21.8: Exercises
- Page ID
- 43126
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(P_{0}=\operatorname{diag}\left(P_{11}, P_{22}\right)\).
- Set-up the problem as a stability robustness problem, i.e., put the problem in the \(M - \Delta\) form.
- 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).
- 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.