Robust Control For Two-dimensional Discrete-time Systems

Over the past several decades, a considerable interest has been devoted to problems involving signals and systems that depend on more than one variable. 2-D signals and systems have been studied in relation to several modern engineering fields such as process control, image enhancement, signal processing, etc. On the other hand, 2-D singular systems have received much interest due to their extensive applications in many practical areas. A great number of fundamental results on 2-D regular systems have been extended to 2-D singular systems.In the light of recent work in the theory of 2-D singular systems and that of robust control for 2-D regular systems, this dissertation provides a systematic study on the theory of robust control for 2-D singular systems. Some new design techniques are proposed, and results of robust control for 2-D regular systems have been extended to 2-D singular systems. The main contents and results in this dissertation are as follows:First, the problem of robust stabilization for linear discrete time 2-D singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainty is studied. The purpose is the design of static output feedback controller such that the resulting closed-loop system is acceptable, jump modes free, stable for all admissible uncertainties. A sufficient condition for the solvability of the robust stabilization problem is derived in terms of matrix inequalities. However, this approach requires that system input matrix B is of full row rank. In order to solve the problem, approach in terms of bilinear matrix inequalities is adopted, and an iterative procedure for solving the bilinear matrix inequalities is proposed.Then, for uncertain linear discrete time 2-D singular Roesser models, the problem of robust H_∞ control is investigated. The problem addressed is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a specified H_∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H_∞ control problem is solved, and adesired output feedback controller can be constructed by solving a set of matrix inequalities. But this approach requires that system input matrix B is of full row rank. In order to solve the problem, approach in terms of bilinear matrix inequalities is adopted, and an iterative procedure for solving the bilinear matrix inequalities is proposed.Thirdly, we consider the Hx model reduction for 2-D singular systems of Roesser Models. The aim is to find a reduced-order 2-D regular Roesser model for a given 2-DSRM such that the associated model reduction error meets a prescribed Hx norm bound constraint. We first establish a condition of bounded realness for 2-D SRM, which is shown to be an extension of the existing result for 2-D regular systems. Then, in terms of linear matrix inequalities and a set of coupling non-convex rank constraints, sufficient conditions for the existence of solutions are obtained. An explicit parameterization of the desired reduced-order models is given. Furthermore, a simple linear matrix inequality condition without rank constraint is derived for the zeroth-order H w approximation problem.Fourthly, the problem for the Hm filtering of the discrete time 2-D singular Roesser models is discussed. The purpose is the design of the observer-based 2-D singular filter such that the error system is acceptable, jump modes free, stable and satisfies a specified Hn performance level. Using the two approaches of general Riccati inequality and bilinear matrix inequalities, a sufficient condition for the solvability of the observer-based HK filtering problem for 2-D SRM is solved.Then, for uncertain linear discrete time 2-D singular Roesser models, the problem of positive real control is considered. The purpose of this study is to design a state feedback controller such that the resulting closed-loop system is acceptable, jump modes free and stable, and achieves the extended strictly positive realness for all admissible uncertainties. A version of positive real lemma for the 2-D SRM is given in terms of linear matrix inequalities (LMIs). Based on the lemma, a sufficient condition for the solvability of the positive real control problem is derived in terms of bilinear matrix inequalities (BMIs) and an iterative procedure for solving the BMIs is proposed.Next, we are concerned with the problem of robust H? filtering for 2-D discrete systems in the Fornasini-Marchesini second local state-space model...