Elsevier

Automatica

Volume 107, September 2019, Pages 398-405
Automatica

Brief paper
Semi-global stabilization of linear systems with distributed infinite input delays and actuator saturations

https://doi.org/10.1016/j.automatica.2019.06.005Get rights and content

Abstract

This paper addresses the semi-global stabilization problem of linear systems with simultaneous presence of distributed infinite input delays and actuator saturations. Two low gain controllers are proposed, respectively, for two classes of linear systems with both infinite input delays and actuator saturations. It is shown that under our first controller, the semi-global stabilization problem can be solved for the first class of systems where all eigenvalues of the open loop system have no positive real parts. It is also shown that under our second controller, which is much simpler, the semi-global stabilization problem can be solved for the second class of systems where all eigenvalues of the open loop system are zero. Another contribution of this paper is that the bounds for the peak magnitude of the control signals can be explicitly given. Furthermore, our results include relevant results on bounded distributed delays and constant delays as their special cases. Simulation examples are finally provided to illustrate the effectiveness of the proposed controllers.

Introduction

Time-delay systems have been extensively studied for the past decades, see, for example Chen and Latchman (1995), Gu, Chen, and Kharitonov (2003), Hale and Lunel (2013), Liu, Sun, and Krstic (2018), Lu and Huang (2015), Niculescu (2001) and Richard (2003) and the references therein. Most of the existing works consider bounded delays. More recently, we study the stabilization problem of linear systems with infinite delays in Xu, Liu, and Feng (2018). Infinite delays, also known as unbounded delays, are more general than bounded delays, often including bounded delays as their special cases. In practical systems, there are many scenarios where infinite delays exist or modeling of such infinite delays is needed. For instance, in Michiels, Morarescu, and Niculescu (2009) and Sipahi, Atay, and Niculescu (2007), for the car following system, distributed infinite delays are used to model the memory effects of drivers. In Josić, López, Ott, Shiau, and Bennett (2011), the processes of transcription, translation and propagation of genetic materials are described by a model with distributed infinite delays. Moreover, infinite delays can also be used to model coupled oscillators (Atay, 2003), neural networks (Gopalsamy & He, 1994), wireless communication networks (Roesch & Roth, 2005) and so on. As discussed in Xu et al. (2018), it is much more difficult to deal with infinite delays than bounded delays in general. The major challenges include limitations of analysis tools, the sensitivity of solutions to initial conditions and mathematical complexity caused by infinite delays.

On the other hand, actuator saturation also exists in many practical control systems (Chen et al., 2003, Lin, 1999). Like time delays, actuator saturation often leads to instability and poor performance. Thus actuator saturation has also been a topic of continuous interest for many years, see Sussmann, Sontag, and Yang (1994) and Teel (1996) and references therein. More recently, the stabilization problem of linear systems with simultaneous presence of input delays and actuator saturations has been considered. In Lin and Fang (2007), the linear system with a constant input delay and actuator saturation is discussed. In Zhou, Gao, Lin, and Duan (2012), the linear system with distributed bounded input delay and actuator saturation is studied. However, all those existing works only consider bounded delays which motivate our study in this work.

