Control of unknown plants in reduced state space

Abstract

A method is proposed in this paper for the synthesis of an adaptive controller for a class of model reference systems in which the plant is not known exactly, but which is of the following type: single variable, time varying, either linear or nonlinear, of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th order, and capable of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> th order input differentiation. The model is linear, stable, and of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> 'th order, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n - m) \leq n' \leq n</tex> . The only knowledge of the plant that is required in this synthesis procedure is the form of the plant equation and the bounds of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b_{m}(t)</tex> , the coefficient of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> th order plant input derivative. The synthesis procedure makes use of an unique function, called the characteristic variable, and Lyapunov type synthesis. The introduction of the characteristic variable reduces the synthesis problem to one that involves a known, linear time-invariant lower order plant. The control signal is generated by measuring the plant and model outputs, and their first <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n - m)</tex> derivative signals. This ensures that the norm of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n - m)-</tex> dimensional error vector is ultimately bounded by ε, an arbitrarily small positive number provided <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\xi(t)</tex> , the characteristic variable, is bounded. Two nontrivial simulation examples are included.