H_∞ Control for 2D Singular Continuous Systems

Abstract

This paper considers the problem of admissibility and admissibilization of 2D singular continuous systems described by the Roesser model. A necessary and sufficient admissibility condition is first proposed for 2D singular continuous systems in terms of a strict Linear Matrix Inequality (LMI). Then, a necessary and sufficient condition is established for the closed-loop system to be admissible (i.e., stable, regular, and impulse-free). Moreover, the stability condition is completed to give a sufficient condition to ensure a specified H_∞ disturbance attenuation level for the state feedback closed loop. To illustrate the effectiveness of the proposed methodology, a numerical example is given to illustrate an admissibilization of a state feedback closed-loop system.

1. Introduction

During the last few decades, much attention has been devoted to singular systems, thanks to their capacity to represent the dynamic structure of physical systems involving algebraic constraints.

Many findings on stability and controller/filter design for singular systems in discrete and continuous frameworks have been reported in the literature [1,2,3,4,5,6,7,8,9,10,11]. For example, stability and stabilization of continuous descriptor systems using an LMI approach has been studied in [12]. Robust admissibilization of descriptor systems by static output feedback has been established in [13]. A solid criterion based on strict LMI without invoking equality constraint for stabilization of continuous singular systems has been proposed in [14]. On the other hand, two-dimensional (2D) systems have also attracted considerable attention from researchers since they have many applications in areas such as water stream heating, seismographic data processing, thermal processes, multidimensional digital filtering, process control, image and signal processing, etc. [15,16,17,18,19]. There are several works on 2D continuous systems [15,20,21,22,23,24]. For instance, the stability and stabilization of 2D continuous state-delayed systems have been established in [25]. The problem of $H_{\infty }$ performance analysis for 2D continuous time-varying delay systems has been addressed in [24]. The robust $H_{\infty }$ filtering for uncertain 2D continuous systems based on a polynomial parameter-dependent Lyapunov function has been studied in [26]. Furthermore, 2D singular systems have received great interest from the academic community due to their wide applications to describe physical systems in several practical areas [27,28]. However, a great number of fundamental results on 1D singular systems have been extended to 2D singular systems, but the majority of existing results have been devoted to the discrete case [27,29,30,31,32,33,34,35,36]. It should be pointed out that in [36], we established necessary and sufficient conditions for the admissibility and the admissibilization using strict LMIs for 2D singular systems described by the Roesser model in the discrete form. Recently, based on this result, we proposed sufficient conditions for the regularity, causality, and stability for 2D discrete singular systems in the stochastic case in [37]. Furthermore, we have derived a sufficient mean square asymptotic stability condition with a prescribed $H_{\infty }$ disturbance attenuation level bound. The majority of papers addressing 2D singular systems thus far have focused on the discrete case, with the exception of [31]. In [31], nonstrict LMI conditions for the admissibility of 2D singular continuous systems, derived from existing results for 1D singular systems, have been established. The authors proposed then necessary and sufficient conditions for admissibility and admissibilization expressed in terms of strict LMIs. To the best of our knowledge, there is no significant progress reported on the $H_{\infty }$ control of 2D singular continuous systems which motivates the investigations here.

This paper deals with admissibility, admissibilization, and $H_{\infty }$ control of 2D singular continuous systems. The main contributions of this paper can be summarized as follows:

(1)

The regularity and impulse-freeness of 2D singular continuous systems is investigated.

(2)

Necessary and sufficient conditions dealing with the admissibility and admissibilization of 2D singular continuous systems are established in terms of a strict LMI condition.

(3)

A state feedback controller is designed to ensure the admissibility of the closed-loop system and guarantee a prescribed $H_{\infty }$ disturbance attenuation level bound.

This paper is structured as follows: Section 2 presents the problem formulation. Section 3 studies the regularity and impulse-freeness of 2D singular continuous systems. Section 4 introduces necessary and sufficient admissibility condition in terms of a strict LMI. Section 5 discusses the case of a nonseparable Roesser model. Section 6 provides the design of a state feedback controller that ensures the stability of the closed-loop system. Section 7 proposes sufficient conditions ensuring the admissibility of the closed loop with a specified $H_{\infty }$ disturbance attenuation level. Section 8 gives a numerical example to highlight the usefulness of the proposed approach. The last section provides conclusions drawn from this paper.

Notation 1.

Throughout this paper, we use the following notations:

*

$\mathrm{I}\mathrm{R}$ denotes the set of real values.

*

$\mathrm{Sym}\left\{A\right\}$ stands for the addition of a matrix and its transpose:

$$\mathrm{Sym}\left\{A\right\}=A^T+A.$$

*

$diag\{.\}$ represents a block-diagonal matrix.

*

$int\left(x\right)$ represents the integer part of the real x.

2. Problem Formulation

The class of 2D singular continuous systems under consideration is represented by a Roesser model of the following form:

$$\begin{array}{ccc}\hfill E\left[\begin{array}{c}{\displaystyle \frac{\partial x^h(t_1,t_2)}{\partial t_1}}\\ {\displaystyle \frac{\partial x^v(t_1,t_2)}{\partial t_2}}\end{array}\right]& =& \left[\begin{array}{cc}{A}^{hh}& A^{hv}\\ A^{vh}& A^{vv}\end{array}\right]\left[\begin{array}{c}{x}^h(t_1,t_2)\\ x^v(t_1,t_2)\end{array}\right]+\left[\begin{array}{c}{B}^h\\ B^v\end{array}\right]u(t_1,t_2)\hfill \\ \hfill \end{array}$$

(1)

where $x^h(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{n_h}$ is the horizontal state, $x^v(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{n_v}$ is the vertical state, $u(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^q$ is the control input, $A^{ij}$, $B^i$, ($i,j=h,v$) are real constant matrices of appropriate dimensions, and E is eventually singular.

The boundary conditions are given by

$$\begin{array}{ccc}\hfill x^h(0,t_2)& =& {\varphi }^h(0,t_2),\forall t_2\in {\mathrm{I}\mathrm{R}}_{+}\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill x^v(t_1,0)& =& {\varphi }^v(t_1,0),\forall t_1\in {\mathrm{I}\mathrm{R}}_{+}\hfill \end{array}$$

(3)

where ${\varphi }^h$ and ${\varphi }^v$ are bounded and have compact support, that is, there exist non-negative $L_1$, $L_2$, $T_1$, and $T_2$ such that

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill ∥{\varphi }^h(0,t)∥\le L_1& \mathrm{if}& 0\le t\le T_2\hfill \\ \hfill ∥{\varphi }^h(0,t)∥=0& \mathrm{if}& t>T_2\hfill \end{array}\right.\end{array}$$

(4)

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill ∥{\varphi }^v(t,0)∥\le L_2& \mathrm{if}& 0\le t\le T_1\hfill \\ \hfill ∥{\varphi }^v(t,0)∥=0& \mathrm{if}& t>T_1\hfill \end{array}\right.\end{array}$$

(5)

Definition 1

([34]).

