\subsection{Homework 7 - Due: 03/07}
\subsubsection{Problem 1}
Recall that for an LTI system, \(\dot x = Ax + Bu\), the controllability
Gramian has the form \[
W (0, t) = \int \limits _{0} ^{t} e^{-As}BB^Te^{-A^T s} ds
\]
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
Prove that the matrix \(\overline{W}(0,t)\) defined below is
nonsingular for some \(t>0\) if and only if \(W(0, t)\) is.
\end{enumerate}
\[ \overline{W}\left(0, t\right) := \int \limits _{0} ^{t} e^{As} B B^T e^{A^Ts} ds \]
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
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.
\end{enumerate}
\subsubsection{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 \(t\).
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
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.
\item
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
\(t\)).
\item
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.
\item
Answer the same questions as in part (c) but for the system
\(\dot x=x+u\) (again with \(|u|\le 1\)).
\end{enumerate}
\subsubsection{Problem 3}
Let us revisit our \href{hw06.html\#problem-4}{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 \emph{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.
\subsubsection{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.
\subsubsection{Problem 5}
Do BMP Problem 5.5.6
\subsubsection{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.