section{Case Study: Securing a Building} \\label{sec:securing}\nThe objective of this section is to describe the {\\it Securing a Building} mission, which will be used as testing scenario for the composition framework. We describe now the main structure and objective of the mission, while we deconstruct it into coordinated behaviors in the next subsection.\n\n\\subsection{Mission Overview}\nIn the Securing a Building mission, a group of robots are deployed in an urban environment to identify an unknown target building and rescue a subject located inside. Based on \\cite{FieldManual}, we\ndecompose this mission into the following 4 phases:\n\nFIND - First, the robots are tasked with identifying the target building by means of surveillance of the perimeters of all the buildings. For efficient exploration, robots can be broken into sub-teams. Each team reports collected information at the base after each building has been investigated. Once the target building has been identified, the robots reunite and prepare for the next phase.\n\nISOLATE - The robots isolate the target building by patrolling its perimeter. To achieve this, the robots are divided into two subgroups - the {\\it security agents} responsible for boundary protection and the {\\it maneuvering agents} tasked with entering the building.\n\nRESCUE - During the rescue phase, the security agents keep patrolling around the building. In the meanwhile, the maneuvering agents enter the building, clear the rooms, and seize positions as they maneuver through the building to find the subject to be rescued. Once the subject has been located, the robots transport it to the safe zone.\n\nFOLLOW-THROUGH - As the interior of the building is being cleared, individual robots are left inside as beacons, while the remaining robots from the maneuvering agents leave the building, gather on the outside with the security agents, and report back to the base station.\n\nA number of arguments support the choice of the Securing a Building mission as an ideal scenario for testing multi-robot techniques and algorithms. First, the requirement of spatially diverse functionalities that cannot be provided by single robots naturally requires the use of multi-robot systems. Second, the final goal of the mission, namely rescuing the subjects of interest, reflect the fact that general real-world missions cannot be accomplished with single controllers. Lastly, thanks to its modularity, techniques focusing on specific aspects of the mission can be integrated and tested without influencing the overall structure of the mission (see the Appendix for details).\n\n\\subsection{Securing a Building Through Composition of Behaviors}\n\n\\begin{figure*}[t]\n\\centerline{ \\includegraphics[width=1.9\\columnwidth]{graphics\/missionChart1.pdf}}\n\\caption{Mission design chart showing how coordinated behaviors are composed together to tackle the Securing a Building mission. The four bold titles are the mission phases and the large boxes below them indicate specific agent roles\nand associated behaviors. The arrows in the chart indicate the transitions between different behaviors. We note that he choice of controllers that produces the behaviors in the chart is not unique.\\label{fig:missionChart}}\n\\end{figure*}\n\nWe deconstruct the Securing a Building mission through ordered sequences of coordinated behaviors. The process is summarized in Fig.~\\ref{fig:missionChart}. We refer to behaviors in terms of their main objectives, acknowledging that different implementations can be used to achieve the same results. We highlight these behaviors in parenthesis.\n\n\\paragraph{FIND} Robots initially coordinate with the operator at the base station ({\\it rendezvous}). After that, robots are divided into different search teams, each assigned with a list of buildings to investigate ({\\it task allocation}). Subsequently, all the teams investigate their own lists of buildings. First team of robots travels to the vicinity of a building ({\\it leader-follower}), then start to survey the exterior of the building ({\\it perimeter patrol}), and return to the base ({\\it leader-follower}). This process repeats until the target building is discovered.\n\n\\paragraph{ISOLATE} Robots gather near the base ({\\it rendezvous}), then are divided into {\\it security} and {\\it maneuvering} agents ({\\it task allocation}). After traveling from the base to the vicinity of the target building ({\\it go-to-goal}), security agents protect the building's perimeter ({\\it cyclic pursuit}), until the end of the RESCUE phase. Meanwhile, the maneuvering agents locate the building's entrance, by following its perimeter ({\\it perimeter patrol}). Once the entrance has been found, the maneuvering agents gather at the entrance ({\\it rendezvous}) and create a formation ({\\it formation control}) before entering.\n\n\\paragraph{RESCUE} The maneuvering agents enter the building in formation ({\\it formation control}) and cover the interior area ({\\it area coverage}). Once the location of the subject to rescue is identified, the robots form a circular closure around the subject ({\\it cyclic pursuit}). Then, the robots transport the subject to the safety zone, while maintaining the circular closure around the subject ({\\it containment control}).