(i) 

The system (1) is said to be regular if $det\left(EI\left(z_1,z_2\right)-A\right)$ is not identically zero, where $I\left(z_1,z_2\right)=diag\left\{z_1{I}_{r_h},z_2{I}_{r_v}\right\}$.

(ii) 

The system (1) is said to be impulse-free if the degree of $det\left(EI\left(z_1,z_2\right)-A\right)$ is equal to $rank\left(E\right)$.

(iii) 

The system (1) is said to be stable if $\lambda \left(E,A\right)\subset D_{int}\left(0,1\right)$.

(iv) 

The system (1) is said to be admissible if it is regular, impulse-free, and stable.

Definition 2

([38]). A 2D singular Roesser model is said to be of the separated standard form if it is of the form (1) with $E=\mathrm{diag}\left\{E^h,E^v\right\}$, where $E^h\in {\mathbb{R}}^{n_h\times n_h}$ and $E^v\in {\mathbb{R}}^{n_v\times n_v}$.

Remark 1.

Note that asymptotic stability requires the horizontal and vertical components $x^h(t_1,t_2)$ and $x^v(t_1,t_2)$ to converge to the equilibrium state $x^h\equiv 0$, $x^v\equiv 0$ as $t\to \infty$ with $t=t_1+t_2$.

3. Regularity and Impulse-Freeness of 2D Singular Continuous Systems

For a singular system to have a smooth solution, it must be both regular and impulse-free. This means the system can be transformed into a standard system, subject to an algebraic constraint on the system state vector.

Let matrices A and B be defined as follows:

$$A=\left[\begin{array}{cc}{A}^{hh}& A^{hv}\\ A^{vh}& A^{vv}\end{array}\right],B=\left[\begin{array}{c}{B}^h\\ B^v\end{array}\right]u(t_1,t_2).$$

Let $U=\mathrm{diag}\left\{U^h,U^v\right\}$, $V=\mathrm{diag}\left\{V^h,V^v\right\}$, where

$U^i$ and $V^i$, $i=h,v$ are some nonsingular matrices such that

$${\overline{E}}^i=U^i{E}^i{V}^i=\left[\begin{array}{cc}{I}_{r^i}& 0\\ 0& 0\end{array}\right]$$

for $i=h,v$.

The same transformation applied to matrix A yields

$$\overline{A}=UAV=\left[\begin{array}{cc}{U}^h{A}^{hh}{V}^h& U^h{A}^{hv}{V}^v\\ U^v{A}^{vh}{V}^h& U^v{A}^{vv}{V}^v\end{array}\right]=\left[\begin{array}{cc}{\overline{A}}^{hh}& {\overline{A}}^{hv}\\ {\overline{A}}^{vh}& {\overline{A}}^{vv}\end{array}\right].$$

Let also $x^i=V^i\left[\begin{array}{c}{\overline{x}}_1^i\\ {\overline{x}}_2^i\end{array}\right]$ with ${\overline{x}}_1^i\in {\mathrm{I}\mathrm{R}}^{r^i}$, for $i=h,v$.

System (1) with $u(t_1,t_2)=0$ can be transformed as follows

$$\begin{array}{ccc}\hfill \left[\begin{array}{c}{\overline{E}}^h{\displaystyle \frac{\partial {\overline{x}}^h(t_1,t_2)}{\partial t_1}}\\ {\overline{E}}^v{\displaystyle \frac{\partial {\overline{x}}^v(t_1,t_2)}{\partial t_2}}\end{array}\right]& =& \left[\begin{array}{cc}{\overline{A}}^{hh}& {\overline{A}}^{hv}\\ {\overline{A}}^{vh}& {\overline{A}}^{vv}\end{array}\right]\left[\begin{array}{c}{\overline{x}}^h(t_1,t_2)\\ {\overline{x}}^v(t_1,t_2)\end{array}\right]\hfill \end{array}$$

(6)

The differential equation in (6) concerning ${\overline{x}}^h(t_1,t_2)$ is then as follows:

$$\begin{array}{ccc}\hfill {\overline{E}}^h{\displaystyle \frac{\partial {\overline{x}}^h(t_1,t_2)}{\partial t_1}}& =& {\overline{A}}^{hh}{\overline{x}}^h(t_1,t_2)+{\overline{A}}^{hv}{\overline{x}}^v(t_1,t_2)\hfill \end{array}$$

(7)

and taking into account the partitioning implied by the structure of ${\overline{E}}^h$, we obtain

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\displaystyle \frac{\partial {\overline{x}}_1^h(t_1,t_2)}{\partial t_1}}\\ 0\end{array}\right]& =\left[\begin{array}{cc}{\overline{A}}_{11}^{hh}& {\overline{A}}_{12}^{hh}\\ {\overline{A}}_{21}^{hh}& {\overline{A}}_{22}^{hh}\end{array}\right]\left[\begin{array}{c}{\overline{x}}_1^h(t_1,t_2)\\ {\overline{x}}_2^h(t_1,t_2)\end{array}\right]+\left[\begin{array}{cc}{\overline{A}}_{11}^{hv}& {\overline{A}}_{12}^{hv}\\ {\overline{A}}_{21}^{hv}& {\overline{A}}_{22}^{hv}\end{array}\right]\left[\begin{array}{c}{\overline{x}}_1^v(t_1,t_2)\\ {\overline{x}}_2^v(t_1,t_2)\end{array}\right]\end{array}$$

(8)

The same can be performed for ${\overline{x}}_1^v(t_1,t_2)$, obtaining

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{\displaystyle \frac{\partial {\overline{x}}_1^v(t_1,t_2)}{\partial t_2}}\\ 0\end{array}\right]& =\left[\begin{array}{cc}{\overline{A}}_{11}^{vh}& {\overline{A}}_{12}^{vh}\\ {\overline{A}}_{21}^{vh}& {\overline{A}}_{22}^{vh}\end{array}\right]\left[\begin{array}{c}{\overline{x}}_1^h(t_1,t_2)\\ {\overline{x}}_2^h(t_1,t_2)\end{array}\right]+\left[\begin{array}{cc}{\overline{A}}_{11}^{vv}& {\overline{A}}_{12}^{vv}\\ {\overline{A}}_{21}^{vv}& {\overline{A}}_{22}^{vv}\end{array}\right]\left[\begin{array}{c}{\overline{x}}_1^v(t_1,t_2)\\ {\overline{x}}_2^v(t_1,t_2)\end{array}\right]\end{array}$$

