# Stabilization and *H*_{∞} Control of 2-D Discrete Systems in FM LSS Model

## Abstract

Most of the existing work on 2-D output feedback control synthesis is based on solving certain linear equations for polynomials or polynomial matrices in two variables; see, for example, [41][77]. In this chapter, we shall study the 2-D output feedback control problem using an LMI approach. We are concerned with the problem of designing dynamic output feedback controllers for 2-D linear discrete systems to achieve asymptotic stability. Based on a 2-D Lyapunov lemma [48], we establish conditions for output feedback stabilizability in terms of LMIs. We shall also study the *H* _{∞} control problem of 2-D discrete systems described by the FM LSS model. The 2-D *H* _{∞} control problem studied here is to design a controller for a linear discrete time 2-D system, such that the closed-loop 2-D system is asymptotically stable and its *H* _{∞} norm from the disturbance input to the controlled output is bounded by a specified constant. Similar to the result of 1-D systems, the 2-D bounded realness property derived in Chapter 2 plays a key role in obtaining a 2-D *H* _{∞} control solution. Using this property, the 2-D *H* _{∞} controller design is formulated into a feasibility problem of an LMI.

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