Non-fragile guaranteed cost control of nonlinear switched systems with actuator saturation

Abstract

This paper considers a non-fragile guaranteed cost control problem for a class of nonlinear switched systems with actuator saturation. For nonlinear switched systems, we derive some sufficient conditions to satisfy simultaneously the stabilization and the performance index of guaranteed cost control. To ensure the asymptotic stability of the nonlinear switched systems and minimize the upper bound of cost function, a switching law and non-fragile state feedback controllers are designed using the multiple Lyapunov functions method. On this basis, an optimization problem is solved by adopting linear matrix inequality (LMI) constraints, and the minimum upper bound of cost function is determined. At the end of the paper, a numerical example is given to prove the effectiveness of the proposed method.

Keywords

Actuator saturation nonlinear switched systems non-fragile multiple Lyapunov functions LMI guaranteed cost control

Introduction

As a special case of hybrid system, a switched system is composed of several continuous-time or discrete-time subsystems and a switching law determining which subsystem can be activated at any time. In practical engineering, many systems, such as traffic management systems, chemical process, and computer network control systems, can be modeled as a switched system. In the process of switching, various performance indexes of the switching system become particularly important, so the research methods of each performance index have attracted extensive attention recently (Wang et al., 2021; Yang et al., 2021; Zhao et al., 2020a, 2020b; Zong et al., 2021). For example, in Aleksandrov (2021) and Gao et al. (2019), the method of common Lyapunov function used to study the stability of switched systems is proposed. The common Lyapunov function is difficult to obtain, so it is seldom applied in practical systems. However, there is more than common Lyapunov function method to study the switched system. The single Lyapunov function method is used in Zhang et al. (2011) to investigate the anti-interference of the system, and in Song and Zhai (2018) to study the practical output tracking of the system. In addition, there are the average dwell-time method and the multiple Lyapunov functions method, which are, respectively, used in Ma and Cai (2016) and Göksu and Başer (2021) and Wang et al. (2020) and Zhang (2019) to study the stability of various kinds of switched systems. These two methods are widely recognized as the more practical approaches for studying switched system.

What is more, actuator saturation is a common phenomenon in the control system. The reason is that the physical properties of the actuator determine its own saturation characteristics (Zhang et al., 2020). Actuator saturation will affect the performance of all aspects of the system and even affect the stability of the system. If the actuator saturation occurs in the switched system, the original switching law may be invalid, resulting in the failure of normal switching between subsystems. Therefore, switched systems with actuator saturation have become a hot research problem in recent years, and some achievements have been made (Li et al., 2020; Luo et al., 2020; Wang and Zhao, 2016; You et al., 2019; Zhang et al., 2012). It boils down to two ways of dealing with saturation. The first approach is to consider actuator saturation at the beginning of the control design and then design a linear controller to stabilize the system. The second approach is to ignore the actuator saturation, design a linear controller to meet the performance index in the first stage of control design, and then add an anti-windup compensator to reduce the influence of saturation. The main purpose of these two strategies is to obtain a larger estimation of the domain of attraction in the presence of saturation.

Although the researchers in the above literature have made in-depth exploration of the switched system with actuator saturation, they have all ignored a situation that the parameters of the controller may change. The parameters may be affected by the limitation of the computer word length, the aging of components, the calculation of truncation error, and so on. As Zhang et al. (2021) pointed out, if the controller of a closed-loop system is uncertain or the parameters of the controller change subtly, the stability and other characteristics of the system will be dramatically affected. Such a controller is said to be fragile or inelastic. To overcome this vulnerability, scholars have done a lot of research. Based on linear matrix inequality (LMI) and Lyapunov functional methods, Kavikumar et al. (2019) researched the reliable non-fragile H∞ control design problem for a class of discrete-time interval-valued fuzzy systems with actuator faults. Hu et al. (2013) researched the problem of non-fragile $H_{2}$ reliable control for switched linear systems with actuator faults. Li et al. (2018) discussed the non-fragile reliable control for positive switched systems with actuator faults by proposing linear co-positive Lyapunov functions with a linear programming approach.

Although scholars have done a lot of research on non-fragile control, almost no scholar has considered the non-fragile control of the nonlinear switched system with actuator saturation. The reason is that the interaction between switching and actuator saturation makes it difficult to stabilize the system. If the vulnerability of the controller is also taken into account, the performance analysis of various aspects of the system will also be more complex. However, it is well known that actuator saturation and vulnerability of the controller are often simultaneously encountered in almost all practical engineering systems, which are also a significant cause of instability and poor performance. Thus, it is of great significance to consider the non-fragile control problem of switched systems subject to actuator saturation. However, to the best of our knowledge, almost no results on the non-fragile control have been reported in the existing literature for switched systems with actuator saturation. This is the motivation for this study.

Based on the above idea, the issue of non-fragile guaranteed cost control for a class of nonlinear switched systems with actuator saturation is studied in this paper. First, for the closed-loop system, the sufficient conditions are derived to ensure that the system is non-fragile stabilization and satisfies the guaranteed cost control performance index by adopting the method of multiple Lyapunov functions. Then, we come up with the switching law and the non-fragile state feedback control law which can minimize the upper bound of the cost function and solve it as an optimization problem with a set of matrix inequality constraints. Finally, a concrete numerical example is given to demonstrate the effectiveness of the design procedure.

