Homework 7 - Due: 03/07
Problem 1
Recall that for an LTI system, \(\dot x = Ax + Bu\), the controllability Gramian has the form \[ W (0, t) = \int \limits _{0} ^{t} e^{-As}BB^Te^{-A^T s} ds \]
- Prove that the matrix \(\overline{W}(0,t)\) defined below is nonsingular for some \(t>0\) if and only if \(W(0, t)\) is.
\[ \overline{W}\left(0, t\right) := \int \limits _{0} ^{t} e^{As} B B^T e^{A^Ts} ds \]
- The result of (a) implies that the pair \((A, B)\) is controllable if and only if the pair \((-A, B)\) is controllable. Is this true for LTV systems? Prove or give a counter example.
Problem 2
For the scalar system
\[\dot x=-x+u\]
consider the problem of steering its state from \(x=0\) at time 0 to \(x=1\) at some given time \(t\).
Since the system is controllable, we know that this transfer is possible for every value of \(t\). Verify this by giving an explicit formula for a control that solves the problem.
Is the control you obtained in part (a) unique? If yes, prove it; if not, find another control that achieves the transfer (in the same time \(t\)).
Now suppose that the control values must satisfy the constraint \(|u|\le 1\) at all times. Is the above problem still solvable for every \(t\)? for at least some \(t\)? Prove or disprove.
Answer the same questions as in part (c) but for the system \(\dot x=x+u\) (again with \(|u|\le 1\)).
Problem 3
Let us revisit our consensus problem from the last homework. Assume that agent 1 is a the leader and knows a desired location \(p \in \mathbb{R}\) to which all agents should converge. Agents 2 and 3 do not know \(p\) and all agents see each other. Based on this information, write down modified consensus equations for which you can prove that all three agents asymptotically converge to \(p\) from arbitrary initial positions.
Problem 4
Consider the system \(\dot x=Ax+Bu\) with \[ A= \begin{bmatrix} -1&0&3\\0&1&1\\0&0&2 \end{bmatrix},\qquad B= \begin{bmatrix} 1\\1\\1 \end{bmatrix} \] Compute its Kalman controllability decomposition. Identify controllable and uncontrollable modes.
Problem 5
Do BMP Problem 5.5.6
Problem 6
Recall that the controllable subspace of an LTI system \(\dot x = A x + Bu\) is the range of its controllability matrix \(\mathcal{C}\left(A, B \right)\). Consider a pair of matrices \((A, B)\) with \(A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}\) and let a matrix \(K \in \mathbb{R}^{m \times n}\) be given. Prove that the controllability subspaces of \((A, B)\) and \((A+BK, B)\) are equal and thus \((A+BK, B)\) is controllable if and only if \((A,B)\) is.