linearizebmi & solvebmi

Manuals

1. solvebmi
2. solvebmiOptions
3. linearizebmi
4. linearizebmiOptions

solvebmi

Syntax

   [gg,variables,outopts] = solvebmi(Fstr,{'X','Y'},options)

Description

Example

1. Define data matrices and their size in the problem.

             n = ...;
             m = ...;
             a = ...;
             b1 = ...;
             .
             .
             .
           

2. Define the decision matrices \(P\), \(K\) and the objective function \(\gamma\).

             P = sdpvar(n,n);
             K = sdpvar(m,m);
             g = sdpvar(1,1);
           

3. (Optional) Assign initial values for decision matrices if you know some feasible solutions.
   For example, if \(P=I, K=O\) are feasible solutions, you can assign them as below.

             assign(P,eye(n,n));
             assign(K,zeros(n,n));
           

4. Describe the BMI and LMI constraints as strings.

             Fstr1 = "-P";
             Fstr2 = "[P*(a+b2*K*c2)+(P*(a+b2*K*c2))', P*(b1+b2*K*d21), (c1+d12*K*c2)';"   +...
                     "(P*(b1+b2*K*d21))',              -g*eye(nw,nw),   (d11+d12*K*d21)';" +...
                     "c1+d12*K*c2,                     d11+d12*K*d21,   -g*eye(nz)]"; 
             Flist = {Fstr1,Fstr2};
           

5. Use solvezebmi to solve the BMI problem.

             [gg,vars] = solvebmi(Flist,{'P','K'},g,options);
           

   solvezebmi can use different dilation types (with options) to solve the BMI problem. See examples in detail.


6. Get optimal solutions for decision matrices.

             vars.P
             vars.K
             vars.g
           

   In this problem, vars.g is same as gg.

solvebmiOptions

Syntax

     opts = solvebmiOptions
     opts = solvebmiOptions('NAME1',VALUE1,'NAME2',VALUE2,...)
     opts = solvebmiOptions(OLDOPTIONS,'NAME1',VALUE1,'NAME2',VALUE2,...)

Description

Example

  yalmipopts = sdpsettings;
  yalmipopts = sdpsettings(yalmipopts,'solver','mosek');
  opts = solvebmiOptions;
  opts = solvebmiOptions(opts,'lcmax',1e3);
  opts = solvebmiOptions(opts,'yalmip',yalmipopts);

linearizebmi

Syntax

   LMI = linearizebmi(Fstr,{'X','Y'},{'X0','Y0'})
   LMI = linearizebmi(Fstr,{'X','Y'},{'X0','Y0'},'G')
   LMI = linearizebmi(Fstr,{'X','Y'},{'X0','Y0'},'G',opts)
   [LMI, LMIstr] = linearizebmi(Fstr,{'X','Y'},{'X0','Y0'},'G',opts)

Description

Example

1. Problem definition

         n = 4;
         m = 2;
         A = rand(n,n);
         B = rand(n,m);
         C = rand(m,n);
       

2. Define the decision matrices $P$, $K$ and the constant decomposition matrix $G$.

         % decision matrices
         P = sdpvar(n,n);
         K = sdpvar(m,m);
         % current feasible solutions
         P0 = zeros(n,n);
         K0 = zeros(m,m);
         % constant decomposition matrix
         G = eye(m,m);
     

   In case of using linearizebmi in a iteration loop, one can pass sdpvar object as current feasible solutions. See examples in detail.
   
   
3. Describe the BMI constraint $P(A+BKC)+(A+BKC)^\mathrm{T}\prec O$ by a character string.

         Fstr = "P*(A+B*K*C)+(A+B*K*C)'*P";
     

4. Use linearizebmi to generate a dilated LMI.

         % matrix sdp variable
         [LMI,Lstr] = linearizebmi(Fstr,{'P','K'},{'P0','K0'},'G');
         % constraint in YALMIP format
         constraints = [LMI <= 0];
     

5. Return values of linearizebmi.

         LMI
   
         Linear matrix variable 6x6 (symmetric, real, 13 variables)
         Coeffiecient range: 0.043024 to 3.4622
   
   
         Lstr
   
         LMI:
             "(P-P0)*A+comp*C"    "(P-P0)*B"
             "G*(K-K0)*C"         "-G"      
         comp:
             "P0*B*K0+(P-P0)*B*K0+P0*B*(K-K0)"
     

   Lstr'+Lstr corresponds to the left-hand side of the dilated LMI.

linearizebmiOptions

Syntax

     opts = linearizebmiOptions
     opts = linearizebmiOptions('NAME1',VALUE1,'NAME2',VALUE2,...)
     opts = LinearizebmiOptions(OLDOPTIONS,'NAME1',VALUE1,'NAME2',VALUE2,...)

Description

Example

  opts = solvebmiOptions;
  opts = solvebmiOptions(opts,'method',1);