Specify LMI System at the Command Line
This tutorial example shows how to specify LMI systems at the command line using the LMI Lab tools.
Specify LMI System
Consider a stable transfer function,
$$G\left(s\right)=C{\left(sIA\right)}^{1}B.$$
Suppose that G has four inputs, four outputs, and six states. Consider also a set of input/output scaling matrices D with blockdiagonal structure given by:
$$D=\left(\begin{array}{cccc}{d}_{1}& 0& 0& 0\\ 0& {d}_{1}& 0& 0\\ 0& 0& {d}_{2}& {d}_{3}\\ 0& 0& {d}_{4}& {d}_{5}\end{array}\right).$$
The following problem arises in the robust stability analysis of systems with timevarying uncertainty [4]. Find, if any, a scaling D with the specified structure, such that the largest gain across frequency of $$DG(s){D}^{1}$$ is less than 1.
This problem has a simple LMI formulation: There exists an adequate scaling D if the following feasibility problem has solutions. Find two symmetric matrices $$X\in {R}_{6\times 6}$$ and $$S={D}^{T}D\in {R}_{4\times 4}$$ such that:
$$\left(\begin{array}{cc}{A}^{T}X+XA+{C}^{T}SC& XB\\ {B}^{T}X& S\end{array}\right)<0,$$
$$X>0,$$
$$S>1.$$
You can use the LMI Editor to specify the LMI problem described by these expressions, as shown in Specify LMIs with the LMI Editor GUI. Alternatively, define it at the command line using lmivar
and lmiterm
, as follows.
For this example, use the following values for A, B, and C.
A = [ 0.8715 0.5202 0.7474 1.0778 0.9686 0.1005; 0.5577 1.0843 1.8912 0.2523 1.0641 0.0345; 0.2615 1.7539 1.5452 0.2143 0.0923 2.4192; 0.6087 1.0741 0.1306 2.5575 2.3213 0.2388; 0.7169 0.3582 1.4195 1.7043 2.6530 1.4276; 1.2944 0.6752 1.6983 1.6764 0.3646 1.7730 ]; B = [ 0 0.8998 0.2130 0.9835; 0 0.3001 0 0.2977; 1.0322 0 1.0431 1.1437; 0 0.3451 0.2701 0.5316; 0.4189 1.0128 0.4381 0; 0 0 0.4087 0]; C = [ 0 2.0034 0 1.0289 0.1554 0.7135; 0.9707 0.9510 0.7059 1.4580 1.2371 0.3174; 0 0 1.4158 0.0475 2.1935 0.4136; 0.4383 0.6489 1.6045 1.7463 0.3334 0.5771];
Define the LMI variables X
and S
, and then specify the terms of each LMI.
setlmis([]) X = lmivar(1,[6 1]); S = lmivar(1,[2 0;2 1]); % 1st LMI lmiterm([1 1 1 X],1,A,'s'); lmiterm([1 1 1 S],C',C); lmiterm([1 1 2 X],1,B); lmiterm([1 2 2 S],1,1); % 2nd LMI lmiterm([2 1 1 X],1,1); % 3rd LMI lmiterm([3 1 1 S],1,1); lmiterm([3 1 1 0],1); LMISYS = getlmis;
The lmivar
commands define the two matrix variables, X and S. The lmiterm
commands describe the terms in each LMI. getlmis
returns the internal representation LMISYS
of this LMI problem.
For more details on how to use these commands, see:
For more information about how lmivar
updates the internal representation of the LMI problem, see How lmivar and lmiterm Manage LMI Representation.
