1.7 How to check that state space system $x^{\prime }=Ax+Bu$ is controllable?

A system described by

$$\begin{array}{rl}{x}^{\prime }& =Ax+Bu\\ y& =Cx+Du\end{array}$$

Is controllable if for any initial state $x_0$ and any final state $x_f$ there exist an input $u$ which moves the system from $x_0$ to $x_f$ in finite time. Only the matrix $A$ and $B$ are needed to decide on controllability. If the rank of

$$[BAB{A}^{2}B\dots A^{n-1}B]$$

is $n$ which is the number of states, then the system is controllable. Given the matrix

$$A=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\\ 0& 0& 5& 0\end{array}\right)$$

And

$$B=\left(\begin{array}{c}0\\ 1\\ 0\\ -2\end{array}\right)$$

Mathematica A0 = {{0, 1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, 1}, {0, 0, 5, 0}}; B0 = {{0}, {1}, {0}, {-2}}; sys = StateSpaceModel[{A0, B0}]; m = ControllabilityMatrix[sys]	$$\left(\begin{array}{cccc}0& 1& 0& 2\\ 1& 0& 2& 0\\ 0& -2& 0& -10\\ -2& 0& -10& 0\end{array}\right)$$
ControllableModelQ[sys]	True
MatrixRank[m]	4

 

Matlab A0 = [0 1 0 0; 0 0 -1 0; 0 0 0 1; 0 0 5 0]; B0 = [0 1 0 -2]'; sys = ss(A0,B0,[],[]); m = ctrb(sys)	m = 0 1 0 2 1 0 2 0 0 -2 0 -10 -2 0 -10 0
rank(m)	4

 

Maple restart: alias(DS=DynamicSystems): A:=Matrix( [ [0,1,0,0], [0,0,-1,0], [0,0,0,1], [0,0,5,0] ] ); B:=Matrix([[0],[1],[0],[-2]]); sys:=DS:-StateSpace(A,B); m:=DS:-ControllabilityMatrix(sys);	$$\left[\begin{array}{cccc}0& 1& 0& 2\\ 1& 0& 2& 0\\ 0& -2& 0& -10\\ -2& 0& -10& 0\end{array}\right]$$
DS:-Controllable(sys,method=rank);	true
DS:-Controllable(sys,method=staircase);	true
LinearAlgebra:-Rank(m);	4