lmivar

Specify matrix variables in LMI problem

Syntax

X = lmivar(type,structure)

[X,n,sX] = lmivar(type,structure)

Description

X = lmivar(type,structure) defines a new matrix variable in the LMI system currently described. The array structure specifies the structure of the variable. The optional output X is an identifier that can be used for subsequent reference to the variable. Before using lmivar to create any variables, initialize the LMI system using setlmis.

[X,n,sX] = lmivar(type,structure) returns the total number of decision variables in the problem thus far, and a matrix indicating the dependence of X on the decision variables.

Examples

Type 1 and Type 2 Matrix Variables

Consider an LMI system with three matrix variables $$X_1$$, $$X_2$$, and $$X_3$$ such that:

• $$X_1$$ is a 3-by-3 symmetric matrix (unstructured)

• $$X_2$$ is a 2-by-4 rectangular matrix (unstructured)

• $$X_3$$ =

$$\left(\begin{array}{ccc}\Delta & 0& 0\\ 0& {\delta }_1& 0\\ 0& 0& {\delta }_2{I}_2\end{array}\right),$$

where Δ is an arbitrary 5-by-5 symmetric matrix, $${\delta }_1$$ and $${\delta }_2$$ are scalars, and $$I_2$$ denotes the identity matrix of size 2.

Define these three variables using lmivar.

setlmis([]) 
X1 = lmivar(1,[3 1]);          % Type 1 
X2 = lmivar(2,[2 4]);          % Type 2 of dimension 2-by-4 
X3 = lmivar(1,[5 1;1 0;2 0]);  % Type 1

The last command defines $$X_3$$ as a variable of Type 1 with one full block of size 5 and two scalar blocks of sizes 1 and 2.

Type 3 Matrix Variables

Combined with the extra outputs n and sX of lmivar, Type 3 allows you to specify fairly complex matrix variable structures. For instance, consider a matrix variable X with structure given by

$$X=\left(\begin{array}{cc}{X}_1& 0\\ 0& X_2\end{array}\right)$$,

where $$X_1$$ and $$X_2$$ are 2-by-3 and 3-by-2 rectangular matrices, respectively. Specify this structure as follows.

Define the rectangular variables $$X_1$$ and $$X_2$$.

setlmis([]) 
[X1,n,sX1] = lmivar(2,[2 3]); 
[X2,n,sX2] = lmivar(2,[3 2]);

The outputs sX1 and sX2 give the decision variable content of $$X_1$$ and $$X_2$$.

sX1

sX1 = 2×3

     1     2     3
     4     5     6

sX2

sX2 = 3×2

     7     8
     9    10
    11    12

For instance, sX2(1,1) = 7 means that the (1,1) entry of $$X_2$$ is the seventh decision variable.

Next, use Type 3 to specify the matrix variable X and define its structure in terms of the structures of $$X_1$$ and $$X_2$$.

[X,n,sX] = lmivar(3,[sX1,zeros(2);zeros(3),sX2]);

Confirm that the result X has the desired structure.

sX

sX = 5×5

     1     2     3     0     0
     4     5     6     0     0
     0     0     0     7     8
     0     0     0     9    10
     0     0     0    11    12

Input Arguments

type — Matrix variable type

1 | 2 | 3

Matrix variable type, specified as 1, 2, or 3.

• 1 — Symmetric matrix with a block diagonal structure. Use structure to specify the size and structure of the diagonal blocks.

• 2 — Full rectangular matrix. Use structure to specify the dimensions of the matrix.

• 3 — Other structure. Use structure to provide further information about the structure of the variable.

structure — Matrix variable structure

array

Matrix variable structure, specified as an array. How you specify the structure depends on the variable type.

Type	Value
type = 1	For a matrix variable with N diagonal blocks, specify structure as an N-by-2 matrix, where the ith row describes the ith diagonal block. structure(i,1) specifies the block size. structure(i,2) specifies the block type. structure(i,2) = 0 for a scalar block (a scalar multiple of the identity matrix). structure(i,2) = 1 for a full block (arbitrary symmetric matrix). structure(i,2) = -1 for a zero block. For instance, if the first block of X is a single scalar value and the second block of X is a 3-by-3 full block, then specify structure as [1 0; 3 1].
type = 2	For a full rectangular m-by-n matrix variable, specify structure = [m,n].
type = 3	You can use Type 3 to specify sophisticated matrix variable structures. To specify a variable X of Type 3, first identify how many free independent entries are involved in X. These entries constitute the set of decision variables associated with X. If the problem already involves n decision variables, label the new free variables as xn+1, . . ., xn+p. You then specify the structure of X in terms of those free variables. To do so, specify structure as a matrix of the same dimensions as X, where structure(i,j) is: 0, if X(i,j) = 0. k, if X(i,j) is the kth decision variable. -k, if X(i,j) is (-1) times the kth decision variable. To help specify matrix variables of Type 3, use the n and sX output variables of lmivar to check the number of decision variables and entry-wise dependence of X on the decision variables, respectively. For examples specifying Type 3 LMI variables, see Type 3 Matrix Variables and Advanced LMI Techniques.

Output Arguments

X — Identifier for matrix variable

positive integer

Identifier for the matrix variable, returned as a positive integer. The value of this identifier is k, where k-1 is the number of matrix variables previously declared in this LMI problem. The value of this identifier is not affected by subsequent modifications of the LMI system.

n — Number of decision variables

positive integer

Number of decision variables in the system so far, returned as a positive integer. This value includes the decision variables in X and those in previously defined LMI variables.

sX — Dependence of X on decision variables

matrix

Dependence of X on decision variables, returned as a matrix. Each entry in sX is either 0, the index of a decision variable, or minus the index of a decision variable. For instance, suppose that X depends on two decision variables, x₁ and x₁, as follows:

$$X=\left(\begin{array}{cc}0& -x_1\\ x_2& 0\end{array}\right)$$

When you create the variable X, the lmivar command returns the following value for sX.

sX = 2×2

     0   -1
     2    0

Version History

Introduced before R2006a