A survey on anti-disturbance control of switched systems with input saturation

Abstract

This paper reviews the latest developments in anti-disturbance control of switched systems subject to input saturation. Firstly, the research significance of switched system, input saturation and anti-disturbance control is discussed, respectively. The significance of these researches is to realize the solution of the problems existing in practical engineering. Then it introduces the research status and development of dynamic analysis at home and abroad, including the research status of system control under input saturation constraints and the research status of anti-disturbance control of switched systems with input saturation.

Keywords:

• Switched system
• input saturation
• multiple disturbances
• anti-disturbance control

1. Introduction

Hybrid system is a class of complex systems that contains the interaction of the dynamics of discrete time and continuous variables. As an important and typical class of hybrid systems, switched systems can be used effectively to describe a variety of practical engineering systems, such as power electronics, manipulator robots, networked haptic systems, flight control systems, and in other fields (Zhang et al., 2015). Switched systems are composed of a finite number of subsystems, described by differential equations (or difference equations), and a switching law designed to orchestrate the switching among the subsystems. The overall motion trajectory of the system presents the migration of discrete positions, while the local representation is asymptotic evolution of the continuous state, which makes its dynamic characteristics very complex and the system stability analysis very difficult. The problem of performance analysis and control synthesis of switched systems is not a simple superposition between subsystems. For example, the stability of subsystems cannot guarantee the stability of switched system, and viceversa. Therefore, for studying the whole control problem of the switched system, we should not only design a controller with good performance but also consider the selection of switching law. Since the research on the control strategy of switched system was proposed in the 1960s, it has been widely concerned by the control science community (Greco et al., 2015; Yang & Zhao, 2016; Zhao et al., 2012a). The stability analysis and anti-windup design problem for the linear switched systems with saturating actuators is investigated, in which the single Lyapunov function method and a sector condition are used to design the anti-windup compensation gains which aim at maximizing the estimation of domain of attraction of the closed-loop systems (Zhang et al., 2017). Liu and Wang (2017) studies the regional stability for positive switched linear systems with multi-equilibrium points (PSLS-MEP). Compared with other results, the region of stability given in this paper is much less conservative. The problem of switched dynamical systems modeling especially the DC-DC converters is considered (Khoudiri et al., 2016) and two techniques can describe normal and abnormal behaviours of the DC-DC converters. In Jia and Zhao (2018) the cooperative output regulation problem of a class of multi-agent systems (MASs) is addressed in which each agent is a switched linear system. The International Federation of Automation Control (IFAC) has listed switched system research as one of the five top topics that have emerged in the past three decades.

With the development of the research on control theory and practical engineering applications, it has been recognized that there are many constraints in the actual control system, such as the linear interval of the amplifier circuit operation, the range of motor speed and torque variation, the rotation amplitude of the steering gear and the rate change interval, as well as the input current signal strength of the hydraulic electro-hydraulic proportional flow valve in large components has a certain constraint range condition (Tarbouriech et al., 2011). Due to the limitations of physics, security or technology, actuators that generate input signals in the control system are often constrained by amplitude or transmission rate, which may lead to saturation of the system input. This phenomenon is called actuator saturation or input saturation. When studying the control problem of the system, the actuator saturation will result in the state of the controller being updated incorrectly. Therefore, when designing the controller, ignoring the saturation effect may cause the controller to fail, destroying the performance of the closed-loop system, even resulting in loss of stability of the closed-loop system (Hu & Lin, 2001; Hu et al., 2018), such as the Chernobyl nuclear power plant accident in 1986 (Stein, 1989, 13–15 December), and the YF-22 fighter collision accident in 1992 (Bernstein & Michel, 1995). In modern control theory, input saturation constraints have been extensively studied as system nonlinear factors (Li & Lin, 2013, 2015; Nguyen et al., 2014; Wang & Duan, 2016; Wang & Zhao, 2016; Zhang et al., 2008). In Shahri and Balochian (2015), local stability and performance analysis of fractional-order linear systems with saturating elements are shown, which lead to less conservative on the region of stability and the disturbance rejection. By using the parametric Lyapunov equation and the gain scheduling technique, the output feedback dynamic gain scheduled control for stabilizing a spacecraft rendezvous system subject to actuator saturation is investigated (Wang & Duan, 2016; Wang & Zhao, 2016).

