Suggested languages for you:

Americas

Europe

Problem 12

A continuous-time time-invariant system with outputs (but no inputs) \(\dot{x}=f(x), y=h(x)\), having state space \(x=\mathbb{R}^{n}\), and output- value space \(y=\mathbb{R}^{p}\), is said to be output-to-state stable (OSS) if there exist a function \(\beta \in \mathcal{K} \mathcal{L}\) and a function \(\gamma \in \mathcal{K}_{\infty}\) (terminology is as in Exercise 7.3.11) so that the following property holds: For each \(T>0\), if \(\xi\) is a solution of \(\dot{\xi}=f(\xi)\) on the interval \([0, T]\), and \(\eta(t)=h(\xi(t))\) for $t \in[0, T]$, then $$ \|\xi(T)\| \leq \max \left\\{\beta(\|\xi(0)\|, T), \gamma\left(\|\eta\|_{\infty}\right)\right\\} . $$ Show: 1\. A continuous-time time-invariant linear system is OSS if and only if it is detectable. 2\. In general, a system may be detectable (with respect to \(x^{0}=0\) and \(y^{0}=0\) ) yet not be OSS.

Expert verified

In summary, we proved that a continuous-time time-invariant linear system is output-to-state stable (OSS) if and only if it is detectable. We also showed that, in general, a system may be detectable but not be OSS by providing an example of a detectable system for which a Lyapunov function does not exist, demonstrating that the system is not OSS.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 7

For linear time-invariant continuous-time or discrete-time systems, \((A, C)\) is asymptotically observable if and only if $\left(A^{\prime}, C^{\prime}\right)$ is asymptotically controllable.

Chapter 7

The functions \(f, g \in \mathbf{R H}_{\infty}\) are coprime if and only if there exist \(\tilde{\alpha}, \tilde{\beta} \in \mathbf{R H}_{\infty}\) such that $$ \tilde{\alpha} f+\tilde{\beta} g=1 . $$

Chapter 7

The behavior \(\Lambda_{\Sigma}\) is UBIBO if and only if there exists a rational representation \(W_{\Sigma}=q^{-1} P\) in which \(q\) is a Hurwitz (or convergent, in discrete-time) polynomial.

Chapter 7

When the system \(\Sigma\) is discrete-time and observable, one may pick a matrix \(L\) so that \(A+L C\) is nilpotent, by the Pole-Shifting Theorem. Thus, in that case a deadbeat observer results: The estimate becomes exactly equal to the state after \(n\) steps.

Chapter 7

The behavior \(\Lambda\) is UBIBO if and only if its impulse response \(K\) is integrable.

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner