Robust Control Toolbox™

Describe how entries of matrix variable X relate to decision variables

Syntax

• decinfo(lmisys) 
  decX = decinfo(lmisys,X) 
  

Description

The function decinfo expresses the entries of a matrix variable X in terms of the decision variables x1, . . ., xN. Recall that the decision variables are the free scalar variables of the problem, or equivalently, the free entries of all matrix variables described in lmisys. Each entry of X is either a hard zero, some decision variable xn, or its opposite -xn.

If X is the identifier of X supplied by lmivar, the command

• decX = decinfo(lmisys,X)
  

returns an integer matrix decX of the same dimensions as X whose (i, j) entry is

• 0 if X(i, j) is a hard zero
• n if X(i, j) = xn (the n-th decision variable)
• -n if X(i, j) = -xn

decX clarifies the structure of X as well as its entry-wise dependence on
x1, . . ., xN. This is useful to specify matrix variables with atypical structures (see lmivar).

decinfo can also be used in interactive mode by invoking it with a single argument. It then prompts the user for a matrix variable and displays in return the decision variable content of this variable.

Example 1

Consider an LMI with two matrix variables X and Y with structure:

• X = x I3 with x scalar
• Y rectangular of size 2-by-1

If these variables are defined by

• setlmis([]) 
  X = lmivar(1,[3 0]) 
  Y = lmivar(2,[2 1]) 
      : 
      : 
  lmis = getlmis
  

the decision variables in X and Y are given by

• dX = decinfo(lmis,X)
  
  dX = 
      1   0   0 
      0   1   0 
      0   0   1
  
  dY = decinfo(lmis,Y)
  
  dY = 
      2 
      3
  

This indicates a total of three decision variables x1, x2, x3 that are related to the entries of X and Y by

Note that the number of decision variables corresponds to the number of free entries in X and Y when taking structure into account.

Example 2

Suppose that the matrix variable X is symmetric block diagonal with one 2-by-2 full block and one 2-by-2 scalar block, and is declared by

• setlmis([]) 
  X = lmivar(1,[2 1;2 0]) 
          : 
  lmis = getlmis
  

The decision variable distribution in X can be visualized interactively as follows:

• decinfo(lmis)
  
  There are 4 decision variables labeled x1 to x4 in this problem.
  
  Matrix variable Xk of interest (enter k between 1 and 1, or 0 to 
  quit):
  
  ?> 1
  
  The decision variables involved in X1 are among {-x1,...,x4}. 
  Their entry-wise distribution in X1 is as follows
          (0,j>0,-j<0 stand for 0,xj,-xj, respectively):
  
  X1 :
  
      1   2   0    0 
      2   3   0    0 
      0   0   4    0 
      0   0   0    4
      
               *********
  
  Matrix variable Xk of interest (enter k between 1 and 1, or 0 to 
  quit):
  
  ?> 0
  

See Also
lmivar      Specify the matrix variables in an LMI problem

mat2dec     Return the vector of decision variables corresponding to particular values of the matrix variables

dec2mat     Given values of the decision variables, derive the corresponding values of the matrix variables decnbr

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