On the other hand, in practical systems such as industrial processes, advanced manufacturing, and aerospace, the existence of the disturbances is inevitable. Generally, the disturbances are divided into two categories. One is the external disturbance caused by the environmental change of the system, such as gust disturbance in flight control system and missile system (Li et al., 2012). The other is the internal disturbance caused by generalized unmodeled dynamics, perturbation of system model parameters or structural perturbation, such as uncertainty of aerodynamic parameters in aircraft system (Chen et al., 2003). Internal disturbance often results in a certain gap between the actual dynamic system and the mathematical model of the system, which is often referred as the uncertainty of the model. Long before the American mathematician Norbert Wiener founded cybernetics, the French mathematician J. V. Poncelet proposed that the purpose of control was to eliminate the effects of disturbance. In the mid-20th century, the British scholar W. Ross Ashby, the pioneer of cybernetics, pointed out that the control idea means that the system can still maintain its original performance when disturbance exists. It can be seen that people have realized the importance of the anti-disturbance control of the systems, since the establishment of the control theory. With the continuous innovation of modern control technology and the continuous advancement of industrial information technology, many scholars home and abroad have been working on the research of the anti-disturbance control for the system, and proposed many anti-disturbance control methods, such as adaptive control (Sun et al., 2013), robust control, sliding mode control (Fridman et al., 2011), disturbance observer-based control, active-disturbance-rejection controller. The problem of load transportation and robust mitigation of payload oscillations in uncertain tower-cranes is addressed (Coral-Enriquez et al., 2019) and a control scheme based on the philosophy of active-disturbance-rejection is used to tackle this problem. A novel nonlinear energy-based coupling control for an underactuated offshore ship-mounted crane is proposed, which guarantees both precise trolley positioning and payload swing suppressing performances under external sea wave disturbance (Qian et al., 2018). However, with the high-precision and high-reliability requirements of system control in the field of practical engineering, people have to intensify the research on anti-disturbance control. Therefore, how to improve the conclusions of existing methods or propose more effective theory of the anti-disturbance technology has become a research hotspot and difficulty in the field of control.

In practical engineering operation, switched system is inevitably affected by saturation constraints, model uncertainty and external disturbance, which is described as the model of nonlinear uncertain switched system subject to input saturation and disturbances. Due to the coupling switching, saturation nonlinearity, model uncertainty and external environment disturbance in the system, the model structure and system dynamics are very complicated, and its stability, robustness analysis and controller synthesis are more difficult. Through continuous efforts, some excellent results have been obtained for the stability analysis and anti-disturbance control research of uncertain switched system under input saturation constraint (Wang & Zhao, 2015; Zhao & Zhao, 2015). Such as a hybrid particle swarm optimization (PSO) algorithm with differential evolution (DE) is proposed for numerical benchmark problems and optimization of active disturbance rejection controller (ADRC) parameters (Lin et al., 2018) and a chaotic map with greater Lyapunov exponent are introduced into PSO for balancing the exploration and exploitation abilities of the proposed algorithm. These results generally adjust the control quantity through the error between the feedback quantity and the set value to realize the rejection of disturbance, but usually can not accurately eliminate the impact of disturbance on the system control performance. In view of the above limitations of anti-disturbance control, a kind of anti-disturbance control method based on disturbance estimation and compensation is proposed, which is characterized in that the disturbance of the system can be cancelled or compensated in the controller design according to the disturbance estimation value. Considering that the system is subject to input saturation constraints, the disturbance compensation process will be affected by saturation, and the dynamic characteristics of the switched system will bring challenges to the research of disturbance estimation and compensation control of the system.

