Volume 14 Number 5
October 2017
Article Contents
Xin-Quan Zhang, Xiao-Yin Li and Jun Zhao. Stability Analysis and Anti-windup Design of Switched Systems with Actuator Saturation. International Journal of Automation and Computing, vol. 14, no. 5, pp. 615-625, 2017. doi: 10.1007/s11633-015-0920-z
Cite as: Xin-Quan Zhang, Xiao-Yin Li and Jun Zhao. Stability Analysis and Anti-windup Design of Switched Systems with Actuator Saturation. International Journal of Automation and Computing, vol. 14, no. 5, pp. 615-625, 2017. doi: 10.1007/s11633-015-0920-z

Stability Analysis and Anti-windup Design of Switched Systems with Actuator Saturation

Author Biography:
  • Xiao-Yin Li received the B. Sc. degree in applied mathematics from Shenyang Normal University, China in 2007. She received the M. Sc. degree in applied mathematics in 2012 at Liaoning Technical University of Fuxin, China. Since 2012, as a research intern, she has been with School of Foreign Languages, Liaoning Shihua University, China.
         Her research interests include systems optimization, constrained systems and intelligent control.
         E-mail:lxy1982_@163.com

    Jun Zhao received the B. Sc. and M. Sc. degrees in mathematics in 1982 and 1984 respectively, both from Liaoning University, China. He received the Ph. D. in control theory and applications in 1991 at Northeastern University, China. From 1992 to 1993, he was a postdoctoral fellow at the same University. Since 1994, as a professor, he has been with College of Information Science and Engineering, Northeastern University, China. From 1998 to 1999, he was a senior visiting scholar at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, USA. From November 2003 to May 2005, he was a research fellow at Department of Electronic Engineering, City University of Hong Kong. During 2007-2010, he was a fellow at School of Engineering, the Australian National University.
         His research interests include switched systems, nonlinear systems and network synchronization.
         E-mail:zhaojun@mail.neu.edu.cn

  • Corresponding author: Xin-Quan Zhang received the B. Sc. degree in automation and M. Sc. degree in control theory & engineering from Liaoning Technical University, China in 2003 and 2007, respectively. He received the Ph. D. degree in control theory and control engineering in 2012 at the College of Information Science & Engineering, of the Northeastern University of Shenyang, China. Since 2012, as a lecturer, he has been with School of Information and Control Engineering, Liaoning Shihua University, China.
         His research interests include switched systems, robust control and systems control under constraints.
         E-mail:zxq_19800126@163.com(Corresponding author)
        ORCID ID:0000-0003-0712-3671
  • Received: 2014-08-07
  • Accepted: 2014-11-18
  • Published Online: 2017-10-07
Fund Project:

Scientific Research Fund of Education Department of Liaoning Province L2014159

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Stability Analysis and Anti-windup Design of Switched Systems with Actuator Saturation

  • Corresponding author: Xin-Quan Zhang received the B. Sc. degree in automation and M. Sc. degree in control theory & engineering from Liaoning Technical University, China in 2003 and 2007, respectively. He received the Ph. D. degree in control theory and control engineering in 2012 at the College of Information Science & Engineering, of the Northeastern University of Shenyang, China. Since 2012, as a lecturer, he has been with School of Information and Control Engineering, Liaoning Shihua University, China.
         His research interests include switched systems, robust control and systems control under constraints.
         E-mail:zxq_19800126@163.com(Corresponding author)
        ORCID ID:0000-0003-0712-3671
Fund Project:

Scientific Research Fund of Education Department of Liaoning Province L2014159

Abstract: The stability analysis and anti-windup design problem is investigated for two linear switched systems with saturating actuators by using the single Lyapunov function approach. Our purpose is to design a switching law and the anti-windup compensation gains such that the maximizing estimation of the domain of attraction is obtained for the closed-loop system in the presence of saturation. Firstly, some sufficient conditions of asymptotic stability are obtained under given anti-windup compensation gains based on the single Lyapunov function method. Then, the anti-windup compensation gains as design variables are presented by solving a convex optimization problem with linear matrix inequality (LMI) constraints. Two numerical examples are given to show the effectiveness of the proposed method.

Xin-Quan Zhang, Xiao-Yin Li and Jun Zhao. Stability Analysis and Anti-windup Design of Switched Systems with Actuator Saturation. International Journal of Automation and Computing, vol. 14, no. 5, pp. 615-625, 2017. doi: 10.1007/s11633-015-0920-z
Citation: Xin-Quan Zhang, Xiao-Yin Li and Jun Zhao. Stability Analysis and Anti-windup Design of Switched Systems with Actuator Saturation. International Journal of Automation and Computing, vol. 14, no. 5, pp. 615-625, 2017. doi: 10.1007/s11633-015-0920-z
  • A switched system as an important hybrid dynamical system is composed of a family of continuous-time or discrete-time subsystems and a switching law that governs which subsystem is activated along the system trajectory during a certain interval of time. Many practical systems such as computer controlled systems, power systems and network control systems can be modeled as switched systems. As a result, the analysis and synthesis of switched systems have attracted considerable attention in the past few years [1-7]. As pointed out by [1], stability is of great importance in the analysis and control for switched systems. Because stability under arbitrary switchings is a preferable property which allows us to pursue for other performances with stability maintained. It is well known that the stability under arbitrary switchings can be guaranteed by a common Lyapunov function [8-10], however, to find such a common Lyapunov function is often difficult, yet switched system still may be stable under certain switching laws. The multiple Lyapunov functions method [11, 12], the single Lyapunov function method [13, 14] and the average dwell-time technique [15] are effective tools for choosing such switching laws.

    On the other hand, almost every physical actuator is subject to saturation for practical control systems. It is generally known that actuator saturation can lead to performance deterioration of the system and even make the otherwise stable system unstable. Thus, more and more attention has been focused on the analysis and synthesis for systems subject to actuator saturation and many methods have been developed to deal with actuator saturation [16-21]. In general, there are two major methods to deal with actuator saturation. The first method is to take actuator saturation into account at the outset of the control design, and then design a linear controller which drives the system to stability [19]. The second method is to neglect actuator saturation and design a controller which meets the performance specifications in the first stage of the control design process, and then add an anti-windup compensator that weakens the influence of saturation [21]. However, the anti-windup approach is a popular method and efficient technique to deal with actuator saturation from the practical perspective.

    The study of switched systems subject to actuator saturation becomes more difficult due to the phenomena of interacting switching and actuator saturation nonlinearities. Thus, the stability results about such switched systems are few [22-25]. For a class of saturated switched linear systems, Zhang et al. [22] addressed the robust stabilization problem based on the multiple Lyapunov functions method. Via the multiple quadratic Lyapunov functions method, a switching anti-windup design is proposed for a linear system with actuator saturation in [23]. For a class of discrete-time saturated switched systems, Benzaouia et al. [24] investigated the stabilization problem via the switched Lyapunov function method. In [25], by using multiple Lyapunov functions method, the design of switching scheme is considered for a class of switched linear systems in the presence of actuator saturation. To the best of the authors' knowledge, for the problem of the stability analysis and anti-windup design with resort to the single Lyapunov function method nearly no results have been reported for discrete-time saturated switched systems in the existing literature, which motivates the present study.

    In this paper, we study the stability analysis and anti-windup design problem for two switched linear systems with actuator saturation based on the single Lyapunov function approach. We first obtain some sufficient conditions of asymptotic stability when anti-windup compensation gains are given. Then, the anti-windup compensators are designed which aim to enlarge the domain of attraction of the considered system. Finally, all the results are formulated and solved as a convex optimization problem with linear matrix inequality (LMI) constraints.

    In this paper, compared with the existing results on switched systems with actuator saturation, the results have two features. First of all, the stability analysis and anti-windup design problem is addressed for the switched systems with saturating actuator, while most existing works considered only the stabilization problem; second, we use the single Lyapunov functions method for designing a switching law, while the existing works mostly aimed at arbitrary switchings for discrete-time switched systems subject to actuator saturation.

