Robust Invariance Conditions of Uncertain Linear Discrete Time Systems Based on Semidefinite Programming Duality

Abstract

This article proposes a novel robust invariance condition for uncertain linear discrete-time systems with state and control constraints, utilizing a method of semidefinite programming duality. The approach involves approximating the robust invariant set for these systems by tackling the dual problem associated with semidefinite programming. Central to this method is the formulation of a dual programming through the application of adjoint mapping. From the standpoint of semidefinite programming dual optimization, the paper presents a novel linear matrix inequality (LMI) conditions pertinent to robust positive invariance. Illustrative examples are incorporated to elucidate the findings.

1. Introduction

Invariant sets are a fundamental concept in the realm of dynamic systems and play a pivotal role in the comprehensive analysis of such systems [1,2] as well as in the design of controllers [3]. Problems related to different systems with disturbance rejection can be analyzed and solved with the help of positively invariant sets. Similarly, many constrained control problems of dynamical systems can also be represented and solved by positively invariant sets. To explore further examples of positive invariant set applications across various constrained systems, consult more references in [1,3]. Consequently, investigating invariant sets is of paramount importance for understanding the control constraints. Over recent decades, substantial research has been undertaken on the invariant sets pertaining to linear systems [4] and nonlinear systems [5]. Positive invariance is frequently employed as both an analytical tool and a design instrument for systems that are subjected to bounded restrictions and disturbances. The construction of an approximation for the minimum robust positive invariant set in linear systems was addressed under the conditions of compressed set iteration in [6]. An algorithm devised in [7] facilitates the computation of the RPI set along with the corresponding controller K in a single-step system that is subject to (norm-bounded) model uncertainties and input and state polyhedral restrictions due to additive disturbances. Furthermore, it has been demonstrated that, in scenarios with a fixed K and no uncertainty (merely interference), the algorithm ascertains the optimal set by addressing a singular linear programming (LP) problem. In systems characterized by polynomial dynamics and compact semi-algebraic state and control constraints, our maximum control invariant set is articulated as the solution to an infinite-dimensional linear programming challenge [8]. Novel methodologies for determining the robust control invariant set for linear discrete systems affected by additive bounded disturbances have been introduced [9]. These methods ascertain the robust control invariant set for discrete systems by resolving linear matrix inequalities as stipulated by logarithmic norm and difference inequalities. Chen Y, Peng H, et al. proposed an iterative algorithm rooted in robust optimization, which approximates the value of the minimal robust control invariant set for uncertain dynamical systems while selecting the most suitable model from an acceptable set [10]. A computational technique for the polyhedral robust positive invariant set associated with a class of linear uncertain systems is elucidated in [11,12]. This approach necessitates solving a singular LP to derive the smallest set of robust positive invariants, delineated by a finite number of inequalities featuring predetermined normal vectors.

When encountering difficulties with linear programming methods, semidefinite programming can be utilized as an alternative. If the original problem of semidefinite programming remains unsolvable, it can also be transformed into its dual problem for resolution. Semidefinite programming serves as a numerical tool for the analysis and design of control systems, and has recently been the subject of extensive research. Several significant problems in control theory can be rephrased as semidefinite programming problems, specifically the minimization problem of linear objectives constrained by linear matrix inequalities (LMI). From the perspective of convex optimization duality theory, the infeasible conditions for LMI and the dual optimization problem are provided. The study of semi-definite and semi-infinite programming problems (SDSIP) [13,14], which includes both semi-infinite linear programming and semi-definite programming, was conducted. The uniform duality of homogeneous (SDSIP) and its Lagrangian-type duality problem is equivalent to the closure condition of a certain cone. The dual optimization method was used to study the positive invariance conditions of convex sets in discrete dynamical systems. Three equivalent necessary and sufficient conditions for polyhedra and polyhedral cones to be positive invariant sets of nonlinear discrete-time systems, as well as two types of convex sets to be positive invariant sets of linear discrete-time systems, are given [15]. The Farkas lemma is a fundamental result of linear programming, which provides a linear proof of the infeasibility of linear inequality systems. In semidefinite programming, this linear proof only exists in strongly infeasible linear matrix inequalities. Starting from the proof of the sum of real roots and squares in real algebraic geometry, a new exact dual theory for semidefinite programming is presented [16]. Balakrishnan and Vandenberghe provided a proof of the KYP lemma using semidefinite programming duality theory. On the basis of these theorems, a generalized proof of the discrete-time KYP lemma is provided [17]. The computation of robust invariant sets for state constrained perturbed polynomial systems under the Hamiltonian Jacobi reachable framework was investigated [18]. Express the maximum robust invariant set as the zero level set of the unique Lipschitz viscosity solution of the HJB equation. In addition, based on the derived HJB equation, a new semidefinite programming method is proposed, which can calculate robust invariant sets by solving a single semidefinite programming problem.

This article employs semidefinite programming and its corresponding dual optimization method to investigate the approximation technique for the maximum robust invariant set pertinent to uncertain linear discrete systems. The primary contribution of this paper initially delves into unconstrained uncertain linear discrete systems, presenting a sufficient condition that establishes the ellipsoid set as the maximum robust invariant set for these systems. Building upon this foundation, the study then addresses uncertain linear discrete systems with constraints. Through the application of the dual theory of semidefinite programming, optimization problems characterized by invariance and constraint conditions are transformed, ultimately yielding the system’s maximum robust invariant set.