(9)

Stacking up the different Equations (8) and (9), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\overline{x}}_1(t_1,t_2)}{\partial (t_1,t_2)}}& =& {\overline{\overline{A}}}_{11}{\overline{x}}_1(t_1,t_2)+{\overline{\overline{A}}}_{12}{\overline{x}}_2(t_1,t_2)\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill 0& =& {\overline{\overline{A}}}_{21}{\overline{x}}_1(t_1,t_2)+{\overline{\overline{A}}}_{22}{\overline{x}}_2(t_1,t_2)\hfill \end{array}$$

(11)

with

$$\begin{array}{ccc}\hfill {\overline{x}}_i(t_1,t_2)& =& \left[\begin{array}{c}{\overline{x}}_i^h(t_1,t_2)\\ {\overline{x}}_i^v(t_1,t_2)\end{array}\right],i=1,2\hfill \\ \hfill {\displaystyle \frac{\partial {\overline{x}}_1(t_1,t_2)}{\partial (t_1,t_2)}}& =& \left[\begin{array}{c}{\displaystyle \frac{\partial {\overline{x}}_1^h(t_1,t_2)}{\partial t_1}}\\ {\displaystyle \frac{\partial {\overline{x}}_1^v(t_1,t_2)}{\partial t_2}}\end{array}\right]\hfill \\ \hfill {\overline{\overline{A}}}_{ij}& =& \left[\begin{array}{cc}{\overline{A}}_{ij}^{hh}& {\overline{A}}_{ij}^{hv}\\ {\overline{A}}_{ij}^{vh}& {\overline{A}}_{ij}^{vv}\end{array}\right],\mathrm{for}i=1,2\mathrm{and}j=1,2\hfill \end{array}$$

Equation (10) is a differential equation for a standard system. Its solution is unique and depends on the fast component ${\overline{x}}_2(t_1,t_2)$. Equation (11) shows then that the regularity of matrix ${\overline{\overline{A}}}_{22}$ is necessary and sufficient to obtain a unique solution to system (1).

This development is summarized by the following theorem.

Theorem 1.

System (1) is regular and impulse-free if and only if ${\overline{\overline{A}}}_{22}$ is nonsingular.

If system (1) is regular, then matrix ${\overline{\overline{A}}}_{22}$ is invertible which, with the help of (11), allows one to write (10) as follows

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial {\overline{x}}_1(t_1,t_2)}{\partial (t_1,t_2)}}& =& \left({\overline{\overline{A}}}_{11}-{\overline{\overline{A}}}_{12}{\overline{\overline{A}}}_{22}^{-1}{\overline{\overline{A}}}_{21}\right){\overline{x}}_1(t_1,t_2)\hfill \end{array}$$

(12)

which means that the system will be stable if the matrix

$$\begin{array}{ccc}\hfill \mathcal{A}& =& {\overline{\overline{A}}}_{11}-{\overline{\overline{A}}}_{12}{\overline{\overline{A}}}_{22}^{-1}{\overline{\overline{A}}}_{21}\hfill \end{array}$$

is stable.

4. Admissibility of 2D Singular Continuous Systems

To check the stability of the system, consider the Lyapunov inequality

$$\begin{array}{c}\hfill \mathrm{Sym}\left\{AP{E}^{\top }\right\}+Z<0\end{array}$$

(13)

with $P=\mathrm{diag}\left\{P^h,P^v\right\}$. Matrix Z has to be chosen in order to satisfy (13).

Let

$$\begin{array}{ccc}\hfill J& =& \left[\begin{array}{cccc}{I}_{r^h}& 0& 0& 0\\ 0& 0& I_{n^h-r^h}& 0\\ 0& I_{r^v}& 0& 0\\ 0& 0& 0& I_{n^v-r^v}\end{array}\right]\hfill \end{array}$$

satisfying $J{J}^{\top }=J^{\top }J=I$

$$\begin{array}{ccc}\hfill \mathrm{Sym}\left\{J^{\top }UAP{E}^{\top }{U}^{\top }J\right\}& =& \mathrm{Sym}\left\{\left(J^{\top }UAVJ\right)\left(J^{\top }{V}^{-1}P{V}^{-\top }J\right)\left(J^{\top }{V}^{\top }{E}^{\top }{U}^{\top }J\right)\right\}\hfill \\ & =& \mathrm{Sym}\left\{\left(J^{\top }\overline{A}J\right)\left(J^{\top }\overline{P}J\right)\left(J^{\top }{\overline{E}}^{\top }J\right)\right\}\hfill \\ & =& \mathrm{Sym}\left\{\overline{\overline{A}}\overline{\overline{P}}{\overline{\overline{E}}}^{\top }\right\}\hfill \end{array}$$

Notice that

$$\begin{array}{ccc}\hfill \overline{\overline{A}}& =& J^{\top }\overline{A}J=\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\hfill \\ \hfill \overline{\overline{E}}& =& J^{\top }\overline{E}J=\left[\begin{array}{cccc}{I}_{r^h}& 0& 0& 0\\ 0& I_{r^v}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\hfill \\ \hfill \overline{\overline{P}}& =& J^{\top }\overline{P}J=\left[\begin{array}{cccc}{\overline{P}}_{11}^{hh}& 0& {\overline{P}}_{12}^{hh}& 0\\ 0& {\overline{P}}_{11}^{vv}& 0& {\overline{P}}_{12}^{vv}\\ {\overline{P}}_{21}^{hh}& 0& {\overline{P}}_{22}^{hh}& 0\\ 0& {\overline{P}}_{21}^{vv}& 0& {\overline{P}}_{22}^{vv}\end{array}\right]=\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& {\overline{\overline{P}}}_{12}\\ {\overline{\overline{P}}}_{21}& {\overline{\overline{P}}}_{22}\end{array}\right]\hfill \end{array}$$

It comes then that

$$\begin{array}{ccc}\hfill \mathrm{Sym}\left\{\overline{\overline{A}}\overline{\overline{P}}{\overline{\overline{E}}}^{\top }\right\}& =& \mathrm{Sym}\left\{\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& 0\\ {\overline{\overline{P}}}_{21}& 0\end{array}\right]\right\}\hfill \end{array}$$

