_{∞}

_{∞}

_{∞}

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 _{∞}_{∞}

_{∞}control

The issue of stability analysis and controller synthesis is a hot research topic. Reference Medvedeva et al. [

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 [

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 _{∞}_{∞}_{∞}

In this paper, we confine our attention to the robust _{∞}_{∞}

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 _{∞}

The following notations are used throughout the paper: the superscript “_{i}, ^{-1} denotes the inverse of ^{n} denotes the ^{+} represents the set of all non-negative integers. The _{2} norm of a 2-D signal

The uncertain Roesser model for a 2-D switched system

with

where ^{i}, ^{i}, ^{i}, ^{i}, ^{i}, ^{i}, ^{i}, are constant matrices with appropriate dimensions.

The boundary condition satisfies:

It is easy to know from the above formula

with

holds for all D ≥ j [

(1) when

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

where 0 <

holds for given

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.

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

where

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

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

By Definition 1, we know that if

The proof is completed.

This section focuses on the Robust _{∞}_{∞} disturbance attenuation level

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

_{∞} performance γ for any nonzero

To establish the weighted _{∞} 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

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 _{∞} disturbance attenuation level γ.

The proof is completed.

This subsection will deal with the Robust _{∞} 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 _{∞} 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 _{∞} 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 _{∞} disturbance attenuation level γ. The controller design procedure is provided in the following theorem.

with

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

where

and the controller parameters can be obtained as follows:

_{∞} 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.

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

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.

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 _{∞} disturbance attenuation level

This paper has investigated the problems of stability and weighted robust

_{∞}control and filtering of two-dimensional Systems

_{∞}control for 2D fuzzy systems

_{1}stability of switched positive singular systems with time-varying delay

_{∞}filtering for discrete-time 2-D switched systems: An extended average dwell time approach

_{-}/

_{∞}fault detection observer design for two-dimensional Roesser systems

_{∞}filtering for uncertain 2-D Roesser systems

_{2}-gain and asynchronous

_{∞}control of discrete-time switched systems with average dwell time