15.3: Exercises
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- 24328
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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 15.1 Non-causal Systems
In this chapter, we only focused on causal operators, although the results derived were more general. As an example, consider a particular CT LTI system with a bi-lateral Laplace transform:
\[G(s)=\frac{s+2}{(s-2)(s+1)}\nonumber\]
(a) Check the p-stability and causality of the system in the following cases:
(i) the ROC (Region of Convergence) is \(R_{1}=\{s \in \mathbb{C} \mid \operatorname{Re}(s)<-1\}\) where \(Re(s)\) denotes the real part of s;
(ii) the ROC is \(R_{2}=\{s \in \mathbb{C} \mid-1<\operatorname{Re}(s)<2\}\)
(iii) the ROC is \(R_{3}=\{s \in \mathbb{C} \mid \operatorname{Re}(s)>2\}\)
(b) In the cases where the system is not p-stable for \(p = 2\) and \(p = \infty\), find a bounded input that makes the output unbounded, i.e., find an input \(u \in L_{p}\) that produces an output \(y \notin L_{p}\), for \(p=2, \infty\).
Exercise 15.2
In nonlinear systems, \(p\)-stability may be satisfied in only a local region around zero. In that case, a system will be locally \(p\)-stable if:
\[\|G u\|_{p} \leq C\|u\|_{p}, \quad \text { for all } u \text { with }\|u\|_{p} \leq \delta\nonumber\]
Consider the system:
\[\begin{array}{l}
\dot{x}=A x+B u \\
z=C x+D u \\
y=g(y)
\end{array}\nonumber\]
Where \(g\) is a continuous function on \([-T, T ]\). Which of the following systems is \(p\)-stable, locally \(p\)-stable or unstable for \(p \geq 1\):
(a) \(g(x)=\cos x\)
(b) \(g(x)=\sin x\)
(c) \(g(x)=\operatorname{Sat}(x)\) where
\[\operatorname{Sat}(x)=\left\{\begin{array}{ll}
x & |x| \leq 1 \\
1 & |x| \geq 1
\end{array}\right.\nonumber\]