Elsevier

Automatica

Volume 127, May 2021, 109515
Automatica

Sliding mode control for singularly perturbed Markov jump descriptor systems with nonlinear perturbation

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Abstract

This paper develops a stochastic integral sliding mode control strategy for singularly perturbed Markov jump descriptor systems subject to nonlinear perturbation. The transition probabilities (TPs) for the system modes are considered to switch randomly within a finite set. We first present a novel mode and switch-dependent integral switching surface, based upon which the resulting sliding mode dynamics (SMD) only suffers from the unmatched perturbation that is not amplified in the Euclidean norm sense. To overcome the difficulty of synthesizing the nominal controller, we rewrite the SMD into the equivalent descriptor form. By virtue of the fixed-point principle and stochastic system theory, we give a rigorous proof for the existence and uniqueness of the solution and the mean-square exponential admissibility for the transformed SMD. A generalized framework that covers arbitrary switching and Markov switching of the TPs as special cases is further achieved. Then, by analyzing the stochastic reachability of the sliding motion, we synthesize a mode and switch-dependent SMC law. The adaptive technique is further integrated to estimate the unavailable boundaries of the matched perturbation. Finally, simulation results on an electronic circuit system confirm the validity and benefits of the developed control strategy.

Introduction

Descriptor systems, also referred to as singular systems, are commonly utilized to model physical systems containing both differential and algebraic equations. Typical examples include space vehicles, micro-grids, and circuits, where the state constraints or boundary conditions are integrated. Different from the standard state–space systems, impulsive behaviors might occur during the time response of a descriptor system (Dai, 1989). To efficiently eliminate the impulses, an appropriate controller needs to be designed to reduce the nilpotent index to one for the strangeness index to zero (Chadli & Darouach, 2014). Besides the descriptor systems, singularly perturbed systems (SPSs) are another kind of extensively investigated systems that are adopted to describe the physical systems containing multiple time scales (Karimi et al., 2006, Nguang et al., 2007, Song and Niu, 2020, Song et al., 2019). The corresponding singular perturbation approach has been developed to deal with the multiple time phenomena, which is usually the root of high dimensionality and stiffness of the systems (Shen et al., 2018, Wang, Shi et al., 2018).

In practice, the dynamics of SPSs may also contain algebraic equations, which can be referred to as singularly perturbed descriptor systems (SPDSs) (Brenan, Campbell, & Petzold, 1989). The analysis and synthesis for SPDSs are much more complicated than those for SPSs or descriptor systems due to the substantial challenges brought by their couplings. Up to now, only a scattered of literatures has been reported on this topic (Du and Linh, 2005, Lu et al., 2008, Zhou and Liu, 2011). To be specific, by virtue of linear algebra and classical analysis, the asymptotic stability problem and complicated stability radius of the SPDSs were considered by Du and Linh (2005). By resorting to the singular system method, the issue of robust D-stability for delayed SPDSs was addressed in Lu et al. (2008). Perturbation often occurs in actual systems and how to tolerate or alleviate the effect of perturbation while ensuring system stability is an important research topic. In Zhou and Liu (2011), the nonlinear perturbation was investigated, and the robust stability for SPDSs was studied by adopting the fixed-point principle. It is noted that the mode of SPDSs studied in the aforementioned results is deterministic. In some practical situations such as the occurrence of abrupt actuator faults (Mahmoud, Jiang, & Zhang, 2001) or stochastic communication imperfection induced by network environment (Zhang, Shi, Chen, & Huang, 2005), the system modes may display stochastic variations subject to a Markov process, where the involved systems are referred to as Markov jump systems (MJSs) (Shen et al., 2019, Todorov et al., 2018, Zhang et al., 2019). Promising methodologies have been emerged for various MJSs such as Zhang, Leng, and Colaneri (2016). By following the result achieved in Zhou and Liu (2011), the authors in Wang, Zhang, and Yang (2014) further considered the Markovian switching of the system modes and studied the stability problem for singularly perturbed Markov jump descriptor systems (SPMJDSs) under perturbation. It is worth mentioning, however, that this topic is not well investigated, and the stability for SPMJDSs is much different from that for SPDSs due to the mode switching (detailed illustrations are provided in Remark 6). More importantly, only stability results are achieved in the existing referees (Du and Linh, 2005, Lu et al., 2008, Wang et al., 2014, Zhou and Liu, 2011), and the stabilization problem is unsolved due to the highly complicated system dynamics.