    The paper is organized as follows. The system description and relevant preliminaries are given in Section 2. Section 3 provides the stability conditions for the considered systems. The design problem of anti-windup compensators is proposed in Section 4. Two examples illustrate the effectiveness of the proposed method in Section 5. Conclusions are in Section 6.

    Notations.  Throughout this paper, the following notations are used. $I$ denotes the identity matrix with compatible dimension. $A^{\rm T}$ is the transpose of the matrix $A$ . $ * $ denotes the symmetry elements in symmetric matrices, that is

    $\left[{\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } \\ * & {Q_{22} } \\ \end{array}} \right] = \left[{\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } \\ {Q_{12}^{\rm T} } & {Q_{22} } \\ \end{array}} \right]. $

  • We consider the following class of switched linear systems with actuator saturation.

    $ \left\{ \begin{array}{l} \dot x = {A_\sigma }x + {B_\sigma }sat(u)\\ y = {C_\sigma }x \end{array} \right. $

    (1)

    where $x \in {\bf R}^n$ is the state vector, $u\in {\bf R}^m $ is the control input vector and $y \in {\bf R}^p $ is the measured output vector. The function $sat:\;{\bf R}^m \to {\bf R}^m $ is the standard, vector-valued, saturation function:

    $\left\{ \begin{array}{*{35}{l}} sat(u)\ ={{\left[ sat({{u}^{1}})\ sat({{u}^{2}})\ \cdots \ sat({{u}^{m}}) \right]}^{\text{T}}} \\ sat({{u}^{j}})=\text{sgn}({{u}^{j}})\min \left\{ 1,\ \left| {{u}^{j}} \right| \right\} \\ \qquad ~~~~~~~~\forall j\in {{Q}_{m}}=\left\{ 1,\ \cdots ,\ m \right\}. \\ \end{array} \right.$

    It is well known that it is without loss of generality to assume unity saturation level [16]. Function $\sigma:[0, \;\infty)\rightarrow I_{N}=\{1, \;\cdots, \;N\}$ is a piecewise constant switching signal; $\sigma= i$ means that the i-th subsystem is active. $A_i, \;B_i $ and $C_i $ are constant matrices with appropriate dimensions.

    For system (1), we will consider that a set of $n_c$ -order dynamic output feedback controllers are of the form

    $ \left\{ \begin{array}{l} {{\dot x}_c} = {A_{ci}}{x_c} + {B_{ci}}{u_c}\\ {v_c} = {C_{ci}}{x_c} + {D_{ci}}{u_c} \\ \qquad ~~\forall i \in {I_N} \end{array} \right. $

    (2)

    where $x_c \in {\bf R}^{n_c }$ , $u_c = y$ and $v_c = u$ are the vector of state, input and output of the controller respectively. Due to our focus on analysis and design of anti-windup compensation gains, as commonly adopted in the literature (see, for example [20]), we assume that the dynamic compensators have been designed that stabilize the system (1) and (2) without input saturation and satisfy performance requirements.

    In order to alleviate the undesirable effects of the windup caused by saturating actuators, a typical anti-windup compensator involves adding to the controller dynamics a "correction" term of the form $E_{ci} (sat(v_c ) - v_c )$ . Then, the final controller structure has the form

    $\left\{ \begin{array}{l} {{\dot x}_c} = {A_{ci}}{x_c} + {B_{ci}}{u_c} + {E_{ci}}(sat({v_c}) - {v_c})\\ {v_c} = {C_{ci}}{x_c} + {D_{ci}}{u_c} \\ \qquad ~~\forall i \in {I_N}. \end{array} \right. $

    (3)

    Obviously, by using such the correction terms, the dynamic controllers (3) continue to operate in the linear domain in the absence of saturation, which does not affect the considered systems performance, and the controller state of the system under input saturation are able to be corrected through the anti-windup compensators which recover the nominal performance of the system as much as possible.

    Now, under the above dynamic controllers and anti-windup strategy, the closed-loop system can be written as

    $\left\{ \begin{array}{l} \dot x = {A_i}x + {B_i}sat({v_c})\\ y = {C_i}x\\ {{\dot x}_c} = {A_{ci}}{x_c} + {B_{ci}}{C_i}x + {E_{ci}}(sat({v_c}) - {v_c})\\ {v_c} = {C_{ci}}{x_c} + {D_{ci}}{C_i}x\\ \qquad ~~\forall i \in {I_N}. \end{array} \right. $

    (4)

    Then, we define a new state vector as

    $\zeta = \left[ {\begin{array}{*{20}c} {x} \\ {x_c } \\ \end{array}} \right] \in {\bf R}^{n + n_c } $

    (5)

    and the matrices

    $\begin{align} & {{{\tilde{A}}}_{i}}=\left[ \begin{matrix} {{A}_{i}}+{{B}_{i}}{{D}_{ci}}{{C}_{i}} & {{B}_{i}}{{C}_{ci}} \\ {{B}_{ci}}{{C}_{i}} & {{A}_{ci}} \\ \end{matrix} \right],\ {{{\tilde{B}}}_{i}}=\left[ \begin{matrix} {{B}_{i}} \\ 0 \\ \end{matrix} \right] \\ & \quad \quad G=\left[ \begin{matrix} 0 \\ {{I}_{{{n}_{c}}}} \\ \end{matrix} \right],{{K}_{i}}=\left[ \begin{matrix} {{D}_{ci}}{{C}_{i}} & {{C}_{ci}} \\ \end{matrix} \right]. \\ \end{align}$

    Therefore, according to (4) and (5), the closed-loop system can be rewritten as

    $\dot{\zeta} = \tilde A_i \zeta - (\tilde B_i + GE_{ci} )\psi (v_c ), \;\forall i \in I_N $

    (6)

    where $v_c = K_i \zeta, \;\psi (v_c ) = v_c - sat(v_c ).$

  • We consider the following class of discrete-time switched systems subject to actuator saturation

    $ \label{e1-1} \left\{ \begin{array}{l} x(k + 1) = {A_\sigma }x(k) + {B_\sigma }sat(u(k))\\ y(k) = {C_\sigma }x(k) \end{array} \right. $

    (7)

    where $k \in {\pmb Z}^ + $ , $x(k) \in {\bf R}^n$ is the state vector, $u(k) \in {\bf R}^m $ is the control input vector and $y(k) \in {\bf R}^p $ is the measured output vector. The function $sat:\;{\bf R}^m \to {\bf R}^m $ is the standard, vector-valued, saturation function

    $\left\{ {\begin{array}{*{20}{l}} {sat(u)\; = {{\left[ {sat({u^1})\;sat({u^2})\; \cdots \;sat({u^m})} \right]}^{\rm T}}}\\ {sat({u^j}) = {\rm sgn}({u^j})\min \left\{ {1,\;\left| {{u^j}} \right|} \right\}}\\ {\qquad ~~~~~~~\forall j \in {Q_m} = \left\{ {1,\; \cdots ,\;m} \right\}.} \end{array}} \right.$

    The function $\sigma (k)$ is a switching law that takes its values in the finite set $I_N = \{1, \;\cdots, \;N\} $ ; $\sigma (k) = i$ means that the i-th subsystem is active. $A_i, \;B_i $ and $C_i $ are constant matrices of appropriate dimensions.

    For system (7), we will assume that a set of $n_c$ -order dynamic compensators are of the form

    $\left\{ \begin{array}{l} {x_c}(k + 1) = {A_{ci}}{x_c}(k) + {B_{ci}}{u_c}(k)\\ {v_c}(k) = {C_{ci}}{x_c}(k) + {D_{ci}}{u_c}(k)\;\\ \qquad ~~~~~\forall i \in {I_N} \end{array} \right. $

    (8)

    where $x_c (k) \in {\bf R}^{n_c }$ , $u_c (k) = y(k)$ and $v_c (k) = u(k)$ are the state, input and output of the controller respectively.

    We assume that the dynamic compensators have been designed which can stabilize the systems (7) and (8) without actuator saturation.