The remainder of this article is structured as follows. Section 2 presents foundational knowledge on uncertain linear discrete-time systems, semidefinite programming, and the definition of robust control invariant sets. In Section 3, we explore the robust control invariant set for uncertain linear discrete-time systems with state and control constraints, utilizing the dual theory of semidefinite programming, a dual method to compute the ellipsoid set for uncertain linear discrete systems under a saturated controller, considering the constraints. Section 4 offers simulation examples, while Section 5 provides a comprehensive summary of the text.

In this paper, we use the following notations. Uppercase letters represent real matrices, while lowercase letters signify the column vectors of scalars. The symbol R refers to the set of real numbers. The notation $R^n$ is used for the n-dimensional Euclidean space, and $R^{m\times n}$ represents the set of real matrices. The superscript T denotes the transpose of a matrix. It is implicitly understood that all vectors and matrices are of compatible dimensions, without the need for explicit specification.

2. Preliminaries

In this section, some key definitions and lemmas related to robust invariant sets of uncertain linear discrete-time systems are introduced. Consider a uncertain linear discrete-time dynamical system described by a difference equations of the following form,

$$x\left(k+1\right)=A\left(k\right)x\left(k\right)+B\left(k\right)u\left(k\right),$$

(1)

where $x\left(k\right)\in R^n$ is the system state, $u\left(k\right)\in R^m$ is the input vector, and matrix $A\left(k\right)\in R^{n\times n}$, $B\left(k\right)\in R^{n\times m}$ satisfies the following equation

$$\left\{\begin{array}{c}A\left(k\right)=\sum _{i=1}^q{\alpha }_i\left(k\right)A_i,\hfill \\ B\left(k\right)=\sum _{i=1}^q{\alpha }_i\left(k\right)B_i,\hfill \\ \sum _{i=1}^q{\alpha }_i\left(k\right)=1,{\alpha }_i\left(k\right)\ge 0.\hfill \end{array}\right.$$

(2)

where the matrices $A_i$, $B_i$, $i=1,2,\dots ,q$ are the extreme realizations of $A\left(k\right)$ and $B\left(k\right)$.

The state and control are constrained by the following

$$\left\{\begin{array}{c}x\left(k\right)\in X,X=\left\{x\in R^n:F_{x}x\le g_x\right\},\hfill \\ u\left(k\right)\in U,U=\left\{u\in R^m:F_{u}u\le g_u\right\}.\hfill \end{array}\right.$$

Definition 1 

(Robustly controlled invariant set [19]). The set $\mathsf{\Omega }\subset X$ is a robustly controlled invariant for the system (1) if there exists a control value $u\left(k\right)\in U$, such that for all $x\left(k\right)\in \mathsf{\Omega }$, $x\left(k+1\right)\in \mathsf{\Omega }$.

$S^n$ denotes the set of Hermitian $n\times n$ matrices. The standard inner product related to $S^n$ is defined as ${\langle A,B\rangle }_{S^n}=tr\left(A\times B\right)=tr\left(AB\right)$. Let $\mathcal{S}$ be the set of Hermitian matrices and $\mathcal{V}$ is a finite-dimensional vector space. Given a linear mapping $\mathcal{A}:\mathcal{V}\to \mathcal{S}$, ${\mathcal{A}}^{adj}$ represents the adjoint mapping of $\mathcal{A}$, which is a linear mapping from $\mathcal{S}$ to $\mathcal{V}$, such that for all $x\in \mathcal{V}$ and $Z\in \mathcal{S}$, ${\langle \mathcal{A}\left(x\right),Z\rangle }_{\mathcal{S}}={\langle x,{\mathcal{A}}_{adj}\left(Z\right)\rangle }_{\mathcal{V}}$.

Lemma 1 

([17]). The semidefinite programming problem, which is the minimization problem of linear objectives constrained by linear matrix inequality (LMI) is as follows:

$$\begin{array}{cc}\hfill & min{\langle c,x\rangle }_{\mathcal{V}}\hfill \\ \hfill & subjectto\mathcal{A}\left(x\right)+A_0\ge 0.\hfill \end{array}$$

(3)

Then, the dual problem of semidefinite programming is

$$\begin{array}{cc}\hfill & max-{\langle A_0,Z\rangle }_{\mathcal{S}}\hfill \\ \hfill & subjectto{\mathcal{A}}^{adj}\left(Z\right)=c,Z\ge 0\hfill \end{array}$$

(4)

3. Results

3.1. Robust Invariant Set with Symmetric Constraints

We present three theorems where we derive three transformations of a robust invariant set of the system (1).

Lemma 2 

([19]). The set $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ is a robust invariant set of the system (1) if and only if for all $x\in E\left(P\right)$, there exists an input $u=Kx\in U$ such that

$$\left[\begin{array}{cc}P& A_{i}P+B_{i}KP\\ P{A}_i^T+P{K}^T{B}_i^T& P\end{array}\right]\ge 0,\forall i=1,2,\dots ,q.$$

(5)

Theorem 1. 