Therefore, the research on anti-disturbance control of uncertain switched systems with input saturation has certain complexity and openness. How to use the advanced disturbance estimation and compensation control, and the composite hierarchical anti-disturbance method to achieve the fine anti-disturbance control of the saturated nonlinear switched system has become one of the urgent problems to be solved. The problem mainly includes the research of compensation control with matching and mismatching model disturbance under the premise of the input saturation of the switched system, and the research of composite hierarchical anti-disturbance control with multiple disturbances for the switched systems subject to input saturation. The research of these problems is challenging, and it will provide new research ideas for the anti-disturbance control problem of switched systems, which has positive scientific significance.

2. Research status of system control under input saturation constraints

As we all know that stability is the most basic property of control system. Therefore, when studying the control system with input saturation, the stabilization problem of the system should be considered first. Stabilizability and controllability of the system are closely related. For example, for a linear system, the system can be stability if and only if the unstable mode is controllable. In the 1980s, American scholar Schmitendorf first proposed the concept of asymptotic null controllability by bounded control (Schmitendorf & Barmish, 1980). As sufficient and necessary conditions for a system to realize global stabilization by bounded control, asymptotic null controllability lays a foundation for the stabilization of a nonlinear system subject to input saturation. Famous scholar Sontag further pointed out that this system is controllable without saturated nonlinear actuator, and all the open-loop poles are located in the closed left half plane (Sontag & Sussmann, 1990, 5–7 December). Based on the above conditions, well-known experts and scholars at home and abroad are committed to the design and study of globally stabilizing controller for the system with input saturation, professors Zongli Lin, h. j. Sussmann, A. R.Teel, Bin Zhou and others have made outstanding contributions in this study (Sussmann et al., 1994; Teel, 1992).

However, there are certain limitations to the use of system global stabilization conditions under input saturation constraints. This kind of control method is difficult to be realized by linear feedback and required that the open-loop system is asymptotically null controllability. For this purpose, Professor Zongli Lin used the low-gain feedback method to prove that linear feedback can achieve semi-global stabilization for the control system with input saturation (Lin & Saberi, 1993). However, the conclusion of semi-global stabilization cannot get rid of the requirement for the system itself to be asymptotically null controllability. Therefore, professor Tingshu Hu et al. proposed a control method that can both realize linear feedback control and get rid of the above requirement, which is the local stabilization condition of the control system with input saturation. It is necessary to characterize the stable attractive region of the closed-loop system when studying the problem of local stabilization for the system with input saturation constraints. However, it is almost impossible to accurately characterize the stable attraction region of a closed-loop system, and it can only be effectively estimated (Hu & Lin, 2001).

At present, the problem about local stabilization of control system with input saturation has become one of the research hotspots in the field of control. Professor Tingshu Hu et al. proposed a polytope differential inclusion method to deal with saturation nonlinear locally and used Lyapunov level set to estimate the attraction region of the closed-loop system (Hu & Lin, 2001). Yuanlong Li has realized the optimization of the attraction domain of the closed-loop system by improving the polytope differential inclusion theorem and dividing input space (Li & Lin, 2013, 2015). Professor Lixian Zhang has studied the controller design conditions of saturation nonlinear time-delay system, and used Lyapunov-Krasovskii functional level set to estimate the attraction region (Zhang et al., 2008). Professor Mou Chen has studied the robust tracking problem of uncertain multiple-input multiple-output nonlinear systems with RWNNDO method (Chen et al., 2014). Professor Zhiqiang Zuo has studied the event-driven control of Markov systems subject to input saturation constraints (Li et al., 2016). For singular switched systems subject to input saturation constraints, Jie Lian has given the conditions of exponential stabilization (Lian & Wang, 2015). Gongfei Song has studied the quantized control of nonlinear systems with input saturation and bounded disturbances (Song et al., 2016). Yonggang Chen has studied the problem of robust stabilization for distributed time-delay saturation nonlinear uncertain systems (Chen et al., 2016). With the development of control science, how to improve the saturation nonlinear processing method, obtain more advanced control strategies and less conservative estimation of the attractive region is an important and challenging topic.

