Finite-time control of multiagent networks as systems with time transformation and separation principle

Abstract

In this paper, we study finite-time control of multiagent networks as systems, where they involve floating agents that exchange local information and driver agents that not only exchange local information but also take input and output roles. For this class of multiagent networks, control algorithms are applied to the driver agents based on the measurements collected from them for the purpose of influencing the overall behavior of the resulting system. Specifically, we consider time-critical applications in the control of multiagent networks as systems and propose a finite-time control approach predicated on a recent time transformation method. The proposed method guarantees execution of control algorithms over a prescribed time interval $[0,T)$ with $T$ being a user-defined convergence time based on analysis performed over a stretched, infinite-time interval $[0,\infty )$. We analytically show that the resulting system achieves user-defined finite-time convergence regardless of the initial conditions of agents, where we also discuss the separation principle obtained with the proposed method. The presented theoretical contributions are not only illustrated by numerical examples but also experimentally validated using ground mobile robots.

Introduction

The last decades have witnessed a considerable attention and growth in theory and application of multiagent networks (see, e.g., Lewis et al., 2014, Mesbahi and Egerstedt, 2010, Ren and Beard, 2008 and references therein). In the near future, these systems will play a key role for enabling network-centric operations that range from collaborative surveillance and reconnaissance to guidance and control of autonomous vehicle teams. This paper contributes to the studies in control of multiagent networks as systems. Specifically, the behavior of multiagent systems involving large agent teams fall into the studies in this category. Examples include large underwater, ground, aerial, and space vehicle teams; air, water, and ground transportation systems; thermal building processes; and physio-social networks; to name but a few examples. These engineered and natural agent teams are subject to local information exchange, where additional control signals can only be inserted to a few subset of nodes in practice for influencing the overall multiagent system behavior. Here, a common consideration is to model local information exchange using first-order integrator dynamics since some of the aforementioned examples satisfy this consideration as they are and the remaining ones can have inner-loop control algorithms to allow feedback regulation in the presence of high system orders (see, e.g., Chapter 10 Mesbahi & Egerstedt, 2010, Rahmani, Ji, Mesbahi, & Egerstedt, 2009, and Liu, Slotine, & Barabási, 2011). For this class of multiagent systems, the agents that only exchange consensus and consensus-like algorithms are called floating agents, where the ones not only exchange local information but also take input and output roles in the system for executing additional control signals are called driver agents (that is, additional control algorithms of interest are applied to the driver agents based on the measurements collected from them). An example multiagent network as a system is depicted in Fig. 1.

Time-critical applications in control of multiagent networks as systems such as engagement of a large number of vehicles with a target and sequential execution of complex tasks often require finite-time control algorithms for completing given operations of interest at a user-defined time. A representative example scenario is the salvo attack (see, e.g., Jeon et al., 2006, Li et al., 2020, Saleem and Ratnoo, 2016 and references therein), where several interceptors are aimed for a simultaneous impact against a common target (or targets). Another example is the formation tracking problem, where a group of vehicles is required to form a formation and track a target. A finite-time control algorithm can be used to achieve the formation in finite-time before switching to tracking the target. Motivated by these and other similar time-critical applications, this paper focuses on finite-time control over a prescribed time interval $[0,T)$, where $T$ is the user-defined convergence time. In particular, we first propose an approach based on a recent time transformation method (Kan et al., 2017, Yucelen et al., 2019), which guarantees execution of control algorithms over $[0,T)$ based on analysis performed over a stretched, infinite-time interval $[0,\infty )$. We then analytically show (i) separation principle of the proposed time-critical algorithm and (ii) user-defined finite-time convergence of the resulting system regardless of the initial conditions of agents. Note that the preliminary conference version of this paper has appeared in Tran, Yucelen, and Sarsilmaz (2018). This present paper considerably expands on Tran et al. (2018) by providing detailed proofs; additional informative discussions, remarks, and examples; practical consideration of the proposed algorithms; and experimental results with ground mobile robot platforms.

