Computer Systems Science & Engineering DOI:10.32604/csse.2021.016475 | |

Article |

Output Feedback Robust H∞ Control for Discrete 2D Switched Systems

1School of Electronic Information and Engineering, Changchun University of Science and Technology, Changchun, 130000, China

2Department of Computer Science and Engineering, University of California, Riverside, CA92521, USA

*Corresponding Author: Yang Yang. Email: cloneyang@126.com

Received: 03 January 2021; Accepted: 17 February 2021

Abstract: The two-dimensional (2-D) system has a wide range of applications in different fields, including satellite meteorological maps, process control, and digital filtering. Therefore, the research on the stability of 2-D systems is of great significance. Considering that multiple systems exist in switching and alternating work in the actual production process, but the system itself often has external perturbation and interference. To solve the above problems, this paper investigates the output feedback robust H∞ stabilization for a class of discrete-time 2-D switched systems, which the Roesser model with uncertainties represents. First, sufficient conditions for exponential stability are derived via the average dwell time method, when the system’s interference and external input are zero. Furthermore, in the case of introducing the external interference, the weighted robust H∞ disturbance attenuation performance of the underlying system is further analyzed. An output feedback controller is then proposed to guarantee that the resulting closed-loop system is exponentially stable and has a prescribed disturbance attenuation level γ. All theorems mentioned in the article will also be given in the form of linear matrix inequalities (LMI). Finally, a numerical example is given, which takes two uncertain values respectively and solves the output feedback controller’s parameters by the theorem proposed in the paper. According to the required controller parameter values, the validity of the theorem proposed in the article is compared and verified by simulation.

Keywords: 2-D systems; robust H∞ control; LMI; output feedback; switched systems; Roesser model

The issue of stability analysis and controller synthesis is a hot research topic. Reference Medvedeva et al. [1–3] investigated the stability of one-dimensional (1-D) continuous-time or discrete-time systems. Considering the complexity of many manufacturing processes and physical phenomena, a 2-D continuous-time or discrete-time system that depends on two independent variables has its irreplaceable application area. 2-D systems have attracted considerable research attention in control theory and practice over the past few decades due to their wide applications. Reference Du et al. [4–7] showed multi-dimensional digital filtering, linear image processing, signal processing, and process control. Different models such as the Roesser model and Fornasini–Marchesini model can represent 2-D systems, and the stability issues concerning these two models can be found [8,9].

On the other hand, considerable interest has been devoted to the research of switched systems during the recent decades. A switched system comprises a family of subsystems described by continuous or discrete-time dynamics and a switching law that specifies the active subsystem at each instant of time. The switching strategy improves control performance [10–13] and arise many engineering applications, such as in motor engine control, constrained robotics, and satellite image control systems [14]. As far as time-dependent switching is concerned, the average dwell time (ADT) switching is employed in most references owing to its flexibility [15,16].

However, perturbations and uncertainties widely exist in practical systems. In some cases, the perturbations can be merged into the disturbance, which can be bounded in the appropriate norms. The main advantage of robust H∞ control is that its performance specification considers the system’s worst-case performance in terms of energy gain. This is more appropriate for system robustness analysis and robust control under modeling disturbances than other performance specifications. Recently, the problems of robust H∞ control and filtering for 2-D systems have been studied by many researchers [17–20], and so do the same problems of switched systems [21–23]. However, to the best of our knowledge, the output feedback robust H∞ control problem of 2-D switched systems in the Roesser model with uncertainties has not yet been thoroughly investigated, which motivates this present study.

In this paper, we confine our attention to the robust H∞ control problem of discrete 2-D switched systems described by the Roesser model with uncertainties. The main theoretical contributions are threefold: (1) We contribute to the development of stabilization for a class of 2-D switched systems that are exponentially stable, which the Roesser model with uncertainties represents. (2) A sufficient condition is presented to ensure a 2-D switched system’s exponential stability at a given disturbance attenuation level robust weighted H∞. (3) Based on the above two points, this article further designs the output feedback controller of the closed-loop system.

The paper is organized as follows. According to the current research results, we first study the exponential stability of 2-D switched systems described by the Roesser model with uncertainties. Further, when the system contains perturbations, we analyze the robust H∞ performance index of the system. Finally, we design an output feedback controller for the open-loop system, and an example is given to illustrate the effectiveness of the proposed method.

The following notations are used throughout the paper: the superscript “T” denotes the transpose, and the notation X ≥ Y (X > Y) means that matrix X − Y is positive semi-definite (positive definite, respectively).

3 Problem Formulation and Preliminaries

The uncertain Roesser model for a 2-D switched system

with

where

The boundary condition satisfies:

It is easy to know from the above formula

Remark 1. “In this paper, it is assumed that switching occurs only at each sampling point of k or l. The switching sequence can be described as

with

Definition 1. System (1) is said to be exponentially stable under

holds for all D ≥ j [26].

Remark 2. From Definition 1, it is easy to see that when j is given,

Definition 2. For a given scalar

(1) when

(2) under the zero-boundary condition, we have

where 0 <

Definition 3. For any

holds for given

Lemma 1. For a given matrix

Lemma 2. Assuming that

Proof.

