The maximal contractively invariant ellipsoids for discrete-time linear systems under saturated linear feedback

doi:10.1016/j.automatica.2016.10.007

Automatica

Volume 76, February 2017, Pages 336-344

https://doi.org/10.1016/j.automatica.2016.10.007 Get rights and content

Abstract

In this paper, we consider the problem of determining the maximal contractively invariant ellipsoids for discrete-time linear systems with multiple inputs under saturated linear feedback. We propose an algebraic computational approach to determining such maximal contractively invariant ellipsoids. We divide the state space into several regions according to the saturation status of each input, and compute the possible maximal contractively invariant ellipsoids on each region except the region where none of inputs saturate and on their intersections. The minimal one among these possible maximal contractively invariant ellipsoids is the maximal contractively invariant ellipsoids of the system. Simulation results demonstrate the effectiveness of our methods.

Introduction

For a linear system asymptotically null controllable by bounded controls, except for very special cases, saturated linear feedback can only achieves local asymptotical stabilization (Sussmann & Yang, 1991). A linear system is said to be asymptotically null controllable with bounded controls if it is stabilizable in the usual linear systems theory sense and all its open loop poles are in the closed left-half plane. For such a system under saturated linear feedback, its domain of attraction in general cannot be analytically characterized. This raises an important problem of how to estimate its domain of attraction. Much effort has been made on this problem (Alamo et al., 2005, Alamo et al., 2006a, Alamo et al., 2006b, Dai et al., 2009, Gomes da Silva and Tarbouriech, 1999, Gomes da Silva and Tarbouriech, 2001, Hu and Lin, 2002a, Hu and Lin, 2005, Milani, 2002, Pittet et al., 1997, Tarbouriech et al., 2011). For example, the authors of Alamo et al. (2006a) used a polyhedral SNS invariant set, which embeds the characteristics of saturation functions, as an estimate of the domain of attraction of discrete-time saturated linear systems. In Gomes da Silva and Tarbouriech (1999), a necessary and sufficient condition to determine the contractivity of polyhedral regions has been presented for linear discrete-time systems with saturating controls. Moreover, a Lyapunov function is composed from a group of quadratic Lyapunov functions (Hu & Lin, 2005). The level set of this composite Lyapunov function, which is the convex hull of the ellipsoids corresponding to the level sets of the individual quadratic Lyapunov functions, leads to a larger estimate of the domain of attraction (Hu & Lin, 2005).

As one of the most commonly used invariant sets, the ellipsoid, as the level set of the quadratic Lyapunov function, has been widely used in estimating the domain of attraction of saturated linear systems (Alamo et al., 2005, Gomes da Silva and Tarbouriech, 2001, Hu and Lin, 2001, Hu and Lin, 2002a, Tarbouriech et al., 2011). Based on the convex hull representation of saturated linear feedback, an optimization problem with a set of LMI constraints has been formulated to obtain a large contractively invariant ellipsoid (Alamo et al., 2005, Hu and Lin, 2002a). Since these constraints are only sufficient conditions for an ellipsoid to be contractively invariant, the resulting optimal ellipsoid is generally not the maximal contractively invariant ellipsoid. However, for both continuous-time and discrete-time linear systems with a single input under saturated linear feedback, it is proven in Hu and Lin, 2002b, Hu and Lin, 2005 that these sufficient LMI conditions are also necessary, and hence the optimal ellipsoid resulting from the optimization problem is actually the maximal contractively invariant ellipsoid. With multiple inputs under saturated linear feedback, the optimal ellipsoid can be the maximal contractively invariant ellipsoid only when certain additional conditions are satisfied (Hu & Lin, 2003). For general continuous-time linear systems with multiple inputs under saturated linear feedback, an algebraic computational approach was developed for determining the maximal contractively invariant ellipsoid (Li & Lin, 2015). For general discrete-time linear systems under saturated linear feedbacks, a set of necessary and sufficient conditions were presented in Fiacchini, Prieur, and Tarbouriech (2013) that characterize the invariance and contractivity of convex sets.