Suppose there exits a feedback control law K and the optimization problem in Lemma 2 is feasible.The maximum robust invariant set $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ of the system (1) is transformed into solving the following optimization problems:

$$\begin{array}{cc}\hfill & maxd\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_{i}Q+A_i^T{Z}_{12}+Q^T{B}_i^T+Z_{22}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right]\ge 0\hfill \end{array}$$

(6)

where $Q=K{Z}_{12}^T$, variables $Z_{11}\in S^n$, $Z_{12}\in R^{n\times m}$, $Z_{22}\in S^m$.

Proof of Theorem 1. 

The size of the ellipsoid $E\left(P\right)$ can be measured by the determinant or the trace of matrix P. According to Lemma 2, the problem of finding the maximum robust invariant ellipsoid with the trace of a matrix as the objective function can be written as follows:

$$\begin{array}{cc}\hfill & maxtr\left(P\right)\hfill \\ \hfill & subjectto\left[\begin{array}{cc}P& A_{i}P+B_{i}KP\\ P{A}_i^T+P{K}^T{B}_i^T& P\end{array}\right]\ge 0.\hfill \end{array}$$

(7)

where $P\in S^n$.

The above problem (7) can be written as

$$\begin{array}{cc}\hfill & min{\langle c,x\rangle }_{\mathcal{V}}\hfill \\ \hfill & subjectto\mathcal{A}\left(x\right)+A_0\ge 0.\hfill \end{array}$$

where $\mathcal{A}:{\mathcal{S}}^n\to {\mathcal{S}}^{n+m}\times {\mathcal{S}}^n$, $A_0$, and c are, respectively,

$$\begin{array}{c}\hfill \mathcal{A}\left(P\right)=diag\left(\left[\begin{array}{cc}P& A_{i}P+B_{i}KP\\ P{A}_i^T+P{K}^T{B}_i^T& P\end{array}\right],P\right),A_0=diag\left(\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],0\right),c=-I.\end{array}$$

The dual problem of the above semidefinite programming problem is

$$\begin{array}{cc}\hfill & max-{\langle A_0,Z\rangle }_{\mathcal{S}}\hfill \\ \hfill & subjectto{\mathcal{A}}^{adj}\left(Z\right)=c,Z\ge 0\hfill \end{array}$$

${\mathcal{A}}^{adj}:{\mathcal{S}}^{n+m}\times {\mathcal{S}}^n\to {\mathcal{S}}^n$ is given by

$${\mathcal{A}}^{adj}\left(Z\right)=Z_{11}+A_i{Z}_{12}^T+B_{i}K{Z}_{12}^T+A_i^T{Z}_{12}+K^T{B}_i^T{Z}_{12}+Z_{22}+Z_2,$$

where $Z=diag\left(\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right],Z_2\right)\in {\mathcal{S}}^{n+m}\times {\mathcal{S}}^n$, with $Z_{11}\in {\mathcal{S}}^n$.

Therefore, the dual problem can be rewritten as

$$\begin{array}{cc}\hfill & maxd\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_{i}Q+A_i^T{Z}_{12}+Q^T{B}_i^T+Z_{22}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right]\ge 0\hfill \end{array}$$

where d is constant. By setting $Q=K{Z}_{12}^T$, condition becomes

$$Z_{11}+A_i{Z}_{12}^T+B_{i}K{Z}_{12}^T+A_i^T{Z}_{12}+K^T{B}_i^T{Z}_{12}+Z_{22}+I\le 0.$$

So, $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ is the maximum robust invariant set of the system (1), which can be transformed to solve the following optimization problems:

$$\begin{array}{cc}\hfill & maxd\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_{i}K{Z}_{12}^T+A_i^T{Z}_{12}+K^T{B}_i^T{Z}_{12}+Z_{22}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{12}^T& Z_{22}\end{array}\right]\ge 0\hfill \end{array}$$

where variables $Z_{11}\in S^n$, $Z_{12}\in R^{n\times m}$, $Z_{22}\in S^m$. □

Consider the following uncertain linear discrete-time systems

$$x\left(k+1\right)=A\left(k\right)x\left(k\right)+B\left(k\right)u\left(k\right),$$

where matrix $A\left(k\right)\in R^{n\times n}$ and $B\left(k\right)\in R^{n\times m}$ satisfy (2). Assuming that the state constraints and control constraints satisfy the following equation

$$\left\{\begin{array}{c}x\left(k\right)\in X,X=\left\{x\in R^n:\mid F_i^{T}x\mid \le 1\right\},\forall i=1,2,\dots ,n.\hfill \\ u\left(k\right)\in U,U=\left\{u\in R^m:\mid u_j\mid \le u_{jmax}\right\},\forall j=1,2,\dots ,m.\hfill \end{array}\right.$$

(8)

where $u_{jmax}$ is the j-component of vector $u_{max}\in R^m$.