Problem formulation and preliminaries

Consider a nonlinear switched system with actuator saturation

{\begin{matrix} \dot{x} = A_{σ} x + B_{σ} s a t (u_{σ}) + E_{σ} f_{σ} (x) \\ x (0) = x_{0} \end{matrix}

(1)

where $x \in R^{n}$ is the state vector, $u \in R^{m}$ is the control input vector, $f_{σ} (x)$ is an unknown nonlinear function. $A_{σ}, B_{σ}, E_{σ}$ are constant matrices with appropriate dimension. $σ : [0, + \infty) \to I_{N} = {1, 2, \dots, N}$ serves as a switching signal, which is a piecewise constant function dependent on time or state. $σ = i$ means the ith subsystem is activated. $sat : R^{m} \to R^{m}$ is the standard vector-valued saturation function defined as

\begin{matrix} sat (u) = [sat (u^{1}), \dots, sat (u^{m})]^{T} \\ sat (u^{j}) = sign (u^{j}) min {1, | u^{j} |} \\ \forall j \in Q_{m} = {1, \dots, m} \end{matrix}

(2)

Obviously, it is no loss of generality to assume the unit-saturation limit since the nonstandard saturation function can always be obtained by altering the matrix $B$ with the appropriate transformation, and it is no harm to slightly abuse the symbol $sat (•)$ to represent scalar and vector saturation functions.

Considering the cost function of system (1), it can be expressed as

J = \int_{0}^{\infty} [x^{T} (t) Q x (t) + s a t^{T} (u (t)) R s a t (u (t))] d t

(3)

where $Q$ and $R$ are given as positive-definite weighting matrices.

The following definitions are made for system (1).

Definition 1 (Hakimzadeh and Ghaffari, 2020). There are $N$ known constant matrices $G_{i}$ such that for $\forall x \in R^{n}$ , the unknown nonlinear function $f_{i} (x)$ satisfies the following constraints

‖ f_{i} (x) ‖ \leq ‖ G_{i} x ‖, i \in I_{N}

(4)

Definition 2 (Chen and Sun, 2019). Consider the following non-fragile controllers with gain perturbation

\begin{matrix} u = (K_{i} + Δ K_{i}) x, i \in I_{N} \\ Δ K_{i} = M_{1 i} F_{1 i} (t) N_{1 i}, F_{1 i}^{T} (t) F_{1 i} (t) \leq I \end{matrix}

(5)

where $M_{1 i}$ and $N_{1 i}$ are constant matrices of appropriate dimensions. Then, the closed-loop system can be rewritten as

{\begin{matrix} \overset{\cdot}{x} = A_{i} x + B_{i} sat ((K_{i} + Δ K_{i}) x) + E_{i} f_{i} (x), i \in I_{N} \\ x (0) = 0 \end{matrix}

(6)

Definition 3 (Zhang and Sun, 2022). For nonlinear switched system (1) in the presence of actuator saturation, if there exists a state feedback controller $u^{*}$ of each subsystem and a positive scalar $J^{*}$ under the certain designed switching law $σ$ such that the considered system (6) is asymptotically stable and the cost function value (3) is satisfied with $J \leq J^{*}$ , then $J^{*}$ is called a guaranteed cost and $u^{*}$ is called the non-fragile guaranteed cost control law.

To derive the main results, we give the following lemmas.

Lemma 1 (Zhang et al., 2015) (Schur’s complements). For the symmetric matrix $A = [\begin{matrix} A_{11} & A_{12} \\ A_{12}^{T} & A_{22} \end{matrix}]$ , the following three conditions are equivalent:

$A < 0;$

$A_{11} < 0, A_{22} - A_{12}^{T} A_{11}^{- 1} A_{12} < 0;$

$A_{22} < 0, A_{11} - A_{12} A_{22}^{- 1} A_{12}^{T} < 0 .$

Lemma 2 (Qiu et al., 2011). Let $U$ , $W$ , and $X$ be matrices of given appropriate dimensions, $X$ satisfying $X = X^{T}$ , then

X + UVW + W^{T} V^{T} U^{T} < 0

for all $V V^{T} \leq I$ , if and only if there exists a scalar $ε > 0$ such that

X + ε^{- 1} U U^{T} + ε W^{T} W < 0 .

Lemma 3 (Petersen, 1987). Given any constant $λ > 0$ and any matrix $M, Γ, U$ with compatible dimension, for all $x \in R^{n}$ , there is

2 x^{T} M Γ Ux \leq λ x^{T} M M^{T} x + λ^{- 1} x^{T} U^{T} Ux

where $Γ$ is an uncertain matrix, satisfying $Γ^{T} Γ \leq I$ .

Let $P \in R^{n \times n}$ be a positive-definite matrix, an ellipsoid is defined as

Ω (P, ρ) = {x \in R^{n} : x^{T} Px \leq ρ, ρ > 0}

(7)

Representing row $j$ of matrix $F \in R^{m \times n}$ by $F^{j}$ , we define the following symmetric polyhedron

L (F) = {x \in R^{n} : | F^{j} x | \leq 1, j \in Q_{m}}

(8)

Let $D$ be the $m \times m$ diagonal matrix, where the diagonal elements are either 1 or 0. For example, if $m = 2$ , then

D = {[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]}

It is easy to know that there are $2^{m}$ elements in $D .$ Suppose that each element is labeled $D_{s},$ and denote $D_{s}^{-} = I - D_{s} \in D, s \in Q = {1, 2, \dots 2^{m}} .$