(14)

Notice that in (14), matrix ${\overline{\overline{A}}}_{22}$ is eliminated, and as a consequence, matrix Z in (13) should include matrix ${\overline{\overline{A}}}_{22}$ in order to constraint its invertibility, which can be insured if there exists a symmetric matrix ${\overline{\overline{Y}}}_{22}$ such as ${\overline{\overline{A}}}_{22}{\overline{\overline{Y}}}_{22}{\overline{\overline{A}}}_{22}^{\top }<0$. Notice that

$$\begin{array}{cc}\hfill \left[\begin{array}{cc}\ast & \ast \\ \ast & {\overline{\overline{A}}}_{22}{\overline{\overline{Y}}}_{22}{\overline{\overline{A}}}_{22}^{\top }\end{array}\right]& =\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& {\overline{\overline{Y}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}^{\top }& {\overline{\overline{A}}}_{21}^{\top }\\ {\overline{\overline{A}}}_{12}^{\top }& {\overline{\overline{A}}}_{22}^{\top }\end{array}\right]\hfill \\ \hfill & =\overline{\overline{A}}\left(I-\overline{\overline{E}}\right)\left(\overline{\overline{Y}}\right){\left(I-\overline{\overline{E}}\right)}^{\top }{\overline{\overline{A}}}^{\top }\hfill \\ \hfill & =J^{\top }UZ{U}^{\top }J\hfill \end{array}$$

(15)

with

$$\begin{array}{ccc}\hfill Z& =& A{E}^{†}Y{E^{†}}^{\top }{A}^{\top }\hfill \end{array}$$

(16)

and $E^{†}=V\left(I-UEV\right)U$. As a consequence, condition (13) becomes

$$\begin{array}{c}\hfill \mathrm{Sym}\left\{AP{E}^{\top }\right\}+A{E}^{†}Y{E^{†}}^{\top }{A}^{\top }\end{array}$$

(17)

If we complete (14) with (15), we obtain

$$\begin{array}{c}{\displaystyle \mathrm{Sym}\left\{\overline{\overline{A}}\overline{\overline{P}}\overline{\overline{E}}\right\}+\overline{\overline{A}}\left(I-\overline{\overline{E}}\right)\left(\overline{\overline{Y}}\right){\left(I-\overline{\overline{E}}\right)}^{\top }{\overline{\overline{A}}}^{\top }}\hfill \\ \hspace{1em}=\mathrm{Sym}\left\{\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& 0\\ {\overline{\overline{P}}}_{21}& 0\end{array}\right]\right\}+\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& {\overline{\overline{Y}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}^{\top }& {\overline{\overline{A}}}_{21}^{\top }\\ {\overline{\overline{A}}}_{12}^{\top }& {\overline{\overline{A}}}_{22}^{\top }\end{array}\right]\hfill \end{array}$$

(18)

which implies that, if (12) holds, matrix ${\overline{\overline{A}}}_{22}$ satisfies ${\overline{\overline{A}}}_{22}{\overline{\overline{Y}}}_{22}{\overline{\overline{A}}}_{22}^{\top }<0$ which, in fact, yields the invertibility of matrix ${\overline{\overline{A}}}_{22}$.

Notice that the invertibility of matrix ${\overline{\overline{A}}}_{22}$ implies that the system is regular and impulse-free.

At this step, we have to check the stability of system (12). For this aim, let us left-multiply condition (18) by

$$S_1=\left[\begin{array}{cc}I& -{\overline{\overline{A}}}_{12}{\overline{\overline{A}}}_{22}^{-1}\\ 0& I\end{array}\right]$$

and right-multiply by $S_1^{\top }$ as follows:

$$\begin{array}{c}\hfill {\displaystyle \mathrm{Sym}\left\{S_1\overline{\overline{A}}\overline{\overline{P}}\overline{\overline{E}}{S}_1^{\top }\right\}+S_1\overline{\overline{A}}\left(I-\overline{\overline{E}}\right)\left(\overline{\overline{Y}}\right){\left(I-\overline{\overline{E}}\right)}^{\top }{\overline{\overline{A}}}^{\top }{S}_1^{\top }}\\ \hspace{1em}=\mathrm{Sym}\left\{S_1\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& 0\\ {\overline{\overline{P}}}_{21}& 0\end{array}\right]S_1^{\top }\right\}\hfill \\ \hspace{1em}+S_1\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}& {\overline{\overline{A}}}_{12}\\ {\overline{\overline{A}}}_{21}& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& {\overline{\overline{Y}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{A}}}_{11}^{\top }& {\overline{\overline{A}}}_{21}^{\top }\\ {\overline{\overline{A}}}_{12}^{\top }& {\overline{\overline{A}}}_{22}^{\top }\end{array}\right]S_1^{\top }\hfill \end{array}$$

(19)

In order to transform $\overline{\overline{A}}$ into a block diagonal matrix, we introduce matrix

$$S_2=\left[\begin{array}{cc}I& 0\\ -{\overline{\overline{A}}}_{22}^{-1}{\overline{\overline{A}}}_{21}& I\end{array}\right]$$

into (19) to obtain

$$\begin{array}{c}{\displaystyle \mathrm{Sym}\left\{S_1\overline{\overline{A}}{S}_2{S}_2^{-1}\overline{\overline{P}}\overline{\overline{E}}{S}_1^{\top }\right\}+S_1\overline{\overline{A}}{S}_2{S}_2^{-1}\left(I-\overline{\overline{E}}\right)\left(\overline{\overline{Y}}\right){\left(I-\overline{\overline{E}}\right)}^{\top }{S}_2^{-\top }{S}_2^{\top }{\overline{\overline{A}}}^{\top }{S}_1^{\top }}\hfill \\ \hspace{1em}=\mathrm{Sym}\left\{\left[\begin{array}{cc}\mathcal{A}& 0\\ 0& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& 0\\ {\overline{\overline{A}}}_{22}^{-1}{\overline{\overline{A}}}_{21}{\overline{\overline{P}}}_{11}+{\overline{\overline{P}}}_{21}& 0\end{array}\right]\right\}+\left[\begin{array}{cc}\mathcal{A}& 0\\ 0& {\overline{\overline{A}}}_{22}\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& {\overline{\overline{Y}}}_{22}\end{array}\right]{\left[\begin{array}{cc}\mathcal{A}& 0\\ 0& {\overline{\overline{A}}}_{22}\end{array}\right]}^{\top }\hfill \end{array}$$