    Similarly, in order to weaken the effects of the windup caused by actuator saturation, a typical anti-windup compensator involves adding to the controller dynamics a "correction" term of the form $E_{ci} (sat(v_c (k)) - v_c (k))$ . Then, the revised compensators have the form

    $\left\{ \begin{array}{l} {x_c}(k + 1) = {A_{ci}}{x_c}(k) + {B_{ci}}{u_c}(k)+\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {E_{ci}}(sat({v_c}(k)) - {v_c}(k))\\ {v_c}(k) = {C_{ci}}{x_c}(k) + {D_{ci}}{u_c}(k)\;, \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\forall i \in {I_N}. \end{array} \right. $

    (9)

    Clearly, the compensators (9) would continue to run in the linear domain without saturation by using such modified terms which does not affect the systems nominal performance, and the anti-windup compensators can also amend the controller state in order to recover the nominal performance of the system with input saturation as much as possible.

    Then, in combination with the above dynamic controllers and anti-windup strategy, the closed-loop system can be written as

    $\left\{ \begin{array}{l} x(k + 1) = {A_i}x(k) + {B_i}sat({v_c}(k))\\ y(k) = {C_i}x(k)\\ {x_c}(k + 1) = {A_{ci}}{x_c}(k) + {B_{ci}}{C_i}x(k)+\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{E_{ci}}(sat({v_c}(k)) - {v_c}(k))\\ {v_c}(k) = {C_{ci}}{x_c}(k) + {D_{ci}}{C_i}x(k)\\\;\;\;\;\;\;\;\;\;\;\; \; \forall i \in {I_N}. \end{array} \right. $

    (10)

    Now, define a new state vector

    $ \zeta (k) = \left[ {\begin{array}{*{20}c} {x(k)} \\ {x_c (k)} \\ \end{array}} \right] \in {\bf R}^{n + n_c } $

    (11)

    and the matrices

    $\begin{align} & {{{\tilde{A}}}_{i}}=\left[ \begin{matrix} {{A}_{i}}+{{B}_{i}}{{D}_{ci}}{{C}_{i}} & {{B}_{i}}{{C}_{ci}} \\ {{B}_{ci}}{{C}_{i}} & {{A}_{ci}} \\ \end{matrix} \right],\ {{{\tilde{B}}}_{i}}=\left[ \begin{matrix} {{B}_{i}} \\ 0 \\ \end{matrix} \right] \\ & \quad \quad G=\left[ \begin{matrix} 0 \\ {{I}_{{{n}_{c}}}} \\ \end{matrix} \right],\ {{K}_{i}}=\left[ \begin{matrix} {{D}_{ci}}{{C}_{i}} & {{C}_{ci}} \\ \end{matrix} \right]. \\ \end{align}$

    Thus, from (10) and (11), the closed-loop system can be rewritten as

    $\left\{ \begin{array}{l} \zeta (k + 1) = {{\tilde A}_i}\zeta (k) - ({{\tilde B}_i} + G{E_{ci}})\psi ({v_c})\;\\ \;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\forall i \in {I_N} \end{array} \right. $

    (12)

    where $v_c = K_i \zeta (k), \;\psi (v_c ) = v_c - sat(v_c ).$

    The objective of this paper is to design the anti-windup compensation gains and the switching law such that the system (6) or (12) is locally asymptotically stable in the origin of the state space and meanwhile the estimation of domain of attraction of the closed-loop system (6) or (12) is maximized.

    The following lemmas will be needed in the development of the main results.

    For a positive definite matrix $P \in {\bf R}^{(n + n_c ) \times (n + n_c )}$ and a scalar ρ > 0, an ellipsoid $\Omega (P,\;\rho )$ is defined as

    $\Omega (P,\;\rho ) = \left\{ {\zeta \in {\bf R}^{n + n_c } :\;\zeta ^{\rm T} P\zeta \le \rho } \right\}.$

    Consider matrices $K_i ,\;H_i \in {\bf R}^{m \times (n + n_c )}$ and define the following polyhedral set:

    $L(K_i ,\;H_i )=\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\nonumber\\ \;\;\;\;\;\;\;\;\;\;\left\{ {\zeta \in {\bf R}^{n + n_c } :\;\left| {(K_i^j - H_i^j )\zeta } \right| \le 1,\;i \in I_N ,\;j \in Q_m } \right\}\nonumber$

    where $K_i^j, \;H_i^j$ are the j-th row of matrices $K_i$ and $H_i$ respectively.

    Lemma 1. [20  21] Consider the function $\psi (v_c )$ defined above. If $\zeta \in L(K_i, \;H_i )$ , then the relation

    $\left\{ \begin{array}{l} {\psi ^{\rm T}}({K_i}\zeta ){J_i}[\psi ({K_i}\zeta )-{H_i}\zeta] \le 0\;\\ \qquad \forall i \in {I_N} \end{array} \right. $

    (13)

    holds for any diagonal and positive definite matrix $J_i \in {\bf R}^{m \times m}$ .

    Lemma 2 (Schur's complements). Given the symmetric matrix $A = \left[{\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ {A_{12}^{\rm T} } & {A_{22} } \\ \end{array}} \right], $ the following statements are equivalent:

    1) $A < 0$

    2) $A_{11} < 0, \;A_{22} - A_{12}^{\rm T} A_{11}^{ - 1} A_{12} < 0$

    3) $A_{22} < 0, \;A_{11} - A_{12} A_{22}^{ - 1} A_{12}^{\rm T} < 0.$

  • In this section, under the given anti-windup compensation gains, sufficient conditions for stability of the closed-loop system (6) or (12) are derived by way of the single Lyapunov function approach.

  • Theorem 1. Suppose that there exist symmetric positive definite matrices $P \in {\bf R}^{(n + n_c ) \times (n + n_c )}$ , matrices $H_i \in {\bf R}^{m \times (n + n_c )},\;E_{ci} \in {\bf R}^{n_c \times m}$ , diagonal positive definite matrices $J_i \in {\bf R}^{m \times m}$ and a set of scalars $\xi_{i} \geq 0$ such that

    $\sum\limits_{i = 1}^N {{\xi _i}} \left[{\begin{array}{*{20}{c}} {\tilde A_i^{\rm T}{P} + {P}{{\tilde A}_i}} & \begin{array}{l}-{P}({{\tilde B}_i} + G{E_{Ci}})+\\ H_i^{\rm T}{J_i} \end{array}\\ * & {-2{J_i}} \end{array}} \right] <0 $

    (14)

    and

    $\left\{ \begin{array}{l} \Omega (P, \;1) \subset L({K_i}, \;{H_i})\;\\ \qquad \forall i \in {I_N}. \end{array} \right. $

    (15)

    Then the closed-loop switched system (6) is asymptotically stable at the origin with $ \Omega ({P}, \;1) $ contained in the domain of attraction under the switching law

    $ \sigma = \arg \min \left\{ \begin{array}{l} {\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]^{\rm T}} \times \\[4mm] \left[{\begin{array}{*{20}{c}} \begin{array}{l} \tilde A_i^{\rm T}{P}+\\ {P}{{\tilde A}_i} \end{array} & \begin{array}{l}-{P}({{\tilde B}_i} + G{E_{Ci}})+\\ H_i^{\rm T}{J_i} \end{array}\\ * & {-2{J_i}} \end{array}} \right] \times \\[6mm] \left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right] \end{array} \right\}. $

    (16)

    Proof.  In view of condition (15), if $\forall \zeta \in \Omega (P, \;1) $ , then $\zeta \in L(K_i, \;H_i ).$ Thus, in view of Lemma 1, for $\forall \zeta \in \Omega (P, \;1)$ it follows that $\psi (K_i \zeta ) = K_i \zeta - sat(K_i \zeta )$ satisfies the sector condition (13).

