# 21.8: Exercises

Exercise $$\PageIndex{21.1}$$

In decentralized control, the plant is assumed to be diagonal and controllers are de- signed independently for each diagonal element. If however, the real process is not completely decou- pled, the interactions between these separate subsystems can drive the system to instability.

Consider the $$2 \times 2$$ plant

$P(s)=\left(\begin{array}{ll} P_{11} & P_{12} \\ P_{21} & P_{22} \end{array}\right)\nonumber$

Assume that $$P_{12}$$ and $$P_{21}$$ are stable and relatively small in comparison to the diagonal elements, and only a bound on their frequency response is available. Suppose a controller $$K=\operatorname{diag}\left(K_{1}, K_{2}\right)$$ is designed to stabilize the system $$P_{0}=\operatorname{diag}\left(P_{11}, P_{22}\right)$$.

1. Set-up the problem as a stability robustness problem, i.e., put the problem in the $$M - \Delta$$ form.
2. Derive a non-conservative condition (necessary and sufficient) that guarantees the stability robustness of the above system. Assume the off-diagonal elements are perturbed independently. Reduce the result to the simplest form (an answer like $$\mu(M)<1$$ is not acceptable; this problem has an exact solution which is computable).
3. How does your answer change if the off-diagonal elements are perturbed simultaneously with the same $$\Delta$$.

Exercise $$\PageIndex{21.2}$$

Consider the rank 1 $$\mu$$ problem. Suppose $$\Delta_{0}$$, contains only real perturbations. Compute the exact expression of $$\mu(M)$$.

Exercise $$\PageIndex{21.3}$$

Consider the set of plants characterized by the following sets of numerators and denominators of the transfer function:

$N(s)=N_{0}(s)+N_{\delta}(s) \delta, \quad D(s)=D_{0}(s)+D_{\delta}(s) \delta\nonumber$

Where both $$N_{0}$$ and $$D_{0}$$ are polynomials in $$s, \delta \in \mathbb{R}^{n}$$, and $$N_{\delta}, D_{\delta}$$ are polynomial row vectors. The set of all plants is then given by:

$\Omega=\left\{\frac{N(s)}{D(s)}\left|\delta \in \mathbb{R}^{n},\right| \delta_{i} \mid \leq \gamma\right\}\nonumber$

Let $$K$$ be a controller that stabilizes $$N_{0}/D_{0}$$ . Compute the exact stability margin; i.e., compute the largest $$\gamma$$ such that the system is stable.