Among the existing stabilization methodologies, sliding mode control (SMC) is featured with excellent transient performance and strong robustness to perturbations or disturbances (Basin et al., 2012, Basin et al., 2018, Edwards et al., 2006). As a promising SMC strategy, the integral SMC is usually composed of a continuous nominal part aiming to stabilize the sliding mode dynamics (SMD), and the discontinuous compensating part is utilized to ensure sliding motion (Rubagotti, Estrada, Castanos, Ferrara, & Fridman, 2011). The appealing features of integral SMC lie in the elimination of reaching phase required in traditional SMC methods, and the maintenance of the original order of the system. The past years have witnessed the advances on integral SMC for various complex dynamical systems including stochastic systems (Basin and Ramirez, 2014, Wu et al., 2017), descriptor systems (Han et al., 2017, Wu and Zheng, 2009), singularly perturbed systems (Wang, Gao et al., 2018, Yang et al., 2019), Markov jump systems (Cao et al., 2020, Chen et al., 2013), and several kinds of hybrid systems among them (Feng and Shi, 2017, Li et al., 2018).

Motivated by the above discussion, we aim to address the stabilization issue for perturbed SPMJDSs by developing a stochastic integral SMC strategy. The key challenges to be confronted are summarized as two folds:

  • (1)

    In the existing integral SMC results on various descriptor systems and SPSs (Feng and Shi, 2017, Han et al., 2017, Li et al., 2018, Wang, Shi et al., 2018, Wu and Zheng, 2009, Yang et al., 2019), the designed nominal controller can be solved by analyzing the sliding motion, where the equivalent transformation for the yielding matrix inequalities is needed. For an SPDS or SPMJDS, however, it is challenging to get the inverse matrix expression for the involved Lyapunov matrix due to its complicated and coupling structure. As a consequence, it is difficult to make an equivalent matrix transformation and get the explicit solution for the designed nominal controller.

  • (2)

    During the sliding motion analysis in previous result Zhou and Liu (2011), one important step is to introduce two common nonsingular matrices to make a matrix transformation. For SPMJDSs, how-ever, it is difficult to find common nonsingular matrices to make a similar transformation since the system matrices are usually different under different modes. This fact brings substantial challenges to the unique solution and stability analysis. Besides the above challenges, we consider a more complicated situation, that is, the transition probabilities (TPs) for the system modes are assumed to switch randomly within a finite set (namely finite-time homogeneous), which is characterized by a high-level Markov process. This situation covers the deterministic TPs as a special case, and may reflect more reality for the case that the TPs of the system modes are affected by some random trigger signals (Niri, Motlagh, & Yazdi, 2017). To deal with the partially or completely unknown high-level TPs, we also establish a generalized framework that covers the random switching and arbitrary switching as special cases. The achieved results can be easily extended to SPMJDSs with uncertain TPs (additive norm-bounded uncertain terms are involved into the two-level TP matrices).

The achieved contributions of the paper are summarized as follows:

(1) In contrast with the most existing results on SPMJDSs and SPDs where only the stability problem are addressed, we provide a solution to the stabilization problem for SPMJDSs by developing a novel stochastic integral SMC strategy, which is the key contribution of the work.

(2) The ISSs in most of the existing integral SMC results on various MJSs are simply treated as mode-independent functions. As a consequence, some related terms in the corresponding equivalent controller are lost. We correct this issue for the first time and present a novel mode and switch-dependent ISS such that the corresponding SMD only suffers from unmatched nonlinear perturbation that is not enlarged in the Euclidean norm sense. To address the issue of synthesizing the nominal controller, we propose to rewrite the SMD into the equivalent descriptor form.

(3) In the sliding motion phase, we give rigorous proof for the existence and uniqueness of the solution (EUS) and the mean-square exponential admissibility (MSED) for the transformed SMD subject to nonlinear perturbation and switched TPs. A generalized framework to cover the random switching and arbitrary switching of the TPs as special cases is further obtained.

The remainder of the work is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, the ISS is introduced and sliding motion is analyzed. During Section 4, a detailed design procedure for the stochastic integral SMC strategy is provided. In Section 5, simulation results are performed on an electronic circuit system. Concluding comments are finally presented in Section 6.

Notation: Rm×n specifies the set of all the m×n real matrices and Rn stands for the n-dimensional Euclidean space. Superscript “T” specifies the matrix transposition. The asterisk “*” is adopted to specify the asymmetric terms.symM represents the term of M+MT. For the notation (Ψ,ϒ,Pr), Ψ specifies the sample space, ϒ specifies the σ-algebra of the subsets for the sample space and Pr specifies the probability measure on ϒ. EM specifies the mathematical expectation of the matrix M.