Lemma 4 (Hu et al., 2002). Given matrices $F$ and $H \in R^{m \times n}$ , for $x \in R^{n}$ , if $x \in L (H)$ , then

sat (Fx) \in co {D_{s} Fx + D_{s}^{-} Hx, s \in Q}

(9)

where $co {•}$ denotes the convex hull of a set. So the corresponding $sat (Fx)$ can be expressed as

sat (Fx) = \sum_{s = 1}^{2^{m}} η_{s} (D_{s} F + D_{s}^{-} H) x

(10)

$η_{s}$ in the above equation is a function of state $x$ , and $\sum_{s = 1}^{2^{m}} η_{s} = 1, 0 \leq η_{s} \leq 1 .$

Main results

In this section, several sufficient conditions are derived for system (6) to solve the non-fragile guaranteed cost control problem, and an approach to minimize the upper bound of the cost function is given using the multiple Lyapunov functions method.

Theorem 1. If there are positive-definite matrices $P_{i}$ , matrices $H_{i}$ , and a set of positive numbers $λ_{1 i}$ , $λ_{2 i}$ , $λ_{3 i}$ , $λ_{4 i}$ , $β_{ir}$ , the following matrix inequalities

[\begin{matrix} Ω_{1} & 0 \\ * & - R^{- 1} + λ_{4 i} D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} \end{matrix}] < 0, \forall i \in I_{N}

(11)

\begin{array}{l} Ω_{1} = {[A_{i} + B_{i} (D_{s} K_{i} + D_{s}^{-} H_{i})]}^{T} P_{i} \\ + P_{i} [A_{i} + B_{i} (D_{s} K_{i} + D_{s}^{-} H_{i})] + Q \\ + {(D_{s} K_{i} + D_{s}^{-} H_{i})}^{T} R (D_{s} K_{i} + D_{s}^{-} H_{i}) \\ + λ_{1 i}^{- 1} G_{i}^{T} G_{i} + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} + λ_{2 i}^{- 1} N_{1 i}^{T} N_{1 i} \\ + λ_{2 i} P_{i} B_{i} D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} B_{i}^{T} P_{i} \\ + λ_{3 i}^{- 1} {(D_{s} K_{i} + D_{s}^{-} H_{i})}^{T} R D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} R (D_{s} K_{i} + D_{s}^{-} H_{i}) \\ + λ_{3 i} N_{1 i}^{T} N_{1 i} + λ_{4 i}^{- 1} N_{1 i}^{T} N_{1 i} + \sum_{r = 1, r \neq i}^{N} β_{i r} (P_{r} - P_{i}) \end{array}

where

x_{0}^{T} P_{i} x_{0} \leq β, \forall i \in I_{N}

(12)

hold and

Ω (P_{i}, β) \cap Φ_{i} \subset L (H_{i}), \forall i \in I_{N}

(13)

are satisfied, where

Φ_{i} = {x \in R^{n} : x^{T} (P_{r} - P_{i}) x \geq 0, \forall r \in I_{N}, r \neq i} .

Therefore, under the action of state-dependent switching law

σ = \arg min {x^{T} P_{i} x, i \in I_{N}}

(14)

The closed-loop system (6) is asymptotically stable at the origin, and the set $\cup_{i = 1}^{N} (Ω (P_{i}, β) \cap Φ_{i})$ is contained in the domain of attraction. Specifically, $u_{i} = K_{i} x$ is the control law of non-fragile guaranteed cost for system (6) and the upper bound of cost function (3) satisfies $J < β .$

Proof. In consideration of Lemma 4, there exists $sat ((K_{i} + Δ K_{i}) x) \in co {D_{s} (K_{i} + Δ K_{i}) x + D_{s}^{-} H_{i} x, s \in Q}$ for any

x \in Ω (P_{i}, β) \cap Φ_{i} \subset L (H_{i})

It follows that $A_{i} x + B_{i} sat ((K_{i} + Δ K_{i}) x) + E_{i} f_{i} (x) \in co {A_{i} x + B_{i} (D_{s} (K_{i} + Δ K_{i}) + D_{s}^{-} H_{i}) x + E_{i} f_{i} (x), s \in Q}$

According to switching law (14), the ith subsystem is activated for $x \in Ω (P_{i}, β) \cap Φ_{i} \subset L (H_{i}) .$ The multiple Lyapunov functions of the closed-loop system (6) are selected as

V (x) = V_{σ} (x) = x^{T} P_{σ} x .

When $σ (t) = i$ , for

\forall x (t) \in Ω (P_{i}, β) \cap Φ_{i} \subset L (H_{i})

\begin{matrix} Λ = D_{s} (K_{i} + Δ K_{i}) + D_{s}^{-} H_{i} \\ Π = D_{s} K_{i} + D_{s}^{-} H_{i} \end{matrix}

we obtain

\begin{matrix} {\overset{\cdot}{V}}_{i} + x^{T} Qx + sa t^{T} (u_{i}) Rsat (u_{i}) \\ = {\overset{\cdot}{x}}^{T} P_{i} x + x^{T} P_{i} \overset{\cdot}{x} + x^{T} Qx + sa t^{T} (u) Rsat (u) \\ = 2 x^{T} P_{i} {\sum_{s = 1}^{2^{m}} η_{is} [A_{i} x + B_{i} Λ x + E_{i} f_{i} (x)]} \\ + x^{T} Qx + {(\sum_{s = 1}^{2^{m}} η_{is} Λ x)}^{T} R (\sum_{s = 1}^{2^{m}} η_{is} Λ x) \\ \leq max_{s \in Q} 2 x^{T} {P_{i} [A_{i} + B_{i} Λ] x + P_{i} E_{i} f_{i} (x)} + x^{T} Qx + x^{T} Λ^{T} R Λ x \end{matrix}

