Adaptive stabilization of continuous‐time systems through a controllable modified estimation model

M De La Sen

doi:10.1155/s1024123x0431001x

Abstract

This paper presents an indirect adaptive control scheme of continuous‐time systems. The estimated plant model is controllable and then the adaptive scheme is free from singularities. Such singularities are avoided through a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a positive threshold. An alternative modification scheme based on the achievement of a modifieddiagonally dominant Sylvester matrix of the parameter estimates is also proposed. This diagonal dominance is achieved through estimates modification as a way to guarantee the controllability of the modified estimated model when a controllability measure of the estimation model without modification fails. In both schemes, the use of an explicit hysteresis switching function for the modification of the estimates is not required to ensure the controllability of the modified estimated model. Both schemes ensure that chattering due to switches associated with the modification is not present.

Highlights

The adaptive stabilization and control of linear continuous and discrete systems have been successfully developed in the last decades [10, 17, 18, 19]
The solution is unique since the modified plant parameter estimated model is controllable at all time, which implies that the time-varying polynomials A(D,t) and B(D,t) are coprime for all time [5] in the sense that the absolute value of the Sylvester matrix of the modified plant estimates is nonzero and lower-bounded by a positive real constant
An adaptive stabilizer possibly possessing an unstable inverse has been proposed for a continuous-time plant without assuming the inverse stability of the plant, a priori knowledge on the plant parameters and knowledge of the high-frequency gain sign

Summary

Introduction

The mechanism which guarantees the controllability of the modified estimated plant model consists basically in the online perturbation of some of the estimated plant parameters prior to the modification In this way, the resulting modified Sylvester matrix becomes nonsingular, while chattering is avoided since the control law based on the modified estimates is nonsingular and solvable for all time [14, 15] when the controllability of the estimation model fails against some appropriate numerical test about nonsingularity. The estimation scheme has suitable stability and convergence properties, and the resulting closed-loop scheme is asymptotically stable in the large in the absence of noise and unmodeled dynamics This modification is an alternative method in the case when a sufficiency test on maintenance of controllability of the unmodified estimated model fails. The mathematical proofs of the results are developed in the appendix

Adaptive stabilizer for a continuous-time plant

Stability results

Numerical example

Conclusions