    Then, Choose the Lyapunov functional candidate for the system (6) as

    $ V (\zeta) = \zeta^{\rm T} P \zeta. $

    (17)

    Then, the time derivative of $V(\zeta)$ along the trajectory of the system (6) is

    $ \begin{align} & \dot{V}(\zeta )={{{\dot{\zeta }}}^{\text{T}}}P\zeta +{{\zeta }^{\text{T}}}P\dot{\zeta }= \\ & \quad \quad {{[{{{\tilde{A}}}_{i}}\zeta -({{{\tilde{B}}}_{i}}+G{{E}_{ci}})\psi ({{v}_{c}})]}^{\text{T}}}P\zeta + \\ & \quad \quad {{\zeta }^{\text{T}}}P[{{{\tilde{A}}}_{i}}\zeta -({{{\tilde{B}}}_{i}}+G{{E}_{ci}})\psi ({{v}_{c}})]. \\ \end{align} $

    Thus, using Lemma 1 and condition (15) gives

    $ \begin{array}{l} \dot V(\zeta ) \le {[{{\tilde A}_i}\zeta-({{\tilde B}_i} + G{E_{ci}})\psi ({K_i}\zeta )]^{\rm T}}P\zeta+ \\ {\rm{ }} \;\;\;\;\;\;\;\;\;\;\;{\zeta ^{\rm T}}P[{{\tilde A}_i}\zeta-({{\tilde B}_i} + G{E_{ci}})\psi ({K_i}\zeta )]-\\ {\rm{ }} \;\;\;\;\;\;\;\;\;\;\; 2{\psi ^{\rm T}}({K_i}\zeta ){J_i}[\psi ({K_i}\zeta )-{H_i}\zeta]=\\ {\rm{ }} \;\;\;\;\;\;\;\;\;\;\;{\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]^{\rm T}}\left[{\begin{array}{*{20}{c}} \begin{array}{l} \tilde A_i^T{P}+\\ {P}{{\tilde A}_i} \end{array} & \begin{array}{l}-{P}({{\tilde B}_i} + G{E_{Ci}})+\\ H_i^{\rm T}{J_i} \end{array}\\ * & {-2{J_i}} \end{array}} \right]\times\\ \qquad \quad \left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]. \end{array} $

    Multiplying (14) from the left by $\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]^{\rm T}$ and then from the right by $\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]$ , we have

    $ \begin{array}{l} \sum\limits_{i = 1}^N {{\xi _i}} {\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]^{\rm T}} \times \\[5mm] \;\;\;\;\;\;\;\left[{\begin{array}{*{20}{c}} {\tilde A_i^{\rm T}{P} + {P}{{\tilde A}_i}} & {\begin{array}{*{20}{l}} {-{P}({{\tilde B}_i} + G{E_{Ci}})}+\\ { H_i^{\rm T}{J_i}} \end{array}}\\ * & {-2{J_i}} \end{array}} \right]\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right] < 0. \end{array} $

    (18)

    Thus, using the switching law (16) results in

    $\begin{array}{l} \dot V(\zeta ) \le {\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]^{\rm T}}\left[{\begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {\tilde A_i^{\rm T}P}+\\ { P{{\tilde A}_i}} \end{array}} & {\begin{array}{*{20}{l}} {-P({{\tilde B}_i} + G{E_{Ci}})}+\\ { H_i^{\rm T}{J_i}} \end{array}}\\ * & {-2{J_i}} \end{array}} \right]\times\\[3mm] \;\; \;\; \;\; \;\; \;\; \;\; \;\; \left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right] <0 \end{array}$

    Additionally, according to (17), adjacent Lyapunov functions at switching points are equal, namely at the switching instant

    ${V_i}(\zeta) = {V_j}(\zeta)$

    which meets with the nonincreasing requirement on any Lyapunov function over the "switched on" time sequence of the corresponding subsystem.

    Therefore, the switched system (6) is asymptotically stable for all initial states $\zeta_{0}\in {\Omega ({P}, \;1)} $ .

  • Theorem 2. Suppose there exist symmetric positive definite matrix $P \in {\bf R}^{(n + n_c ) \times (n + n_c )}$ , matrices $H_i \in {\bf R}^{m \times (n + n_c )},\;E_{ci} \in {\bf R}^{n_c \times m} ,$ a set of scalars $\xi_{i}>0$ and diagonal positive definite matrices $J_i \in {\bf R}^{m \times m}$ satisfying

    $\;\;\;\;\;\;\;\sum\limits_{i=1}^{N}\xi_{i}\begin{array}{l} \left[ {\begin{array}{*{20}c} { - P} & {H_i^{\rm T} J_i } & {\tilde A_i^{\rm T} P } \\ * & { - 2J_i } & { - (\tilde B_i + GE_{ci} )^{\rm T} P } \\ * & * & { - P } \\ \end{array}} \right] < 0 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall i \in I_N \\ \end{array} $

    (19)

    and

    $\left\{ \begin{array}{l} \Omega (P, \;1) \subset L({K_i}, \;{H_i})\;\\ \qquad \forall i \in {I_N}. \end{array} \right. $

    (20)

    Then the closed-loop switched system (12) is asymptotically stable at the origin with $\Omega (P, \;1)$ contained in the domain of attraction under the switching law

    $\begin{array}{l} \;\;\;\;\;\;\sigma (k) = \\ \arg \min \left\{ \begin{array}{l} {\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]^{\rm T}} \times \\[5mm] \left[{\begin{array}{*{20}{c}} {\tilde A_i^{\rm T}P{{\tilde A}_i}-P} & \begin{array}{l}-\tilde A_i^{\rm T}P({{\tilde B}_i} + \\ G{E_{ci}}) + H_i^{\rm T}{J_i} \end{array}\\ * & \begin{array}{l} {({{\tilde B}_i} + G{E_{ci}})^{\rm T}}P \times \\ ({{\tilde B}_i} + G{E_{ci}})-2{J_i} \end{array} \end{array}} \right]\times\\[9mm] \left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right] \end{array} \right\}. \end{array} $

    (21)

    Proof.  By condition (20), if $\forall \zeta \in \Omega (P, 1)$ , then $\zeta \in L(K_i, H_i ).$ Therefore, by Lemma 1, for $\forall \zeta \in \Omega (P, \;1)$ it follows that $\psi (K_i \zeta (k)) = K_i \zeta (k) - sat(K_i \zeta (k))$ satisfies the condition (13).

    We choose the following Lyapunov function candidate for the system (12)

    $V(\zeta (k)) = \zeta ^{\rm T} (k)P\zeta (k). $

    (22)

    The difference of the Lyapunov function candidate (22) along the trajectories of the considered system (12) is given by

    $\begin{array}{l} \Delta V(\zeta(k)) = \zeta ^{\rm T} (k + 1) P \zeta (k + 1) - \zeta ^{\rm T} (k) P \zeta (k)= \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {[\tilde A_i \zeta (k)-(\tilde B_i + GE_{ci} )\psi (K_i \zeta (k))]^{\rm T} } \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; P {[\tilde A_i \zeta (k)-(\tilde B_i + GE_{ci} } )\psi (K_i \zeta (k))] - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \zeta ^{\rm T} (k) P\zeta (k). \\ \end{array} $

    (23)

    Therefore, for $\forall \zeta (k) \in \Omega (P, \;1), $ by using Lemma 1 and condition (20), we have

    $\begin{array}{l} \Delta V(\zeta(k)) \le {[\tilde A_i \zeta (k)-(\tilde B_i + GE_{ci} )\psi (K_i \zeta (k))]^{\rm T} } \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; P {[\tilde A_i \zeta (k)-(\tilde B_i + GE_{ci} } )\psi (K_i \zeta (k))]- \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \zeta ^{\rm T} (k) P\zeta (k) - 2\psi ^{\rm T} (K_i \zeta )J_i [\psi (K_i \zeta )-H_i \zeta] \\ \end{array}$

    or equivalently

    $\begin{array}{l} \Delta V(\zeta(k)) \le \left[\!\!\! {\begin{array}{*{20}c} \zeta \\ \psi \\ \end{array}} \right]^{\rm T} \times\\ \;\;\;\;\;\left[\!\!\! {\begin{array}{*{20}c} \begin{array}{l} \tilde A_i ^{\rm T} P \tilde A_i-P \\ \end{array} & \begin{array}{l}-\tilde A_i^{\rm T} P (\tilde B_i +GE_{ci} )+ \\ H_i^{\rm T} J_i \\ \end{array} \\ * & \begin{array}{l} (\tilde B_i + GE_{ci} )^{\rm T} P (\tilde B_i +\\ GE_{ci} )-2J_i \\ \end{array} \\ \end{array}} \!\!\!\right]\left[\!\!\! {\begin{array}{*{20}c} \zeta \\ \psi \\ \end{array}} \!\!\!\right]. \end{array} $