(15)

Because

\begin{array}{l} 2 x^{T} {P_{i} [A_{i} + B_{i} Λ] x + P_{i} E_{i} f_{i} (x)} = x^{T} \\ {{[A_{i} + B_{i} Λ]}^{T} P_{i} + P_{i} [A_{i} + B_{i} Λ]} + 2 x^{T} P_{i} E_{i} f_{i} (x) \end{array}

In view of Definition 1 and Lemma 3, we get

2 x^{T} P_{i} E_{i} f_{i} (x) \leq λ_{1 i} x^{T} P_{i} E_{i} E_{i}^{T} P_{i} x + λ_{1 i}^{- 1} x^{T} G_{i}^{T} G_{i} x

Therefore, inequality (15) can be transformed into

\begin{matrix} {\overset{\cdot}{V}}_{i} + x^{T} Qx + sa t^{T} (u_{i}) Rsat (u_{i}) \\ \leq max_{s \in Q} x^{T} {\begin{matrix} [A_{i} + B_{i} Λ]^{T} P_{i} + P_{i} [A_{i} + B_{i} Λ] \\ + Q + Λ^{T} R Λ + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} + λ_{1 i}^{- 1} G_{i}^{T} G_{i} \end{matrix}} x \end{matrix}

(16)

where

\begin{matrix} [A_{i} + B_{i} Λ]^{T} P_{i} + P_{i} [A_{i} + B_{i} Λ] + Q + Λ^{T} R Λ + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} \\ + λ_{1 i}^{- 1} G_{i}^{T} G_{i} \\ = [A_{i} + B_{i} Π]^{T} P_{i} + P_{i} [A_{i} + B_{i} Π] + Q + Π^{T} R Π \\ + P_{i} B_{i} D_{s} Δ K_{i} + (B_{i} D_{s} Δ K_{i})^{T} P_{i} \\ + Π^{T} R D_{s} Δ K_{i} + (D_{s} Δ K_{i})^{T} R Π + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} + λ_{1 i}^{- 1} G_{i}^{T} G_{i} \\ + (D_{s} Δ K_{i})^{T} R D_{s} Δ K_{i} \end{matrix}

If we expand condition (11), we can write it as

\begin{matrix} [\begin{matrix} \begin{matrix} [A_{i} + B_{i} Π]^{T} P_{i} + P_{i} [A_{i} + B_{i} Π] + Q \\ + Π^{T} R Π + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} + λ_{1 i}^{- 1} G_{i}^{T} G_{i} \\ + \sum_{r = 1, r \neq i}^{N} β_{ir} (P_{r} - P_{i}) \end{matrix} & 0 \\ * & - R^{- 1} \end{matrix}] \\ + λ_{2 i}^{- 1} [\begin{matrix} N_{1 i}^{T} \\ 0 \end{matrix}] [\begin{matrix} N_{1 i} & 0 \end{matrix}] + λ_{2 i} [\begin{matrix} P_{i} B_{i} D_{s} M_{1 i} \\ 0 \end{matrix}] [\begin{matrix} M_{1 i}^{T} D_{s}^{T} B_{i}^{T} P_{i} & 0 \end{matrix}] \\ + λ_{3 i}^{- 1} [\begin{matrix} Π^{T} R D_{s} M_{1 i} \\ 0 \end{matrix}] [\begin{matrix} M_{1 i}^{T} D_{s}^{T} R Π & 0 \end{matrix}] + λ_{3 i} [\begin{matrix} N_{1 i}^{T} \\ 0 \end{matrix}] [\begin{matrix} N_{1 i} & 0 \end{matrix}] \\ + λ_{4 i} [\begin{matrix} 0 \\ D_{s} M_{1 i} \end{matrix}] [\begin{matrix} 0 & M_{1 i}^{T} D_{s}^{T} \end{matrix}] + λ_{4 i}^{- 1} [\begin{matrix} N_{1 i}^{T} \\ 0 \end{matrix}] [\begin{matrix} N_{1 i} & 0 \end{matrix}] < 0 \end{matrix}

Then, according to Lemma 2, we can make the following deduction

\begin{matrix} [\begin{matrix} \begin{matrix} [A_{i} + B_{i} Π]^{T} P_{i} + P_{i} [A_{i} + B_{i} Π] + Q \\ + Π^{T} R Π + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} + λ_{1 i}^{- 1} G_{i}^{T} G_{i} \\ + \sum_{r = 1, r \neq i}^{N} β_{ir} (P_{r} - P_{i}) \end{matrix} & 0 \\ * & - R^{- 1} \end{matrix}] \\ + [\begin{matrix} N_{1 i}^{T} \\ 0 \end{matrix}] F_{1 i}^{T} (t) [\begin{matrix} M_{1 i}^{T} D_{s}^{T} B_{i}^{T} P_{i} & 0 \end{matrix}] \\ + [\begin{matrix} P_{i} B_{i} D_{s} M_{1 i} \\ 0 \end{matrix}] F_{1 i} (t) [\begin{matrix} N_{1 i} & 0 \end{matrix}] \\ + [\begin{matrix} Π^{T} R D_{s} M_{1 i} \\ 0 \end{matrix}] F_{1 i} (t) [\begin{matrix} N_{1 i} & 0 \end{matrix}] \\ + [\begin{matrix} N_{1 i}^{T} \\ 0 \end{matrix}] F_{1 i}^{T} (t) [\begin{matrix} M_{1 i}^{T} D_{s}^{T} R Π & 0 \end{matrix}] \\ + [\begin{matrix} 0 \\ D_{s} M_{1 i} \end{matrix}] F_{1 i} (t) [\begin{matrix} N_{1 i} & 0 \end{matrix}] \\ + [\begin{matrix} N_{1 i}^{T} \\ 0 \end{matrix}] F_{1 i}^{T} (t) [\begin{matrix} 0 & M_{1 i}^{T} D_{s}^{T} \end{matrix}] < 0 \end{matrix}

