Compute solution to given system of LMIs
[tmin,xfeas] = feasp(lmisys,options,target)
[tmin,xfeas] = feasp(lmisys,options,target) computes a solution
xfeas (if any) of the system of LMIs described by
lmisys. The vector
xfeas is a particular value
of the decision variables for which all LMIs are satisfied.
Given the LMI system
xfeas is computed by solving the auxiliary convex program:
Minimize t subject to NTL(x)N–MTR(x)M≤tI.
The global minimum of this program is the scalar value
tmin returned as first output argument by
feasp. The LMI constraints are feasible if
tmin ≤ 0 and strictly feasible if
tmin < 0. If the problem is feasible but not strictly feasible,
tmin is positive and very small. Some post-analysis may then be required to decide whether
xfeas is close enough to feasible.
The optional argument
target sets a target value for
tmin. The optimization code terminates as soon as a value of t below this target is reached. The default value is
target = 0.
xfeas is a solution in terms of the decision variables and not in terms of the matrix variables of the problem. Use
dec2mat to derive feasible values of the matrix variables from
The optional argument
options gives access to certain control parameters for the optimization algorithm. This five-entry vector is organized as follows:
options(1)is not used.
options(2)sets the maximum number of iterations allowed to be performed by the optimization procedure (100 by default).
options(3)resets the feasibility radius. Setting
options(3)to a value R > 0 further constrains the decision vector x = (x1, . . ., xN) to lie within the ball
In other words, the Euclidean norm of
xfeasshould not exceed R. The feasibility radius is a simple means of controlling the magnitude of solutions. Upon termination,
feaspdisplays the f-radius saturation, that is, the norm of the solution as a percentage of the feasibility radius R.
The default value is R = 109. Setting
options(3)to a negative value activates the “flexible bound” mode. In this mode, the feasibility radius is initially set to 108, and increased if necessary during the course of optimization
options(4)helps speed up termination. When set to an integer value J > 0, the code terminates if t did not decrease by more than one percent in relative terms during the last J iterations. The default value is 10. This parameter trades off speed vs. accuracy. If set to a small value (< 10), the code terminates quickly but without guarantee of accuracy. On the contrary, a large value results in natural convergence at the expense of a possibly large number of iterations.
options(5) = 1turns off the trace of execution of the optimization procedure. Resetting
options(5)to zero (default value) turns it back on.
option(i) to zero is equivalent to setting the corresponding control parameter to its default value. Consequently, there is no need to redefine the entire vector when changing just one control parameter. To set the maximum number of iterations to 10, for instance, it suffices to type
options=zeros(1,5) % default value for all parameters options(2)=10
When the least-squares problem solved at each iteration becomes ill conditioned, the
feasp solver switches from Cholesky-based to QR-based linear algebra (see Memory Problems for details). Since the QR mode typically requires much more memory, MATLAB® may run out of memory and display the message
??? Error using ==> feaslv Out of memory. Type HELP MEMORY for your options.
You should then ask your system manager to increase your swap space or, if no additional swap space is available, set
options(4) = 1. This will prevent switching to QR and
feasp will terminate when Cholesky fails due to numerical instabilities.
Solve System of LMIs
Consider the problem of finding P > I such that:
This problem arises when studying the quadratic stability of the polytope of the matrices, .
To assess feasibility using
feasp, first enter the LMIs.
setlmis() p = lmivar(1,[2 1]); A1 = [-1 2;1 -3]; A2 = [-0.8 1.5; 1.3 -2.7]; A3 = [-1.4 0.9;0.7 -2.0]; lmiterm([1 1 1 p],1,A1,'s'); % LMI #1 lmiterm([2 1 1 p],1,A2,'s'); % LMI #2 lmiterm([3 1 1 p],1,A3,'s'); % LMI #3 lmiterm([-4 1 1 p],1,1); % LMI #4: P lmiterm([4 1 1 0],1); % LMI #4: I lmis = getlmis;
feasp to a find a feasible decision vector.
[tmin,xfeas] = feasp(lmis);
Solver for LMI feasibility problems L(x) < R(x) This solver minimizes t subject to L(x) < R(x) + t*I The best value of t should be negative for feasibility Iteration : Best value of t so far 1 0.972718 2 0.870460 3 -3.136305 Result: best value of t: -3.136305 f-radius saturation: 0.000% of R = 1.00e+09
tmin = -3.1363 means that the problem is feasible. Therefore, the dynamical system is quadratically stable for
To obtain a Lyapunov matrix
P proving the quadratic stability, use
P = dec2mat(lmis,xfeas,p)
P = 2×2 270.8553 126.3999 126.3999 155.1336
It is possible to add further constraints on this feasibility problem. For instance, the following command bounds the Frobenius norm of
P by 10 while asking
tmin to be less than or equal to –1.
options = [0,0,10,0,0]; [tmin,xfeas] = feasp(lmis,options,-1);
Solver for LMI feasibility problems L(x) < R(x) This solver minimizes t subject to L(x) < R(x) + t*I The best value of t should be negative for feasibility Iteration : Best value of t so far 1 0.988505 2 0.872239 3 -0.476638 4 -0.920574 5 -0.920574 *** new lower bound: -3.726964 6 -1.011130 *** new lower bound: -1.602398 Result: best value of t: -1.011130 f-radius saturation: 91.385% of R = 1.00e+01
The third entry of
options sets the feasibility radius to 10 while the third argument to
-1, sets the target value for
tmin. This constraint yields
tmin = -1.011 and a matrix
P with largest eigenvalue = 8.4653.
P = dec2mat(lmis,xfeas,p); e = eig(P)
e = 2×1 3.8875 8.4653
The feasibility solver
feasp is based on Nesterov and Nemirovski's Projective Method described in:
Nesterov, Y., and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994.
Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, Baltimore, Maryland, p. 840–844.
The optimization is performed by the C-MEX file