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This paper is concerned with control synthesis of uncertain Roesser-type discrete-time two-dimensional (2D) systems. The mathematical model of the 2D system’s parameter uncertainty, which may appear typically in many actual environment, is modeled as a convex bounded uncertain domain. By using the Lyapunov stability theory, stabilization conditions is proposed in with the purpose of ensuring the robust asymptotical stability of the underlying closed-loop uncertain Roesser-type discrete-time 2D systems. Furthermore, the obtained result of this paper is formulated in the form of linear matrix inequalities (LMIs), which can be easily solved via standard numerical software. Finally, a numerical example is also provided to demonstrate the effectiveness of the proposed result.

Over the past several decades, the 2D systems have attracted considerable research interests due to their wide applications in many areas such as water stream heating, thermal processes, biomedical imaging, data processing and transmission, multidimensional digital filters, image processing, grid based wireless sensor networks [

As is well known, the Lyapunov stability theory has become an efficient tool for addressing the problem of stability analysis and control synthesis of linear or nonlinear uncertain systems. Indeed, the earlier results on stability analysis and control synthesis of linear uncertain systems were obtained by applying the common quadratic Lyapunov function (CQLF). However, the CQLF applies a single Lyapunov matrix for all uncertain submodels and the obtained results are often very conservative. With the purpose of further reducing the conservatism, an efficient affine parameter-dependent Lyapunov function (APDLF) is proposed in [

Motivated by the above analysis, the problem of control synthesis of uncertain Roesser-type discrete-time two-dimensional systems will be investigated based on the Lyapunov stability theory in this paper. In particular, the used mathematical model of the system’s parameter uncertainty, which often appears typically in most practical environments, is modeled as a convex bounded uncertain domain. Because the Roesser-type discrete-time 2D system’s information is propagated along two independent directions, and thus the derivation of control synthesis would be more complicated than the usual 1D models. Then, LMI-based stabilization conditions are proposed by applying both a Lyapunov function for the uncertain Roesser-type discrete-time 2D system and a state-feedback control law for the uncertain Roesser-type discrete-time 2D system. Under the control of obtained state-feedback control law, the uncertain Roesser-type discrete-time 2D system could be ensured to be robust asymptotically stable. More importantly, the obtained result of this paper is formulated by means of linear matrix inequalities (LMIs), which can be easily solved via standard MATLAB software. Finally, the effectiveness of the proposed approach in this paper is demonstrated by some numerical examples.

The rest of this paper is organized as follows: following the introduction, some preliminaries are provided in Section

For simplicity, the notations used are fair standard. For example,

Consider a class of uncertain Roesser-type discrete-time 2D systems which are described as follows:

Let us denote

The uncertain Roesser-type discrete-time 2D systems (

Then, we end this section with a well-known lemma which plays an important role in the proof of our main result.

Given matrices

Considering the characteristics of the uncertain Roesser-type discrete-time 2D systems (

Therefore, the closed-loop uncertain Roesser-type discrete-time 2D systems (

Under the control of the state-feedback control law (

Consider a Lyapunov function for closed-loop uncertain Roesser-type discrete-time 2D systems (

Then, the variation of

Then, it is easy to see that the closed-loop uncertain Roesser-type discrete-time 2D systems (

This completes the proof.

By using the Lyapunov stability theory, LMI-based stabilization conditions are given in Theorem

It is worth noting that the used Lyapunov matrix in (

Consider a class of uncertain Roesser-type discrete-time two-dimensional systems which could be described as follows:

Let

The state trajectory of

The state trajectory of

Using Theorem

Under the control of (

The state trajectory of

The state trajectory of

In this paper, the problem of control synthesis of uncertain Roesser-type discrete-time 2D systems has been investigated via the well-known Lyapunov stability theory. The mathematical model of the system’s parameter uncertainty, which often appears typically in practical environment, is modeled by a convex bounded uncertain domain. With the purpose of conceiving the robust asymptotic stability of the closed-loop uncertain Roesser-type discrete-time 2D systems, stabilization conditions have been developed by proposing a Lyapunov function for uncertain Roesser-type discrete-time 2D systems. The obtained stabilization conditions are in terms of LMIs and can be easily solved via standard MATLAB software. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed result.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Nature Science Foundation of China under Grants 61304069 and 61372195, the National Nature Science Foundation of Liaoning Province under Grants 2013020124, 2012201010, and 201102160, the Key Project of Chinese Ministry of Education under Grant 212033, the Key Technologies R&D Program of Liaoning Province under Grant 2011224006, the Program for Liaoning Excellent Talents in University under Grant LJQ2011136, the Scientific Research Fund of Liaoning Provincial Education Department under Grant L2013494, and the Science and Technology Program of Shenyang under Grant F11-264-1-70.