By Definition 2, we have

[\begin{matrix} \begin{matrix} [A_{i} + B_{i} Π]^{T} P_{i} + P_{i} [A_{i} + B_{i} Π] \\ + Q + Π^{T} R Π + P_{i} B_{i} D_{s} Δ K_{i} \\ + (B_{i} D_{s} Δ K_{i})^{T} P_{i} + Π^{T} R D_{s} Δ K_{i} \\ + (D_{s} Δ K_{i})^{T} R Π + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} \\ + λ_{1 i}^{- 1} G_{i}^{T} G_{i} + \sum_{r = 1, r \neq i}^{N} β_{ir} (P_{r} - P_{i}) \end{matrix} & {(D_{s} Δ K_{i})}^{T} \\ * & - R^{- 1} \end{matrix}] < 0

Then according to Lemma 1, the above equation can be rewritten as the following equation

\begin{matrix} [A_{i} + B_{i} Π]^{T} P_{i} + P_{i} [A_{i} + B_{i} Π] + Q \\ + Π^{T} R Π + P_{i} B_{i} D_{s} Δ K_{i} + (B_{i} D_{s} Δ K_{i})^{T} P_{i} \\ + Π^{T} R D_{s} Δ K_{i} + (D_{s} Δ K_{i})^{T} R Π + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} \\ + λ_{1 i}^{- 1} G_{i}^{T} G_{i} + (D_{s} Δ K_{i})^{T} R D_{s} Δ K_{i} + \sum_{r = 1, r \neq i}^{N} β_{ir} (P_{r} - P_{i}) < 0 \end{matrix}

From switching law (14), we get

\begin{matrix} {\overset{\cdot}{V}}_{i} + x^{T} Qx + sa t^{T} (u_{i}) Rsat (u_{i}) \\ \underset{s \in Q}{\leq max} x^{T} {\begin{matrix} [A_{i} + B_{i} Π]^{T} P_{i} + P_{i} [A_{i} + B_{i} Π] \\ + Q + Π^{T} R Π + P_{i} B_{i} D_{s} Δ K_{i} \\ + (B_{i} D_{s} Δ K_{i})^{T} P_{i} + Π^{T} R D_{s} Δ K_{i} \\ + (D_{s} Δ K_{i})^{T} R Π + (D_{s} Δ K_{i})^{T} R D_{s} Δ K_{i} \\ + λ_{1 i} P_{i} E_{i} E_{i}^{T} P_{i} + λ_{1 i}^{- 1} G_{i}^{T} G_{i} \end{matrix}} x \\ < - \sum_{r = 1, r \neq i}^{N} β_{ir} x^{T} (P_{r} - P_{i}) x \leq 0 \end{matrix}

(17)

Since $Q$ and $R$ matrices are positive-definite matrices, we get

{\overset{\cdot}{V}}_{i} < 0

The above equation shows that under any switching signal, the closed-loop system (6) is asymptotically stable at the origin, and the set $\cup_{i = 1}^{N} (Ω (P_{i}, β) \cap Φ_{i})$ is contained in the domain of attraction.

Next, we will explain that system (6) satisfies the upper bound of the cost function. In accordance with inequality (17), we get

J < - \sum_{k \in Z^{+}} \int_{t_{k}}^{t_{k + 1}} {\overset{\cdot}{V}}_{ik} dt = - [V_{i} (x (\infty)) - V_{i} (x (0))]

Since $V (\infty) \geq 0$ , it is not difficult to conclude that

$J < V_{i} (x (0)) = x_{0}^{T} P_{i} x_{0} \leq β$

The proof of Theorem 1 is completed.

However, it is easy to see that the parameters in formulas (11), (12), and (13) in Theorem 1 cannot be easily solved because they are not derived based on the LMI method. Then, using Theorem 1, a non-fragile state feedback controller design method based on LMI is proposed for system (6) so that the system can satisfy guaranteed cost control.

