1.26 Check continuous system stability in the Lyapunov sense

Problem: Check the stability (in Lyapunov sense) for the state coefficient matrix \[ A=\begin {bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ -1 & -2 & -3 \end {bmatrix} \]

The Lyapunov equation is solved using lyap() function in MATLAB and LyapunovSolve[] function in Mathematica, and then the solution is checked to be positive definite (i.e. all its eigenvalues are positive).

We must transpose the matrix \(A\) when calling these functions, since the Lyapunov equation is defined as \(A^TP+PA=-Q\) and this is not how the software above defines them. By simply transposing the \(A\) matrix when calling them, then the result will be correct.

Mathematica

Remove["Global`*"]; (mat = {{0,1,0},{0,0,1},{-1,-2,-3}})	\[ \left ( {\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & -3 \\ \end {array}} \right ) \]
p = LyapunovSolve[Transpose@mat, -IdentityMatrix[Length[mat]]]; MatrixForm[N[p]]	\[ \left ( {\begin {array}{ccc} 2.3 & 2.1 & 0.5 \\ 2.1 & 4.6 & 1.3 \\ 0.5 & 1.3 & 0.6 \\ \end {array}} \right ) \]
N[Eigenvalues[p]]	\(\{6.18272,1.1149,0.202375\}\)

 

Matlab

clear all; A=[0 1 0 0 0 1 -1 -2 -3]; p=lyap(A.',eye(length(A)))	p = 2.3 2.1 0.5 2.1 4.6 1.3 0.5 1.3 0.6
e=eig(p)	e = 0.20238 1.1149 6.1827

 

Maple

with(LinearAlgebra): A:=<<0,1,0;0,0,1;-1,-2,-3>>; p,s:=LyapunovSolve(A^%T,-<<1,0,0;0,1,0;0,0,1>>); Eigenvalues(p);

\[ \left [ {\begin {array}{c} 6.18272045921436+ 0.0\,i \\ \noalign {\medskip } 1.11490451203192+ 0.0\,i\\ \noalign {\medskip } 0.202375028753723+ 0.0\,i\end {array}} \right ] \]