Use symbolic variable for lyapunov function
4 views (last 30 days)
I am trying to find a value for a lyapunov function but I do not know the numeric values. When I run the lyapunov command, I get an error that only numeric arrays can be used. Is there a way for using only symbolic variable to get the answer.
Sam Chak on 9 Mar 2022
If you are writing for a journal paper or a thesis, the following explanation might be helpful.
Let , , and .
There are a few ways to solve this symbolically.
syms a b c p11 p12 p22 p23 p33 p31
eqns = [1 - 2*a*p12 == 0, - a*p22 - b*p12 + c*p31 + p11 == 0, 1 - 2*b*p22 + 2*c*p23 + 2*p12 == 0, - b*p23 - c*p23 + c*p33 + p31 == 0, 1 - 2*c*p33 == 0, - a*p23 - c*p31 == 0];
S = solve(eqns);
sol = [S.p11; S.p12; S.p22; S.p23; S.p33; S.p31]
The result has been verified numerically:
clear all; clc
A = [0 1 0; -1 -2 0; 0 1 -1]
Q = eye(3)
P = lyap(A', Q)
A'*P + P*A