Advanced LMI Techniques
This last section gives a few hints for making the most out of the LMI Lab. It is directed toward users who are comfortable with the basics, as described in Tools for Specifying and Solving LMIs.
Structured Matrix Variables
Fairly complex matrix variable structures and interdependencies can be specified
lmivar. Recall that the symmetric
block-diagonal or rectangular structures are covered by Types 1 and 2 of
lmivar provided that the matrix
variables are independent. To describe more complex structures or correlations
between variables, you must use Type 3 and specify each entry of the matrix
variables directly in terms of the free scalar variables of the problem (the
so-called decision variables).
With Type 3, each entry is specified as either 0 or
xn is the n-th
decision variable. The following examples illustrate how to specify nontrivial
matrix variable structures with
lmivar. The following examples
show variable structures with uncorrelated and interdependent matrix
Specify Matrix Variable Structures
Suppose that the variables of the problem include a 3-by-3 symmetric matrix X and a 3-by-3 symmetric Toeplitz matrix, Y, given by:
The variable Y has three independent entries, and thus involves three decision variables. Since Y is independent of X, label these decision variables n + 1, n + 2, and n + 3, where n is the number of decision variables involved in X. To retrieve this number, define the Type 1 variable X.
setlmis() [X,n] = lmivar(1,[3 1]); n
n = 6
The second output argument
n gives the total number of decision variables used so far, which in this case is
n = 6. Given this number, you can define Y.
Y = lmivar(3,n+[1 2 3;2 1 2;3 2 1]);
An equivalent expression to define Y uses the MATLAB(R) command
toeplitz to generate the matrix.
Y = lmivar(3,toeplitz(n+[1 2 3]));
To confirm the variables, visualize the decision variable distributions in X and Y using
lmis = getlmis; decinfo(lmis,X)
ans = 3×3 1 2 4 2 3 5 4 5 6
ans = 3×3 7 8 9 8 7 8 9 8 7
Specify Interdependent Matrix Variables
Consider three matrix variables, X, Y, and Z, with the following structure.
where x, y, z, and t are independent scalar variables. To specify such a triple, first define the two independent variables, X and Y, which are both Type 1.
setlmis(); [X,n,sX] = lmivar(1,[1 0;1 0]); [Y,n,sY] = lmivar(1,[1 0;1 0]);
The third output of lmivar gives the entry-wise dependence of X and Y on the decision variables
sX = 2×2 1 0 0 2
sY = 2×2 3 0 0 4
Using lmivar, you can now specify the structure of the Type 3 variable Z in terms of the decision variables and .
[Z,n,sZ] = lmivar(3,[0 -sX(1,1);-sY(2,2) 0]);
sX(1,1) refers to and
sY(2,2) refers to , this expression defines the variable
Confirm this results by checking the entry-wise dependence of
Z on its decision variables.
sZ = 2×2 0 -1 -4 0
The LMI solvers are written for real-valued matrices and cannot directly handle LMI problems involving complex-valued matrices. However, complex-valued LMIs can be turned into real-valued LMIs by observing that a complex Hermitian matrix L(x) satisfies
L(x) < 0
if and only if
This suggests the following systematic procedure for turning complex LMIs into real ones:
Decompose every complex matrix variable X as
X = X1 + jX2
where X1 and X2 are real
Decompose every complex matrix coefficient A as
A = A1 + jA2
where A1 and A2 are real
Carry out all complex matrix products. This yields affine expressions in X1, X2 for the real and imaginary parts of each LMI, and an equivalent real-valued LMI is readily derived from the above observation.
For LMIs without outer factor, a streamlined version of this procedure consists of replacing any occurrence of the matrix variable X = X1 + jX2 by
and any fixed matrix A = A1 + jA2, including real ones, by
For instance, the real counterpart of the LMI system
|MHXM < X, X = XH > I||(1)|
reads (given the decompositions M = M1 + jM2 and X = X1 + jX2 with Mj, Xj real):
Note that X = XH in turn requires that and . Consequently, X1 and X2 should be declared as symmetric and skew- symmetric matrix variables, respectively.
