CDS 212, Homework 5, Fall 2010

J. Doyle	Issued: 26 Oct 2010
CDS 212, Fall 2010	Due: 4 Nov 2010

Reading

• [PD], Chapter 4

Problems

1. [PD 4.1]
   Suppose <amsmath>A, X</amsmath> and <amsmath>C</amsmath> satisfy <amsmath>A^*X+XA+C^*C=0.</amsmath> Show that any two of the following implies the third:
   1. <amsmath>A</amsmath> Hurwitz.
   2. <amsmath>(C,A)</amsmath> observable.
   3. <amsmath>X>0</amsmath>
2. Assume <amsmath>(A,B)</amsmath> is controllable. Show that <amsmath>(F,G)</amsmath> with <amsmath>

   F=\left[\begin{array}{ccc} A&0\\C&0 \end{array}\right], G=\left[\begin{array}{ccc} B\\0 \end{array}\right],

   </amsmath>

   is controllable if and only if

   <amsmath>

   \left[\begin{array}{ccc} A&B\\C&0 \end{array}\right]

   </amsmath>

   is a full row rank matrix.

3. [PD 4.4]
   Controllability gramian vs. controllability matrix. We have seen that the singular values of the controllability gramian <amsmath>X_c</amsmath> can be used to determine "how controllable" the states are. In this problem you will show that the controllability matrix
<amsmath>

M_c=\left[\begin{array}{ccccccc} B&AB&A^2B&\cdots&A^{n-1}B \end{array}\right]

</amsmath>

cannot be used for the same purpose, since its singular values are unrelated to those of <amsmath>X_c</amsmath>. In particular, construct examples (<amsmath>A\in\mathcal{C}^{2\times 2}, B\in\mathcal{C}^{2\times 1}</amsmath> suffices) such that

1. <amsmath>X_c=I</amsmath>, but <amsmath>\underline{\sigma}(M_c)</amsmath> is arbitrarily small.
2. <amsmath>M_c=I</amsmath>, but <amsmath>\underline{\sigma}(X_c)</amsmath> is arbitrarily small.
4. Prove the Schur complement inequality <amsmath>

   \left[\begin{array}{ccccccc} A&B\\B^T&C \end{array}\right]> 0 \Longleftrightarrow A-BC^{-1}B^T>0,\quad C>0

   </amsmath>
5. We know that the discrete-time system <amsmath>x[k+1]=Ax[k]</amsmath> is stable (i.e, <amsmath>x[k]\rightarrow 0</amsmath> as <amsmath>k\rightarrow 0</amsmath>) if and only if all eigenvalues of <amsmath>A</amsmath> are inside the unit circle. Derive a necessary and sufficient LMI condition for stability of this system.
6. Consider the following optimization problem: <amsmath>

   \max_{Q,\alpha}\; \alpha

   </amsmath>

   subject to

   <amsmath>

   AQ+QA^T+\alpha Q< 0, \quad Q>0

   </amsmath> Find an analytical expression (in terms of <amsmath>A</amsmath>) for the maximum value of <amsmath>\alpha</amsmath>.