(20)

from which we obtain

$$\begin{array}{c}\hfill \mathrm{Sym}\left\{\mathcal{A}{\overline{\overline{P}}}_{11}\right\}<0\end{array}$$

(21)

a condition that states the stability of system (12) if matrix ${\overline{\overline{P}}}_{11}$ is positive definite. A condition that imposes the positivity on ${\overline{\overline{P}}}_{11}$ could be given as follows:

$$\begin{array}{c}\hfill -\mathrm{Sym}\left\{\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& 0\\ {\overline{\overline{P}}}_{21}& 0\end{array}\right]\right\}+\left[\begin{array}{cc}0& 0\\ 0& {\overline{\overline{X}}}_{22}\end{array}\right]<0\end{array}$$

Notice that

$$\begin{array}{ccc}& & -\mathrm{Sym}\left\{\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& 0\\ {\overline{\overline{P}}}_{21}& 0\end{array}\right]\right\}+\left[\begin{array}{cc}0& 0\\ 0& {\overline{\overline{X}}}_{22}\end{array}\right]\hfill \\ & =& -\mathrm{Sym}\left\{\left[\begin{array}{cc}{\overline{\overline{P}}}_{11}& {\overline{\overline{P}}}_{12}\\ {\overline{\overline{P}}}_{21}& {\overline{\overline{P}}}_{22}\end{array}\right]\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]\right\}+\left[\begin{array}{cc}0& 0\\ 0& I\end{array}\right]\left[\begin{array}{cc}{\overline{\overline{X}}}_{11}& {\overline{\overline{X}}}_{12}\\ {\overline{\overline{X}}}_{21}& {\overline{\overline{X}}}_{22}\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& I\end{array}\right]\hfill \\ & =& -\mathrm{Sym}\left\{J^{\top }{V}^{-1}P{V}^{-\top }J\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]\right\}+\left[\begin{array}{cc}0& 0\\ 0& I\end{array}\right]\overline{\overline{X}}\left[\begin{array}{cc}0& 0\\ 0& I\end{array}\right]<0\hfill \end{array}$$

The previous condition is equivalent to

$$\begin{array}{c}\hfill -\mathrm{Sym}\left\{P{V}^{-\top }J\left[\begin{array}{cc}I& 0\\ 0& 0\end{array}\right]J^{\top }{V}^{\top }\right\}+VJ\left[\begin{array}{cc}0& 0\\ 0& I\end{array}\right]\overline{\overline{X}}\left[\begin{array}{cc}0& 0\\ 0& I\end{array}\right]J^{\top }{V}^{\top }<0\\ \hfill -\mathrm{Sym}\left\{P{V}^{-\top }{\overline{E}}^{\top }{V}^{\top }\right\}+VJ\left(I-\overline{\overline{E}}\right)\overline{\overline{X}}{\left(I-\overline{\overline{E}}\right)}^{\top }{J}^{\top }{V}^{\top }<0\\ \hfill -\mathrm{Sym}\left\{P{V}^{-\top }{\overline{E}}^{\top }{V}^{\top }\right\}+VJ\left(I-J^{\top }\overline{E}J\right)\overline{\overline{X}}{\left(I-J^{\top }\overline{E}J\right)}^{\top }{J}^{\top }{V}^{\top }<0\\ \hfill -\mathrm{Sym}\left\{P{V}^{-\top }{\overline{E}}^{\top }{V}^{\top }\right\}+V\left(I-\overline{E}\right)J\overline{\overline{X}}{J}^{\top }{\left(I-\overline{E}\right)}^{\top }{V}^{\top }<0\\ \hfill -\mathrm{Sym}\left\{P{V}^{-\top }{\overline{E}}^{\top }{V}^{\top }\right\}+\left(V\left(I-\overline{E}\right)U\right)U^{-1}\overline{X}{U}^{-\top }{\left(V\left(I-\overline{E}\right)U\right)}^{\top }>0\\ \hfill \mathrm{Sym}\left\{P{E}^{‡}\right\}+E^{†}X{E}^{{†}^{\top }}<0\end{array}$$

with

$$\begin{array}{ccc}\hfill E^{‡}& =& {\left(VUE\right)}^{\top }\hfill \\ \hfill \overline{X}& =& UX{U}^{\top }\hfill \end{array}$$

Finally, if there exists a symmetric matrix X such that the condition

$$\begin{array}{c}\hfill -\mathrm{Sym}\left\{P{E}^{‡}\right\}+E^{†}X{E}^{{†}^{\top }}<0\end{array}$$

(22)

holds, then ${\overline{\overline{P}}}_{11}$ in (21) is positive definite.

The previous development is summarized by the following theorem

Theorem 2.

System (1) is admissible if and only if there exist matrices P, Y, and X such that (17) and (22) are satisfied.

Theorem 2 gives a necessary and sufficient condition with two LMIs. These LMIs can, in fact, be transformed into a single LMI thanks to the elimination lemma [39]. Indeed, notice that (13) can be written as

$$\begin{array}{ccc}\hfill \mathrm{Sym}\left\{AP{E}^{\top }\right\}+A{E}^{†}X{E^{†}}^{\top }{A}^{\top }& =& \left[\begin{array}{cc}I& A\end{array}\right]\left[\begin{array}{cc}0& EP\\ P{E}^{\top }& E^{†}X{E^{†}}^{\top }\end{array}\right]\left[\begin{array}{c}I\\ A^{\top }\end{array}\right]\hfill \end{array}$$

(23)

where we used matrix X instead of Y, and condition (22) can also be written as

$$\begin{array}{ccc}\hfill -\mathrm{Sym}\left\{P{E}^{‡}\right\}+E^{†}X{E}^{{†}^{\top }}& =& \left[\begin{array}{cc}VU& -I\end{array}\right]\left[\begin{array}{cc}0& EP\\ P{E}^{\top }& E^{†}X{E^{†}}^{\top }\end{array}\right]\left[\begin{array}{c}{U}^{\top }{V}^{\top }\\ -I\end{array}\right]\hfill \end{array}$$

(24)

and according to the elimination lemma, conditions (23) and (24) hold if and only if there exists a matrix G of appropriate dimension such that

$$\begin{array}{c}\hfill \left[\begin{array}{cc}0& EP\\ P{E}^{\top }& E^{†}X{E^{†}}^{\top }\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}A\\ -I\end{array}\right]G\left[\begin{array}{cc}I& U^{\top }{V}^{\top }\end{array}\right]\right\}<0\end{array}$$

(25)

The development above allows one to state the following Theorem

Theorem 3.

