What is the input of Lyapunov function?

Question

1. I am not sure whether I declare a correct type of matrix in matlab, but it can run. Is it a correct way to derive Lyapunov function? Y is a $3\times3$ matrix.

According to the paper V(z) = z'*inv(Y)*z

1. However where can I get input z or z is what?

2. i assume it is a vector z = [z0 z1 z2], is it correct? or use [z z z]?

   A = [1.0005, 0.0595, 0.0339; 0.0032, 1.2009, 0.8003;-0.0073, -0.3148, -1.6262]; B = [1.0e-008*0.0378;-0.0716*1.0e-008;-0.1925*1.0e-008;]; C = [-2.3726, 0.1557, -0.0053]; D = [1.9591e-009];

   G = ss(A, B, C, D);

   setlmis([])

   Y=lmivar(1,[3 1]) W=lmivar(2,[1 3])

   lmiterm([1 1 1 Y],1,A','s'); % LMI #1: Y*A'+A*Y lmiterm([1 1 1 -W],.5*1,B','s'); % LMI #1: W'*B' (NON SYMMETRIC?) lmiterm([1 1 1 W],.5*B,1,'s'); % LMI #1: B*W (NON SYMMETRIC?)

   LMIs = getlmis;

   [tmin, feas] = feasp(LMIs) Y = dec2mat(LMIs, feas, Y) W = dec2mat(LMIs, feas, W)

   z = [z0; z1; z2]; V = z'*inv(Y)*z

   subs(V, {z0, z1, z2} , {0.5+0.5j, 0.5+0.5j, 0.5+0.5j})

Reference :

lmiedit

Y*A'+W'*B'+A*Y+B*W < 0

Answer 1

You want to compute the expression z'*inv(Y)*z in MATLAB, and you have told us Y is a 3x3 matrix. Then the inverse of Y is a 3x3 matrix. Then z must be a 3x1 for the matrix multiplication to proceed. That is, z has 3 rows and 1 column. The syntax for this in MATLAB is (without the quotes): "z = [z1; z2; z3]". z1, z2, and z3 are numbers.