Chapter 7: 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 \(x, z, \omega\) $$ \mathrm{d}(x, \theta(z, h(x)))<\delta \Rightarrow \mathrm{d}(\xi(t), \hat{\xi}(t))<\varepsilon \text { for all } t \geq 0 $$

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.

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.

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 \(\left(x_{1}^{0}, x_{2}^{0}\right)\).

Let \(\Sigma_{1}\) be a continuous-time system of class \(\mathcal{C}^{1}\), $$ \dot{x}=f(x, u), \quad y=h(x), $$ and assume that \((0,0)\) is an equilibrium pair and \(h(0)=0\). Let \(\Sigma\). be the linearization about the zero trajectory, and assume that $\Sigma_{\text {. is asymptotically }}$ controllable and asymptotically observable. Show that then there exist matrices \(F, D, E\) (of suitable sizes) such that the origin of $$ \begin{aligned} \dot{x} &=f(x, F z) \\ \dot{z} &=D z+E h(x) \end{aligned} $$ is locally asymptotically stable. This corresponds to a notion of local dynamic stabilizability.

Assume that \(w\) is rational and that \(v=k \neq 0\) is a constant. Then the poles of the entries of \(W_{\mathrm{cl}}\) (that is, the elements in the list \((7.14),\), are precisely the zeros of \(w-\frac{1}{k}\).

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 \(\mathrm{cw}\left(\Gamma, \frac{1}{k}\right)=0\).