Lemma 3 

([19]). For a given symmetric matrix $M=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{12}^T& M_{22}\end{array}\right]$, where $M_{11}\in R^{n\times n}$, the following three conditions are equivalent:

$$\begin{array}{cc}\hfill & \left(1\right)\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{12}^T& M_{22}\end{array}\right]>0;\hfill \\ \hfill & \left(2\right)\left\{\begin{array}{c}{M}_{11}>0,\hfill \\ M_{11}-M_{12}{M}_{22}^{-1}{M}_{12}^T>0;\hfill \end{array}\right.\hfill \\ \hfill & \left(3\right)\left\{\begin{array}{c}{M}_{22}>0,\hfill \\ M_{22}-M_{12}^T{M}_{11}^{-1}{M}_{12}>0.\hfill \end{array}\right.\hfill \end{array}$$

As the support function of the ellipsoid $E\left(P\right)$ at vector $f\in R^n$ is ${\varphi }_{E\left(P\right)}\left(z\right)=\sqrt{f^{T}Pf}$. Ellipsoid $E\left(P\right)$ is a subset of the polyhedral set $\simeq \left(f,1\right)=\left\{x\in R^n:\mid f^{T}x\mid \le 1\right\}$ if and only if $f^{T}Pf\le 1$. The condition can be rewritten using the Schur complement theorem as

$$\left[\begin{array}{cc}1& f^{T}P\\ Pf& P\end{array}\right]\ge 0.$$

Theorem 2. 

The set $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ is the maximum robust invariant set of the system (1) with constraint (8), which can be transformed into solving the following optimization problems:

$$\begin{array}{cc}\hfill & max-tr\left(Z_{33}+u_{jmax}{Z}_{55}\right)\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_{i}K{Z}_{12}^T+A_i^T{Z}_{12}+K^T{B}_i^T{Z}_{12}+Z_{22}+Z_{34}^T{F}_i^T\hfill \\ \hfill & +F_i{Z}_{34}+Z_{44}+T_{j}K{Z}_{56}^T+K^T{T}_j^T{Z}_{56}+Z_{66}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cccccc}{Z}_{11}& Z_{12}& 0& 0& 0& 0\\ Z_{12}^T& Z_{22}& 0& 0& 0& 0\\ 0& 0& Z_{33}& Z_{34}& 0& 0\\ 0& 0& Z_{34}^T& Z_{44}& 0& 0\\ 0& 0& 0& 0& Z_{55}& Z_{56}\\ 0& 0& 0& 0& Z_{56}^T& Z_{66}\end{array}\right]\ge 0\hfill \end{array}$$

(9)

where variables $Z_{11},Z_{33},Z_{55}\in S^n$, $Z_{12},Z_{34},Z_{56}\in R^{n\times m}$, $Z_{22},Z_{44},Z_{66}\in S^m$.

Proof of Theorem 2. 

Assuming the ellipsoid $E\left(P\right)\subseteq X$, the state constraints become $\left\{P\in S^n:\mid F_i^{T}P\mid \le 1\right\}$. Ellipsoid $E\left(P\right)$ is a subset of the set $X=\left\{P\in S^n:\mid F_i^{T}P\mid \le 1\right\}$ if and only if $F_i^{T}P{F}_i\le 1$. This condition can be rewritten using the Schur complement as

$$\left[\begin{array}{cc}1& F_i^{T}P\\ P{F}_i& P\end{array}\right]\ge 0,\forall i=1,2,\dots ,n.$$

Assume the ellipsoid $E\left(P\right)\subseteq U$ and $u=Kx$, the input constraints become $\left\{P\in S^n:\mid K_{j}P\mid \le u_{jmax}\right\}$ where $K_j$ is the j-row of the matrix $K\in R^{m\times n}$. Ellipsoid $E\left(P\right)$ is a subset of the set $U=\left\{P\in S^n:\mid K_{j}P\mid \le u_{jmax}\right\}$ if and only if $K_{j}P{K}_j^T\le 1$. This condition can be rewritten using the Schur complement, as follows

$$\left[\begin{array}{cc}{u}_{jmax}^2& K_{j}P\\ P{K}_j^T& P\end{array}\right]\ge 0,\forall j=1,2,\dots ,m.$$

Define a vector $T_j\in R^m$ with its j-th position being one and the rest being zero. As $K_j=T_{j}K$, the above equation can be written as

$$\left[\begin{array}{cc}{u}_{jmax}^2& T_{j}KP\\ P{K}^T{T}_j^T& P\end{array}\right]\ge 0,\forall j=1,2,\dots ,m.$$

The problem of finding the maximum robust invariant set can be written as

$$\begin{array}{cc}\hfill & maxtr\left(P\right)\hfill \\ \hfill & subjectto\left[\begin{array}{cccccc}P& A_{i}P+B_{i}KP& 0& 0& 0& 0\\ P{A}_i^T+P{K}^T{B}_i^T& P& 0& 0& 0& 0\\ 0& 0& 1& F_i^{T}P& 0& 0\\ 0& 0& P{F}_i& P& 0& 0\\ 0& 0& 0& 0& u_{jmax}^2& T_{j}KP\\ 0& 0& 0& 0& P{K}^T{T}_j^T& P\end{array}\right]\ge 0.\hfill \end{array}$$

(10)