Initializing the LMI System
The description of an LMI system should begin with setlmis
and end with getlmis
. The function setlmis
initializes the LMI system
description. When specifying a new system, type
setlmis([])
To add on to an existing LMI system with internal representation
LMIS0
, type
setlmis(LMIS0)
Specifying the LMI Variables
The matrix variables are declared one at a time with lmivar
and are characterized by
their structure. To facilitate the specification of this structure, the LMI Lab
offers two predefined structure types along with the means to describe more general
structures:
Type 1  Symmetric block diagonal structure. This corresponds to matrix variables of the form
$$X=\left(\begin{array}{cccc}{D}_{1}& 0& \dots & 0\\ 0& {D}_{2}& \ddots & \vdots \\ \vdots & \ddots & \ddots & 0\\ 0& \dots & 0& {D}_{r}\end{array}\right)$$
where each diagonal block D_{j} is square and is either zero, a full symmetric matrix, or a scalar matrix
D_{j}= d × I, d ∊ R This type encompasses ordinary symmetric matrices (single block) and scalar variables (one block of size one). 
Type 2 
Rectangular structure. This corresponds to arbitrary rectangular matrices without any particular structure. 
Type 3 
General structures. This
third type is used to describe more sophisticated structures
and/or correlations between the matrix variables. The
principle is as follows: each entry of
X is specified independently as
either 0, xn, or
–xn where xn
denotes the nth decision variable in
the problem. For details on how to use Type 3, see Structured Matrix Variables as well as the

In Specify LMI System, the matrix variables X and S are of Type 1. Indeed, both are symmetric and S inherits the blockdiagonal structure of D. Specifically, S is of the form
$$S=\left(\begin{array}{cccc}{s}_{1}& 0& 0& 0\\ 0& {s}_{1}& 0& 0\\ 0& 0& {s}_{2}& {s}_{3}\\ 0& 0& {s}_{3}& {s}_{4}\end{array}\right).$$
Initialize the description and declare these two matrix variables.
setlmis([]) lmivar(1,[6 1]); % X lmivar(1,[2 0;2 1]); % S
In both lmivar
commands, the first input specifies the
structure type and the second input contains additional information about the
structure of the variable:
For a matrix variable X of Type 1, this second input is a matrix with two columns and as many rows as diagonal blocks in X. The first column lists the sizes of the diagonal blocks and the second column specifies their nature with the following convention:
1: full symmetric block
0: scalar block
–1: zero block
In the second command, for instance,
[2 0;2 1]
means that S has two diagonal blocks, the first one being a 2by2 scalar block and the second one a 2by2 full block.For matrix variables of Type 2, the second input of
lmivar
is a twoentry vector listing the row and column dimensions of the variable. For instance, a 3by5 rectangular matrix variable would be defined bylmivar(2,[3 5])
For convenience, lmivar
also returns a
“tag” that identifies the matrix variable for subsequent reference.
For instance, X and S in Specify LMI System could be defined by
X = lmivar(1,[6 1]); S = lmivar(1,[2 0;2 1]);
The identifiers X and S are integers
corresponding to the ranking of X and S in
the list of matrix variables (in the order of declaration). Here their values would
be X=1
and S=2
. Note that these identifiers
still point to X and S after deletion or
instantiation of some of the matrix variables. Finally, lmivar
can also return the total
number of decision variables allocated so far as well as the entrywise dependence
of the matrix variable on these decision variables (see the lmivar
entry in the reference
pages for more details).
Specifying Individual LMIs
After declaring the matrix variables with lmivar
, we are left with
specifying the term content of each LMI. Recall that LMI terms fall into three
categories:
The constant terms, i.e., fixed matrices like I in the left side of the LMI S > I.
The variable terms, i.e., terms involving a matrix variable. For instance, A^{T}X and C^{T}SC in the expression:
$$\left(\begin{array}{cc}{A}^{T}X+XA+{C}^{T}SC& XB\\ {B}^{T}X& S\end{array}\right)<0$$
Variable terms are of the form PXQ where X is a variable and P, Q are given matrices called the left and right coefficients, respectively.
The outer factors.
When describing the term content of an LMI, specify only the terms in the blocks on or above the diagonal. The inner factors being symmetric, this is sufficient to specify the entire LMI. Specifying all blocks results in the duplication of offdiagonal terms, hence in the creation of a different LMI. Alternatively, you can describe the blocks on or below the diagonal.
