A stability property of nonlinear sampled-data systems with slowly varying inputs

Abstract

A stability analysis is presented that deals with the response of a nonlinear sampled-data system to a slowly varying exogenous input signal. The main result, similar to existing results for purely continuous-time and discrete-time systems, establishes that if the system possesses a manifold of exponentially stable constant operating points (equilibria) corresponding to constant values of the input signal, then an initial state close to this manifold and a slowly varying input signal yield a trajectory that remains close to the manifold. The analysis involves casting the sampled-data system as a continuous-time system with discrete jumps at the sampling instants.

Sampled‐data implementation of a gain scheduled controller

“…For a gain scheduled system, stability may be guaranteed by a imposing a bound on the rate of change of the exogenous input [9]. With either gain update rates or control loop update rates that become increasingly small, the scheduling variable rate bound required for stability may also become increasingly small.…”

“…Nonlinear controllers designed by gain scheduling techniques, such as in References [9,1,2], are typically based on a parameter-varying family of continuous-time linear time-invariant (LTI) controllers. In contrast, gain scheduled controllers are often implemented digitally.…”

Automated Synthesis of Safe Digital Controllers for Sampled-Data Stochastic Nonlinear Systems

“…Proof: It is shown by Lawrence [26], [27] that the equilibrium point x ae is exponentially stable if and only if all eigenvalues of the matrixĜ in (7) are inside the unit circle. Since exponential stability implies Lyapunov stability, the first part of the theorem holds, i.e., x ae is Lyapunov stable if all eigenvalues ofĜ are inside the unit circle.…”

“…Since exponential stability implies Lyapunov stability, the first part of the theorem holds, i.e., x ae is Lyapunov stable if all eigenvalues ofĜ are inside the unit circle. The second part of the theorem cannot be derived from [26], [27] since exponential stability is a stronger notion than Lyapunov stability. For the sake of completeness, the proof of the second part is presented in the appendix and is based on Lyapunov's indirect method.…”