Problem 2
The results of Theorem \(4.1\) do not hold for time-varying systems as shown by the solution of $$ \frac{d}{d t}\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)=\left(\begin{array}{cc} 4 a & -3 a e^{8 a t} \\ a e^{-8 a t} & 0 \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right) $$ The eigenvalues of the system matrix are \(\lambda_{1}=a, \lambda_{2}=3 a\) (they happen to be constants, i.e. they do not depend on time) and hence for \(a<0\) both eigenvalues have real parts less than zero. However, the exact solution is (with initial condition \(\left.x_{1}(0)=x_{10}, x_{2}(0)=x_{20}\right)\) : $$ \begin{aligned} &x_{1}(t)=\frac{3}{2}\left(x_{10}+x_{20}\right) e^{5 a t}-\frac{1}{2}\left(x_{10}+3 x_{20}\right) e^{7 a t} \\ &x_{2}(t)=\frac{1}{2}\left(x_{10}+3 x_{20}\right) e^{-a t}-\frac{1}{2}\left(x_{10}+x_{20}\right) e^{-3 a t} \end{aligned} $$ which is unstable for any nonzero real \(a\).
Problem 6
Investigate whether the following pairs of matrices are controllable. 1\. \(A=\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right), B=\left(\begin{array}{l}1 \\ 1\end{array}\right)\), 2\. \(A=\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right), B=\left(\begin{array}{l}0 \\ 1\end{array}\right)\) 9\. \(A=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right), B=\left(\begin{array}{l}1 \\ 2\end{array}\right)\) 4\. \(A=\left(\begin{array}{ll}a_{1} & 0 \\ a_{2} & 0\end{array}\right), B=\left(\begin{array}{l}1 \\ 1\end{array}\right)\) 5\. \(A=\left(\begin{array}{cc}0 & l \\ -l & 0\end{array}\right), B=\left(\begin{array}{l}1 \\ 0\end{array}\right)\) 6\. \(A=\left(\begin{array}{lll}\lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda\end{array}\right), B=\left(\begin{array}{l}b_{1} \\ b_{2} \\\ b_{3}\end{array}\right)\) 7\. \(A=\left(\begin{array}{lll}\lambda & 0 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda\end{array}\right), B=\left(\begin{array}{l}b_{1} \\ b_{2} \\\ b_{3}\end{array}\right)\)
Problem 9
We are given a single input system \(\dot{x}=A x+B u\) which is controllable. Suppose that a control is applied of the form \(u=K x+v\), where \(K\) is \(a 1 \times n\) and \(v\) the "new" control, which is a scalar also. The new system is the characterized by the pair \((A+B K, B)\). Prove, by using Theorem 4.8, that this new system is also controllable.
Problem 10
A nonsingular coordinate transformation \(x=S \bar{x}\), (such that \(\left.A \rightarrow S^{-1} A S, C \rightarrow C S\right)\) does not destroy observability. Show this. If the observability matrix of the transformed system is denoted by \(\bar{W}\), then \(W S=\bar{W}\).
