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Problem 2

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.

Expert verified

In summary, the pair \((A, C)\) is asymptotically observable if and only if the dual pair \((A^{\prime}, C^{\prime})\) is asymptotically controllable for linear time-invariant continuous-time or discrete-time systems. We proved this by first showing that asymptotic observability of \((A, C)\) implies asymptotic controllability of \((A^{\prime}, C^{\prime})\) and then showing that asymptotic controllability of \((A^{\prime}, C^{\prime})\) implies asymptotic observability of \((A, C)\). Together, these proofs establish the desired equivalence between observability and controllability for the system and its dual.

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Chapter 7

If \(w\) is stable, then for each \(k \neq 0\) such that $\frac{1}{k} \notin \Gamma\(, the closed-loop system with \)v=k$ is stable if and only if \(\mathrm{cw}\left(\Gamma, \frac{1}{k}\right)=0\).

Chapter 7

If \(\Sigma\) is internally stable, then \(\Lambda_{\Sigma}\) is UBIBO.

Chapter 7

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.

Chapter 7

A (strong) observer for \(\Sigma\) consists of a (time-invariant and complete) system \(\hat{\Sigma}\) having input value space \(u \times y\) together with a map $$ \theta: \mathcal{Z} \times y \rightarrow x $$ (where \(\mathcal{Z}\) is the state space of \(\hat{\Sigma}\) ) so that the following property holds. For each \(x \in X\), each \(z \in \mathcal{Z}\), and each \(\omega \in U[0, \infty)\), we let \(\xi\) be the infinite path resulting from initial state \(x\) and control \(\omega\), that is, $$ \xi(t)=\phi\left(t, 0, x,\left.\omega\right|_{[0, t)}\right) $$ for all \(t \in \mathcal{T}_{+}\), and write \(\eta(t)=h(\xi(t))\); we also let \(\zeta\) be the infinite path resulting from initial state \(z\) and control \((\omega, \eta)\) for the system \(\hat{\Sigma}\), $$ \zeta(t)=\hat{\phi}\left(t, 0, z,\left(\left.\omega\right|_{(0, t)},\left.\eta\right|_{[0, t)}\right)\right), $$ and finally we write $$ \hat{\xi}(t):=\theta(\zeta(t), \eta(t)) . $$ Then it is required that, for every \(x, z\) and every \(\omega\), $$ \mathrm{d}(\xi(t), \hat{\xi}(t)) \rightarrow 0 \text { as } t \rightarrow \infty $$ (global convergence of estimate) as well as that for each \(\varepsilon>0\) there be some \(\delta>0\) so that for every \(x, z, \omega\) $$ \mathrm{d}(x, \theta(z, h(x)))<\delta \Rightarrow \mathrm{d}(\xi(t), \hat{\xi}(t))<\varepsilon \text { for all } t \geq 0 $$

Chapter 7

Let \(\Sigma_{1}\) be a topological system, and pick any \(x_{1}^{0} \in X\). The system \(\Sigma_{1}\) is dynamically stabilizable (with respect to \(x_{1}^{0}\) ) by \(\left(\Sigma_{2}, x_{2}^{0}, k_{1}, k_{2}\right)\), where \(\Sigma_{2}\) is a topological system, if the interconnection of \(\Sigma_{1}\) and \(\Sigma_{2}\) through \(k_{1}, k_{2}\) is topologically well-posed and is globally asymptotically stable with respect to \(\left(x_{1}^{0}, x_{2}^{0}\right)\).

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