4 Exponential Stability Analysis

This section focuses on the exponential stability analysis of the 2D switched systems. The following theorem presents sufficient conditions that can guarantee that system (1) is exponentially stable.

Theorem 1. Consider 2D discrete switched system (1) with

Then, the system is exponentially stable for any switching signal with the average dwell time satisfying

where

Proof. Without loss of generality, we assume that the ith subsystem is active. For the ith subsystem, we consider the following Lyapunov function candidate:

where

Using Lemma 1 to (8), we can get the equivalent inequality as follows:

Then using Lemma 2 to (13), we can get

Using Lemma 1 to (14), we can get

Form (15), we know

The equality holds only if

It follows from (16) that

Now, let n =

Using (11) and (12), at switching instant

Also, according to Definition 3, it follows that

Therefore, the following inequality can be obtained easily:

Combining (20), inequality (21) can be written as follows:

There exist two positive constants a and b (a ≤ b) such that

By Definition 1, we know that if

The proof is completed.

Remark 3. Note that when

5 Robust H∞ Performance Analysis

This section focuses on the Robust H∞ stabilization problem for a class of discrete-time 2D switched systems represented by a Roesser model with uncertainties. The following theorem presents sufficient conditions that can guarantee that system (1) is exponentially stable and has a prescribed weighted H∞ disturbance attenuation level

Theorem 2. For given positive scalars

Then, 2D switched system (1) is exponentially stable and has a prescribed weighted H∞ disturbance attenuation level

Proof. It is an obvious fact that (24) implies that inequality (8) holds. By Lemma 2, we can find that system (1) is exponentially stable when

To establish the weighted H∞ performance, we choose the same Lyapunov functional candidate as in (12) for the system (1). Following the proof line of Theorem 1, we can get

if

Using Lemma 1 to (26), we can get the equivalent inequality as follows:

Pre- and post-multiplying (24) by

Using Lemma 1 to (28), we can get

Then using Lemma 2, we find that (29) is equivalent to (27).

Thus it can be obtained from (24) that

Then we have

Let

Summing up both sides of (31) from (D-1) to 0 with respect to l and 0 to (D-1) with respect to k, respectively, and applying the zero-boundary condition, one gets

Under the zero-initial condition, we have

Thus, we have

Multiplying both sides of (35) by

Noting

Thus

According to Definition 3, we can see that system (1) is exponentially stable and has a prescribed weighted robust H∞ disturbance attenuation level γ.

The proof is completed.

This subsection will deal with the Robust H∞ control problem of 2D switched systems via dynamic output feedback. Our purpose is to design a dynamic output feedback controller such that the closed-loop system is exponentially stable and has a specified robust weighted H∞ disturbance attenuation level γ.

Consider the following discrete 2D switched systems in the Roesser model with uncertainties:

where

Introduce the following output feedback controller of order

where

The closed-loop system consisting of the plant (39) and the controller (40) is of the form

with

where

For the closed-loop system (41), we state the 2D Robust H∞ control problem as finding a 2D dynamic output feedback controller of the form in (40) for the 2D systems (39) such that the closed-loop system (41) has a specified weighted robust H∞ disturbance attenuation level γ. The controller design procedure is provided in the following theorem.

Theorem 3. For given positive scalars

with

then 2D switched closed-loop system (41) is exponentially stable and has a prescribed weighted robust H∞ disturbance attenuation level γ for any switching signals with the average dwell time satisfying

where

and the controller parameters can be obtained as follows:

Proof. Given Theorem 2 to the closed-loop system (41), the controller solves the 2D switched robust H∞ control problem if the following matrix inequalities hold

Pre- and post-multiplying (47) by

Definite

Partition

It is easy to show from (50) that

set

Then it follows that

Pre- and post-multiplying (49)

with

then we take

The condition (42) can be obtained.

Remark 4. It is worth noting that in Theorems 1 and 2, the results are derived from the assumption that the switching rule is not known a priori, but its value is available in each sampling period. In other words, the switching sequence considered here does not include the random switching one.

In what follows, we present an algorithm for the design of a dynamic output controller.

Algorithm 1

Step 1. Given

Step 2. The invertible matrices

Step 3. The invertible matrices

Step 4. By solving inequality (45), the constant

Step 5. By solving (46), the remaining controller parameters

This completes the proof.

Remark 5. If there is only one subsystem in the system (39), it will degenerate into being a general 2D Roesser model, which is a special model of 2D switched systems. Theorem 3 is also applicable for 2D Roesser systems, which means that our results are more general than that just for 2D Roesser systems.

In this section, we shall illustrate the results developed earlier via an example.

Subsystem 1

Subsystem 2

Taking

Then,

The positive scalar

can be obtained from (44). And the rest of the controller parameters

Choosing

and

This paper has investigated the problems of stability and weighted robust

Data Availability: The authors declare that they have no conflicts of interest to report regarding the present study.

Funding Statement: Research supported by the Science and Technology Development Program of Jilin Province, the project named: Research on Key Technologies of Intelligent Virtual Interactive 3D Display System (20180201090GX).

Conflicts of Interest: The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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