Theorem 2. If there are positive-definite matrices $X_{i}$ , matrices $J_{i}$ , $L_{i}$ and a set of positive numbers $β_{ir}, δ_{ir}, λ_{1 i}, λ_{2 i}, λ_{3 i}, λ_{4 i},$ the following matrix inequalities hold

[\begin{matrix} \begin{array}{l} X_{i} A_{i}^{T} + A_{i} X_{i} \\ + {ζ_{i}}^{T} B_{i}^{T} + B_{i} ζ_{i} \\ + λ_{2 i} B_{i} D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} B_{i}^{T} \\ + λ_{1 i} E_{i} E_{i}^{T} - \sum_{r = 1, r \neq i}^{N} β_{i r} X_{i} \end{array} & * & * & * \\ 0 & - R^{- 1} I + λ_{4 i} D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} & * & * \\ X_{i} & 0 & - Q^{- 1} & * \\ ζ_{i} & 0 & 0 & - R^{- 1} I \\ G_{i} X_{i} & 0 & 0 & 0 \\ N_{1 i} X_{i} & 0 & 0 & 0 \\ M_{1 i}^{T} D_{s}^{T} R ζ_{i} & 0 & 0 & 0 \\ N_{1 i} X_{i} & 0 & 0 & 0 \\ N_{1 i} X_{i} & 0 & 0 & 0 \\ X_{i} & 0 & 0 & 0 \\ X_{i} & 0 & 0 & 0 \\ X_{i} & 0 & 0 & 0 \end{matrix}

\begin{matrix} * & * & * & * & * & * & * & * \\ * & * & * & * & * & * & * & * \\ * & * & * & * & * & * & * & * \\ * & * & * & * & * & * & * & * \\ - λ_{1 i} I & * & * & * & * & * & * & * \\ 0 & - λ_{2 i} I & * & * & * & * & * & * \\ 0 & 0 & - λ_{3 i} I & * & * & * & * & * \\ 0 & 0 & 0 & - λ_{3 i}^{- 1} I & * & * & * & * \\ 0 & 0 & 0 & 0 & - λ_{4 i} I & * & * & * \\ 0 & 0 & 0 & 0 & 0 & - β_{i 1}^{- 1} X_{1} & * & * \\ 0 & 0 & 0 & 0 & 0 & 0 & ⋱ & * \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - β_{i N}^{- 1} X_{N} \end{matrix}] < 0

(18)

[\begin{matrix} β & x_{0}^{T} \\ * & X_{i} \end{matrix}] \geq 0, \forall i \in I_{N}

(19)

and

[\begin{matrix} X_{i} + \sum_{r = 1, r \neq i}^{N} δ_{ir} X_{i} & * & * & * & * \\ L_{i}^{j} & 1 & * & * & * \\ X_{i} & 0 & δ_{i 1}^{- 1} X_{1} & * & * \\ X_{i} & 0 & 0 & ⋱ & * \\ X_{i} & 0 & 0 & 0 & δ_{iN}^{- 1} X_{n} \end{matrix}] \geq 0

(20)

where $ζ_{i} = D_{s} J_{i} + D_{s}^{-} L_{i}$ and $L_{i}^{j}$ are represented as the jth row of matrices $L_{i}$ , $X_{i} = P_{i}^{- 1},$ $J_{i} = K_{i} X_{i}$ , $L_{i} = H_{i} X_{i}$ . Then, under the switching law

σ = \arg min {x^{T} X_{i}^{- 1} x, i \in I_{N}}

(21)

the set $\cup_{i = 1}^{N} (Ω (X_{i}, β) \cap Φ_{i})$ is inside the attraction domain for system (6). Specifically, the non-fragile guaranteed cost control law is equal to $u = J_{i} X_{i}^{- 1} x$ for system (6) and the corresponding index of system performance is $J < β .$

Proof. Multiplying the left and right sides of equation (11) by the diagonal matrices $diag {P_{i}, I}$ , we get

[\begin{matrix} Ω_{2} & * \\ 0 & - R^{- 1} I + λ_{4 i} D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} \end{matrix}] < 0

(22)

where

\begin{matrix} Ω_{2} = X_{i} A_{i}^{T} + A_{i} X_{i} + {ζ_{i}}^{T} B_{i}^{T} + B_{i} ζ_{i} \\ + λ_{1 i} E_{i} E_{i}^{T} + λ_{1 i}^{- 1} X_{i} G_{i}^{T} G_{i} X_{i} + X_{i} Q X_{i} + {ζ_{i}}^{T} R ζ_{i} \\ + λ_{2 i}^{- 1} X_{i} N_{1 i}^{T} N_{1 i} X_{i} + λ_{2 i} B_{i} D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} B_{i}^{T} \\ + λ_{3 i}^{- 1} {ζ_{i}}^{T} R D_{s} M_{1 i} M_{1 i}^{T} D_{s}^{T} R ζ_{i} \\ + λ_{3 i} X_{i} N_{1 i}^{T} N_{1 i} X_{i} + λ_{4 i}^{- 1} X_{i} N_{1 i}^{T} N_{1 i} X_{i} \\ + \sum_{r = 1, r \neq i}^{N} β_{ir} (X_{i} X_{r}^{- 1} X_{i} - X_{i}) < 0 \end{matrix}

Let $X_{i} = P_{i}^{- 1}$ , $L_{i} = H_{i} P_{i}^{- 1}$ , and $J_{i} = K_{i} P_{i}^{- 1}$ ; according to Lemma 1, inequality (22) can be transformed into inequality (18) equivalently.

Next, inequality (12), condition (13), and switching law (14) are, respectively, converted into inequalities (19) and (20) and switching law (21) using the similar method in Zhang and Sun (2022). So the proof of Theorem 2 is complete.

Assuming the gain perturbations $Δ K_{i} = 0$ in controllers (5), we obtain the following corollary.