System (1) is admissible if there exist matrices P, X, and G such that (25) is satisfied.

5. Case of Nonseparable Roesser Model

In this section, we consider the case where matrix E is not in a separable form, and we let U and V be two nonsingular matrices that satisfy

$$UEV=\left[\begin{array}{cc}{I}_r& 0\\ 0& 0\end{array}\right]$$

with $r<n_h+n_v$. Let $r_h$ and $r_v$ be such that $r=r_h+r_v$. For instance, we can take $r_h=Int(r/2)$ and $r_v=r-r_h$, where $Int\left(x\right)$ represents the integer part of the real x.

Also, let

$$\begin{array}{ccc}\hfill V\left[\begin{array}{c}{\overline{x}}_1^h(t_1,t_2)\\ {\overline{x}}_1^v(t_1,t_2)\\ {\overline{x}}_2^h(t_1,t_2)\\ {\overline{x}}_2^v(t_1,t_2)\end{array}\right]& =& \left[\begin{array}{c}{x}^h(t_1,t_2)\\ x^v(t_1,t_2)\end{array}\right]\hfill \end{array}$$

with ${\overline{x}}_1^h(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{r_h}$, ${\overline{x}}_1^v(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{r_v}$, ${\overline{x}}_2^h(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{n_h-r_h}$, and ${\overline{x}}_2^v(t_1,t_2)\in {\mathrm{I}\mathrm{R}}^{n_v-r_v}$. With these notations, we obtain Equations (10) and (11), and the same argument can be processed on. It is noticeable that the main difference between the case where matrix E is in a separable form and the case where the form is not is the definition of the various vectors ${\overline{x}}_i^j(t_1,t_2)$, $i=1,2$ and $j=h,v$. In the case where E is in a separable form, these vectors are well-defined and stem directly from the transformations that lead to (8) and (9). Besides, for the case where E is not in a separable form, the dimensions of the vectors ${\overline{x}}_i^j(t_1,t_2)$, $i=1,2$, and $j=h,v$ are arbitrary with the constraint that $r=r_h+r_v$, but we can choose any $r_h$ and $r_v$ that satisfy $r=r_h+r_v$. In the presentation above, we chose a trivial solution where $r_h=int(r/2)$. If r is even, then we obtain $r_h=r_v$.

6. Admissibilization

In this section, we address the problem of stabilization by state feedback for the 2D singular continuous system given by (1).

The control law given by a state feedback is then

$$\begin{array}{ccc}\hfill u(t_1,t_2)& =& K\left[\begin{array}{c}{x}^h(t_1,t_2)\\ x^v(t_1,t_2)\end{array}\right]\hfill \end{array}$$

where the matrix gain K, of appropriate dimension, is computed in a way that the singular closed-loop system is admissible.

Theorem 4.

The closed-loop 2D singular system is admissible with the state feedback

$$\begin{array}{ccc}\hfill K& =& R{G}^{-1}\hfill \end{array}$$

(26)

if and only if the following LMI is feasible in variables P, G, and R.

$$\begin{array}{c}\hfill \left[\begin{array}{cc}0& {\left(P{E}^{\top }\right)}^{\top }\\ P{E}^{\top }& E^{†}Y{E}^{{†}^{\top }}\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}AG+BR\\ -G\end{array}\right]\left[\begin{array}{cc}I& U^{\top }{V}^{\top }\end{array}\right]\right\}<0\end{array}$$

(27)

Proof of Theorem 4.

Theorem 6 is a simple application of Theorem 3 to the closed-loop system with A replaced by $A+BK$. A change in variable allows one to express the state feedback gain as in (26). □

7. ${\mathit{H}}_{\infty }$ Performance Analysis

In this section, we give a sufficient condition to insure a $H_{\infty }$ bound on the transfer matrix of the closed-loop system.

The 2D closed-loop singular continuous system is given by the following form:

$$\begin{array}{ccc}\hfill E\left[\begin{array}{c}{\displaystyle \frac{\partial x^h(t_1,t_2)}{\partial t_1}}\\ {\displaystyle \frac{\partial x^v(t_1,t_2)}{\partial t_2}}\end{array}\right]& =& \left[\begin{array}{cc}{A}_c^{hh}& A_c^{hv}\\ A_c^{vh}& A_c^{vv}\end{array}\right]\left[\begin{array}{c}{x}^h(t_1,t_2)\\ x^v(t_1,t_2)\end{array}\right]+\left[\begin{array}{c}{B}_{\omega }^h\\ B_{\omega }^v\end{array}\right]\omega (t_1,t_2)\hfill \\ \hfill z(t_1,t_2)& =& Cx(t_1,t_2)+D\omega (t_1,t_2)\hfill \end{array}$$

(28)

where $A_c=\left[\begin{array}{cc}{A}_c^{hh}& A_c^{hv}\\ A_c^{vh}& A_c^{vv}\end{array}\right]=A+BK$ and $\omega ((t_1,t_2)$ is a bounded exogenous signal.

Theorem 7 states this result.

Theorem 5.

If there exist P, G, Y, X, K for a given γ such that the inequality

$$\begin{array}{ccc}\hfill \Sigma & =& \left[\begin{array}{cccc}\mathrm{Sym}\left\{AG\right\}& EP+\left(AG+BX\right)U^{\top }{V}^{\top }-G^{\top }& EP{C}^{\top }& B_{\omega }\\ P{E}^{\top }+VU{\left(AG+BX\right)}^{\top }-G^{\top }& E^{†}Y{E^{†}}^{\top }-\mathrm{Sym}\left\{G{U}^{\top }{V}^{\top }\right\}& 0& 0\\ CP{E}^{\top }& 0& -\gamma I& D\\ B_{\omega }^{\top }& 0& D^{\top }& -\gamma I\end{array}\right]<0\end{array}$$

(29)

holds, then system (28) is admissible and satisfies a $H_{\infty }$ bound $\overline{\gamma }$ with

$$\begin{array}{c}\hfill {\overline{\gamma }}^2={\gamma }^2+\gamma \left|{\lambda }_{min}\left[C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }\right]\right|\mathrm{and}K=X{G}^{-1}.\end{array}$$

Proof of Theorem 5.

First, let $\Sigma$ be partitioned as

$$\begin{array}{c}\hfill \Sigma =\left[\begin{array}{cc}\hfill {\Sigma }_{11}& {\Sigma }_{12}\\ \hfill {\Sigma }_{21}& {\Sigma }_{22}\end{array}\right]\end{array}$$

and replace $\left(AG+BX\right)$ by $A_{c}G$.