Finite-time control offers an appealing framework for time-critical applications of dynamical systems. We start with the seminal papers Bhat and Bernstein, 1998, Bhat and Bernstein, 2000, where the authors define finite-time stability for non-smooth dynamical systems. There exist many studies in the multiagent networks literature that utilize and generalize the results in these two (and similar) papers, where the finite-time convergence depends on the initial conditions of agents. The studies documented in Basin et al., 2016, Hong et al., 2017, Lu et al., 2016, Oza et al., 2014, Polyakov, 2012, Ríos and Teel, 2016 and Tian, Zuo, and Wang (2017) address this problem by upper bounding the finite-time convergence time. The necessary and sufficient conditions for the fixed-time stability are recently introduced in Lopez-Ramirez, Efimov, Polyakov, and Perruquetti (2019). A method for designing autonomous and non-autonomous systems with a fixed-time stable equilibrium point, where the predefined upper bound of the settling time is considered as a parameter of the system, is proposed in Aldana-López, Gómez-Gutiérrez, Jiménez-Rodríguez, Sánchez-Torres, and Defoort (2019); yet, the result still requires some knowledge of initial conditions. In addition, the studies documented in Harl and Balakrishnan, 2012, Jiménez-Rodríguez et al., 2016, Jiménez-Rodríguez et al., 2017, Kan et al., 2017, Ning et al., 2019, Ning, Han, and Zuo, 2020, Sánchez-Torres et al., 2015, Song et al., 2017, Song et al., 2019, Wang et al., 2018, Wang et al., 2017, Wang et al., 2013, Wen et al., 2016, Yong et al., 2012a, Yong et al., 2012b, Yucelen et al., 2019, Zhao et al., 2019 and Zuo, Han, Ning, Ge, and Zhang (2018) propose system-theoretic tools for guaranteeing user-defined finite-time convergence regardless of the initial conditions of dynamical systems. For example, the authors of Zhao et al. (2019) propose a class of distributed control protocols to solve the consensus problem of linear multiagent systems within a prescribed time by reducing the sampling period as time progresses. The authors of Ning et al., 2019, Ning, Han, and Zuo, 2020 utilize the time base generators to address the fixed-time consensus problems. A group of papers Harl and Balakrishnan, 2012, Wang et al., 2017, Wang et al., 2013, Wen et al., 2016 and Yong et al., 2012a, Yong et al., 2012b also aim to show the stability of the considered systems through their explicit solutions, while the authors of Song et al., 2017, Song et al., 2019 and Wang et al. (2018) use the comparison principle (see, e.g., Section 3.4 of Khalil, 2002) on the Lyapunov function candidate for stability analysis. Finally, Ning, Han, and Lu (2020) and Wang, Tnunay, Zuo, Lennox, and Ding (2019) focus on application aspects by proposing fixed-time algorithms for mobile robots.

As noted in Section 1.1, the contribution of this paper builds on the novel time transformation method introduced in Kan et al. (2017) and Yucelen et al. (2019) that results in smooth control algorithms (the studies in Harl and Balakrishnan, 2012, Jiménez-Rodríguez et al., 2016, Jiménez-Rodríguez et al., 2017, Sánchez-Torres et al., 2015, Wang et al., 2018, Wang et al., 2017, Wang et al., 2013, Wen et al., 2016, Yong et al., 2012a, Yong et al., 2012b are more related than the other aforementioned ones to the contributions documented in these two papers, where we refer to Kan et al., 2017, Yucelen et al., 2019 for important differences). Our motivation in utilizing and generalizing the results in Kan et al. (2017) and Yucelen et al. (2019) is primarily owing to the fact that the time transformation method allows one to use well-established system-theoretical tools proposed over infinite-time intervals $[0,\infty )$ for reaching guarantees over the user-defined prescribed time interval $[0,T)$. Specifically, the time transformation method is used to transform the original system into an equivalent system on a stretched time interval $[0,\infty )$, where a broad spectrum of standard system-theoretical analysis tools proposed over $[0,\infty )$ such as Lyapunov methods can be adopted (in contrast to limited finite-time analysis tools available in the literature) for analyzing the stability of the transformed system and then concluding the stability for the original system (Tran & Yucelen, 2020). This key aspect here allows us to analytically show (ii) outlined in Section 1.1.

The content of this paper is as follows. Section 2 introduces the necessary mathematical preliminaries for the main results of this paper. The proposed finite-time control approach for multiagent networks as systems is introduced and system-theoretically analyzed in Section 3, where we also present an illustrative numerical example in the same section. In addition, Section 4 discusses an important practical consideration as well as validates the efficacy of the proposed architectures with experimental results. Finally, concluding remarks are summarized in Section 5.