Corollary. If there are positive-definite matrices $X_{i}$ , matrices $J_{i}$ , $L_{i}$ , and a set of positive numbers $β_{ir}, δ_{ir}, λ_{1 i}$ , such that equations (19), (20), and the following inequalities hold

[\begin{matrix} \begin{array}{l} X_{i} A_{i}^{T} + A_{i} X_{i} + {ζ_{i}}^{T} B_{i}^{T} + B_{i} ζ_{i} \\ + λ_{1 i} E_{i} E_{i}^{T} - \sum_{r = 1, r \neq i}^{N} β_{i r} X_{i} \end{array} & * & * & * & * & * & * \\ X_{i} & - Q^{- 1} & * & * & * & * & * \\ ζ_{i} & 0 & - R^{- 1} I & * & * & * & * \\ G_{i} X_{i} & 0 & 0 & - λ_{1 i} I & * & * & * \\ X_{i} & 0 & 0 & 0 & - β_{i 1}^{- 1} X_{1} & * & * \\ X_{i} & 0 & 0 & 0 & 0 & ⋱ & * \\ X_{i} & 0 & 0 & 0 & 0 & 0 & - β_{i N}^{- 1} X_{N} \end{matrix}] < 0

(23)

then, under switching law (21), the set $\cup_{i = 1}^{N} (Ω (X_{i}, β) \cap Φ_{i})$ is inside the attraction domain for system (6), the conventional guaranteed cost control law is equal to $u = J_{i} X_{i}^{- 1} x$ for system (6), and the corresponding index of system performance is $J < β .$

Remark. The purpose of our Corollary proposed is to verify the advantage of the non-fragile guaranteed cost control controller designed in EXAMPLE.

In Theorem 2, a sufficient condition is provided for solving the non-fragile guaranteed cost control problem and its performance index satisfies $J < β$ . The solution of non-fragile guaranteed cost control problem can be obtained in terms of the solutions of LMI. However, there are many feasible solutions meeting the cost upper bound of system (6). Thereby, the objective of this paper is to obtain the minimized upper bound of cost function. The problem can be transformed into the following optimization problem

\begin{matrix} Pb 1 inf_{X_{i}, J_{i}, L_{i}, β_{ir}, δ_{ir}, λ_{1 i}, λ_{2 i}, λ_{4 i}} β, \\ s . t . (a) inequality (18), \forall i \in I_{N} \\ (b) inequality (19), \forall i \in I_{N} \\ (c) inequality (20), \forall i \in I_{N}, \forall j \in Q_{m} \end{matrix}

(24)

Once this optimization problem is solved, the minimized cost upper bound $β^{*}$ can be obtained. Then, based on it, the non-fragile guaranteed cost controllers will be computed as $u_{i}^{*} = J_{i} X_{i}^{- 1} x$ . Thus, the non-fragile guaranteed cost control problem is also solved for the system (6). It is noticeable that one can solve the following optimization problem if we want to obtain the conventional guaranteed cost controller without considering the gain perturbations $Δ K_{i}$

\begin{array}{l} Pb 2 \inf_{X_{i}, J_{i}, L_{i}, β_{i r}, δ_{i r}, λ_{1 i}} β, \\ s . t . (a) inequality (19), \forall i \in I_{N} \\ (b) inequality (20), \forall i \in I_{N}, \forall j \in Q_{m} \\ (c) inequality (23), \forall i \in I_{N} \end{array}

(25)

Example

The effectiveness of the proposed method is demonstrated by a numerical example in this part. Consider a nonlinear switched system with actuator saturation for the following two subsystems

{\begin{matrix} \overset{\cdot}{x} = A_{i} x + B_{i} sat (u_{i}) + E_{i} f_{i} (x), i = 1, 2 \\ x (0) = x_{0} \end{matrix}

(26)

where

\begin{matrix} A_{1} = [\begin{matrix} 1 & 0.5 & 1 \\ 0 & - 1 & 0.2 \\ 0 & 0 & 1 \end{matrix}], A_{2} = [\begin{matrix} 1 & 0 & 0 \\ 0.5 & 1 & 0 \\ 0.1 & 0.5 & 1 \end{matrix}], \\ B_{1} = [\begin{matrix} 2 & 1 \\ 0 & 7 \\ 0 & 0 \end{matrix}], B_{2} = [\begin{matrix} 0 & 0 \\ 0 & 10 \\ 8 & 0 \end{matrix}], x (0) = [\begin{matrix} - 0.1 \\ - 0.1 \\ 0.1 \end{matrix}], \end{matrix}

\begin{matrix} E_{1} = [\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0.2 & 0 \\ 0 & 0 & 0.3 \end{matrix}], E_{2} = [\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0.3 & 0 \\ 0 & 0 & 0.3 \end{matrix}], \\ M_{11} = [\begin{matrix} 1.2 & 0 & 0 \\ 0 & 1.4 & 0 \end{matrix}], M_{12} = [\begin{matrix} 0 & 1.6 & 0 \\ 0 & 1.4 & 0 \end{matrix}], \end{matrix}

\begin{matrix} N_{11} = [\begin{matrix} 1.5 & 0 & 0 \\ 0 & 1.6 & 0 \\ 0 & 0 & 0.5 \end{matrix}], N_{12} = [\begin{matrix} 0.9 & 0 & 0 \\ 0 & 1.5 & 0 \\ 0 & 0 & 0.8 \end{matrix}], \\ G_{1} = [\begin{matrix} 0.2 & 0 & 0 \\ 0 & 0.2 & 0 \\ 0 & 0 & 0.3 \end{matrix}], G_{2} = [\begin{matrix} 0.5 & 0 & 0 \\ 0 & 0.2 & 0 \\ 0 & 0 & 0.3 \end{matrix}], \end{matrix}

\begin{matrix} F_{11} (t) = [\begin{matrix} \sin (t) & 0 & 0 \\ 0 & \sin (t) & 0 \\ 0 & 0 & \cos (t) \end{matrix}], \\ F_{12} (t) = [\begin{matrix} \cos (t) & 0 & 0 \\ 0 & \cos (t) & 0 \\ 0 & 0 & \sin (t) \end{matrix}], \end{matrix}

f_{1} (x) = [\begin{matrix} 0.1 \sin x_{1} \\ 0.1 \sin x_{2} \cos x_{2} \\ 0.1 x_{3} \sin x_{3} \end{matrix}], f_{2} (x) = [\begin{matrix} 0.1 x_{1} \cos x_{1} \\ 0.1 \sin x_{2} \\ 0.1 \sin x_{3} \cos x_{3} \end{matrix}]

