Leader-following consensus of linear multi-agent systems via reset control: A time-varying systems approach

Abstract

In this work, we address the leader-following consensus problem of multi-agent systems by developing a novel reset consensus protocol with a time-varying gain matrix. The reset consensus protocol is distinct from existing results in the sense that it integrates a high-dimensional element with the reset actions triggered by a prescribed reset band. After converting the original closed-loop system to an equivalent linear time-varying system, we develop the time-varying system approach for linear multi-agent system over fixed/switching networks. It further shows that the proposed reset consensus protocol helps to improve the transient performance. Finally, two numerical examples are presented to illustrate the theoretical results and effectiveness.

Introduction

With the development of the communication and computation technologies, cooperative control of some networked autonomous dynamic systems such as multi-agent systems (MASs) has become a viable option. As the fundamental cooperative control problems, leaderless consensus  (Li et al., 2010, Olfati-Saber and Murray, 2004), and leader-following consensus problems (Hong et al., 2006, Ni and Cheng, 2010) have been well investigated. Specially, the leader-following configuration was found to be useful in energy saving in some biological systems (Hummel, 1995). Later, the linear MASs under fixed and switching topologies were considered in Ni and Cheng (2010). A distributed proportional–integral (PI) control was proposed for leader-following consensus of heterogeneous MASs over directed graphs in Lv et al. (2020). As an extension of the leader-following consensus problem, the output regulation problem of heterogeneous MASs under switching network was addressed in Hu et al. (2019) and Su and Huang (2012).

However, owing to some application requirements, the transient performance is also important for the controlled systems. Thus, how to design the consensus protocol with the improved transient performance becomes the research hotspots in recent years. Along this research line, some nonlinear control methods were developed such as finite-time control (Li et al., 2011), fixed time control (Parsegov et al., 2013), prescribed performance control (Bechlioulis & Rovithakis, 2017) and a kind of predictive method (Zhang & Chen, 2014). Another interesting alternative is the reset control strategy. J. C. Clegg first proposed a nonlinear integrator for the servomechanism problem in Clegg (1958), where the nonlinear integrator is called the Clegg integrator, and it is shown that the Clegg integrator only introduces 38.1 degrees of phase lag compared to $90$ degrees of phase lag introduced by the conventional linear integrator. Roughly speaking, the reset control has a reset component whose state is “reset” to zero when some certain condition is satisfied. Some early results have been obtained in  Beker et al., 2004, Guo et al., 2012, van Loon et al., 2017 and Saikumar et al. (2021). It is worth mentioning that some researchers used frequency-domain tools to analyze the stability of reset control systems  (van Loon et al., 2017, Saikumar et al., 2021). It implies that reset control has great potential to be used in industry with industry-preferred loop-shaping properties.

Recently, reset control was borrowed to MASs for improving the system transient performance. For example, the authors in Yucelen and Haddad (2014) proposed a quasi-reset control law for the consensus problem of single-integrator MASs. The leader-following consensus problem of single-integrator MASs was also solved by a distributed reset control protocol in Meng et al. (2016), while the leaderless counterpart was addressed in Meng et al. (2019). It is shown in Hu et al. (2022) by comparison that the proposed reset consensus protocol outperforms the existing static and dynamic control protocols in terms of the transient performance and the control effort for the consensus control of double-integrator MASs. Furthermore, it is also found that reset control essentially generates a dynamically changed gain, which contributes to a faster convergence rate. Though it is expected that simply increasing the gain in the static controller can also lead to a faster convergence rate, but in this case, large control effort will be unavoidable. As a contrast, with reset controller, the convergence rate is increased while the control effort remains in a relatively low level. Lately, Zhao et al. studied reset control for leader-following consensus problem of high-dimensional linear MASs over fixed and switching directed networks in Zhao and Hua (2022). In Zhao and Hua (2022), a hybrid-system approach was used for the stability analysis, where some sufficient conditions were obtained in terms of matrix inequality conditions.

In this paper, we further study the leader-following consensus problem of high-dimensional linear MASs over fixed and switching directed networks. The novelty of this paper can be summarized as follows. First, we develop a novel time-varying systems analysis approach for stability of reset controlled MASs. In this paper, by using the property of the reset mechanism, we convert the original closed-loop system to a linear time-varying (LTV) system. In this case, it is difficult or impossible to calculate the solutions directly. Meanwhile the eigenvalue analysis approach is no longer applied to the LTV systems. In general, it is also difficult to find a common Lyapunov function for the stability analysis of the LTV system. Specifically in Moreau (2004), L. Moreau analyzed the stability of a class of LTV systems given by $\dot{x}=A\left(t\right)x$, where the sufficient condition was obtained. It is noted that the matrix $A\left(t\right)$ in Moreau (2004) is assumed to be piecewise continuous and upper-bounded, which is not satisfied in our case. To overcome this difficulty, technically, we further develop one stability lemma on the LTV systems (Lemma 3), without requiring the system matrix to be upper-bounded all the time. Moreover, compared with the so-called hybrid systems approach developed in Zhao and Hua (2022), where the parameters design depends on some matrix inequalities conditions, with the developed time-varying systems analysis approach, controller parameters are explicitly given in this paper. Second, we propose two novel reset mechanisms which are applicable for the fixed and switching topology, respectively. Different from the assumption on switching network that the graph is required to contain a spanning tree all the time in Zhao and Hua (2022), this paper only requires that the union graph jointly contains a spanning tree. Compared with one-dimensional reset element in Meng et al., 2016, Meng et al., 2019, and Hu et al. (2022), the reset element in this paper is high-dimensional, which can be applied to multi-input multi-output linear MASs. Furthermore, the reset mechanisms are novel in the sense that the reset actions are on a zero band (Barreiro et al., 2014), i.e., reset occurs when states error enters a band around the zero error, which is different from the reset mechanisms in  Meng et al., 2016, Meng et al., 2019, and Zhao and Hua (2022).

The rest of this paper is organized as follows. In Section 2, notation, mathematical preliminaries, and problem formulation are presented. Some stability lemmas and consensus analysis are presented in Section 3. Simulation and conclusion are presented in Sections 4 Examples, 5 Conclusions, respectively. Finally, the technical details of some proofs are shown in Appendix.