We define $\mathcal{A}:{\mathcal{S}}^n\to {\mathcal{S}}^{4n+2}\times {\mathcal{S}}^n$, $A_0\in {\mathcal{S}}^{4n+2}$, $c\in \mathcal{V}$ as

$$\begin{array}{cc}\hfill & \mathcal{A}\left(P\right)=diag\left(\left[\begin{array}{cccccc}P& A_{i}P+B_{i}KP& 0& 0& 0& 0\\ P{A}_i^T+P{K}^T{B}_i^T& P& 0& 0& 0& 0\\ 0& 0& 0& F_i^{T}P& 0& 0\\ 0& 0& P{F}_i& P& 0& 0\\ 0& 0& 0& 0& 0& T_{j}KP\\ 0& 0& 0& 0& P{K}^T{T}_j^T& P\end{array}\right],P\right),\hfill \\ \hfill & A_0=diag\left(\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& u_{jmax}^2& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right],0\right),\hfill \\ \hfill & c=-I.\hfill \end{array}$$

The semidefinite programming problem (10) can be rewritten as

$$\begin{array}{cc}\hfill & min{\langle c,x\rangle }_{\mathcal{V}}\hfill \\ \hfill & subjectto\mathcal{A}\left(x\right)+A_0\ge 0.\hfill \end{array}$$

Its duality problem is

$$\begin{array}{cc}\hfill & max-{\langle A_0,Z\rangle }_{\mathcal{S}}\hfill \\ \hfill & subjectto{\mathcal{A}}^{adj}\left(Z\right)=c,Z\ge 0\hfill \end{array}$$

We have obtained that ${\mathcal{A}}^{adj}:{\mathcal{S}}^{4n+2}\times {\mathcal{S}}^n\to {\mathcal{S}}^n$ is given by

$$\begin{array}{cc}\hfill {\mathcal{A}}^{adj}\left(Z\right)=& Z_{11}+A_i{Z}_{12}^T+B_{i}K{Z}_{12}^T+A_i^T{Z}_{12}+K^T{B}_i^T{Z}_{12}+Z_{22}+Z_{34}^T{F}_i^T\hfill \\ \hfill & +F_i{Z}_{34}+Z_{44}+T_{j}K{Z}_{56}^T+K^T{T}_j^T{Z}_{56}+Z_{66}+Z_6.\hfill \end{array}$$

And $-\langle A_0,Z\rangle =-tr\left(Z_{33}+u_{jmax}^2{Z}_{55}\right)$. So, the set $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ is the maximum robust invariant set of the system (1) with constraint (8), which can be transformed into solving the following optimization problems:

$$\begin{array}{cc}\hfill & max-tr\left(Z_{33}+u_{jmax}^2{Z}_{55}\right)\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_{i}K{Z}_{12}^T+A_i^T{Z}_{12}+K^T{B}_i^T{Z}_{12}+Z_{22}+Z_{34}^T{F}_i^T\hfill \\ \hfill & +F_i{Z}_{34}+Z_{44}+T_{j}K{Z}_{56}^T+K^T{T}_j^T{Z}_{56}+Z_{66}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cccccc}{Z}_{11}& Z_{12}& 0& 0& 0& 0\\ Z_{12}^T& Z_{22}& 0& 0& 0& 0\\ 0& 0& Z_{33}& Z_{34}& 0& 0\\ 0& 0& Z_{34}^T& Z_{44}& 0& 0\\ 0& 0& 0& 0& Z_{55}& Z_{56}\\ 0& 0& 0& 0& Z_{56}^T& Z_{66}\end{array}\right]\ge 0\hfill \end{array}$$

where variables $Z_{11},Z_{33},Z_{55}\in S^n$, $Z_{12},Z_{34},Z_{56}\in R^{n\times m}$, $Z_{22},Z_{44},Z_{66}\in S^m$. □

3.2. Robust Invariant Set under Saturated Controller

In this section, we consider the following uncertain linear discrete-time systems

$$x\left(k+1\right)=A\left(k\right)x\left(k\right)+B\left(k\right)sat\left(Kx\left(k\right)\right),$$

(11)

where $u\left(k\right)=sat\left(Kx\left(k\right)\right)$, matrix $A\left(k\right)\in R^{n\times n}$ and $B\left(k\right)\in R^{n\times m}$ satisfy (2). The state constraints and control constraints satisfy the following equation

$$\left\{\begin{array}{c}x\left(k\right)\in X,X=\left\{x\in R^n:F_i^{T}x\le g_i\right\},\forall i=1,2,\dots ,n.\hfill \\ u\left(k\right)\in U,U=\left\{u\in R^m:u_{jl}\le u_j\le u_{ju}\right\},\forall j=1,2,\dots ,m.\hfill \end{array}\right.$$

(12)

where $F_i^T$ is the j-th row of the matrix $F_x$; $g_j$ is the j-th component of the vector $g_x$; and $u_{jl}$ and $u_{ju}$ are the i-th components of the lower bound $u_l$ and upper bound $u_u$ of the input u, respectively. The matrix $F_x$ and the vectors $g_x$, $u_l$, and $u_u$ are constant with $g_x>0$, $u_l<0$ and $u_u>0$.