    (24)

    Then, in view of Lemma 2, (19) is equivalent to

    $\begin{array}{l} \sum\limits_{i=1}^{N}\xi_{i}\left[\!\!\!\!\!{\begin{array}{*{20}c} \begin{array}{l} \tilde A_i ^{\rm T} P \tilde A_i-P \\ \end{array} & \begin{array}{l}-\tilde A_i^{\rm T} P (\tilde B_i +GE_{ci} )+ \\ H_i^{\rm T} J_i \\ \end{array} \\ * & \begin{array}{l} (\tilde B_i + GE_{ci} )^{\rm T} P (\tilde B_i+ \\ GE_{ci} )-2J_i \\ \end{array} \\ \end{array}} \!\!\!\right]<0. \end{array} $

    (25)

    Multiplying (25) from the left by $[\zeta^{\rm T} \;\psi ^{\rm T}]$ and from the right by $[\zeta^{\rm T} \;\psi ^{\rm T}]^{\rm T}$ , we have

    $\begin{array}{l} \sum\limits_{i=1}^{N}\xi_{i}\left[{\begin{array}{*{20}c} \zeta \\ \psi \\ \end{array}} \right]^{\rm T}\times\\ \left[\!\!\!\!\!{\begin{array}{*{20}c} \begin{array}{l} \tilde A_i ^{\rm T} P \tilde A_i-P \\ \end{array} & \begin{array}{l}-\tilde A_i^{\rm T} P (\tilde B_i +GE_{ci} ) +\\ H_i^{\rm T} J_i \\ \end{array} \\ * & \begin{array}{l} (\tilde B_i + GE_{ci} )^{\rm T} P (\tilde B_i +\\ GE_{ci} )-2J_i \\ \end{array} \\ \end{array}} \!\!\!\right]\left[{\begin{array}{*{20}c} \zeta \\ \psi \\ \end{array}} \!\!\!\right]<0. \end{array} $

    (26)

    Therefore, from the switching law (21) and (26), it is easy to have

    $\Delta V(\zeta(k)) < 0. $

    (27)

    Thus, the state trajectory of the considered system (12) starting from inside $\Omega (P_, \;1)$ will remain inside it. Furthermore, the closed-loop system (12) is asymptotically stable at the origin with $ \Omega (P_i, \;1)$ contained in the domain of attraction under the switching law (21).

  • In order to guarantee that the considered closed-loop system (6) or (12) is asymptotically stable and meanwhile the estimated domain of attraction of the system (6) or (12) is maximized under the switching law, we will give the methods of designing the anti-windup compensation gains.

  • Theorem 3. If there exist symmetric positive definite matrices $X \in {\bf R}^{(n + n_c ) \times (n + n_c )}$ , matrices $M_i \in {\bf R}^{m \times (n + n_c )} , N_i \in {\bf R}^{n_c\times m}$ , diagonal positive definite matrices $S_i \in {\bf R}^ {m \times m}$ and a set of scalars $\xi_{i}>0$ such that the following conditions hold:

    $ \begin{align} & \sum\limits_{i=1}^{N}{{{\xi }_{i}}}\left[ \begin{matrix} X\tilde{A}_{i}^{\text{T}}+{{{\tilde{A}}}_{i}}X & -{{{\tilde{B}}}_{i}}{{S}_{i}}-G{{N}_{i}}+M_{i}^{\text{T}} \\ * & -2{{S}_{i}} \\ \end{matrix} \right]<0 \\ & \quad \quad \quad \quad \quad \quad \quad \forall i\in {{I}_{N}} \\ \end{align} $

    (28)

    and

    $\begin{array}{l} \left[{\begin{array}{*{20}{c}} X & {XK_i^{j{\rm T}}-M_i^{j{\rm T}}}\\ * & 1 \end{array}} \right] \ge 0\\ \;\;\;\;\;\;\forall i \in {I_N}, j \in {Q_m} \end{array} $

    (29)

    where $K_i^j, \;M_i^j$ are the j-th row of matrices $K_i$ and $M_i$ respectively, then the closed-loop saturated switched system (6) with anti-windup compensation gains $E_{ci} = N_i S_i^{ - 1}$ is asymptotically stable at the origin with $\Omega (X^{ - 1}, \;1)$ contained in the domain of attraction under the state dependent switching law

    $ \sigma = \arg \min \left\{ {\begin{array}{*{20}{l}} {{{\left[ {\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]}^{\rm T}} \times }\\[4mm] {\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {\tilde A_i^{\rm T}{X^{ - 1}}}+\\ { {X^{ - 1}}{{\tilde A}_i}} \end{array}}\!\!\!\! & \!\!\!\!{\begin{array}{*{20}{l}} \begin{array}{l} - {X^{ - 1}}({{\tilde B}_i}+\\ G{E_{Ci}})+{H_i^{\rm T}{J_i}} \end{array}\\ \end{array}}\\ * & { - 2{J_i}} \end{array}} \right]}\times\!\!\\[6mm] { \left[ {\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]} \end{array}} \right\}. $

    (30)

    Proof. Pre-and post-multiplying both sides of inequality (14) by the matrix

    $ \left[{\begin{array}{*{20}c} {P^{-1} } & 0 \\ * & {J_i^{-1} } \\ \end{array}} \right] $

    we have

    $ \sum\limits_{i = 1}^N {{\xi _i}} \left[{\begin{array}{*{20}{c}} {{P^{-1}}\tilde A_i^{\rm T} + {{\tilde A}_i}{P^{-1}}} & \begin{array}{l}-({{\tilde B}_i} + G{E_{Ci}})J_i^{ - 1}+\\ {P^{ - 1}}H_i^{\rm T} \end{array}\\ * & { - 2J_i^{ - 1}} \end{array}} \right] < 0 $

    and letting $X = P^{ - 1}, \;S_i = J_i^{ - 1}, \;M_i = H_i X, \;N_i = E_{ci} S_i$ , then we have

    $\sum\limits_{i = 1}^N {{\xi _i}} \left[{\begin{array}{*{20}{c}} {X\tilde A_i^{\rm T} + {{\tilde A}_i}X} & {-{{\tilde B}_i}{S_i}-G{N_i} + M_i^{\rm T}}\\ * & {-2{S_i}} \end{array}} \right] < 0$

    which is exactly (28) in Theorem 3.

    Applying a similar method to the inequality (29), we can also obtain

    $ \left[\begin{array}{cc} 1 & K_{i}^{j}-H_{i}^{j}\\ \ast & P\\ \end{array} \right]\geq 0 $

    (31)

    where $K_{i}^{j}, \;H_{i}^{j}$ denotes the $j$ -th row of $K_{i}$ and $H_{i}$ , respectively.

    Then, we can show that $\Omega(P, \;1)\subset L\left(K_{i}, \;H_{i}\right)$ is implied by (31). In fact, since $\zeta^{{\rm T}}P\zeta\leq 1$ and $(K_{i}^{j}-H_{i}^{j})P^{-1}(K_{i}^{j}-H_{i}^{j})^{{\rm T}}\leq 1$ , it holds that

    $\begin{array}{l} 2{\zeta ^{\rm T}}{(K_i^j - H_i^j)^{\rm T}} \le {\zeta ^{\rm T}}P\zeta + \\(K_i^j - H_i^j){P^{ - 1}}{(K_i^j - H_i^j)^{\rm T}} \le 2 \end{array}$

    therefore, (31) implies $\Omega(P_{i}, \;1)\subset L\left(H_{i}, K_{i}\right)$ .

    Since $P=X^{-1}$ , the switching rule (16) is the same as (30) of the theorem 1.