Then, condition (5) implies that

$$\begin{array}{ccc}\hfill {\Sigma }_{11}& =& \left[\begin{array}{cc}0& EP\\ P{E}^{\top }& E^{†}X{E^{†}}^{\top }\end{array}\right]+\mathrm{Sym}\left\{\left[\begin{array}{c}{A}_c\\ -I\end{array}\right]G\left[\begin{array}{cc}I& U^{\top }{V}^{\top }\end{array}\right]\right\}<0\hfill \end{array}$$

which is condition (25), which implies the admissibility of the system. Moreover, applying a Schur complement to condition (5) yields

$$\begin{array}{c}\hfill {\Sigma }_{11}-\left[\begin{array}{cc}EP{C}^{\top }& B_{\omega }\\ 0& 0\end{array}\right]{\left[\begin{array}{cc}-\gamma I& D\\ D^{\top }& -\gamma I\end{array}\right]}^{-1}\left[\begin{array}{cc}CP{E}^{\top }& 0\\ B_{\omega }^{\top }& 0\end{array}\right]<0\end{array}$$

(30)

Multiplying both sides of (30) by $\left[\begin{array}{c}I\\ A_c\end{array}\right]$ and its transpose, we obtain

$$\begin{array}{c}{\displaystyle \mathrm{Sym}\left\{A_{c}P{E}^{\top }\right\}+A_c{E}^{†}Y{E^{†}}^{\top }{A}_c^{\top }+{\gamma }^{-1}{B}_{\omega }{B}_{\omega }^{\top }+}\hfill \\ \hspace{1em}{\gamma }^{-1}{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)}^{\top }{R}^{-1}\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)<0\hfill \end{array}$$

which we write as follows:

$$\begin{array}{c}{\displaystyle \mathrm{Sym}\left\{\gamma A_{c}P{E}^{\top }\right\}+\gamma A_c{E}^{†}Y{E^{†}}^{\top }{A}_c^{\top }+B_{\omega }{B}_{\omega }^{\top }+}\hfill \\ \hspace{1em}\hspace{1em}{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)}^{\top }{R}^{-1}\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)<0\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill R& =& \left({\gamma }^{2}I-D{D}^{\top }\right)\hfill \end{array}$$

In the condition above, we replace $A_c$ by $\left(\mathbb{I}(s_1,s_2)E-A_c\right)+\mathbb{I}(s_1,s_2)E$, which allows one to write

$$\begin{array}{c}{\displaystyle -\mathrm{Sym}\left\{\gamma \left(\mathbb{I}(s_1,s_2)E-A_c\right)P{E}^{\top }\right\}+{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)}^{\top }{R}^{-1}\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)+}\hfill \\ B_{\omega }{B}_{\omega }^{\top }+\gamma \left(\mathbb{I}(s_1,s_2)E-A_c\right)E^{†}Y{E^{†}}^{\top }{\left(\mathbb{I}(-s_1,-s_2)E-A_c\right)}^{\top }<0\hfill \end{array}$$

(31)

where we made use of the equality

$$\begin{array}{ccc}\hfill \mathrm{Sym}\left\{\mathbb{I}(s_1,s_2)EP{E}^{\top }\right\}& =& 0.\hfill \end{array}$$

and the fact that $E{E}^{†}=0$ with $E^{†}=V\left(I-UEV\right)U$.

Now, define

$$\begin{array}{ccc}\hfill \Phi (s_1,s_2)& =& {\left(\mathbb{I}(s_1,s_2)E-A_c\right)}^{-1}\hfill \\ \hfill {\Phi }^{\ast }(s_1,s_2)& =& {\left(\mathbb{I}(-s_1,-s_2)E-A_c\right)}^{-\top }\hfill \end{array}$$

and multiplying (31) on the left by $C\Phi (s_1,s_2)$ and on the right by ${\Phi }^{\ast }(s_1,s_2)C^{\top }$, we obtain

$$\begin{array}{c}{\displaystyle -\mathrm{Sym}\left\{\gamma CP{E}^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}+\gamma C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }+{\gamma }^{-1}C\Phi (s_1,s_2)B_{\omega }{B}_{\omega }^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }}\hfill \\ \hspace{1em}\hspace{1em}+C\Phi (s_1,s_2){\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)}^{\top }{R}^{-1}\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right){\Phi }^{\ast }(s_1,s_2)C^{\top }<0\hfill \end{array}$$

(32)

The first term in (32) can be expanded as

$$\begin{array}{ccc}\hfill -\mathrm{Sym}\left\{\gamma CP{E}^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}& =& -\mathrm{Sym}\left\{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right){\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}+\hfill \\ & & \mathrm{Sym}\left\{D{B}_{\omega }^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}\hfill \end{array}$$

which yields

$$\begin{array}{c}{\displaystyle -\mathrm{Sym}\left\{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right){\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}+\gamma C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }+\mathrm{Sym}\left\{D{B}_{\omega }^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}+}\hfill \\ \hspace{1em}C\Phi (s_1,s_2)\left(B_{\omega }{B}_{\omega }^{\top }+{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)}^{\top }{R}^{-1}\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)\right){\Phi }^{\ast }(s_1,s_2)C^{\top }<0.\hfill \end{array}$$

(33)

Recall that

$$\begin{array}{ccc}\hfill M(s_1,s_2)M^{\ast }(s_1,s_2)& =& D{D}^{\top }+\mathrm{Sym}\left\{D{B}_{\omega }^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}+C\Phi (s_1,s_2)B_{\omega }{B}_{\omega }^{\top }{\Phi }^{\ast }(s_1,s_2)C^{\top }\hfill \end{array}$$

then, condition (33) reads

$$\begin{array}{c}{\displaystyle \left(-{\gamma }^{2}I+M(s_1,s_2)M^{\ast }(s_1,s_2)\right)+\left({\gamma }^{2}I-D{D}^{\top }\right)-\mathrm{Sym}\left\{\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right){\Phi }^{\ast }(s_1,s_2)C^{\top }\right\}+}\hfill \\ \hspace{1em}C\Phi (s_1,s_2){\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)}^{\top }{R}^{-1}\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right){\Phi }^{\ast }(s_1,s_2)C^{\top }+\gamma C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }<0\hfill \end{array}$$

and noticing the true square term in the expression above with

$$\begin{array}{ccc}\hfill L(s_1,s_2)& =& R-C\Phi (s_1,s_2)\left(\gamma CP{E}^{\top }+D{B}_{\omega }^{\top }\right)\hfill \end{array}$$

we obtain

$$\begin{array}{c}\hfill \left(-{\gamma }^{2}I+M(s_1,s_2)M^{\ast }(s_1,s_2)\right)+\gamma C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }+L(s_1,s_2)L^{\ast }(s_1,s_2)<0\end{array}$$

which implies that

$$\begin{array}{c}\hfill \left(-{\gamma }^{2}I+M(s_1,s_2)M^{\ast }(s_1,s_2)\right)+\gamma C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }<0\end{array}$$

or more precisely

$$\begin{array}{c}\hfill M(s_1,s_2)M^{\ast }(s_1,s_2)<{\gamma }^{2}I+\gamma \left|{\lambda }_{min}\left[C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }\right]\right|={\overline{\gamma }}^{2}I\end{array}$$

since $Y\le 0$, this means that ${∥M(s_1,s_2)∥}_{\infty }<\overline{\gamma }$, which implies the $H_{\infty }$ bound $\overline{\gamma }$ on the closed-loop system.