Distributed input delay has been a topic of continuous interest since Artstein (1982) where a reduction of linear systems with delayed control is proposed. More recently, a more rigorous stability analysis result for linear systems is given in Bekiaris-Liberis and Krstic (2011), where an infinite-dimensional forwarding-backstepping transformation of the infinite-dimensional actuator states is introduced. An adaptive control result for linear systems and a nonlinear control result are further developed in Bekiaris-Liberis, Jankovic, and Krstic (2013) and Bekiaris-Liberis and Krstic (2016), respectively. In Zhou et al. (2012), a low gain method is proposed to solve the stabilization problem of linear systems with distributed bounded input delays. Low gain control proves to be effective to many control problems of various systems with input delays and/or actuator saturations, see for example Lin (1999), Lin and Fang (2007), Su, Wei, and Lin (2019), Wei and Lin (2019), Zhou et al. (2012) and Zhou, Li and Lin (2014). However, the low gain method in these papers cannot be used to deal with the case of infinite delays mainly because the design of low gain parameters in those papers depends on the bounds of delays, which do not exist in the case of infinite delays. Therefore, in Xu et al. (2018), we develop a novel low gain method to handle the stabilization problem of linear systems with distributed infinite input delays. In this paper, building on our previous work in Xu et al. (2018), we propose two low gain controllers to solve the semi-global stabilization problem, respectively, for two classes of linear systems with simultaneous presence of distributed infinite input delays and actuator saturations. It is shown that our first controller can be used to solve the semi-global stabilization problem for the first class of linear systems where all eigenvalues of the open loop system have no positive real parts. It is also shown that our second controller, which is much simpler, can be used to solve the semi-global stabilization problem for the other class of linear systems where all eigenvalues of the open loop system are zero. Moreover, bounds are explicitly given for the peak magnitude of control signals, and it is shown that the peak magnitude of control signals goes to zero as the low gain parameter goes to zero. The proofs of our results rely on our previous work (Xu et al., 2018). The uniqueness of solutions of infinite delayed functional differential equations, which has been proved in Hale and Kato (1978), is also essential to our proofs. Our work can include relevant works on bounded distributed delays and constant delays as its special cases.

Compared with our previous work on linear systems with distributed infinite input delays (Xu et al., 2018), the challenges of this work can be summarized into the following aspects. First, the existence of actuator saturation introduces nonlinearity to the concerned linear system which often leads to its poor performance and even instability. The stabilization problem under this scenario is in general more challenging. In fact, linear control laws, when subject to actuator saturations, can only achieve semi-global stabilization, rather than global stabilization. Second, tools for handling a system with simultaneous presence of distributed infinite input delays and actuator saturations are very limited. Due to the existence of input nonlinearity, the frequency domain method used in Xu et al. (2018) cannot be directly applied in this paper. Third, some new technical difficulties arise from the concerned scenario.

The contributions of this paper, compared with existing literatures, can be summarized into three aspects. First, compared with Bekiaris-Liberis and Krstic (2011), Lin and Fang (2007) and Zhou et al. (2012), the distributed infinite input delays are considered. Second, compared with Xu et al. (2018), the actuator saturation is taken into consideration. Last but not least, we provide explicit bounds for the peak magnitude of the control signals.

The remainder of this paper is organized as follows. In Section 2, some preliminaries are given, including problem formulation and some technical lemmas. In Section 3, we present our main results, including design of low gain controllers and stability analysis of the resulting closed loop control systems. Simulation examples are given in Section 4 and conclusions are drawn in 5.

Notations: Throughout this paper, the following notations are used. Rn denotes the n-dimensional Euclidean space. || represents the absolute value of real numbers, the module of complex numbers, the l2 norm of vectors or the induced 2-norm of matrices.

Section snippets

Preliminaries

The problem formulation and some useful technical lemmas are given in this section.

Main results

In this section, we present our main results, including design of two low gain controllers and stability analysis of the resulting closed loop control systems.

Simulation examples

In this section, we introduce two examples to illustrate the effectiveness of our results.

Conclusion

In this paper, we have considered the semi-global stabilization problem of linear systems with simultaneous presence of distributed infinite input delays and actuator saturations. Two low gain controllers are proposed, respectively, for two classes of linear systems. It is shown that under our first controller, the semi-global stabilization problem can be solved for the first class of systems where the open loop system has all its eigenvalues on the imaginary axis. It is also shown that under

Xiang Xu received the Bachelor of Engineering degree from Nanjing University of Science and Technology, China in 2014 and the Ph.D. degree from City University of Hong Kong, Hong Kong in 2018. He is now a postdoctoral fellow in the Department of Biomedical Engineering, City University of Hong Kong. His research interests include multi-agent systems and time-delay systems.

References (34)

  • Bekiaris-LiberisN. et al.

    Lyapunov stability of linear predictor feedback for distributed input delays

    IEEE Transactions on Automatic Control

    (2011)
  • ChenJ. et al.

    Frequency sweeping tests for stability independent of delay

    IEEE Transactions on Automatic Control

    (1995)
  • ChenB.M. et al.

    Composite nonlinear feedback control for linear systems with input saturation: Theory and an application

    IEEE Transactions on Automatic Control

    (2003)
  • GuK. et al.