In this paper, we carry out the characterization of the maximal contractively invariant ellipsoid associated with a given positive definite matrix for discrete-time linear systems with $m$ inputs under saturated linear feedback. An algebraic computational approach to determining such maximal contractively invariant ellipsoids is developed as follows. We first divide the state space into several regions according to the saturation status of each input. We then compute the possible maximal contractively invariant ellipsoid on each region except the one where none of inputs saturate, and then determine the minimal one among these possible maximal contractively invariant ellipsoids whose extreme states reside in the corresponding regions. Next, we compute the possible maximal contractively invariant ellipsoids on intersections between some regions, where there are $k$ inputs that simultaneously critically saturate, $k = 1, 2, \dots, m - 1$ . The smallest one of these ellipsoids determined above is then the maximal contractively invariant ellipsoid for discrete-time linear systems with multiple inputs under saturated linear feedback.

To apply this algebraic computational method, we need to solve certain polynomial equations or compute the eigenvalues of certain matrices. Software tools exist that could easily carry out such computation, for example, the polynomial toolbox of Matlab. However, the amount of computation associated with this algebraic computational method increases exponentially with the dimensions of the state and input. Note that the algebraic computational method applies to any linear system with multiple inputs under saturated linear feedbacks.

Control based on discrete-time saturated system models is extensively adopted in the practice systems subject to actuator saturation. Hence, it is necessary to analyze stability of the discrete-time saturated systems including the maximal contractively invariant ellipsoid. The algebraic computational approach aforementioned to determine the maximal contractively invariant ellipsoid is a counterpart to our recent result in Li and Lin (2015). The algebraic computation in this paper is however not a direct generalization of that in Li and Lin (2015), as the computational details in the discrete-time setting are significantly different from those in continuous-time setting.

The remainder of the paper is organized as follows. In the preliminaries section, we recall some relevant results on ellipsoidal invariant sets of discrete-time linear systems under saturated linear feedback. In Section 3, we propose an algebraic computational method to determine the maximal contractively invariant ellipsoid for discrete-time linear systems with multiple inputs under saturated linear feedback. In Section 4, numerical examples are given to demonstrate the effectiveness of the results in this paper. Section 5 concludes the paper.

We will use standard notation. For a vector $u = [u_{1} u_{2} \dots u_{m}]^{T}$ , ${| u |}_{\infty} ≔ {max}_{i} | u_{i} |$ . For two integers $k_{1}, k_{2}, k_{1} < k_{2}$ , $I [k_{1}, k_{2}] ≔ {k_{1}, k_{1} + 1, \dots, k_{2}}$ . For a positive definite $P \in R^{n \times n}$ and a positive scalar $ρ$ , $E (P, ρ) ≔ {x \in R^{n} : x^{T} P x \leq ρ}$ . For a set $S$ , we use $S^{\circ}$ and $\partial S$ to denote its interior and its boundary. For a matrix $H \in R^{m \times n}$ , $L (H) ≔ {x \in R^{n} : {| H x |}_{\infty} \leq 1}$ . For a matrix $A$ , $He (A) = A^{T} + A$ . $I_{n}$ stands for an $n$ -dimensional identity matrix.

Section snippets

Preliminaries

Consider the following discrete-time system $x^{+} = A x + B sat (F x),$ where $x \in R^{n}$ denotes the state vector, $x^{+}$ is the successor state, $F \in R^{m \times n}$ is the feedback gain, and $sat : R^{m} \to R^{m}$ denotes the vector valued standard saturation function, which is defined as $sat (u) = [sat (u_{1}), sat (u_{2}), \dots, sat (u_{m})]^{T}, sat (u_{i}) = sgn (u_{i}) min {1, | u_{i} |}$ . A signal $u_{i}$ is said to saturate if $| u_{i} | > 1$ and it is said to saturate critically if $| u_{i} | = 1$ . Given a positive definitive matrix $P \in R^{n \times n}$ , let $V (x) = x^{T} P x$ . The ellipsoid $E (P, ρ)$ is said to be

Characterization by algebraic computation

In this section, we will present an algebraic computational method for determining the maximal contractively invariant ellipsoid associated with a given positive definite matrix $P$ for system (1). By Fact 1, the key problem in the determination of this maximal contractively invariant ellipsoid is to determine an extreme state $x_{0} \in \partial E (P, ρ_{c})$ such that $E^{\circ} (P, ρ_{c})$ is contractively invariant. However, in the state space, the saturation function is piecewise linear so that it cannot be considered as a