Due to $F_i^{T}x\le g_i$, which is equivalent to ${\displaystyle \frac{F_i^T}{g_i}}x\le 1$, the state constraints become $\left\{x\in R^n:{\displaystyle \frac{F_i^T}{g_i}}x\le 1\right\}$. By setting $u_{max}=max\left\{-u_l,u_u\right\}$, the control constraints become $\left\{x\in R^n:\mid Hx\mid \le u_{max}\right\}$ for a given matrix $H\in R^{m\times n}$. Define

$$X_u=\left\{x\in R^n:\left[\begin{array}{c}{\displaystyle \frac{F_i^T}{g_i}}\\ H\\ -H\end{array}\right]x\le \left[\begin{array}{c}1\\ u_{max}\\ u_{max}\end{array}\right]\right\}.$$

Define $D_m$ as the set of $m\times m$ diagonal matrices with diagonal elements of 0 or 1. Each element of $D_m$ is denoted as $E_j$, $j=1,2,\dots ,2^m$. $E_j^{-}=I-E_j$.

Lemma 4 

([20]). Consider two vectors $u\in R^m$ and $v\in R^m$, such that $u_{il}<v_i<u_{iu}$ for all $i=1,2,\dots ,m$, then, it holds that

$$sat\left(u\right)\in Conv\left\{E_{j}u+E_j^{-}v\right\},j=1,2,\dots ,2^m.$$

Lemma 5 

([19]). If there exist a positive definite matrix $P\in R^{n\times n}$ and a matrix $H\in R^{m\times n}$, such that, $\forall i=1,2,\dots ,q$, $\forall j=1,2,\dots ,2^m$,

$$\left[\begin{array}{cc}P& \left\{A_i+B_i\left(E_{j}K+E_j^{-}H\right)\right\}P\\ P{\left\{A_i+B_i\left(E_{j}K+E_j^{-}H\right)\right\}}^T& P\end{array}\right]\ge 0,$$

and $E\left(P\right)\subset X_u$, then the ellipsoid $E\left(P\right)$ is a robust invariant set for system (11) with constraints (12).

Theorem 3. 

The set $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ is the maximum robust invariant set of the system (11) under a saturated controller $u\left(k\right)=sat\left(kx\left(k\right)\right)$ with constraints (12), which can be transformed into solving the following optimization problems:

$$\begin{array}{cc}\hfill & max-tr\left(Z_{33}+u_{jmax}^2{Z}_{55}\right)\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_i{E}_{j}K{Z}_{12}^T+B_i{E}_j^{-}H{Z}_{12}^T+A_i^T{Z}_{12}+K^T{E}_j{B}_i^T{Z}_{12}\hfill \\ \hfill & +H^T{E}_j^{-}{B}_i^T{Z}_{12}+Z_{22}+Z_{34}^T{{\displaystyle \frac{F_i}{g_i}}}^T+{\displaystyle \frac{F_i}{g_i}}{Z}_{34}+Z_{44}+T_{j}H{Z}_{56}^T+\hfill \\ \hfill & H^T{T}_j^T{Z}_{56}+Z_{66}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cccccc}{Z}_{11}& Z_{12}& 0& 0& 0& 0\\ Z_{12}^T& Z_{22}& 0& 0& 0& 0\\ 0& 0& Z_{33}& Z_{34}& 0& 0\\ 0& 0& Z_{34}^T& Z_{44}& 0& 0\\ 0& 0& 0& 0& Z_{55}& Z_{56}\\ 0& 0& 0& 0& Z_{56}^T& Z_{66}\end{array}\right]\ge 0\hfill \end{array}$$

(13)

where variables $Z_{11},Z_{33},Z_{55}\in S^n$, $Z_{12},Z_{34},Z_{56}\in R^{n\times m}$, $Z_{22},Z_{44},Z_{66}\in S^m$.

Proof of Theorem 3. 

The state constraints condition can be rewritten using the Schur complement as

$$\left[\begin{array}{cc}1& {{\displaystyle \frac{F_i}{g_i}}}^{T}P\\ P{\displaystyle \frac{F_i}{g_i}}& P\end{array}\right]\ge 0,\forall i=1,2,\dots ,n.$$

As $H_j=T_{j}H$, the control constraints condition can be rewritten using the Schur complement, as

$$\left[\begin{array}{cc}{u}_{jmax}^2& T_{j}HP\\ P{H}^T{T}_j^T& P\end{array}\right]\ge 0,\forall j=1,2,\dots ,m.$$

For the convenience of writing, let $W=A_{i}P+B_i{E}_{j}KP+B_i{E}_j^{-}HP$, $Y=P{A}_i^T+P{K}^T{E}_j{B}_i^T+P{H}^T{E}_j^{-}{B}_i^T$. The problem of finding the maximum robust invariant set can be written as

$$\begin{array}{cc}\hfill & maxtr\left(P\right)\hfill \\ \hfill & subjectto\left[\begin{array}{cccccc}P& W& 0& 0& 0& 0\\ Y& P& 0& 0& 0& 0\\ 0& 0& 1& {{\displaystyle \frac{F_i}{g_i}}}^{T}P& 0& 0\\ 0& 0& P{\displaystyle \frac{F_i}{g_i}}& P& 0& 0\\ 0& 0& 0& 0& u_{jmax}^2& T_{j}HP\\ 0& 0& 0& 0& P{H}^T{T}_j^T& P\end{array}\right]\ge 0.\hfill \end{array}$$