    Stability of time-delay systems

    (2003)
  • HaleJ.K. et al.

    Phase space for retarded equations with infinite delay

    Funkcialaj Ekvacioj

    (1978)
  • HaleJ.K. et al.

    Introduction to functional differential equations, Vol. 99

    (2013)
  • JosićK. et al.

    Stochastic delay accelerates signaling in gene networks

    PLoS Computational Biology

    (2011)
  • Cited by (26)

    • Lyapunov characterizations on input-to-state stability of nonlinear systems with infinite delays

      2022, Automatica
      Citation Excerpt :

      For instance, infinite delays are used to model the transcription of genetic materials (Josić et al., 2011), the cell dynamics (Djema et al., 2018), the car following systems (Michiels et al., 2009; Sipahi et al., 2007), coupled oscillators (Atay, 2003), neural networks (Gopalsamy & He, 1994), wireless communication networks (Roesch & Roth, 2005) and so on. The existence of infinite delays brings several additional challenges in analysis and synthesis of infinite-delayed systems, as pointed out in Hale and Kato (1978), Xu et al. (2018, 2019, 2020b). These challenges include limitations of available analysis tools, the sensitivity of solutions to initial conditions and mathematical complexity caused by the lack of delay bounds.

    • New results on stability of discrete-time systems with infinite delays

      2022, Automatica
      Citation Excerpt :

      For example, infinite input delays happen in teleoperation systems (Bemporad, 1998) and biology models (Weng, 2000); infinite state delays exist in neural networks (Kaslik & Sivasundaram, 2011) and electrodynamics (Iserles, 1994); and infinite transmission delays occur in mechanics (Atay, 2003). More recently, stability and control problems of continuous-time systems with infinite delays have been studied, see, for example, Xu et al. (2018, 2019) and Xu et al. (2020). However, it is noted that the results on stability and control of discrete-time systems with infinite delays are rather limited.

    View all citing articles on Scopus

    Xiang Xu received the Bachelor of Engineering degree from Nanjing University of Science and Technology, China in 2014 and the Ph.D. degree from City University of Hong Kong, Hong Kong in 2018. He is now a postdoctoral fellow in the Department of Biomedical Engineering, City University of Hong Kong. His research interests include multi-agent systems and time-delay systems.

    Lu Liu received the Ph.D. degree from the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong, in 2008.

    From 2009 to 2012, she was an Assistant Professor with the University of Tokyo, Japan, and then a Lecturer with the University of Nottingham, UK. After that, she joined City University of Hong Kong, Hong Kong, where she is currently an Associate Professor. Her current research interests include networked dynamical systems, control theory and applications, and biomedical devices.

    She is an Associate Editor of the IEEE Transactions on Cybernetics, Control Theory and Technology, Transactions of the Institute of Measurement and Control, and Unmanned Systems.

    Gang Feng received the B.Eng. and M.Eng. degrees in automatic control from Nanjing Aeronautical Institute, China, in 1982 and 1984, respectively, and the Ph.D. degree in Electrical Engineering from the University of Melbourne, Australia, in 1992.

    He has been with the City University of Hong Kong, Hong Kong, since 2000, where he is currently a Chair Professor of Mechatronic Engineering. He was a Lecturer/Senior Lecturer with the School of Electrical Engineering, University of New South Wales, Australia, from 1992 to 1999. His current research interests include multi-agent systems and control, intelligent systems and control, and networked systems and control.

    He was a recipient of the Alexander von Humboldt Fellowship in 1997, the IEEE Transactions on Fuzzy Systems Outstanding Paper Award in 2007, and the Changjiang Chair Professorship from Education Ministry of China in 2009. He is an Associate Editor of the Journal of Systems Science and Complexity, and was an Associate Editor of the IEEE Transactions on Automatic Control, the IEEE Transactions on Fuzzy Systems, the IEEE Transactions on Systems, Man & Cybernetics, Part C, Mechatronics, and the Journal of Control Theory and Applications.

    This work was supported by the Research Grants Council of Hong Kong under grants CityU-11206817 and CityU-11274916. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fouad Giri under the direction of Editor Miroslav Krstic

    View full text