Linearization of discrete-time systems

We characterize the equivalence of single-input single-output discrete-time nonlinear systems to linear ones, via a state-coordinate change and with or without feedback. Four cases are distinguished by allowing or disallowing feedback as well as by including the output map or not; the interdependence of these problems is analyzed. An important feature that distinguishes these discrete-time problems from the corresponding problem in continuous-time is that the state-coordinate transformation is here directly computable as a higher composition of the system and output maps. Finally, certain connections are made with the continuous-time case.     

Remarks on the periodic orbits of the matrix Riccati equation

We investigate certain questions concerning the periodic structure of the matrix Riccati differential equation with constant coefficients. A closed-form expression for the periodic solutions is obtained for both the cases involving distinct or repeated eigenvalues in the associated linear hamiltonian system. Previous results are extended by establishing that periodic solutions are bounded if and only if the span of their range does not intersect the orthogonal complement of the controllable subspace of the associated linear system.     

Stabilization of discrete-time linear systems with a time delay in the feedback loop

A discrete-time, linear, time-invariant control system with a fixed time delay in the feedback loop is considered. We investigate the problem of feedback stabilization and present some general criteria. Simple necessary and sufficient conditions for stabilization, which take the form of algebraic inequalities involving the system parameters, are developed for time delays whose values do not exceed 2, as well as for one-dimensional systems with an arbitrary time delay.     

Simple sliding mode control scheme applied to robot manipulators!

We present a simple sliding mode control scheme for robot manipulators that does not rely upon the construction of individually stable discontinuity surfaces, thus greatly reducing the complexity of design. We utilize the structure of the manipulator dynamics and Lyapunov's second method in order to establish a sliding surface on the intersection of the switching surfaces in a direct manner. A simple numerical example accompanies the theoretical development.