Section snippets

Problem formulation

Fix the probability space (Ψ,ϒ,Pr), and consider the following SPMJDS: Eẋ(t)=A1(ϕt)x(t)+A2(ϕt)z(t)+B1(ϕt)u(t)+fm,1(t,ϕt,x,z)+fu,1(t,ϕt,x,z)εż(t)=A3(ϕt)x(t)+A4(ϕt)z(t)+B2(ϕt)u(t)+fm,2(t,ϕt,x,z)+fu,2(t,ϕt,x,z),where x(t)Rn1 and z(t)Rnn1 are the system states for the slow and fast dynamics, respectively; u(t)Rm is the control input; Ai(ϕt)(i=1,,4), B1(ϕt) and B2(ϕt) are appropriately dimensioned constant matrices; ERn1×n1 may be singular with 0<rank(E)=rn1; ε>0 is a small scalar; fm,i(t,ϕt

ISS design

For piecewise-homogeneous SPMJDS (4), we propose a novel mode and switch-dependent ISS as follows: Sα,p(t)=GαEεX(t)GαEεX(0)0tGαAαX(s)+Bαun(s)ds0tβ=1hπαβpGβEεX(s)X(0)ds,with G(ϕt)T(ϕt)BT(ϕt) specifying the projection term. T(ϕt)Rm×m is a state-dependent nonsingular matrix and un(t)Kα,pX(t) with Kα,pRm×n is the nominal controller used to stabilize the SMD.

Let L be the weak infinitesimal operator of the stochastic process (X(t),ϕt,φt),t0. Then, for each ϕt=α, φt=p, we have LSα,p(t)=GαBα

SMC law synthesis

To ensure the stochastic reachability of the sliding motion, the following mode and switch-dependent SMC law is presented: u(t)=Kα,pX(t)GαBα1ρα(t)Sα,p(t)Sα,p(t)ϕt=αHandφt=pD, where ρα(t)ν+GαBαγαX(t)+χα+GαFαX(t),

with ν>0 being a small constant.

Theorem 4

Consider the SPMJDS (1) with nonlinear perturbation and partly unknown variation TPs. If the conditions in Theorem 3 are feasible and the nominal controller un(t) is determined by (61), the stochastic reachability of the sliding motion

Simulation verification

In this section, an electrical circuit system (depicted in Fig. 1) is utilized to verify the validity and benefits of the developed ISMC strategy. The VcIR value of resistor is IR=15Vc and the circuit is governed by the following semi-state equation: LİL(t)=IL(t)R(ϕt)Vc(t)+u(t)0=IR(t)+15Vc(t)CV̇c(t)=IL(t)IR(t)+αu(t),where L=0.1H, α=0.5, u(t) is the voltage input. The value of R(ϕt) is controlled by two-level circuit changers and is assumed to randomly switch between R1=1.8Ω and R2=2.2Ω,

Conclusions

Based upon the proposition of a novel ISS, the SMC problem for a class of SPMJDSs with switched TPs and nonlinear perturbation has been studied. During the analysis of sliding motion, the SMD has been rewritten into the equivalent descriptor form, and rigorous proof for the EUS and MSED of the augmented SMD has been provided. In the phase of reachability analysis, a mode and switch-dependent SMC law has been designed to ensure the stochastic reachability of sliding motion. Finally, an

Acknowledgments

This work was supported partly by the National Natural Science Foundation of China (61973204, 61827812, 91748116),partly by the National Science Fund of China for Distinguished Young Scholars (61525305), partly by the National Key R&D Program of China (2018AAA 0102804), partly by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (No. NRF-2020R1A2C1005449).

Yueying Wang received the B.Sc. degree in mechanical engineering and automation from the Beijing Institute of Technology, Beijing, China, in 2006, the M. Sc. degree in navigation, guidance, and control, and Ph.D. degree in control science and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2010 and 2015, respectively.

He is currently an Associate Professor with the School of Mechatronic Engineering and Automation, Shanghai University, Shanghai. His current research interests

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  • Cited by (0)

    Yueying Wang received the B.Sc. degree in mechanical engineering and automation from the Beijing Institute of Technology, Beijing, China, in 2006, the M. Sc. degree in navigation, guidance, and control, and Ph.D. degree in control science and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2010 and 2015, respectively.