LMI terms are specified one at a time with lmiterm
. For instance, the
LMI
$$\left(\begin{array}{cc}{A}^{T}X+XA+{C}^{T}SC& XB\\ {B}^{T}X& S\end{array}\right)<0$$
is described by
lmiterm([1 1 1 1],1,A,'s');
lmiterm([1 1 1 2],C',C);
lmiterm([1 1 2 1],1,B);
lmiterm([1 2 2 2],1,1);
These commands successively declare the terms A^{T}X + XA, C^{T}SC, XB, and –S. In each command, the first argument is a fourentry vector listing the term characteristics as follows:
The first entry indicates to which LMI the term belongs. The value
m
means “left side of the mth LMI,” and −m
means “right side of the mth LMI.”The second and third entries identify the block to which the term belongs. For instance, the vector [
1 1 2 1
] indicates that the term is attached to the (1, 2) block.The last entry indicates which matrix variable is involved in the term. This entry is
0
for constant terms,k
for terms involving the kth matrix variable X_{k}, and −k
for terms involving $${X}_{k}^{T}$$ (here X and S are first and second variables in the order of declaration).
Finally, the second and third arguments of lmiterm
contain the numerical data
(values of the constant term, outer factor, or matrix coefficients
P and Q for variable terms
PXQ or
PX^{T}Q).
These arguments must refer to existing MATLAB^{®} variables and be realvalued. See ComplexValued LMIs for the specification of LMIs
with complexvalued coefficients.
Some shorthand is provided to simplify term specification. First, blocks are zero by
default. Second, in diagonal blocks the extra argument
's'
allows you to specify the conjugated expression
AXB +
B^{T}X^{T}A^{T}
with a single
lmiterm
command. For instance, the
first command specifies
A^{T}X
+ XA as the “symmetrization” of
XA. Finally, scalar values are allowed as shorthand for
scalar matrices, i.e., matrices of the form
αI with α scalar. Thus, a constant term of the form
αI can be specified as the “scalar” α. This
also applies to the coefficients P and Q
of variable terms. The dimensions of scalar matrices are inferred from the context
and set to 1 by default. For instance, the third LMI S >
I in Specify Matrix Variable Structures is described
by
lmiterm([3 1 1 2],1,1); % 1*S*1 = S lmiterm([3 1 1 0],1); % 1*I = I
Recall that by convention S
is considered as the right side of
the inequality, which justifies the –3 in the first command.
Finally, to improve readability it is often convenient to attach an identifier
(tag) to each LMI and matrix variable. The variable identifiers are returned by
lmivar
and the LMI identifiers are
set by the function newlmi
. These identifiers can be
used in lmiterm
commands to refer to a
given LMI or matrix variable. For the LMI system of Specify LMI System, this would look like:
setlmis([])
X = lmivar(1,[6 1]);
S = lmivar(1,[2 0;2 1]);
BRL = newlmi;
lmiterm([BRL 1 1 X],1,A,'s');
lmiterm([BRL 1 1 S],C',C);
lmiterm([BRL 1 2 X],1,B);
lmiterm([BRL 2 2 S],1,1);
Xpos = newlmi;
lmiterm([Xpos 1 1 X],1,1);
Slmi = newlmi;
lmiterm([Slmi 1 1 S],1,1);
lmiterm([Slmi 1 1 0],1);
When the LMI system is completely specified, get the internal representation of the problem.
LMISYS = getlmis;
This returns the internal representation LMISYS
of this LMI
system. This MATLAB description of the problem can be forwarded to other LMILab functions
for subsequent processing. The command getlmis
must be used
only
once and after declaring all matrix
variables and LMI terms.
Here the identifiers X
and S
point to the
variables X and S while the tags
BRL
, Xpos
, and Slmi
point to the first, second, and third LMI, respectively. Note that
–Xpos
refers to the righthand side of the second LMI.
Similarly, –X
would indicate transposition of the variable
X.
See Also
lmiterm
 lmivar
 setlmis
 getlmis