(14)

We define $\mathcal{A}:{\mathcal{S}}^n\to {\mathcal{S}}^{4n+2}\times {\mathcal{S}}^n$, $A_0\in {\mathcal{S}}^{4n+2}$, $c\in \mathcal{V}$ as

$$\begin{array}{cc}\hfill & \mathcal{A}\left(P\right)=diag\left(\left[\begin{array}{cccccc}P& W& 0& 0& 0& 0\\ Y& P& 0& 0& 0& 0\\ 0& 0& 0& {{\displaystyle \frac{F_i}{g_i}}}^{T}P& 0& 0\\ 0& 0& P{\displaystyle \frac{F_i}{g_i}}& P& 0& 0\\ 0& 0& 0& 0& 0& T_{j}HP\\ 0& 0& 0& 0& P{H}^T{T}_j^T& P\end{array}\right],P\right),\hfill \\ \hfill & A_0=diag\left(\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& u_{jmax}^2& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right],0\right),\hfill \\ \hfill & c=-I.\hfill \end{array}$$

The semidefinite programming problem (14) can be rewritten as

$$\begin{array}{cc}\hfill & min{\langle c,x\rangle }_{\mathcal{V}}\hfill \\ \hfill & subjectto\mathcal{A}\left(x\right)+A_0\ge 0.\hfill \end{array}$$

Its duality problem is

$$\begin{array}{cc}\hfill & max-{\langle A_0,Z\rangle }_{\mathcal{S}}\hfill \\ \hfill & subjectto{\mathcal{A}}^{adj}\left(Z\right)=c,Z\ge 0\hfill \end{array}$$

We have obtained that ${\mathcal{A}}^{adj}:{\mathcal{S}}^{4n+2}\times {\mathcal{S}}^n\to {\mathcal{S}}^n$ is given by

$$\begin{array}{cc}\hfill {\mathcal{A}}^{adj}\left(Z\right)=& Z_{11}+A_i{Z}_{12}^T+B_i{E}_{j}K{Z}_{12}^T+B_i{E}_j^{-}H{Z}_{12}^T+A_i^T{Z}_{12}+K^T{E}_j{B}_i^T{Z}_{12}\hfill \\ \hfill & +H^T{E}_j^{-}{B}_i^T{Z}_{12}+Z_{22}+Z_{34}^T{{\displaystyle \frac{F_i}{g_i}}}^T+{\displaystyle \frac{F_i}{g_i}}{Z}_{34}+Z_{44}+T_{j}H{Z}_{56}^T+\hfill \\ \hfill & H^T{T}_j^T{Z}_{56}+Z_{66}+Z_6.\hfill \end{array}$$

And $-\langle A_0,Z\rangle =-tr\left(Z_{33}+u_{jmax}^2{Z}_{55}\right)$. So, the set $E\left(P\right)=\left\{x\in R^n:x^T{P}^{-1}x\le 1\right\}$ is the maximum robust invariant set of the system (11) under the saturated controller $u\left(k\right)=sat\left(kx\left(k\right)\right)$ with constraint (12), which can be transformed into solving the following optimization problems:

$$\begin{array}{cc}\hfill & max-tr\left(Z_{33}+u_{jmax}^2{Z}_{55}\right)\hfill \\ \hfill & subjectto{Z}_{11}+A_i{Z}_{12}^T+B_i{E}_{j}K{Z}_{12}^T+B_i{E}_j^{-}H{Z}_{12}^T+A_i^T{Z}_{12}+K^T{E}_j{B}_i^T{Z}_{12}\hfill \\ \hfill & +H^T{E}_j^{-}{B}_i^T{Z}_{12}+Z_{22}+Z_{34}^T{{\displaystyle \frac{F_i}{g_i}}}^T+{\displaystyle \frac{F_i}{g_i}}{Z}_{34}+Z_{44}+T_{j}H{Z}_{56}^T+\hfill \\ \hfill & H^T{T}_j^T{Z}_{56}+Z_{66}+I\le 0,\hfill \\ \hfill & \left[\begin{array}{cccccc}{Z}_{11}& Z_{12}& 0& 0& 0& 0\\ Z_{12}^T& Z_{22}& 0& 0& 0& 0\\ 0& 0& Z_{33}& Z_{34}& 0& 0\\ 0& 0& Z_{34}^T& Z_{44}& 0& 0\\ 0& 0& 0& 0& Z_{55}& Z_{56}\\ 0& 0& 0& 0& Z_{56}^T& Z_{66}\end{array}\right]\ge 0\hfill \end{array}$$

where variables $Z_{11},Z_{33},Z_{55}\in S^n$, $Z_{12},Z_{34},Z_{56}\in R^{n\times m}$, $Z_{22},Z_{44},Z_{66}\in S^m$. □

4. Numerical Examples

Example 1. 