\n\n\\paragraph{FOLLOW-THROUGH} Maneuvering agents spread ({\\it scatter}) over the interior of the building. To signify that the area has been cleared, few robots are left inside the building as beacons ({\\it persistent coverage}). The rest of the maneuvering agents and the security agents reunite outside the building ({\\it rendezvous}). At last, they return to the base ({\\it leader-follower}).\n\n\n\\subsection{Results}\nWe tested the behavior composition framework described in Section~\\ref{sec:multAgImp} on the Securing a Building mission, which was executed on the Robotarium~\\cite{pickem2017robotarium}. In Fig.~\\ref{fig:experiment}, we display selected snapshots of the mission obtained by a camera mounted on the ceiling. In the experiment, $8$ differential-drive robots, indexed $1,\\dots,8$ are deployed in a simulated urban environment composed of $6$ buildings, blue polygons indexed $1,\\dots,6$. In this experiment, we simulate a maximum sensor range $\\Delta = 0.5$m. Because of the different spatial scales between FIND\/ISOLATE phases and RESCUE\/FOLLOW-THROUGH phases, the entire mission is divided in two parts. In the first part (Fig.~\\ref{fig2:a} to Fig.~\\ref{fig2:d}) the experiment is performed at a {\\it neighborhood}-level scale. The remaining two phases are executed in a zoomed-in environment, which focuses on the one building of interest (Fig.~\\ref{fig2:d} to Fig.~\\ref{fig2:f}).\n\nDuring FIND phase (Fig.~\\ref{fig2:a} and~\\ref{fig2:b}), two groups of robots $\\text{\\sc team}1:\\{1,2,3,4\\}$ and $\\text{\\sc team}2:\\{5,6,7,8\\}$ investigates preassigned lists of buildings, leaving some agents near the base station (the purple filled dot in the top right corner) if destination building cannot be reached without breaking the connectivity constraints. The red polygon in Fig.~\\ref{fig2:b} and~\\ref{fig2:c} is the target building after being identified by $\\text{\\sc team}1$. During the ISOLATE phase (Fig.~\\ref{fig2:c}), maneuvering agents look for the entrance, while the security agents secure the outer perimeter. \n\nDuring the RESCUE phase (Fig.~\\ref{fig2:d} to~\\ref{fig2:e}), the agents inside the building, i.e. $\\text{\\sc team}1$, localize the target (red dot) using Voronoi coverage (Fig.~\\ref{fig2:d}) and escort it to the safe area (red circle) as shown in (Fig.~\\ref{fig2:e}). Finally, robots $1$ and $2$ are left as beacon inside the building, while remaining robots return to the base (Fig.~\\ref{fig2:f}).\n\n\\begin{figure*}[t]\n\\centerline{ \n\\subcaptionbox{\\label{fig2:a}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation1}}~\n\\subcaptionbox{\\label{fig2:b}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation2}}~\n\\subcaptionbox{\\label{fig2:c}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation4}}\n} \\vspace{0.25cm}\n\\centerline{ \n\\subcaptionbox{\\label{fig2:d}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation7}}~\n\\subcaptionbox{\\label{fig2:e}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation9}}~\n\\subcaptionbox{\\label{fig2:f}}{\\includegraphics[width=0.66\\columnwidth, height=0.39\\columnwidth]{graphics\/simulation11}}\n}\n\\caption{Overhead screen-shots from experiments on the Robotarium. A team of eight robots is divided in $\\text{\\sc team}1:\\{1,2,3,4\\}$ and $\\text{\\sc team}2:\\{5,6,7,8\\}$. Because of the different spatial scales between FIND\/ISOLATE phases and RESCUE\/FOLLOW-THROUGH phases the mission is executed on two different environments. Each team is assigned with a list of three buildings to inspect sequentially. FIND: (a) perimeter patrol of buildings $2$ and $5$; (b) building $4$ is identified as the target building, while $\\text{\\sc team}1$ waits for $\\text{\\sc team}2$ to return to base. ISOLATE: (c) $\\text{\\sc team}2$ secures perimeter of building, while $\\text{\\sc team}1$ inspects exterior of building, searching for the entrance. RESCUE: after entering the building, $\\text{\\sc team}1$ performs domain coverage of the building until target (red dot) is identified (d); after this, (e) robots escort target to safe location (red circle). FOLLOW-THROUGH: finally, two robots are left as beacons inside the building while all remaining robots return to base (f).   \n\\label{fig:experiment}}\n\n\\end{figure*}\n\\section{Related Work} \nThe problem of partitioning complex objectives into simpler tasks can be solved by sequentially composing {\\it primitives}, e.g.,~\\cite{cassandras2009introduction}, or by blending them simultaneously in a {\\it hierarchical} fashion. An example of hierarchical composition for single robot motion control is navigation between points while avoiding obstacles, e.g.,~\\cite{arkin1998behavior}. \\mymod{In general, the problem of controlling a system by composing different modes of operation pertains to hybrid systems and {\\it multi-modal} control domains~\\cite{koutsoukos2000supervisory}.