Instantiate matrix variable and evaluate all LMI terms involving this matrix variable
mnewsys = setmvar(lmisys,X,Xval)
setmvar sets the matrix
variable X with identifier
Xval. All terms involving X are
evaluated, the constant terms are updated accordingly, and X is
removed from the list of matrix variables. A description of the resulting
LMI system is returned in
X is the identifier returned
lmivar when X is
declared. Instantiating X with
not alter the identifiers of the remaining matrix variables.
setmvar is useful to freeze
certain matrix variables and optimize with respect to the remaining
ones. It saves time by avoiding partial or complete redefinition of
the set of LMI constraints.
Consider the system
x˙ = Ax + Bu
and the problem of finding a stabilizing state-feedback law u = Kx where K is an unknown matrix.
By the Lyapunov Theorem, this is equivalent to finding P > 0 and K such that
(A + BK)P + P(A + BKT) + I < 0.
With the change of variable Y := KP, this condition reduces to the LMI
AP + PAT + BY + YTBT + I < 0.
This LMI is entered by the commands
n = size(A,1) % number of states ncon = size(B,2) % number of inputs setlmis() P = lmivar(1,[n 1]) % P full symmetric Y = lmivar(2,[ncon n]) % Y rectangular lmiterm([1 1 1 P],A,1,'s') % AP+PA' lmiterm([1 1 1 Y],B,1,'s') % BY+Y'B' lmiterm([1 1 1 0],1) % I lmis = getlmis
To find out whether this problem has a solution K for the particular Lyapunov matrix P = I, set P to I by typing
news = setmvar(lmis,P,1)
The resulting LMI system
news has only one
variable Y = K. Its feasibility
is assessed by calling
[tmin,xfeas] = feasp(news) Y = dec2mat(news,xfeas,Y)
The computed Y is feasible whenever