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Problem 2

Consider, as an example, the system \(\Sigma\) corresponding to the linearized pendulum \((2.31)\), which was proved earlier to be controllable. In Appendix C.4 we compute $$ e^{t A}=\left(\begin{array}{cc} \cos t & \sin t \\ -\sin t & \cos t \end{array}\right) $$ Thus, for any \(\delta>0\), $$ \mathbf{R}\left(e^{\delta A}, B\right)=\left(\begin{array}{ll} 0 & \sin \delta \\ 1 & \cos \delta \end{array}\right) $$ which has determinant \((-\sin \delta)\). By Lemma \(3.4 .1, \Sigma\) is \(\delta\)-sampled controllable iff $$ \sin \delta \neq 0 \text { and } 2 k \pi i \neq \pm i \delta \text {, } $$ i.e., if and only if \(\delta\) is not a multiple of \(\pi\). Take, for instance, the sampling time \(\delta=2 \pi\). From the explicit form of \(e^{t A}\), we know that \(e^{\delta A}=I\). Thus, $$ A^{(\delta)}=A^{-1}\left(e^{\delta A}-I\right)=0 $$ so \(G=0\). This means that the discrete-time system \(\Sigma_{[\delta]}\) has the evolution equation $$ x(t+1)=x(t) $$ No matter what (constant) control is applied during the sampling interval $[0, \delta]$, the state (position and velocity) is the same at the end of the interval as it was at the start of the period. (Intuitively, say for the linearized pendulum, we are acting against the natural motion for half the interval duration, and with the natural motion during the other half.) Consider now the case when \(\delta=\pi\), which according to the above Lemma should also result in noncontrollability of \(\Sigma_{[\delta]} .\) Here $$ F=e^{\delta A}=-I $$ and $$ A^{(\delta)}=A^{-1}\left(e^{\delta A}-I\right)=-2 A^{-1}=2 A $$ so $$ G=2 A B=\left(\begin{array}{l} 2 \\ 0 \end{array}\right) . $$ Thus, the discrete-time system \(\Sigma_{[6]}\) has the evolution equations: $$ \begin{aligned} &x_{1}(t+1)=-x_{1}(t)+2 u(t) \\ &x_{2}(t+1)=-x_{2}(t) \end{aligned} $$ This means that we now can partially control the system, since the first coordinate (position) can be modified arbitrarily by applying suitable controls \(u\). On the other hand, the value of the second coordinate (velocity) cannot be modified in any way, and in fact at times \(\delta, 2 \delta, \ldots\) it will oscillate between the values \(\pm x_{2}(0)\), independently of the (constant) control applied during the interval.

Problem 20

Show by counterexample that without analyticity the rank condition is not necessary. More precisely, give two examples as follows: (i) A smooth system with \(n=m=1\) that is controllable in some nontrivial interval but for which there is some \(t_{0}\) so that \(M^{(k)}\left(t_{0}\right) \equiv 0\) for all \(k\); and (ii) A smooth system with \(n=m=2\) that is controllable in some nontrivial interval but for which the rank is one for every \(t_{0}\) and \(k\).

Problem 21

Consider the system with \(\mathbb{K}=\mathbb{R}, n=3, m=1\), and matrices $$ A(t)=\left(\begin{array}{ccc} t & 1 & 0 \\ 0 & t^{3} & 0 \\ 0 & 0 & t^{2} \end{array}\right) \quad B(t)=\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right) $$ This system is smoothly (in fact, analytically) varying on $(-\infty, \infty)$. Since $$ \left[B_{0}(0), B_{1}(0), B_{2}(0), B_{3}(0)\right]=\left(\begin{array}{cccc} 0 & 1 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right), $$ and this matrix has rank 3 , the system is controllable on every nontrivial interval \([\sigma, \tau]\).