    By using the single Lyapunov functions method, the sufficient condition is given in Theorem 3 that allows to design the compensation gains $E_{ci}$ such that the closed-loop system is asymptotically stable at the origin with $\Omega (X^{ - 1}, \;1)$ . However, our objective is to design the anti-windup compensation gains which maximize the estimation of domain of attraction of the closed-loop system (6) which means that the set $ \Omega (X^{ - 1}, \;1)$ is maximized. In general, there are two main methods to measure the largeness of a set. The first strategy is to measure the largeness of a set by its volume. The second strategy is to take its shape into consideration. In this paper, by adopting the latter method, the largeness of the set is measured with respect to a given shape reference set $X_R$ .

    Let $X_R \subset {\bf R}^{n + n_c }$ be a prescribed bounded convex set containing the origin. For a set $\Xi \subset {\bf R}^{n + n_c }$ which contains the origin, define [17]:

    $\alpha _R \left( \Xi \right) = \sup \left\{ {\alpha > 0:\;\alpha X_R \subset \Xi } \right\}.$

    Obviously, if $\alpha _R (\Xi ) \ge 1$ , then $X_R \subset \Xi$ . Thus, $\alpha _R (\Xi )$ provides a kind of measure of the estimated domain of attraction. Two typical types of $X_R$ are the ellipsoid

    $X_R = \left\{ {\zeta \in {\bf R}^{n + n_c } :\;\zeta ^{\rm T} {\bf R}\zeta \le 1,\;R > 0} \right\}$

    and the polyhedron

    $X_R = cov\left\{ {\zeta _1, \;\zeta _2, \; \cdots, \;\zeta _l } \right\}$

    where $\zeta _1, \;\zeta _2, \; \cdots, \;\zeta _l$ are a priori given points in ${\bf R}^{n + n_c }$ .

    As a result, the problem of maximizing $\Omega(X^{-1}, \;1)$ with respect to a given shape reference set $X_R$ can be formulated as the following constrained optimization problem:

    $\begin{array}{*{20}c} {} \hfill & {\mathop {\sup }\limits_{X, \;M_i, \;N_i, \;S_i, \;\xi _{i} } \alpha, } \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;{\rm s.t.}\;(a)\;\alpha X_R \subset \Omega (X^{ - 1}, \;1)} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;(b)\;{\rm inequality}\;(28), \;\forall i \in I_N} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;(c)\;{\rm inequality}\;(29), \;\forall i \in I_N, \; j \in Q_m.} \hfill \\ \end{array} $

    (32)

    If $X_R$ is an ellipsoid, then, (a) is equivalent to

    $\left[{\begin{array}{*{20}c} {\dfrac{1}{{\alpha ^2 }}R} & I \\ I & {X } \\ \end{array}} \right] \ge 0 . $

    (33)

    If $X_R$ is a polyhedron of the form, then, (a) is equivalent to

    $\left[{\begin{array}{*{20}c} {\dfrac{1}{{\alpha ^2 }}} & {\zeta _q^{\rm T} } \\ {\zeta _q } & {X } \\ \end{array}} \right] \ge 0, \;\forall q \in [1, \;l] . $

    (34)

    Let $\gamma = \frac{1}{{\alpha ^2 }}$ . If $X_R$ is an ellipsoid, the optimization problem (34) can be rewritten as

    $\begin{array}{*{20}c} {} \hfill & {\mathop {\inf }\limits_{X, \;M_i, \;N_i, \;S_i, \;\xi _{i} } \gamma } \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;{\rm s.t.}\;(a)\;\left[{\begin{array}{*{20}c} {\gamma R} & I \\ I & {X } \\ \end{array}} \right] \ge 0} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(b)\;{\rm inequality}\;(28), \;\forall i \in I_N} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(c)\;{\rm inequality}\;(29), \;\forall i \in I_N, \; j \in Q_m .} \hfill \\ \end{array} $

    (35)

    If $X_R$ is a polyhedron of the form, we just need to replace (a) in (35) with

    $\left[{\begin{array}{*{20}c} \gamma & {\zeta _q^{\rm T} } \\ {\zeta _q } & {X } \\ \end{array}} \right] \ge 0, \;\forall q \in [1, \;l]. $

    (36)
  • Theorem 4.  Suppose that there exist symmetric positive definite matrix $X \in {\bf R}^{(n + n_c ) \times (n + n_c )}$ , matrices $M_i \in {\bf R}^{m \times (n + n_c )} ,$ $N_i \in {\bf R}^{n_c \times m}$ , diagonal positive definite matrices $S_i \in {\bf R}^{m \times m}$ and a set of scalars $\xi_{i} > 0$ such that the following LMIs hold:

    $\begin{array}{l} \sum\limits_{i=1}^{N}\xi_{i}\left[{\begin{array}{*{20}c} {-X} & {M_i^{\rm T} } & {X \tilde A_i^{\rm T} } \\ * & {-2S_i } & {-S_i \tilde B_i^{\rm T} - N_i^{\rm T} G^{\rm T} } \\ * & * & { - X } \end{array}} \right] < 0 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall i \in I_N \end{array} $

    (37)

    and

    $\begin{array}{l} \left[{\begin{array}{*{20}c} {X } & {X K_i^{j{\rm T}}-M_i^{j{\rm T}} } \\ * & 1 \\ \end{array}} \right] \ge 0 \\ \;\;\;\;\;\;\; \forall i \in I_N, \;j \in Q_m, \\ \end{array} $

    (38)

    where $K_i^j, \;M_i^j$ are the j-th row of matrices $K_i$ and $M_i$ respectively. Then the closed-loop switched system (12) with anti-windup compensation gains $E_{ci} = N_i S_i^{ - 1}$ is asymptotically stable at the origin with $ \Omega (X^{ - 1}, \;1)$ contained in the domain of attraction under the switching law

    $ \begin{array}{l} \sigma (k) = \arg \min \\ \left\{ \!\!\!\begin{array}{l} \begin{array}{*{20}{l}} \!\!\!{{{\left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}} \right]}^{\rm T}}} \end{array} \times \\[4mm] \left[{\begin{array}{*{20}{c}} {\begin{array}{*{20}{l}} {\tilde A_i^{\rm T}{X^{-1}}{{\tilde A}_i}}-\\ {{X^{-1}}} \!\!\!\end{array}}\!\!\!\!\! & {\begin{array}{*{20}{l}} {-\tilde A_i^{\rm T}{X^{-1}}({{\tilde B}_i} + G{N_i}S_i^{-1})}+\\ {{X^{-1}}M_i^TS_i^{-1}} \end{array}}\\ * & {\begin{array}{*{20}{l}} {{{({{\tilde B}_i} + G{N_i}S_i^{ - 1})}^{\rm T}}{X^{ - 1}} \times }\\ {({{\tilde B}_i} + G{N_i}S_i^{ - 1}) - 2S_i^{ - 1}} \end{array}} \end{array}} \!\!\!\!\right]\times\\[8mm] \left[{\begin{array}{*{20}{c}} \zeta \\ \psi \end{array}}\!\!\!\! \right] \end{array} \right\}. \end{array} $

    (39)

    Proof. Pre-and post-multiplying both sides of inequality (19) by the matrix

    $ \left[ \begin{matrix} P-1 & 0 & 0 \\ * & J_{i}^{-1} & 0 \\ * & * & {{P}^{-1}} \\ \end{matrix} \right] $

    and letting $X = P^{ - 1}, \;S_i = J_i^{ - 1}, \;M_i = H_i X, \;N_i = E_{ci} S_i$ , we have

    $ \begin{array}{l} \sum\limits_{i=1}^{N}\xi_{i}\left[{\begin{array}{*{20}c} {-X} & {M_i^{\rm T} } & {X \tilde A_i^{\rm T} } \\ * & {-2S_i } & {-S_i \tilde B_i^{\rm T} - N_i^{\rm T} G^{\rm T} } \\ * & * & { - X } \end{array}} \right] < 0 \end{array} $

    which is exactly (37) in Theorem 4.

    In view of [18], it is easy to see that the condition (19) is guaranteed by

    $P - (K_i^j - H_i^j )^{\rm T} (K_i^j - H_i^j ) \ge 0. $

    (40)

    Then from the Lemma 2, (29) is equivalent to

    $\left[{\begin{array}{*{20}c} {P } & {(K_i^j-H_i^j )^{\rm T} } \\ * & 1 \\ \end{array}} \right] \ge 0. $

    (41)

    Applying a similar method to the inequality (41), it can be transformed into (38) equivalently. Additionally, $P=X^{-1}$ , the switching rule (39) is the same as (21) of Theorem 2.

