This tutorial example shows how to specify LMI systems at the command line using the LMI Lab tools.
Consider a stable transfer function,
Suppose that G has four inputs, four outputs, and six states. Consider also a set of input/output scaling matrices D with block-diagonal structure given by:
The following problem arises in the robust stability analysis of systems with time-varying uncertainty . Find, if any, a scaling D with the specified structure, such that the largest gain across frequency of 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 and such that:
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
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
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;
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.
To add on to an existing LMI system with internal representation
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
Symmetric block diagonal structure. This corresponds to matrix variables of the form
where each diagonal block Dj is square and is either zero, a full symmetric matrix, or a scalar matrix
Dj= d × I, d ∊ R
This type encompasses ordinary symmetric matrices (single block) and scalar variables (one block of size one).
Rectangular structure. This corresponds to arbitrary rectangular matrices without any particular structure.
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 n-th 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 block-diagonal structure of D. Specifically, S is of the form
Initialize the description and declare these two matrix variables.
setlmis() lmivar(1,[6 1]); % X lmivar(1,[2 0;2 1]); % S
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
2-by-2 scalar block and the second one a 2-by-2 full block.
For matrix variables of Type 2, the second input of
lmivar is a two-entry
vector listing the row and column dimensions of the variable. For instance,
a 3-by-5 rectangular matrix variable would 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
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 entry-wise dependence
of the matrix variable on these decision variables (see the
lmivar entry in the reference
pages for more details).
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
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, ATX and CTSC in the expression:
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 off-diagonal 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
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 ATX + XA, CTSC, XB, and –S. In each command, the first argument is a four-entry 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
m-th LMI,” and −
“right side of the m-th 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,
for terms involving the k-th matrix variable
k for terms involving (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
These arguments must refer to existing MATLAB® variables and be real-valued. See Complex-Valued LMIs for the specification of LMIs
with complex-valued 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
with a single
lmiterm command. For instance, the
first command specifies
+ 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
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
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 LMI-Lab functions
for subsequent processing. The command
getlmis must be used
once and after declaring all matrix
variables and LMI terms.
Here the identifiers
S point to the
variables X and S while the tags
point to the first, second, and third LMI, respectively. Note that
Xpos refers to the right-hand side of the second LMI.
X would indicate transposition of the variable