3. Research status of anti-disturbance control for switched systems subject to input saturation constraints

In view of the theoretical and engineering value, the problem of switched systems control has always been a research hotspot. The famous American scholar D. Liberzon and the American Academy of Engineering academician Morse made a basic analysis on the stability of switched systems and divided research results of switched systems control problems into three categories:the design of the system controller under arbitrary switching signals, the design of the system controller under the limited switching signal, and the effective switching signal design of a given switched system (Liberzon & Morse, 1999). Handing the problem of switched systems under arbitrary switching signal, it is necessary for all switched subsystems to be stable and have a common Lyapunov function, which limit the application range of conclusions (Sun, 2006). By limiting the arbitrariness of the switching signal and using multiple Lyapunov functions or switching Lyapunov functions, the purpose of relaxing requirements on the subsystems are realized. In the 1990s, the famous scholar A.S. Morse et al. proposed the dwell time and average dwell time switching signals by means of switching Lyapunov function method (Hespanha & Morse, 1999, 7–10th December; Morse, 1996). Professor Xudong Zhao and professor Lixian Zhang et al. promoted it as a mode-dependent average dwell time method (Zhao et al., 2012b). Based on the maximum region function, famous scholar Pettersson proposed a class of state-dependent switching signals (Pettersson, 2003, 9–12 December). The problem of robust stabilization for a class of discrete-time switched large-scale systems with parameter uncertainties and nonlinear interconnected terms is considered (Sun, 2019). The state feedback and Lyapunov function technique are used and a decentralized switching control approach is put forward to guarantee the solutions of large-scale systems converge to the origin globally. Zeroual et al. (2017) regards the traffic modeling through the enhancement of the cell transmission model and it considers the traffic flow as a hybrid dynamic system and proposes a piecewise switched linear traffic model. Recently, famous Israeli scholars L. I. Allerhand and U. Shaked proposed a kind of state-dependent and time-driven hybrid switching strategy to stabilize switched linear systems Allerhand and Shaked (2013). The problem of simultaneous estimation of the system states and the strategy of commutation for a larger class of nonlinear switched systems is investigated (Manaa et al., 2015). First, a hybrid high gain observer is considered to get the exact estimation of the continuous states. Then, an extension to a larger class of nonlinear hybrid systems with arbitrary switching is made. Professor Jun Zhao and others successfully applied this method to the research of anti-disturbance control of switched systems. The above conclusions mainly focus on switched systems, designing switching signal, analysing the stability and stabilization of the switched system. It is worth pointing out that these problems are considered without regard to possible saturation limits on system inputs.

When the input of switched systems is constrained by saturation, the switched system presents nonlinear dynamics. Therefore, the system structure is complex and the control problem is more difficult. By designing a class of state-dependent switching signals, professor Zongli Lin et al. realized the local feedback control of the switched linear system with the input saturation constraints and proposed optimization strategy to obtain the attraction domain estimation of the large closed-loop system (Lu & Lin, 2008). Professor Daizhan Cheng and others used the minimum dwell time method and the linear differential inclusion method to give a design method of the controller for switched linear system under input saturation constraints and gave stable attraction domain estimation for closed-loop system (Ni & Cheng, 2010). In addition, professor Jun Zhao et al. studied the frequency control problem of switched systems subject to input saturation constraints through the method of composite nonlinear feedback and improved the transient performance of the system (Zhang et al., 2008). Although some achievements have been made in the research on the control problem of switched systems under the input saturation constraints, the research on the problem is still open, such as the refined anti-disturbance problem of the nonlinear uncertain switched systems subject to input saturation and multiple disturbances.

In nowadays control theory, anti-disturbance control, as an important research direction, has been paid attention by the majority of experts and scholars (Fridman et al., 2011; Sun et al., 2013). The existing anti-disturbance control methods can be divided into two categories: passive anti-disturbance control and active anti-disturbance control. The former aims at rejecting disturbance and adjust the control value according to the error between the feedback value and the set value to achieve the rejection of disturbance, such as robust control; The latter is for the purpose of compensating (or even cancelling) the disturbance, mainly to compensate (or even cancel) the disturbance directly in the controller control process according to the disturbance observation or estimation value, such as disturbance observer-based control (Li et al., 2012). It is worth noting that for disturbance estimation and compensation control methods, model uncertainty can be treated as disturbance.