    He is currently an Associate Professor with the School of Mechatronic Engineering and Automation, Shanghai University, Shanghai. His current research interests include intelligent and hybrid control systems, control of unmanned marine vehicles. He has served on the editorial board of a number of journals, including IET-Electronics Letters, International Journal of Electronics, International Journal of Fuzzy Systems, International Journal of Control, Automation and Systems, Journal of Electrical Engineering & Technology, and Cyber–Physical Systems.

    Huayan Pu received the M.Sc. and Ph.D. degrees in mechatronics engineering from Huazhong University of Science and Technology, Wuhan, China, in 2007 and 2011, respectively. She is currently a Professor with Shanghai University, Shanghai, China. Her current research interests include modeling, control, and simulation of field robotics and locomotion systems. Dr. Pu was awarded the Best Paper in Biomimetics at the 2013 IEEE International Conference on Robotics and Biomimetics. She was also nominated as a Best Conference Paper Finalist at the 2012 and 2014 IEEE International Conference on Robotics and Biomimetics.

    Peng Shi received the Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia in 1994; the Ph.D. degree in Mathematics from the University of South Australia in 1998. He was awarded the Doctor of Science degree from the University of Glamorgan, Wales in 2006; and the Doctor of Engineering degree from the University of Adelaide, Australia in 2015.

    Dr Shi is now a professor at the University of Adelaide. His research interests include automation and control systems, intelligence systems, and network systems. He has served on the editorial board of a number of journals, including Automatica, IEEE Transactions on Automatic Control; IEEE Transactions on Cybernetics; IEEE Transactions on Circuits and Systems; etc. He now serves as a Member of Board of Governors and a Distinguished Lecturer in IEEE SMC Society. He is a Fellow of IEEE, IET and IEAust.

    Choon Ki Ahn received the B.S. and M.S. degrees in the School of electrical engineering from Korea University, Seoul, Korea, in 2000 and 2002, respectively. He received the Ph.D. degree in the School of electrical engineering and computer science from Seoul National University, Seoul, Korea, in 2006. He is currently a Crimson Professor of Excellence with the College of engineering and a Full Professor with the School of electrical engineering, Korea University, Seoul, Korea. He was the recipient of the Early Career Research Award and Excellent Research Achievement Award of Korea University in 2015 and 2016, respectively. He was awarded the Medal for ‘Top 100 Engineers’ 2010 by IBC, Cambridge, England. In 2016, he was ranked 1 in Electrical/Electronic Engineering and 2 in entire areas of Engineering among Korean young professors based on paper quality. In 2017, he received the Presidential Young Scientist Award from the President of South Korea. In 2019 and 2020, he received the Research Excellence Award from Korea University (Top 3% Professor of Korea University in Research).

    He is a Senior Member of the IEEE. His current research interests are control, estimation, fuzzy systems, neural networks, and nonlinear dynamics. He is a member of the IEEE Technical Committee on Cyber–Physical Systems; IEEE SMCS Technical Committee on Intelligent Learning in Control Systems; and IEEE SMCS Technical Committee on Soft Computing. He has been on the editorial board of leading international journals, including the IEEE Systems, Man, and Cybernetics Magazine; IEEE Transactions on Neural Networks and Learning Systems; IEEE Transactions on Fuzzy Systems; IEEE Transactions on Systems, Man, and Cybernetics: Systems; IEEE Transactions on Automation Science and Engineering; IEEE Transactions on Circuits and Systems I: Regular Papers; IEEE Systems Journal; Nonlinear Dynamics; Aerospace Science and Technology; Artificial Intelligence Review; and other top-tier journals. He was the recipient of the Highly Cited Researcher Award in Engineering by Clarivate Analytics (formerly, Thomson Reuters).

    Jun Luo received the B.S. and M.S. degrees in mechanical engineering from Henan Polytechnic University, Jiaozuo, China, in 1994 and 1997, respectively, and the Ph.D. degree in mechanical engineering from Shanghai Jiao Tong University, Shanghai, China, in 2000. He is currently a Professor with the School of Mechatronic Engineering and Automation, Shanghai University, Shanghai, where he is also the Head of the Precision Mechanical Engineering Department and the Vice Director of the Shanghai Municipal Key Laboratory of Robotics. His current research interests include robot sensing, sensory feedback, mechatronics, man–machine interfaces, and special robotics.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michael V. Basin under the direction of Editor Ian R. Petersen.

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