}\n\nBecause of the complexity emerging from the composition of distinct controllers, guarantees on the safety and correctness of the final results need to be established~\\cite{kress2018synthesis}. Provable correct composition of control policies is investigated in the formal methods literature. Recently, compositional strategies inspired from formal methods have been used for the development of control strategies for multi-robot systems~\\cite{srinivasan2018control,garg2019control,meyer2019hierarchical,chen2018verifiable}. In particular, the authors of~\\cite{srinivasan2018control} use tools from linear temporal logic (LTL) for the specification of behaviors to be executed by the system. The solution is based on a sequence of constrained reachability problems, each consisting of a target set to be reached in finite time and a safety set within which the system must stay at all times. \\mymod{A related approach is developed in~\\cite{garg2019control}, where the problem of prescribed-time convergence to spatio-temporal specifications is formulated using control barrier functions.} The authors in~\\cite{meyer2019hierarchical} discuss a hierarchical decomposition method for controller synthesis given LTL specifications.\n\nIn the context of controllers composition for multi-robot systems, in~\\cite{belta2007symbolic} the authors use symbolic methods in order to generate high-level instructions from form of {\\it human-like} language. The authors of~\\cite{klavins2000formalism} introduce a framework for the composition of controllers in robotic systems using Petri Nets. In~\\cite{marino2009behavioral}, behaviors from the Null-Space-Behaviors framework are combined in order to solve ad-hoc tasks, such as perimeter patrol. A supervisor, represented as a finite state automata, selects high-level behaviors by assembling low-level behaviors. In~\\cite{nagavalli2017automated}, a revised version of the $A^*$ algorithm is used to generate an optimal path of behaviors, such that the overall cost of the mission is minimized. Similarly, in~\\cite{vukosavljev2019hierarchically} motion planning for a team of quadcopters is solved by defining higher level motion primitives obtained by a spatial partition of the environment. However, none of these approaches specifically address the problem of correct composition between primitives, which is the focus of this paper.\n\nAs discussed in the previous section, coordination between agents is possible only if particular interactions exist between the robots. In multi-robot systems, interaction requirements are commonly investigated in terms of connectivity maintenance, i.e., a certain graph or node-connectivity needs to be guaranteed at all times. Methods employed in the solution to this problem include edge weight functions~\\cite{ji2007distributed}, control rules based on estimate of algebraic connectivity~\\cite{sabattini2013distributed}, hybrid control~\\cite{zavlanos2009hybrid}, passivity~ \\cite{igarashi2009passivity}, and barrier functions~\\cite{wang2016multi}. If connectivity between agents needs to be guaranteed in non-nominal circumstances, resilient solutions must be in place as well, e.g.,\n\\cite{ramachandran2019resilience}, \\cite{panerati2019robust}, and~\\cite{varadharajan2019unbroken}. Notably, a technique based on graph process specifications for the sequential composition of different multi-agent controllers is discussed in~\\cite{twu2010graph}. Similar to our work, the authors in~\\cite{twu2010graph} bridge the gap between composition of controllers and the topology requirements by encoding requisites for each controller in terms of graphs.\nHowever, while in \\cite{twu2010graph} {\\it incompatible} controllers are combined through the introduction of a {\\it bridging} controller, in our approach controllers are minimally modified by the robots in order to satisfy upcoming requirements. Our approach significantly reduces the complexity of the composition process, \\mymod{minimizes the energy spent by the robots to switch between behaviors}, and can accommodate additional constraints, such as inter-robot collisions and obstacles avoidance.\n\n\\section{Introduction} \\label{sec:intro}\n\\IEEEPARstart{A}{s} our understanding of how to structure control and coordination protocols for teams of robots increases, a number of application domains have been identified, such as entertainment~\\cite{ackerman2014flying}\\cite{du2018fast}, surveillance~\\cite{santos2018coverage}~\\cite{shishika2018local}, manipulation~\\cite{han2018hybrid}, and search-and-rescue~\\cite{suarez2011survey}. Along with a decrease in the production and manufacturing costs associated with the platforms themselves, these applications have been enabled by a number of theoretical results that have emerged at the intersection of different disciplines such as robotics, controls, computer science, and graph theory~\\cite{zelazo2018graph}. \n\nFrom a motion controls perspective, one notable requirement is given by the need to define actions capable to solve team-wise objectives on the basis of locally available information. For instance, different extensions of the consensus equation have been used to arrive at locally defined controllers with provable, global convergence properties~\\cite{cortes2017coordinated}. In this way, it is possible to construct coordinated controllers for the solution of motion control problems, such as rendezvous~\\cite{lin2003multi}~\\cite{ren2005coordination}, cyclic pursuit~\\cite{ramirez2009cyclic}, formation control~\\cite{lawton2003decentralized}~\\cite{buckley2017infinitesimally}, coverage~\\cite{cortes2004coverage}~\\cite{santos2018coverage}, leader-based control~\\cite{mesbahi2010graph}, and flocking~\\cite{tanner2007flocking}. Particular instantiations of some of these behaviors are shown in Fig.