Let $ϵ$ be chosen as the minimal positive solution to

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\gamma }}\left(2\gamma ϵ+{ϵ}^2\right)-\left|{\lambda }_{min}\left[C{E}^{†}Y{E^{†}}^{\top }{C}^{\top }\right]\right|>0\end{array}$$

(34)

and $\overline{\gamma }$ could be chosen as $\overline{\gamma }=\gamma +ϵ$. If matrix Y is chosen in such a way that $ϵ$ is very small, then $\overline{\gamma }$ will be in the vicinity of $\gamma$. □

Remark 2.

It should be pointed out that we are interested, in this paper, in the $H_{\infty }$ control of a 2D singular continuous system in the nominal case. Indeed, the study of the robustness of the model against parametric uncertainties could be achieved by considering parameter uncertainties, which can be assumed to be norm-bounded [40], polytopic [28], or described by a linear fractional representation [41].

8. Example

To highlight the effectiveness of the proposed approach, a numerical example is now developed. Consider the 2D singular continuous system in (1) with the following system matrices:

$$\begin{array}{ccc}A& =& \left[\begin{array}{ccccc}-0.7500& 0.4250& -0.1750& 0.4050& 0.3950\\ 0.0750& -0.2250& 0.4275& -0.3750& 0.4250\\ -0.2950& 1.2150& 0.3450& -0.1750& 0.4000\\ -0.0250& 0.5750& 0.4250& 0.0750& 0.2500\\ 0.2500& 0.8750& 0.3450& 0.0750& 0.4000\end{array}\right]\\ \hfill E& =& \left[\begin{array}{ccccc}\hfill 1& 0& 0\hfill & 0& 0\\ \hfill 0& 1& 0\hfill & 0& 0\\ \hfill 0& 0& 0\hfill & 0& 0\\ \hfill 0& 0& 0\hfill & 1& 0\\ \hfill 0& 0& 0\hfill & 0& 0\end{array}\right],U=\left[\begin{array}{ccccc}\hfill 0& 1& 0\hfill & 0& 0\\ \hfill 1& 0& 0\hfill & 0& 0\\ \hfill 0& 0& 1\hfill & 0& 0\\ \hfill 0& 0& 0\hfill & 1& 0\\ \hfill 0& 0& 0\hfill & 0& 1\end{array}\right]\hfill \\ \hfill V& =& \left[\begin{array}{ccccc}\hfill 0& 1& 0\hfill & 0& 0\\ \hfill 1& 0& 0\hfill & 0& 0\\ \hfill 0& 0& 1\hfill & 0& 0\\ \hfill 0& 0& 0\hfill & 1& 0\\ \hfill 0& 0& 0\hfill & 0& 1\end{array}\right]\hfill \\ \hfill B& =& {\left[\begin{array}{ccccc}\hfill 1.00& -0.10& 2.20\hfill & 0.50& 0.70\end{array}\right]}^{\top }.\hfill \end{array}$$

The open-loop system is not regular, with the corresponding matrix ${\overline{\overline{A}}}_{22}$ being not invertible.

By solving the LMI (27), the following feasible solution can be found:

$$\begin{array}{ccc}P& =& \left[\begin{array}{ccccc}1.0018& -0.3591& -0.1420& 0& 0\\ -0.3591& 1.0482& 0.0508& 0& 0\\ -0.1420& 0.0508& 0& 0& 0\\ 0& 0& 0& 0.2649& 0.1804\\ 0& 0& 0& 0.1804& 0\end{array}\right]\\ G& =& \left[\begin{array}{ccccc}-0.1079& 0.5117& -0.1550& 0.1530& -0.0364\\ 0.5311& -0.0514& 0.4227& -0.1214& -0.0478\\ -0.4449& 0.0317& -0.5058& -0.3204& -0.1699\\ -0.0463& 0.0046& 0.1783& 0.2721& 0.0015\\ -0.2428& -0.0595& -0.0878& -0.0251& -0.3444\end{array}\right]\\ R& =& \left[\begin{array}{ccccc}-0.0444& 0.0215& -0.3406& 0.1722& 0.0741\end{array}\right]\\ K& =& \left[\begin{array}{ccccc}0.2712& -2.6693& -3.6971& -4.8858& 1.9290\end{array}\right]\end{array}$$

Some simulated results are now presented, with null boundary values for $T_1>50$ or $T_2>50$. To facilitate the visualization, we plot in Figure 1 the evolution of the two following quantities, which correspond to the norm of the generalized vector along the diagonal $t_1+t_2=t$, that is,

$$\begin{array}{c}\hfill X^h\left(t\right)={\int }_0^t{∥x^h(\theta ,t-\theta )∥}^{2}d\theta \mathrm{and}{X}^v\left(t\right)={\int }_0^t{∥x^v(\theta ,t-\theta )∥}^{2}d\theta .\end{array}$$

This makes it possible to compress the evolution of the 2D system using 1D plots. It can be seen in Figure 1 that $X^h\left(t\right)$ and $X^v\left(t\right)$ go to zero as $\theta$ goes to infinity, indicating that the system is stable.

9. Conclusions

This paper tackles the problem of admissibility and admissibilization of 2D Roesser singular continuous systems. The proposed results are expressed in terms of a strict LMI. A necessary and sufficient admissibility condition is elaborated. Then, a necessary and sufficient condition is established for the design of a state feedback controller that ensures the admissibility of the closed system. Moreover, a state feedback controller is designed to achieve a prescribed $H_{\infty }$ performance level for the 2D singular continuous system. Finally, a numerical example is provided to illustrate the effectiveness of the obtained results.