Problem 23

Let \(\Sigma\) be a continuous-time linear system, analytically varying on \(\mathcal{I}\). Prove that if \(\Sigma\) is controllable on any nontrivial subinterval \([\sigma, \tau]\) then $$ \operatorname{rank}\left[B_{0}(t), B_{1}(t), \ldots, B_{n-1}(t)\right]=n $$ for almost all \(t \in \mathcal{I}\). (Hint: First prove that if rank \(M^{(k)}(t)=\operatorname{rank} M^{(k+1)}(t)\) for \(t\) in an open interval $J \subseteq \mathcal{I}\(, then there must exist another subinterval \)J^{\prime} \subseteq J$ and analytic matrix functions $$ V_{0}(t), \ldots, V_{k}(t) $$ on \(J^{\prime}\) such that $$ M_{k+1}(t)=\sum_{i=0}^{k} M_{i}(t) V_{i}(t) $$ on \(J^{\prime} .\) Conclude that then rank $M^{(k)}(t)=\operatorname{rank} M^{(l)}(t)\( for all \)l>k\( on \)J^{\prime}$. Argue now in terms of the sequence $\left.n_{k}:=\max \left\\{\operatorname{rank} M^{(k)}(t), t \in \mathcal{I}\right\\} .\right)$

Problem 24

Exercise 3.5.24 Consider the continuous-time linear system over \(\mathbb{K}=\mathbb{R}\) with \(n=2, m=1\), and matrices $$ A(t)=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right) \quad B(t)=\left(\begin{array}{c} \cos t \\ -\sin t \end{array}\right) $$ Prove that the system is not controllable over any fixed interval, but that for every fixed \(\sigma\) in \(\mathbb{R}\), the ("frozen") time-invariant linear system \((A(\sigma), B(\sigma))\) is controllable. Show noncontrollability of the time-varying system in two different ways: (i) Via the rank test; and (ii) Alternatively by explicitly calculating solutions and noticing that the reachable set from the origin at any fixed instant is a line.

Problem 3

The following statements are equivalent for \(L, W\) as above: (a) \(L\) is onto. (b) \(L^{*}\) is one-to-one. (c) \(W\) is onto. (d) \(\operatorname{det} W \neq 0\). (e) \(W\) is positive definite. Consider again the situation in Example 3.5.1. Here \(L\) is onto iff the matrix $$ W=\int_{\sigma}^{\tau} k(t)^{*} k(t) d t>0 . $$ Equivalently, \(L\) is onto iff \(L^{*}\) is one-to-one, i.e., there is no \(p \neq 0\) in \(X\) with \(k(t) p=0\) for almost all $t \in[\sigma, \tau)$, (3.18) or, with a slight rewrite and \(k_{i}:=i\) th column of \(k^{*}\) : \(\left\langle p, k_{i}\right\rangle=0\) for all \(i\) and almost all $t \Rightarrow p=0 .$ (3.19)

Problem 3

Consider the system (with \(\mathcal{U}=\mathbb{R}, x=\mathbb{R}^{2}\) ) $$ \begin{aligned} &\dot{x}_{1}=x_{2} \\ &\dot{x}_{2}=-x_{1}-x_{2}+u \end{aligned} $$ which models a linearized pendulum with damping. Find explicitly the systems \(\Sigma_{[\delta]}\), for each \(\delta\). Characterize (without using the next Theorem) the \(\delta\) 's for which the system is \(\delta\)-sampled controllable. The example that we discussed above suggests that controllability will be preserved provided that we sample at a frequency \(1 / \delta\) that is larger than twice the natural frequency (there, \(1 / 2 \pi\) ) of the system. The next result, sometimes known as the "Kalman-Ho-Narendra" criterion, and the Lemma following it, make this precise.

Problem 4

Let \(\Sigma\) be a continuous-time system as in Definition \(2.6 .7\) and let \(\sigma<\tau\). With the present terminology, Lemma \(2.6 .8\) says that $(x, \sigma) \sim\( \)(z, \tau)\( for the system \)\Sigma\( iff \)(z, \sigma) \sim(x, \tau)\( for the system \)\Sigma_{\sigma+\tau}^{-}$. This remark is sometimes useful in reducing many questions of control to a given state to analogous questions (for the time-reversed system) of control from that same state, and vice versa. Recall that a linear system is one that is either as in Definition 2.4.1 or as in Definition 2.7.2.

Problem 4

If \(C\) is an open convex subset of \(\mathbb{R}^{n}\) and \(L\) is a subspace of \(\mathbb{R}^{n}\) contained in \(C\), then \(C+L=C\).

Problem 5

If \(x\) is an equilibrium state, then $$ \mathcal{R}^{S}(x) \subseteq \mathcal{R}^{S+T}(x) $$ for each \(S, T \in \mathcal{T}_{+}\).

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