~\\ref{fig:coordBehExamples} on a group of six simulated differential drive robots.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[trim={2cm 2cm 1cm 2cm},width=0.35\\columnwidth]{graphics\/redezvous.pdf}~\n\t\t\\includegraphics[trim={3cm 2cm 2.5cm 2cm},width=0.32\\columnwidth]{graphics\/cyclicPurs.pdf}~\n\t\t\\includegraphics[trim={2cm 2cm 2cm 2cm},width=0.32\\columnwidth]{graphics\/leadFollow.pdf}\n\t\t\\caption{Simulation of three distributed multi-agent behaviors on a group of differential drive robots. From the left: rendezvous, cyclic-pursuit, and leader-follower. Solid lines indicate the past trajectories of the robots. \\label{fig:coordBehExamples}}\n\t\\end{center}\n\\end{figure}\n\nFor the correct execution of the controllers mentioned, a sufficiently rich set of information needs to be available to the robots. Representing the flow of information between the robots through {\\it graphs}, with vertices and edges being respectively the robots and the pair-wise ability of sharing information, those conditions can be encoded in terms of particular graphs that need to exist between the robots. For example, rendezvous requires a spanning out-branching tree~\\cite{mesbahi2010graph}, cyclic-pursuit requires a cyclic graph~\\cite{ramirez2009cyclic}, formation control a rigid graph~\\cite{mesbahi2010graph}, and a Delaunay graph is required for most of coverage control problems~\\cite{cortes2004coverage}. \n\nEven though the coordinated behaviors mentioned above can address a number of different tasks, they have limited utility in the context of real-world missions, which can rarely be represented as single tasks. However, the utility of these behaviors can be greatly expanded if they are sequenced together, which is the primary consideration in this paper. But, for a construction like this to work, it is necessary that the required information is available to the robots as they transition from one behavior to the next.\n\nAs such, the problem of composing different behaviors, can be recast in terms of the ability of the robots to establish the interactions needed at each stage of a mission. In particular, when the communication between agents depends on their relative configurations (e.g. relative distance or orientation), realizing a certain communication structure directly affects the configuration of the system, which in turn, affects the execution of the mission itself. In order to overcome this coupling, we separate the problem of generating a sequence of behaviors that corresponds to the solution of a mission objective (e.g.~\\cite{nagavalli2017automated}) from their composition. In this work we focus on the problem of designing a composition framework given a sequence of coordinated behaviors. \\mymod{Although the focus of this paper is confined to motion control tasks, our framework is applicable to other forms of autonomous collaboration where desired interaction structures between the robots are required by the mission, e.g., sharing of resources in heterogeneous teams~\\cite{ramachandran2019resilience} or coordinated manipulation~\\cite{culbertson2018decentralized}}.\n\nThe contribution of this paper is twofold. Firstly, extending the results in~\\cite{li2018formally}, we propose a fully decentralized framework for composing a given sequence of multi-robot coordinated behaviors. Secondly, responding to the lack of established large-scale scenarios for the testing of multi-robot techniques, we propose a scenario called {\\it Securing a Building}, which is rich and complex enough to capture many challenges and objectives of real-world implementations. \\mymod{The significance of our framework is demonstrated through implementation of the Securing a Building scenario on a team of mobile robots.}\n\nThe remaining of this paper is organized as follows. In Section~\\ref{sec:fcbf} we review the definition of finite-time convergence barrier functions, while in Section~\\ref{sec:problem} we present a centralized multi-robot composition framework, which is extended to a \\mymod{fully decentralized} formulation in Section~\\ref{sec:multAgImp}. In Section~\\ref{sec:securing}, we describe the {\\it Securing a Building} case study and its implementation. Finally, motivated by the lack of well-established scenarios for testing and comparing multi-agent robotics techniques, \\mymod{in Appendix~\\ref{sec:appendixA} we discuss supportive arguments for considering the Securing a Building as a multi-robot benchmark scenario.}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}