This thesis concerns the development of output-based sliding mode control schemes for discrete time, linear time invariant systems. Unlike most of the work given in the literature in this area, the work is concemed with the development of static output feedback based discrete time sliding mode control schemes for non-minimum phase, non-square systems with arbitrary relative degree and which include unmatched uncertainties. The key concept of extended outputs in discrete time will be introduced. It will be shown that by identifying a minimal set of present and past outputs an augmented system can be obtained which permits the design of a sliding manifold based upon output information only, which renders the sliding manifold stable. Any transmission zeros of the augmented plant will also be shown to be among the transmission zeros of the original plant. It will also be shown that if the extended outputs chosen span the state zero directions of an invariant zero of the system, then the invariant zero disappears from the augmented system. Linear matrix inequalities are then used for sliding surface design. For non-minimum phase, non-square systems with unmatched uncertainties, it will be shown that in some cases the extended outputs can be chosen such that the effect of the disturbance on the sliding surface can be nullified. If this is possible, a procedure to obtain a Lyapunov matrix, which simultaneously satisfies a Riccati inequality and a structural constraint and which is used to formulate the control law that satisfies the reachability condition has been given. For the general case, where the sliding surface is a function of the disturbance, a control law will be chosen such that the effect of the disturbance on the augmented outputs and the sliding manifold will be minimized. Another key contribution of this work is the use of extended outputs for reconfigurable control under sensor loss. The reconfigurable control methodology presented in this work is in discrete time and is a static output feedback based control scheme, unlike most of the reconfigurable control schemes given in the literature which require an estimator and which are continuous time based schemes. Suitable examples, which include multiple sensor failures and a benchmark problem taken from the literature which represents the lateral dynamics of the F-14 aircraft, have been chosen to show the effectiveness of the proposed control design methodologies. - L Abstract T his thesis concerns the development of out put-based sliding mode control schemes for discrete time, linear time invariant systems. Unlike most of the work given in the literature in this area, the work is concerned with the development of static output feedback based discrete time sliding mode control schemes for non-minimum phase, non-square systems with arbitrary relative degree and which include unmatched uncertainties. The' key concept of extended outputs in discrete time will be introduced. It will be shown that by identifying a minimal set of present and past outputs an augmented system can be obtained which permits the design of a sliding manifold based upon output information only, which renders the sliding manifold stable. Any transmission zeros of the augmented plant will also be sho,wn to be among the transmission zeros of the original plant. It will also be shown that- if the extended outputs chosen span the state zero directions of an invariant zero of the system, then the invariant zero disappears from the augment.ed system. Linear matrix inequalities are then used for sliding surface design. For nonminimurn phase, non-square systems with unmatched uncertainties, it will be shown that in some cases the extended outputs can be chosen such ,that the effect of the disturbance on the sliding surface can be nullified. If this is possible, a procedure to obtain a Lyapunov matrix, which simultaneously satisfies a Riccati inequality and a structural constraint and which is used to formulate the control law t hat satisfies the reachability condit ion has been given. For the general case, where the sliding surface is a function of the disturbance, a control law will be chosen such that the effect of the disturbance on the augmented outputs and the sliding manifold will be minimized. Another key contribution of t his work is the use of extended outputs for reconfigurable control under sensor loss. The reconfigura~le control methodology presented in this work is in discrete time and is a static output feedback based control scheme, unlike most of t he reconfigurable control schemes given in the literature which require an estimator and which are continuous time based schemes. Suitable examples, which include multiple sensor failures and a benchmark problem taken from the literature which represents the lateral dynamics of the F-14 aircraft have been chosen to show the effectiveness of the proposed control design methodologies.

Research into observer design for non-linear discrete-time systems has produced many design methods. There is no general design method however and that provides the motivation to search for a new simple and realizable design method. In this thesis, an observer for non-linear discrete-time systems is designed using the sliding mode technique. The equation of the observer error is split into two parts; the first part being a linearized model of the system and the second part an uncertain vector. The sliding mode technique is introduced to eliminate the uncertainty caused by the uncertain vector in the observer error equation. By choosing the sliding surface and the boundary layer, the observer error is attracted to the sliding surface and stays within the sliding manifold. Therefore, the observer error converges to zero. The proposed technique is applied to two cases of observers for nonlinear discrete-time systems. The second case is chosen to be a particular practical system, namely the non-linear discrete-time ball and beam system. The simulations show that the sliding mode technique guarantees the convergence of the observer error for both systems.