Approximately solve constant-matrix, upper bound µ-synthesis problem

[QOPT,BND] = cmsclsyn(R,U,V,BlockStructure); [QOPT,BND] = cmsclsyn(R,U,V,BlockStructure,opt); [QOPT,BND] = cmsclsyn(R,U,V,BlockStructure,opt,qinit); [QOPT,BND] = cmsclsyn(R,U,V,BlockStructure,opt,'random',N)

approximately solves the constant-matrix, upper bound µ-synthesis
problem by minimization,`cmsclsyn`

$${\mathrm{min}}_{Q\in {C}^{r\times t}}{\mu}_{\Delta}\left(R+UQV\right)$$

for given matrices *R* ∊ **C**^{n}x_{m}, *U* ∊ **C**^{n}x_{r}, *V* ∊ **C**^{t}x_{m}, and a set Δ ⊂ **C**^{m}x_{n}. This applies to constant matrix data
in *R*, *U*, and *V*.

`[QOPT,BND] = cmsclsyn(R,U,V,BlockStructure)`

minimizes,
by choice of Q. `QOPT`

is the optimum value of Q,
the upper bound of `mussv(R+U*Q*V,BLK), BND`

. The
matrices `R,U`

and` V`

are constant
matrices of the appropriate dimension.` BlockStructure`

is
a matrix specifying the perturbation blockstructure as defined for `mussv`

.

`[QOPT,BND] = cmsclsyn(R,U,V,BlockStructure,OPT)`

uses
the options specified by `OPT`

in the calls to `mussv`

.
See `mussv`

for more information. The default value
for `OPT`

is `'cUsw'`

.

`[QOPT,BND] = cmsclsyn(R,U,V,BlockStructure,OPT,QINIT)`

initializes
the iterative computation from Q = `QINIT`

. Because
of the nonconvexity of the overall problem, different starting points
often yield different final answers. If `QINIT`

is
an N-D array, then the iterative computation is performed multiple
times - the `i`

'th optimization is initialized at
Q = `QINIT(:,:,i)`

. The output arguments are associated
with the best solution obtained in this brute force approach.

`[QOPT,BND] = cmsclsyn(R,U,V,BlockStructure,OPT,'random',N)`

initializes
the iterative computation from `N`

random instances
of `QINIT`

. If `NCU`

is the number
of columns of `U`

, and `NRV`

is
the number of rows of `V`

, then the approximation
to solving the constant matrix µ synthesis problem is two-fold:
only the upper bound for µ is minimized, and the minimization
is not convex, hence the optimum is generally not found. If `U`

is
full column rank, or `V`

is full row rank, then the
problem can (and is) cast as a convex problem, [Packard, Zhou, Pandey
and Becker], and the global optimizer (for the upper bound for µ)
is calculated.

Packard, A.K., K. Zhou, P. Pandey, and G. Becker, "A
collection of robust control problems leading to LMI's," *30th
IEEE Conference on Decision and Control,* Brighton, UK,
1991, p. 1245–1250.

Was this topic helpful?