Assuming, for instance, that M ∊ C5×5, the LMI system (Equation 1) would be specified as follows:
M1=real(M), M2=imag(M) bigM=[M1 M2;-M2 M1] setlmis() % declare bigX=[X1 X2;-X2 X1] with X1=X1' and X2+X2'=0: [X1,n1,sX1] = lmivar(1,[5 1]) [X2,n2,sX2] = lmivar(3,skewdec(5,n1)) bigX = lmivar(3,[sX1 sX2;-sX2 sX1]) % describe the real counterpart of (1.12): lmiterm([1 1 1 0],1) lmiterm([-1 1 1 bigX],1,1) lmiterm([2 1 1 bigX],bigM',bigM) lmiterm([-2 1 1 bigX],1,1) lmis = getlmis
Note the three-step declaration of the structured matrix variable
Specify X1 as a (real) symmetric matrix variable and save its structure description
sX1as well as the number
n1of decision variables used in X1.
Specify X2 as a skew-symmetric matrix variable using Type 3 of
lmivarand the utility
skewdec. The command
skewdec(5,n1)creates a 5-by–5 skew-symmetric structure depending on the decision variables
Define the structure of
bigXin terms of the structures
sX2of X1 and X2.
See Structured Matrix Variables for more details on such structure manipulations.
Specifying cTx Objectives for mincx
The LMI solver
mincx minimizes linear objectives
of the form cTx where x is the
vector of decision variables. In most control problems, however, such objectives are
expressed in terms of the matrix variables rather than of x.
Examples include Trace(X) where X is a
symmetric matrix variable, or uTXu where u is a given
defcx facilitates the derivation
of the c vector when the objective is an affine function of the
matrix variables. For the sake of illustration, consider
the linear objective
where X and P are two symmetric
variables and x0 is a given vector. If
lmsisys is the internal representation of the LMI system and
if x0, X,
P have been declared by
x0 = [1;1] setlmis() X = lmivar(1,[3 0]) P = lmivar(1,[2 1]) : : lmisys = getlmis
the c vector such that can be computed as follows:
n = decnbr(lmisys) c = zeros(n,1) for j=1:n, [Xj,Pj] = defcx(lmisys,j,X,P) c(j) = trace(Xj) + x0'*Pj*x0 end
The first command returns the number of decision variables in the problem and the
second command dimensions c accordingly. Then the
for loop performs the following operations:
Evaluate the matrix variables X and P when all entries of the decision vector x are set to zero except xj: = 1. This operation is performed by the function
defcx. Apart from
j, the inputs of
defcxare the identifiers
Pof the variables involved in the objective, and the outputs
Pjare the corresponding values.
Evaluate the objective expression for
P:= Pj. This yields the j-th entry of
In our example the result is
c = 3 1 2 1
Other objectives are handled similarly by editing the following generic skeleton:
n = decnbr( LMI system ) c = zeros(n,1) for j=1:n, [ matrix values ] = defcx( LMI system,j, matrix identifiers) c(j) = objective(matrix values) end
When solving LMI problems with
gevp, it is possible to constrain
the solution x to lie in the ball
xTx < R2
where R > 0 is called the feasibility radius. This specifies a maximum (Euclidean norm) magnitude for x and avoids getting solutions of very large norm. This may also speed up computations and improve numerical stability. Finally, the feasibility radius bound regularizes problems with redundant variable sets. In rough terms, the set of scalar variables is redundant when an equivalent problem could be formulated with a smaller number of variables.
The feasibility radius R is set by the third entry of the options vector of the LMI solvers. Its default value is R = 109. Setting R to a negative value means “no rigid bound,” in which case the feasibility radius is increased during the optimization if necessary. This “flexible bound” mode may yield solutions of large norms.
The LMI solvers used in the LMI Lab are based on interior-point optimization techniques. To compute feasible solutions, such techniques require that the system of LMI constraints be strictly feasible, that is, the feasible set has a nonempty interior. As a result, these solvers may encounter difficulty when the LMI constraints are feasible but not strictly feasible, that is, when the LMI
L(x) ≤ 0
has solutions while
L(x) < 0
has no solution.
