Saturated impulsive control of nonlinear systems with applications

Abstract

The problem of saturated impulsive control of nonlinear systems is studied, where the saturation structure of impulse action is fully considered. Based on impulsive control theory and average dwell time (ADT) method, sufficient conditions for local exponential stabilization (LES) are derived. An optimization problem on designing controller is presented in order to maximize the domain of attraction. To get a less conservative estimate of the domain of attraction, a linear matrix inequality (LMI) based algorithm is proposed to solve such kind of optimization problems. The applications of the developed results are illustrated through examples from a Chua’s circuit and a problem of pulse transmission of a simple ball motion model.

Introduction

As is well known, impulsive control strategy is deemed as an effective control technique which permits systems to possess discontinuous inputs. In many cases, impulsive control is more efficient to deal with plants which cannot guarantee continuous control input. For examples, spacecraft cannot leave its engine on continuously with limited fuel supply and the population of bacterium can be instantaneously controlled by changing the density of a bactericide, see Lakshmikantham, Simeonov, et al. (1989) and Yang (2001). Much efforts have been devoted to impulsive control problem due to its broad applications (Dashkovskiy and Kosmykov, 2013, Han et al., 2020, Hespanha et al., 2008 and Li, Song and Wu, 2019).

On the other hand, saturation nonlinearity is widespread in control systems. Due to the physical constraints, the magnitude of the signal that an actuator or sensor can transmit is limited. Such limitation can be easily identified in many common devices of industry, such as electrical actuator (Yang, de Queiroz, & Li, 2019), hydraulic actuator (Wang et al., 2021), etc. Moreover, such limitation obviously restricts the system performance and feasibility. If saturation nonlinearity is ignored or not treated appropriately in the design of the controller, then some undesirable behaviors may be encountered (Stein, 2003). Hence, it is necessary and important to consider the design of the controller when systems subject to input saturation. The study of saturated systems has attracted considerable attention of many researchers over the past decades. Kalman (1957) originally discussed control problems of saturated linear systems. Pittet, Tarbouriech, and Burgat (1997) presented a method of stability regions analysis via both circle and Popov criteria in which conditions for local stability were expressed in terms of (nonlinear) matrix inequalities. For such kind of systems, Hu, Lin, and Chen (2002) proposed a less conservative linear matrix inequality (LMI) based technique for estimating the domain of attraction, where saturation nonlinearity was expressed as the form of linear differential inclusion. As another treatment of saturation nonlinearity, deadzone nonlinearity function was utilized as control modifications to recover the performance deteriorated by input saturation in Li and Lin, 2014, Tarbouriech et al., 2011 and Zaccarian and Teel (2011). Recently, various control problems associated with input saturation have been extensively studied, such as spacecraft attitude control (Zhou, 2019), event-triggered control (You, Hua, & Guan, 2018), sliding mode control (Ma, Shao, Liu, & Wu, 2019), and adaptive control (Tohidi, Yildiz, & Kolmanovsky, 2020).

Although much efforts have been made to develop effective control strategies to stabilize a plant with input saturation, only few works take into account of impulsive control. In fact, saturated impulsive control is encountered in many practical problems, such as impulsive consensus of multi-agent systems in which the transmission of signals among individual systems is limited, impulsive harvesting and releasing of fishing industry with limited fishing capacity, and impulsive accelerating or decelerating of ball motion process with limited actual transmission capacity. However, the problem on saturation structure of impulse action, even for linear systems, was tackled in only a few works (Li et al., 2020, Wei and Liu, 2020) due to the complicated coupling effects between impulses and input saturation. Some sufficient conditions for asymptotical stabilization and the estimate of the domain of attraction were derived in Li et al. (2020) and Wei and Liu (2020) via linear differential inclusion method in the framework of impulse saturation. However, one may observe that the continuous dynamics considered in above two works were both stabilizing, which leads to a fact that the stabilization results allow for the non-existence of impulses. It implies that the control effect of impulse action was not well considered. Moreover, the relation between impulse action and the estimate of the domain of attraction which is important to estimate the domain of attraction was also not addressed. Hence, a basic and important issue on how designing the impulsive controller to stabilize a plant and estimate the domain of attraction in the framework of impulse saturation remains unresolved and deserves in-depth thought, which is the main motivation of the present paper.

In this paper, we shall consider the analysis and design of nonlinear impulsive systems in the framework of impulse saturation. Based on impulsive control theory, some average dwell time (ADT) based sufficient conditions for local exponential stabilization (LES) are obtained, where a relationship between impulse action and the estimate of the domain of attraction is implicitly established. We provide a convex optimization algorithm on the design of the controller that maximize the contractively invariant ellipsoids, which is formalized in terms of LMI and hence can easily be solved for controller synthesis. The remainder of this paper is organized as follows. Preliminaries are presented in Section 2. Then the main results are proposed in Section 3. Finally, two examples are given in Section 4 and Section 5 concludes the paper.

Notations. Let $\mathbb{R}$ denote the set of real numbers, ${\mathbb{R}}^n$ the $n$-dimensional Euclidean space with form $\left|\cdot \right|$, and $I$ the unit matrix. For a matrix $A$, we use ${\lambda }_{max}\left(A\right)$, ${\lambda }_{min}\left(A\right)$ and $A^T$ to denote its maximum eigenvalue, minimum eigenvalue and transposition, respectively. $A>0$ means that matrix $A$ is symmetric and positive definite. For given integers $a<b\in {\mathbb{Z}}_{+}$, let $\mathcal{J}[a,b]=\{a,a+1,\dots ,b\}$. Given matrix $P\in {\mathbb{R}}^{n\times n}>0$ and constant $\varrho >0$, $\mathcal{B}(P,\varrho )$ is defined as $\mathcal{B}(P,\varrho )≔\{x\in {\mathbb{R}}^n:x^{T}Px\le \varrho \}$. Given matrix $H\in {\mathbb{R}}^{m\times n}$, let $h_i$ be the $i$th row of the matrix $H$ and define $ℒ\left(H\right)≔\{x\in {\mathbb{R}}^n:\left|h_{i}x\right|\le 1,i\in \mathcal{J}[1,m]\}$. $\mathfrak{D}$ denotes the set of $m\times m$ diagonal matrices whose diagonal elements are either 1 or 0. Suppose that each element of $\mathfrak{D}$ is labeled as $D_i$ and denote $D_i^{-}=I-D_i$, $i\in \mathcal{J}[1,2^m]$.