15.3: Exercises

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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) \(g(x)=\sin x\)

\[\operatorname{Sat}(x)=\left\{\begin{array}{ll}
x & |x| \leq 1 \\
1 & |x| \geq 1
\end{array}\right.\nonumber\]