Find the maximal switching period that ensures asymptotic stability of the switching system.

Question

I have a time-dependent switched system $\mathbf{\dot{x}} = \mathbf{A}_i\mathbf{x}$. With

$$\mathbf{A}_1 = \begin{bmatrix} -0.5 & 1 \\ 100 & -1 \end{bmatrix} \quad \mathbf{A}_2 = \begin{bmatrix} -1 & -100 \\ -0.5 & -1 \end{bmatrix}$$

Now I have to find the maximal switching period which ensures asymptotic stability i.e. the eigenvalues are inside the unitdisk.

I already found that the period is approximate $T = \frac{1}{f} \leq 0.04$.

Now I wanted to find a exact solution. Using Matlab I have...

% define symbolic period
syms T;

% define A matrices
A{1} = [-0.5 1; 100 -1];
A{2} = [-1 -100; -0.5 -1];
Aeig = eig(expm(T/2*A{2})*expm(T/2*A{1}))

% solve
solve(abs(Aeig(1)) <= 1,abs(Aeig(2)) <= 1,T)

Unfortunatly Matlab gives

Warning: 2 equations in 1 variables. In /usr/local/MATLAB/R2012a/toolbox/symbolic/symbolic/symengine.p>symengine at 54 In mupadengine.mupadengine>mupadengine.evalin at 97 In mupadengine.mupadengine>mupadengine.feval at 150 In solve at 160
In S02E01 at 14 Warning: Explicit solution could not be found. in solve at 169 In S02E01 at 14

What goes wrong here? I guess the problem lies in the fact that Aeig has two pretty huge equations?

Answer 1

Switched stability is an open problem in control theory. For a good discussion and some pointers, see problem 6.4, page 203, in Blondel, V., & Megretski, A. (2004). Unsolved problems in mathematical systems and control theory‎, 334.

So the problem must be that the code is asking Matlab to do more than it is capable of.....