Numerical examples

Example 1

Consider system (1) with $\begin{matrix} A & = [\begin{matrix} 0 & 2 & - 6.5 \\ 0 & - 0.5 & 0 \\ - 6 & 10 & 4 \end{matrix}], B = [\begin{matrix} - 3 & - 8.5 & - 4 \\ 3 & 0 & 4 \\ 9 & 1 & - 2 \end{matrix}], \\ F & = [\begin{matrix} 0.5737 & - 0.8983 & - 0.2415 \\ - 0.0235 & 0.0824 & - 0.7647 \\ - 0.4303 & 0.9987 & 0.1811 \end{matrix}] . \end{matrix}$ Let $P = I_{3}$ . We compute $ρ^{*}$ and $ρ^{l ♮}$ for the maximal contractively invariant ellipsoid. By the algebraic computational method in Section 3, we obtain that $ρ^{*} = 0.9330$ . On the other hand, we can compute $ρ^{1 ♮} = 0.9736$ and $ρ^{2 ♮} = + \infty$ . Then we have $ρ_{c} = min {ρ^{*}, ρ^{1 ♮}} = min {0.9330, 0.9736} = 0.9300$ .

The difference $Δ V (x)$ along $\partial E (P, ρ_{c})$ is plotted in Fig. 1. In the right plot we can observe

Conclusions

This paper revisited the problem of determining the maximal contractively invariant ellipsoid for a discrete-time linear system with multiple inputs under saturated linear feedback. We developed an algebraic computational approach to determining the maximal contractively invariant ellipsoids. This algebraic computational approach can be used for any linear system with multiple inputs under saturated linear feedback. It is illustrated by simulation examples that the methods developed in this

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Yuanlong Li was born in Hebei Province, China, on December 24, 1984. He received his B.S. and M.S. degrees from School of Communication and Control Engineering from Jiangnan University, Wuxi, China, in 2007 and 2009, respectively, and his Ph.D. degree in Control Theory and Control Engineering from Shanghai Jiao Tong University, Shanghai, China, in 2015. He is now a postdoctoral research fellow at Shanghai Jiao Tong University. He was a visiting graduate student with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, USA, from September 2011 to August 2012. His current research interests include nonlinear control theory and constrained control systems.

Zongli Lin is the Ferman W. Perry Professor in the School of Engineering and Applied Science and a Professor of Electrical and Computer Engineering at the University of Virginia. He received his B.S. degree in Mathematics and Computer Science from Xiamen University, Xiamen, China, in 1983, his Master of Engineering degree in automatic control from Chinese Academy of Space Technology, Beijing, China, in 1989, and his Ph.D. degree in electrical and computer engineering from Washington State University, Pullman, Washington, in 1994. His current research interests include nonlinear control, robust control, and control applications. He was an Associate Editor of the IEEE Transactions on Automatic Control (2001–2003), IEEE/ASME Transactions on Mechatronics (2006–2009) and IEEE Control Systems Magazine (2005–2012). He was an elected member of the Board of Governors of the IEEE Control Systems Society (2008–2010) and chaired the IEEE Control Systems Society Technical Committee on Nonlinear Systems and Control (2013–2015). He has served on the operating committees of several conferences and will be the program chair of the 2018 American Control Conference. He currently serves on the editorial boards of several journals and book series, including Automatica, Systems & Control Letters, Science China Information Sciences, and Springer/Birkhauser book series Control Engineering. He is a Fellow of the IEEE, a Fellow of the IFAC, and a Fellow of AAAS, the American Association for the Advancement of Science.

^☆: Work supported in part by the National Natural Science Foundation of China under Grant Nos. 61221003 and 61273105, and in part by the China Postdoctoral Science Foundation under Grant Nos. 2015M580332 and 2016T90373. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor Ian R. Petersen.

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Brief paperThe maximal contractively invariant ellipsoids for discrete-time linear systems under saturated linear feedback☆

Abstract

Introduction

Section snippets

Preliminaries

Characterization by algebraic computation

Numerical examples

Conclusions

Automatica

Systems & Control Letters

Systems & Control Letters

Systems & Control Letters

Automatica

Eatimation of the domain of attraction for saturated discrete-time systems

International Jounal of Systems Science

Polyhedral regions of local stability for linear discrete-time systems with saturating controls

IEEE Transactions on Automatic Control

Brief paper
The maximal contractively invariant ellipsoids for discrete-time linear systems under saturated linear feedback☆