Problem: Disturbance decoupling

                                                                                                                         

Consider the system
 

 

 \begin{displaymath} \begin{array} {rcl} \dot x&=&Ax+Bu+Ew \\ y&=&Cx,\end{array}\end{displaymath}

(4)

where u is the control signal and w is an external disturbance that cannot be measured. The question is if there exits a state feedback

u=Fx+v

such that the output y is unaffected by the disturbance w.

Suppose such a feedback exists. Then by plugging in the control, we have

 

\begin{displaymath} \begin{array} {rcl} \dot x&=&(A+BF)x+Bv+Ew \\ y&=&Cx.\end{array}\end{displaymath}

(5)

The fact that the output y is unaffected by the disturbance w implies that the nth derivative y(n)(t) for any n.>=1and any t does not depend on w. Since

y(n)(t)=C(A+BF)nx(t)+C(A+BF)n-1Bv(t)+C(A+BF)n-1Ew(t),

(here we assume v and w are constants for the sake of simplicity) we must have

\begin{displaymath} C(A+BF)^{n-1}E=0,~~\forall n\ge 1.\end{displaymath}

In other words, if we can find an F that satisfies the above equations, then the problem is solved.

However, the equations are highly nonlinear, thus difficult to solve. In Chapter 3 we will use the idea of ``controlled invariance'' to reduce the problem into a linear one.