Consider the following uncertain linear discrete-time system

$$x\left(k+1\right)=A\left(k\right)x\left(k\right)+B\left(k\right)u\left(k\right)$$

and $A_1=\left[\begin{array}{cc}1& 0.1\\ 0& 1\end{array}\right],A_2=\left[\begin{array}{cc}1& 0.2\\ 0& 1\end{array}\right],B_1=\left[\begin{array}{c}0\\ 1\end{array}\right],B_2=\left[\begin{array}{c}0\\ 1.5\end{array}\right].$

A robust invariant set was constructed using Theorem 1. By solving the optimization problem (6), obtaining an optimal value of 0.5658, and

$$Q=\left[\begin{array}{cc}-102.2080& -173.8163\end{array}\right],$$

$$Z_{12}=\left[\begin{array}{cc}-133.4669& -98.9062\\ -98.9062& -80.9254\end{array}\right].$$

Thus,

$$P=\left[\begin{array}{cc}0.3687& -0.1504\\ -0.1504& 0.1971\end{array}\right],$$

$$K=Q{Z}_{12}^{-1}=\left[\begin{array}{cc}-8.7520& 12.8629\end{array}\right].$$

Then, the maximum robust invariant set of the system (1) is

$$E\left(P\right)=\left\{x\in R^n:x^T\left[\begin{array}{cc}3.9380& 3.0050\\ 3.0050& 7.3665\end{array}\right]x\le 1\right\}.$$

The robust invariant set yielded from Theorem 1 is shown in Figure 1.

Example 2. 

Consider the following uncertain linear discrete-time system

$$x\left(k+1\right)=A\left(k\right)x\left(k\right)+B\left(k\right)u\left(k\right)$$

with $A_1=\left[\begin{array}{cc}1& 0.1\\ 0& 1\end{array}\right],A_2=\left[\begin{array}{cc}1& 0.2\\ 0& 1\end{array}\right],B_1=\left[\begin{array}{c}0\\ 1\end{array}\right],B_2=\left[\begin{array}{c}0\\ 1.5\end{array}\right].$

And $F_x=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right],u_{jmax}=1,T_j=\left[\begin{array}{c}1\\ 0\end{array}\right]$.

By solving the optimization problem (9), obtain the optimal value 40.2817.

And,

$$P=\left[\begin{array}{cc}10.2147& -7.4640\\ -7.4640& 30.0670\end{array}\right].$$

$$K=\left[\begin{array}{cc}-4.0370& 6.3310\end{array}\right].$$

Then, the maximum robust invariant set of the system (1) with constraint (8) is

$$E\left(P\right)=\left\{x\in R^n:x^T\left[\begin{array}{cc}0.1196& 0.0297\\ 0.0297& 0.0406\end{array}\right]x\le 1\right\}.$$

The solid ellipsoid in Figure 2 demonstrates the robust invariant set derived from Theorem 2. Furthermore, the invariant set obtained by solving the LMI problem (2.30), as mentioned in reference [19], is depicted by the dashed ellipsoid in Figure 2.

Example 3. 

Consider the following uncertain linear discrete-time systems

$$x\left(k+1\right)=A\left(k\right)x\left(k\right)+B\left(k\right)sat\left(kx\left(k\right)\right),$$

with $A_1=\left[\begin{array}{cc}1& 0.1\\ 0& 1\end{array}\right],A_2=\left[\begin{array}{cc}1& 0.2\\ 0& 1\end{array}\right],B_1=\left[\begin{array}{c}0\\ 1\end{array}\right],B_2=\left[\begin{array}{c}0\\ 1.5\end{array}\right].$

And $F_x=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\end{array}\right],g_i=1,u_{max}=1,E_j=0,E_j^{-}=I,T_j=\left[\begin{array}{c}1\\ 0\end{array}\right]$.

Then, the optimization problem (13) is transformed into problem (9) in Example 2, which will not be further elaborated below.

The robust invariant set yielded from Theorem 3 is shown by the solid ellipsoid, the invariant set given by solving the LMI problem (2.55) in [19] is represented by the dashed-dotted ellipsoid in Figure 3.

Validity of Results

The uncertain discrete time in Examples 1–3 are sourced from [19]. Example 1 demonstrates the validity of the approach proposed in this paper. The results from Examples 2 and 3 indicate that the invariant set derived from our method is more extensive than that obtained from [19]. This suggests that the conservatism of our technique is less pronounced compared with that of article [19]. As our proposed method converts a semi-definite program into its dual form, which is essentially a semi-definite program, it does not change the computational complexity or increase the computational cost.

5. Conclusions

This article explores approximating the maximum robust invariant set for uncertain linear discrete-time systems through an optimization viewpoint. It introduces a novel approach to compute the maximum robust invariant set for these systems by leveraging the dual optimization theory of semidefinite programming. The dual problem associated with the original linear matrix inequality optimization challenge is formulated using semidefinite programming, and, subsequently, the LMI optimization issue is addressed. Our findings offer a broader array of methodologies for approximating the maximum robust invariant set of ellipsoids from a dual optimization standpoint. For uncertain discrete linear systems, due to the conservatism of the positive invariant set of ellipsoids generated by quadratic Lyapunov functions, exploring suitable Lyapunov functions to obtain polyhedral positive invariant sets using the dual method proposed in this paper is a future research direction.