Assuredly, neither of these subsystems is stable, and neither subsystem can be stabilized by state feedback alone. However, system (26) with actuator saturation can be stabilized and the upper bound of the cost function can be minimized by the designed switching law and non-fragile guaranteed cost controller.

If $Q = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], R = 1$ , $β_{1} = β_{2} = 10, δ_{1} = δ_{2} = 1, λ_{31} = 12.31, λ_{32} = 12.72$ , Pb 1 can be solved

β^{*} = 0.1548

\begin{matrix} K_{1}^{nc} = [\begin{matrix} - 30.6770 & 0.9347 & - 0.0579 \\ - 11.5528 & - 17.7349 & - 0.2658 \end{matrix}], \\ K_{2}^{nc} = [\begin{matrix} - 0.1173 - 0.3159 - 19.1497 \\ 0.8519 - 21.3865 - 0.4709 \end{matrix}] \end{matrix}

where $K_{i}^{nc}$ denotes the gain of the non-fragile guaranteed cost controller. Then, the state response curves of subsystems 1 and 2 are shown in Figures 1 and 2, respectively, and the state response curve of switched system (26) is shown in Figure 3. It is undeniable that although the two subsystems are unstable, switched system (26) is asymptotically stable when the perturbations $Δ K_{i}$ occur by adopting the switching law and the non-fragile guaranteed cost controller designed in this paper. In addition, the cost function curve of closed-loop system (26) is shown in Figure 4, and it is easy to see that the performance index of the system $J < β^{*}$ . Therefore, the method designed in this paper is effective.

Next, to demonstrate the advantage of the proposed method in this paper, solving Pb 2 by choosing the same parameters as above, we obtain $β^{*} = 0.0305$

K_{1}^{c} = [\begin{matrix} - 7.3927 & - 0.0384 & - 0.0392 \\ - 3.3818 & - 1.4044 & - 0.1453 \end{matrix}]

and

K_{2}^{c} = [\begin{matrix} - 0.1160 - 0.1547 - 5.6671 \\ - 0.2454 - 1.6954 - 0.1845 \end{matrix}]

where $K_{i}^{c}$ denotes the gain of the conventional guaranteed cost controller. However, when the gain perturbations occur in the controller, the state response curve of closed-loop system (26) is shown in Figure 5. Clearly, the use of conventional guaranteed cost controller does not stabilize closed-loop system (26) within the same running time. Similarly, the cost function curve is shown in Figure 6, where it can be seen that the system is not meeting the performance index $J < β^{*}$ . Consequently, it can be demonstrated that small perturbation in the controller can largely destroy the original performance of the system. If this perturbation is not considered in the process of designing the controller and the conventional guaranteed cost controller is used, not only the system stability is destroyed but also the performance index of guaranteed cost control cannot be met. So the importance of designing non-fragile guaranteed cost controllers cannot be overstated.

Conclusion

In this article, the issue of non-fragile guaranteed cost control for a class of nonlinear switched systems with actuator saturation is studied. We derive the sufficient conditions to satisfy both the non-fragile stabilization and guaranteed cost control performance indexes using the multiple Lyapunov function method. Then, the switching law and the non-fragile state feedback control law which can minimize the upper bound of the cost function are proposed and solved as an optimization problem with a set of matrix inequality constraints. Finally, a concrete numerical example is given to demonstrate the effectiveness of the design procedure.

In comparison with existing results on nonlinear switching systems with input saturation, the study in this paper has the following three remarkable features. First, the non-fragile control problem is addressed, whereas the extant results only design conventional state feedback controller and do not consider the perturbation phenomenon in the controller. Second, we provide an in-depth and synthetic analysis of the problem of guaranteed cost control and system stabilization, rather than just focusing on stability as in existing works. Third, this paper adopts the multiple Lyapunov function approach to research the nonlinear switched systems subject to saturation, and no solvability of the control problem for subsystem is required, while the existing works aimed at arbitrary switching and the solvability for each subsystem are required.

This paper focuses on non-fragile guaranteed cost control problem for nonlinear switched systems subject to actuator saturation and the external disturbances of the system are not considered. However, it is well known that many practical systems are inevitably affected by exogenous disturbances. Thus, it is more meaningful to study the problem of non-fragile $L_{2}$ -gain analysis and design. In particular, how to design the non-fragile controllers and the switched law to improve the $L_{2}$ -gain performance is a challenging issue for switched systems with actuator saturation, which deserves further study in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Liaoning Province of China (Grant No. 2020-MS-283), the Scientific Research Fund of Education Department of Liaoning Province of China (Grant No. LJKMZ20220731).

ORCID iDs

Jianlin Wang

Xinquan Zhang

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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