State Space Approach

Abstract

In this chapter we shall consider a system of the following state space form

$$ \begin{gathered} \dot x = Ax + Bu, \hfill \\ y = Cx. \hfill \\ \end{gathered} $$

(4.1)

If x(0) = 0 and the input and the output vectors have the same dimension m, these are related by the transfer function matrix:

$$ y(s) = G(s)u(s) = C(sI - A)^{ - 1} Bu(s), $$

(4.2)

which may be expanded into

$$ \begin{gathered} y_1 (s) = g_{11} (s)u_1 (s) + g_{12} (s)u_2 (s) + \cdots + g_{1m} (s)u_m (s), \hfill \\ y_2 (s) = g_{21} (s)u_1 (s) + g_{22} (s)u_2 (s) + \cdots + g_{2m} (s)u_m (s), \hfill \\ \vdots \hfill \\ y_m (s) = g_{m1} (s)u_1 (s) + g_{m2} (s)u_2 (s) + \cdots + g_{mm} (s)u_m (s), \hfill \\ \end{gathered} $$

(4.3)

These equations are said to be coupled, since each individual input influences all of the outputs. If it is necessary to adjust one of the outputs without affecting any of the others, determining appropriate inputs u ₁, u ₂, . . . , u m will be a difficult task in general. Consequently, there is considerable interest in designing control laws which remove this coupling, so that each input control only the corresponding output. A system of the form (4.1) is said to be (dynamically) decoupled (or non-interacting) if its transfer function matrix G(s) is diagonal and non-singular, that is,

$$ \begin{gathered} y_1 (s) = g_{11} (s)u_1 (s), \hfill \\ y_2 (s) = g_{22} (s)u_2 (s), \hfill \\ \vdots \hfill \\ y_m (s) = g_{mm} (s)u_m (s). \hfill \\ \end{gathered} $$

(4.4)

and none of the hii(s) are identically zero. Such a system may be viewed as consisting of m independent subsystems. Note that this definition depends upon the ordering of inputs and outputs, which is of course quite arbitrary. The definition will be extended to block-decoupling case in Section 4.5.