Stability Of Discrete-time Nonlinear Systems Research Articles

Passivity and absolute stabilization of a class of discrete-time nonlinear systems

By using the theory of discrete-time passive systems and the concept of feedback equivalence, we present an approach toward deriving sufficient conditions for the global stabilization of discrete-time nonlinear systems in the form x(k + 1) = tf(x(k)) + g(x(k))u(k).The central feature of the approach is that, a unifying set of stability criteria can be obtained and the stability results recover the previous stabilization schemes proposed in the literature.

Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems

In this paper a necessary and sufficient condition for a nonlinear system of the form /spl Sigma/, given by x(k+1)=f(x(k))+g(x(k))u(k), y(k)=h(x(k))+J(x(k))u(k), to be lossless is given, and it is shown that a lossless system can be globally asymptotically stabilized by output feedback if and only if the system is zero-state observable. Then, we investigate conditions under which /spl Sigma/ can be rendered lossless via smooth state feedback. In particular, we show that this is possible if and only if the system in question has relative degree /spl lcub/0,...,0/spl rcub/ and has lossless zero dynamics. Under suitable controllability-like rank conditions, we prove that nonlinear systems having relative degree /spl lcub/0,...,0/spl rcub/ and lossless zero dynamics can be globally stabilized by smooth state feedback. As a consequence, we obtain sufficient conditions for a class of cascaded systems to be globally stabilizable. The global stabilization problem of the nonlinear system /spl Sigma/ without output is also investigated in this paper by means of feedback equivalence. Some of the results are parallel to analogous ones in continuous-time, but in many respects the theory is substantially different and many new phenomena appear.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stabilization of discrete-time nonlinear systems by smooth state feedback

For a class of nonlinear discrete-time systems of the form Σ: x(k+1)=ƒ(x(k))+g(x(k))u(k) , we investigate conditions under which a nonlinear system can be rendered globally asymptotically stable via smooth state feedback. Our main result is that any nonlinear system with Lyapunov-stable unforced dynamics can always be globally stabilized by smooth state feedback if suitable controllability-like rank conditions are satisfied. Several examples are presented to demonstrate the applications of the stability results developed in this paper.

Stabilization of nonlinear discrete-time systems using state detection

Some recent results concerning stabilization of continuous-time systems using state detection are extended to the case of discrete-time nonlinear systems. It is shown that if a discrete-time locally detectable system can be stabilized by a state feedback law, then it can also be stabilized by a feedback law that depends on the output of a weak detector. >

A note on the absolute stability of non-linear discrete-time systems

This note presents two absolute stability theorems which arc versions for discrete-time systems of a theorem by S. K. Persidskii (1969).

Stability of non-linear discrete-time systems

Converse theorems for Liapunov stability of non-linear difference equations are presented, which improve upon past results in this area. The main improvement consists in not assuming that solutions can be extended backward in time. When such assumptions are made, however, it is shown that necessary conditions for finite-time stability of non-linear difference equations can be readily derived.