8.4: Exercises

Exercise 8.1

Suppose we wish to realize a two-input differential equation of the form

\[\begin{aligned}
y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{0} y=& b_{01} u_{1}+b_{11} \dot{u}_{1}+\cdots+b_{n-1,1} u_{1}^{(n-1)} \\
&+b_{02} u_{2}+b_{12} \dot{u}_{2}+\cdots+b_{n-1,2} u_{2}^{(n-1)}
\end{aligned}\nonumber\]

Show how you would modify the observability canonical realization to accomplish this, still using only \(n\) integrators.

Exercise 8.2

How would reachability canonical realization be modified if the linear differential equation that we started with was time varying rather than time invariant?

Exercise 8.3

Show how to modify the reachability canonical realization- but still using only \(n\) integrators - to obtain a realization of a two-output system of the form

\[\begin{array}{ll}
y_{1}^{(n)}+a_{n-1} y_{1}^{(n-1)}+\cdots+a_{0} y_{1} & =c_{10} u+c_{11} \dot{u}+\cdots+c_{1, n-1} u^{(n-1)} \\
y_{2}^{(n)}+a_{n-1} y_{2}^{(n-1)}+\cdots+a_{0} y_{2} & =c_{20} u+c_{21} \dot{u}+\cdots+c_{2, n-1} u^{(n-1)}
\end{array}\nonumber\]

Exercise 8.4

Consider the two-input two-output system:

\[\begin{array}{l}
\dot{y}_{1}=y_{1}+\alpha u_{1}+u_{2} \\
\dot{y}_{2}=y_{2}+u_{1}+u_{2}
\end{array}\nonumber\]

(a) Find a realization with the minimum number of states when \(\alpha \neq 1\).

(b) Find a realization with the minimum number of states when \(\alpha = 1\).