Design of multivariable feedback systems with stable plant

Abstract

This paper considers, in a general algebraic framework, the design of a unity-feedback multivariable system with a stable plant. The method is based on a simple parameterization of the four closed-loop transfer functions in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> , the plant transfer function, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q=H_{y_{1}u_{1}}</tex> . In particular, the I/O transfer function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q=H_{y_{2},u_{1}}=PQ</tex> . Using the framework of rational transfer functions, we show that the closed-loop system will be exponentially stable if and only if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> is exponentially stable. Furthermore, if both <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> are strictly proper then the controller is also strictly proper. The basic result is Design Theorem 2. An algorithm is given for obtaining strictly proper controllers such that the resulting I/O map is decoupled, all its poles can be chosen by the designer, and the same holds for zeros except, of course, for the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</inf> -zeros prescribed by the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</inf> -zeros of the plant. A discussion is included to temper these results by the constraints imposed by noise and plant saturation.