    Then, the problem of the maximal $ \Omega (X^{ - 1}, \;1)$ with respect to a given shape reference set $X_R$ can be presented as the following constrained optimization problem:

    $\begin{array}{*{20}c} {} \hfill & {\mathop {\sup }\limits_{X, \;M_i, \;N_i, \;S_i, \;\xi_{i} } \alpha, } \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;{\rm s.t.}\;(a)\;\alpha X_R \subset \Omega (X^{ - 1}, \;1), \;\forall i \in I_N } \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;(b)\;{\rm inequality}\;(37), \;\forall i \in I_N \:} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;(c)\;{\rm inequality}\;(38), \;\forall i \in I_N, \; j \in Q_m .} \hfill \\ \end{array} $

    (42)

    If we choose $X_R$ as an ellipsoid, then (a) is equivalent to

    $\left[{\begin{array}{*{20}c} {\dfrac{1}{{\alpha ^2 }}R} & I \\ I & {X } \\ \end{array}} \right] \ge 0, \;\forall i \in I_N . $

    (43)

    If we chooose $X_R$ as a polyhedron of the form, then, (a) is equivalent to

    $\left[{\begin{array}{*{20}c} {\dfrac{1}{{\alpha ^2 }}} & {\zeta _q^{\rm T} } \\ {\zeta _q } & {X } \\ \end{array}} \right] \ge 0, \;\forall q \in [1, \;l], \; i \in I_N . $

    (44)

    Let $\gamma = \frac{1}{{\alpha ^2 }}$ . If $X_R$ is an ellipsoid, the optimization problem (42) can be rewritten as

    $\begin{array}{*{20}c} {} \hfill & {\mathop {\inf }\limits_{X, \;M_i, \;N_i, \;S_i, \;\xi _{i} } \gamma, } \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;{\rm s.t.}\;(a)\;\left[{\begin{array}{*{20}c} {\gamma R} & I \\ I & {X } \\ \end{array}} \right] \ge 0} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(b)\;{\rm inequality}\;(37), \;\forall i \in I_N \:} \hfill \\ {} \hfill & {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(c)\;{\rm inequality}\;(38), \;\forall i \in I_N, \; j \in Q_m .} \hfill \\ \end{array} $

    (45)

    If $X_R$ is a polyhedron of the form, we only need to replace (a) in (25) with

    $\left[{\begin{array}{*{20}c} \gamma & {\zeta _q^{\rm T} } \\ {\zeta _q } & {X } \\ \end{array}} \right] \ge 0, \;\forall q \in [1, \;l], \; i \in I_N . $

    (46)

    Remark 1. If $\psi=0$ , namely without actuator saturation, then the switching law (30) or (39) will be reduced to the general switching law of switched system in the absence of actuator saturation. Therefore, these results can be viewed as an extension from the normal switched systems to the switched systems subject to actuator saturation.

    Remark 2. If the parameters $\xi_{i}$ are given in advance, the anti-windup compensation gains and the estimation of domain of attraction are formulated and solved as a set of linear matrix inequality (LMI) optimization problem.

  • In the section, we give the following examples to show the validity of the results in Section 4.

    Example 1.

    $\left\{ \begin{array}{l} \dot x = {A_i}x + {B_i}sat({v_c})\\ y = {C_i}x \end{array} \right. $

    (47)

    and the dynamic controllers with the anti-windup terms are given as

    $\left\{ \begin{array}{l} \mathop {{x_c}}\limits^. = {A_{ci}}{x_c} + {B_{ci}}{C_i}x+\\ ~~~~~~{E_{ci}}(sat({v_c}) - {v_c})\\ {v_c} = {C_{ci}}{x_c} + {D_{ci}}{C_i}x \end{array} \right. $

    (48)

    where $\sigma (k) \in I_2 = \{ 1, \;2\}$

    $\begin{align} & {{A}_{1}}=\left[ \begin{matrix} 0.2 & 0 \\ 1.4 & -0.9 \\ \end{matrix} \right],\ {{A}_{2}}=\left[ \begin{matrix} -1.3 & -0.3 \\ 0 & 0.6 \\ \end{matrix} \right] \\ & \quad \quad {{B}_{1}}=\left[ \begin{matrix} 0.5 \\ -0.7 \\ \end{matrix} \right],\ {{B}_{2}}=\left[ \begin{matrix} -0.5 \\ 0.8 \\ \end{matrix} \right] \\ & \quad \quad {{C}_{1}}={{\left[ \begin{matrix} -0.3 \\ 0.6 \\ \end{matrix} \right]}^{\text{T}}},\ {{C}_{2}}={{\left[ \begin{matrix} 0.3 \\ -0.8 \\ \end{matrix} \right]}^{\text{T}}} \\ & {{A}_{c1}}=\left[ \begin{matrix} -0.9 & 0.3 \\ -0.6 & 0.7 \\ \end{matrix} \right],\ {{A}_{c2}}=\left[ \begin{matrix} 0.6 & -0.3 \\ 0.3 & -0.9 \\ \end{matrix} \right] \\ & \quad \quad {{B}_{c1}}=\left[ \begin{matrix} 0.6 \\ -0.8 \\ \end{matrix} \right],\ {{B}_{c2}}=\left[ \begin{matrix} -0.6 \\ 1.5 \\ \end{matrix} \right] \\ & \quad \quad {{C}_{c1}}={{\left[ \begin{matrix} 0.8 \\ -0.7 \\ \end{matrix} \right]}^{\text{T}}},\ {{C}_{c2}}={{\left[ \begin{matrix} -0.6 \\ 0.9 \\ \end{matrix} \right]}^{\text{T}}} \\ & \quad \quad \quad \quad {{D}_{c1}}=15,\ {{D}_{c2}}=6.8. \\ \end{align}$

    Then, the anti-windup compensation gains $E_{ci}$ will be designed to stabilize the switched system (47) -(48) with actuator saturation with the maximized estimation of domain of attraction in terms of the proposed method.

    Let ${\bf R} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right]$ . We Solve the optimization problem (35), then the optimal solutions is obtained as follows:

    $\gamma = 0.103\, 2, S_1 = 175.482\, 5, \;S_2 = 53.269\, 1$

    $X = \left[{\begin{array}{*{20}c} {11.200\, 0} & {-2.900\, 0} & {-4.200\, 0} & {3.700\, 0} \\ * & {95.100\, 0} & {18.800\, 0} & {-9.300\, 0} \\ * & * & {135.700\, 0} & {23.5000} \\ * & * & * & {672.300\, 0} \\ \end{array}} \right]$

    $P = \left[{\begin{array}{*{20}{c}} {0.091\, 0} & {0.002\, 2} & {0.002\, 6} & {-0.000\, 6}\\ * & {0.010\, 9} & {-0.001\, 5} & {0.000\, 2}\\ * & * & {0.007\, 7} & {-0.000\, 3}\\ * & * & * & {0.001\, 5} \end{array}} \right]$

    $N_1 = \left[{\begin{array}{*{20}c} {-119.302\, 8} \\ {352.734\, 2} \\ \end{array}} \right], \;N_2 = \left[{\begin{array}{*{20}c} {-78.476\, 9} \\ {-35.592\, 6} \\ \end{array}} \right]$

    $M_1 = \left[{\begin{array}{*{20}c} { 23.437\, 3} & {145.902\, 2} & {63.928\, 1} & {-124.664\, 3} \\ \end{array}} \right]$

    $M_2 = \left[{\begin{array}{*{20}c} {-17.338\, 3} & {-12.619\, 3} & {48.973\, 9} & {164.549\, 3} \\ \end{array}} \right]$

    $E_{c1} = \left[{\begin{array}{*{20}c} {-1.389\, 5} \\ {2.394\, 3} \\ \end{array}} \right], \;E_{c2} = \left[{\begin{array}{*{20}c} {-2.982\, 2} \\ { 4.278\, 9} \\ \end{array}} \right].$

    Fig. 1 gives the state response of the considered system (47) -(48) under the switching law and anti-windup compensation designed gains with the initial state $x(0) = [-0.5\; 0.7]^\mathrm{T}$ . The response of controller state with $x_c (0) = [0.5{\rm{ }}-0.5]^\mathrm{T}$ is shown in Fig. 2. Fig. 3 provides the control input signal of the switched system (47)-(48).