For feasibility problems, this difficulty is automatically circumvented by
feasp, which reformulates the
Find x such that
|L(x) ≤ 0||(2)|
Minimize t subject to
Lx < t × I.
In this modified problem, the LMI constraint is always strictly feasible in x, t and the original LMI Equation 2 is feasible if and only if the global minimum tmin of Equation 2 satisfies
tmin ≤ 0
For feasible but not strictly feasible problems, however, the computational effort
is typically higher as
feasp strives to approach the
global optimum tmin = 0 to a high
For the LMI problems addressed by
gevp, nonstrict feasibility
generally causes the solvers to fail and to return an “infeasibility”
diagnosis. Although there is no universal remedy for this difficulty, it is
sometimes possible to eliminate underlying algebraic constraints to obtain a
strictly feasible problem with fewer variables.
Another issue has to do with homogeneous feasibility problems such as
ATP + P A < 0, P > 0
While this problem is technically well-posed, the LMI optimization is likely to produce solutions close to zero (the trivial solution of the nonstrict problem). To compute a nontrivial Lyapunov matrix and easily differentiate between feasibility and infeasibility, replace the constraint P > 0-by-P > αI with α > 0. Note that this does not alter the problem due to its homogeneous nature.
Semi-Definite B(x) in gevp Problems
Consider the generalized eigenvalue minimization problem
Minimize λ subject to
|A(x) < λB(x), B(x) > 0, C(x) <0.||(3)|
Technically, the positivity of B(x) for some x ∊ Rn is required for the well-posedness of the problem and the applicability of polynomial-time interior-point methods. Hence problems where
with B1(x) > 0 strictly feasible, cannot be directly solved with
gevp. A simple remedy consists of
replacing the constraints
A(x) < B(x), B(x) > 0
where Y is an additional symmetric variable of proper
dimensions. The resulting problem is equivalent to Equation 3 and can be solved directly with
Efficiency and Complexity Issues
As explained in Tools for Specifying and Solving LMIs, the term-oriented description of LMIs used in the LMI Lab typically leads to higher efficiency than the canonical representation
|A0 + x1A1 + ... + xNAN < 0.||(4)|
This is no longer true, however, when the number of variable terms is nearly equal to or greater than the number N of decision variables in the problem. If your LMI problem has few free scalar variables but many terms in each LMI, it is therefore preferable to rewrite it as Equation 4 and to specify it in this form. Each scalar variable xj is then declared independently and the LMI terms are of the form xjAj.
If M denotes the total row size of the LMI system and
N the total number of scalar decision variables, the flop
count per iteration for the
mincx solvers is proportional
N3 when the least-squares problem is solved via Cholesky factorization of the Hessian matrix (default) .
M-by-N2 when numerical instabilities warrant the use of QR factorization instead.
While the theory guarantees a worst-case iteration count proportional to
M, the number of iterations actually performed grows slowly
with M in most problems. Finally, while
mincx are comparable in
gevp typically demands more
computational effort. Make sure that your LMI problem cannot be solved with
mincx before using
Solving M + PTXQ + QTXTP < 0
In many output-feedback synthesis problems, the design can be performed in two steps:
Compute a closed-loop Lyapunov function via LMI optimization.
Given this Lyapunov function, derive the controller state-space matrices by solving an LMI of the form
M + PTXQ + QTXTP < 0 (5)
where M, P, Q are given matrices and X is an unstructured m-by-n matrix variable.
It turns out that a particular solution Xc of Equation 5 can be computed via simple linear algebra manipulations . Typically, Xc corresponds to the center of the ellipsoid of matrices defined by Equation 5.
basiclmi returns the “explicit”
Xc = basiclmi(M,P,Q)
Since this central solution sometimes has large norm,
also offers the option of computing an approximate least-norm solution of Equation 5. This is done by
X = basiclmi(M,P,Q,'Xmin')
and involves LMI optimization to minimize ∥X ∥.