At present, for the research of anti-disturbance control of switched systems under input saturation constraint, the main conclusions are focused on the passive anti-disturbance control. Professor Shaocheng Tong has studied robust stabilization problem of fuzzy switched uncertain system under input saturation constraints by designing the state-dependent switching signal and combining with the differential inclusion method (Yang & Tong, 2015). Professor Long Wang and Professor Guangming Xie designed switching signals for several disturbance types, proposed the design method of corresponding saturation nonlinear robust controller, and gave the corresponding estimation of stable attraction region (Yu et al., 2010). Professor Jun Zhao et al. proposed an effective robust controller design method for uncertain switched T-S system subject to input saturation, and the obtained results were applied to the turbo charging system (Wang & Duan, 2016; Wang & Zhao, 2016).

It is well known that compensation control is more effective than feedback control for the case of observable or estimated external disturbance. Professor Lei Guo and Professor Shihua Li used the disturbance estimation compensation technology to achieve the purpose of anti-disturbance control for the system subject to input matching/mismatch disturbance (Li et al., 2012). If the system suffers from multiple disturbances, the system's overall anti-disturbance control goal may not be realized by the single disturbance compensation observation technology. Therefore, by classifying disturbances, Professor Guo proposed a kind of composite hierarchical anti-disturbance control method (Wei & Guo, 2010). However, when the input is saturated, the disturbance compensation process will be affected, the control performance of the system controller will be limited, unable to realize the purpose of anti-disturbance control of the system. Therefore, it is urgent to solve the problem of disturbance estimation and compensation for uncertain systems with input saturation constraints.

Many research results are as follows: The problem of the disturbance observer-based control for a class of continuous-time uncertain systems subject to input saturation and nonlinearity is concerned in Wei et al. (2015). And the problem of the composite control for a class of uncertain Markovian jump systems with partial information on transition rates, saturating actuator and multiple disturbances has been investigated, which is in combination with disturbance observer based control, adaptive neural network control and robust $H_{\mathrm{\infty }}$ control in Wei et al. (2019). Wei et al. (2019) studies the problem of disturbance attenuation and rejection for switched systems with nonlinear uncertainty and input saturation via composite anti-disturbance control technique, in which the exosystem generated disturbance and $H_2$-norm bounded disturbance are considered. In Wei et al. (2019), the problem of composite anti-disturbance control for a class of switched nonlinear systems subject to input saturation and multiple disturbances is investigated, the problem of disturbance attenuation and rejection for a class of switched nonlinear systems subject to input and sensor saturations has also been discussed, and the full-order and reduced-order observers have been designed accordingly. And Wei and Liu (2020) focuses on the composite control problem for a class of uncertain switched impulsive systems with time-varying time delay, actuator saturation and multiple disturbances which are matched and mismatched with continuous control inputs.

4. Conclusion

In this paper, the latest developments in anti-disturbance control of switched systems with input saturation constraints have been reviewed. Firstly, the research significance of switched system, input saturation and anti-disturbance has been discussed, respectively. The significance of these researches is combined with the problems existing in practical engineering. Then the research status and development dynamic analysis at home and abroad has been introducted, including the research status of system control under input saturation constraints and the research status of anti-disturbance control of switched systems under input saturation constraints. The former has been discussed the stability and global stability of the system under input saturation constraints, while the latter mainly has been described the development of anti-disturbance control and its main solutions in the current society. And it has been discribed its application prospects in combination with the key scientific and technological issues that need to be solved urgently in national economic and social development. In future, how to obtain optimize the performance index of passive anti-disturbance control, design more accurate disturbance observer and construct high-precision composite anti-disturbance controller would be challenging research topics for the anti-disturbance control of the switched system subject to input saturation. On the other hand, the most advanced anti-disturbance control method is extended to network system which has been widely studied, such as (Hu et al., 2016, 2019).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by a research grant from the National Natural Science Foundations of China [grant numbers 61703233, 61773144, 61690212], the Postdoctoral Science Foundation of China [grant number 2016M602112], the Natural Science Foundation of Shandong Province [grant number ZR2016FQ09], the Open Fund of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education under Grant: MCCSE2016A04, Science and Technology Planning Project of Qufu Normal University [grant number xkj201513].