    Figure 1.  The state response of system (47) -(48)

    Figure 2.  The response of controller state of system (47) -(48)

    Figure 3.  The input signal of system (47) -(48)

    On the other hand, it is worth noticing that if we let $E_{c1} = E_{c2} = 0$ , the obtained optimal solution is $\gamma = 12.193\, 2$ , which indicates that the anti-windup compensation gains can enlarge the estimation of domain of attraction of the considered system.

    Example 2.

    $\left\{ \begin{array}{l} x(k{\rm{ }} + {\rm{ }}1){\rm{ }} = {\rm{ }}{A_i}{\rm{ }}x(k){\rm{ }} + {\rm{ }}{B_i}{\rm{ }}sat({v_c}{\rm{ }}(k))\\ y(k){\rm{ }} = {C_{i{\rm{ }}}}x(k) \end{array} \right. $

    (49)

    and the dynamic controllers with the anti-windup terms are given as

    $\left\{ \begin{array}{l} {x_{c{\rm{ }}}}(k{\rm{ }} + {\rm{ }}1){\rm{ }} = {\rm{ }}{A_{ci}}{\rm{ }}{x_{c{\rm{ }}}}(k){\rm{ }} + {\rm{ }}{B_{ci}}{\rm{ }}{C_i}{\rm{ }}x(k)+\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {\rm{ }}{E_{ci}}(sat({v_c}{\rm{ }}(k)){\rm{ }} - {\rm{ }}{v_{c{\rm{ }}}}(k))\\ {v_c}{\rm{ }}(k){\rm{ }} = {\rm{ }}{C_{ci}}{\rm{ }}{x_c}{\rm{ }}(k){\rm{ }} + {\rm{ }}{D_{ci}}{\rm{ }}{C_{i{\rm{ }}}}x(k) \end{array} \right. $

    (50)

    where $\sigma (k) \in I_2 = \{ 1, \;2\}$ ,

    $\begin{align} & {{A}_{1}}=\left[ \begin{matrix} 3 & 2 \\ 1.1 & 4 \\ \end{matrix} \right],\ {{A}_{2}}=\left[ \begin{matrix} -2 & -2 \\ -0.2 & -0.6 \\ \end{matrix} \right] \\ & \quad \quad {{B}_{1}}=\left[ \begin{matrix} 0.8 \\ 1.53 \\ \end{matrix} \right],\ {{B}_{2}}=\left[ \begin{matrix} -0.1 \\ -0.5 \\ \end{matrix} \right] \\ & \quad \quad {{C}_{1}}={{\left[ \begin{matrix} 0.1 \\ 0.26 \\ \end{matrix} \right]}^{\text{T}}},\ {{C}_{2}}={{\left[ \begin{matrix} -0.64 \\ 0.1 \\ \end{matrix} \right]}^{\text{T}}} \\ & {{A}_{c1}}=\left[ \begin{matrix} -0.1 & 0.3 \\ -0.4 & 4.3 \\ \end{matrix} \right],\ {{A}_{c2}}=\left[ \begin{matrix} -0.3 & -0.1 \\ -0.1 & -0.4 \\ \end{matrix} \right] \\ & \quad \quad {{B}_{c1}}=\left[ \begin{matrix} 0.4 \\ 0.3 \\ \end{matrix} \right],\ {{B}_{c2}}=\left[ \begin{matrix} 0.5 \\ -2 \\ \end{matrix} \right] \\ & \quad \quad {{C}_{c1}}={{\left[ \begin{matrix} -0.7 \\ 0.5 \\ \end{matrix} \right]}^{\text{T}}},\ {{C}_{c2}}={{\left[ \begin{matrix} 0.4 \\ 0.2 \\ \end{matrix} \right]}^{\text{T}}} \\ & \quad \quad \quad \quad {{D}_{c1}}=24,\ {{D}_{c2}}=2.35. \\ \end{align}$

    It is easy to check that the closed-loop system (49) and (50) without actuator saturation is asymptotically stable at the origin from the conditions (37) and (38). Now, the anti-windup compensation gains $E_{ci}$ are designed to stabilize the system (49) and (50) subject to actuator saturation with the maximal estimation of domain of attraction under the designed switching law.

    Let ${\pmb R} = \left[{\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}} \right].$ By solving the optimization problem (45), we obtain the optimal solutions as follows:

    $\alpha = 19.257\, 0, S_1 = 176.502\, 3, \;S_2 = 25.298\, 4, $

    $X = 10^{3}\times\left[{\begin{array}{*{20}c} {0.021\, 8} & {-0.039\, 2} & {-0.004\, 6} & {0.192\, 0} \\ * & {0.075\, 3} & {0.005\, 5} & {-0.331\, 0} \\ * & * & {0.039\, 3} & {-0.195\, 5} \\ * & * & * & {2.335\, 4} \\ \end{array}} \right]$

    $P = \left[{\begin{array}{*{20}c} {8.208\, 9} & {1.095\, 3} & {-3.050\, 8} & {-0.774\, 9} \\ * & {0.289\, 8} & {-0.268\, 4} & {-0.071\, 4} \\ * & * & {1.311\, 3} & { 0.322\, 5} \\ * & * & * & {0.081\, 0} \\ \end{array}} \right]$

    $N_1 = \left[{\begin{array}{*{20}c} {-3.627\, 6} \\ {42.097\, 2} \\ \end{array}} \right], \;N_2 = \left[{\begin{array}{*{20}c} {16.020\, 9} \\ { 197.315\, 9} \\ \end{array}} \right]$

    $M_1 = \left[{\begin{array}{*{20}c} {-90.022\, 1} & {199.539\, 7} & {-101.969\, 6} & {-273.032\, 0} \\ \end{array}} \right]$

    $M_2 = \left[{\begin{array}{*{20}c} {-1.420\, 2} & {4.695\, 2} & {-15.535\, 2} & {54.867\, 0} \\ \end{array}} \right]$

    $E_{c1} = \left[{\begin{array}{*{20}c} {-0.020\, 6} \\ {0.238\, 5} \\ \end{array}} \right], \;E_{c2} = \left[{\begin{array}{*{20}c} {0.633\, 3} \\ { 7.799\, 5} \\ \end{array}} \right].$

    The state response of the considered system (49) -(50) under the switching law and anti-windup compensation designed gains with the initial state $x(0) = [2\;-1]^\mathrm{T}$ is shown in Fig. 4. Fig. 5 gives the response of controller state with $x_c (0) = [-1{\rm{ }}-1]^\mathrm{T}.$ The input signal is depicted in Fig. 6.

    Figure 4.  The state response of system (49) -(50)

    Figure 5.  The response of controller state of system (49) -(50)

    Figure 6.  The input signal of system (49) -(50)

    In the same way, let $E_{c1} = E_{c2} = 0$ , the obtained optimal solution is only $\alpha = 0.089\, 7$ . This shows that the anti-windup compensation gains are able to enlarge the estimation of domain of attraction of the closed-loop system obviously.

  • The stability analysis and anti-windup design problem has been studied for two switched linear systems subject to actuator saturation in this paper. We introduce the single Lyapunov function method and a sector condition to the design of the anti-windup compensation gains which aim at maximizing the estimation of domain of attraction of the closed-loop systems. Then, the problem of designing the anti-windup compensation gains and the switching law is formulated and solved as a convex optimization problem with a set of LMI constraints.

    It should be noted that due to the complexity of dealing with simultaneous switching and actuator saturation nonlinearity, the proposed method is only suitable for linear switched systems with saturating actuators. Thus, how to design a switching law and the anti-windup compensation gains to improve the performance of nonlinear switched systems is a challenging issue which deserves further study in the future.

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