Compute bounds on structured singular value (µ)
bounds = mussv(M,BlockStructure) [bounds,muinfo] = mussv(M,BlockStructure) [bounds,muinfo] = mussv(M,BlockStructure,Options) [ubound,q] = mussv(M,F,BlockStructure) [ubound,q] = mussv(M,F,BlockStructure,'s')
bounds = mussv(M,BlockStructure)
calculates upper and lower bounds on the structured singular value,
or µ, for a given block structure.
M is a
frd model, or a state-space (
Mis an N-D array (with
N≥ 3), then the computation is performed pointwise along the third and higher array dimensions.
frdmodel, then the computations are performed pointwise in frequency (as well as any array dimensions).
ssmodel, the computations are performed using state-space algorithms. Frequencies are adaptively selected, and upper bounds are guaranteed to hold over each interval between frequencies.
Mmust be a single system, without array dimensions.
BlockStructure is a matrix specifying the
perturbation block structure.
2 columns, and as many rows as uncertainty blocks in the perturbation
structure. The i-th row of
the dimensions of the i'th perturbation block.
BlockStructure(i,:) = [-r 0], then the i-th block is an
rrepeated, diagonal real scalar perturbation;
BlockStructure(i,:) = [r 0], then the i-th block is an
rrepeated, diagonal complex scalar perturbation;
BlockStructure(i,:) = [r c], then the i-th block is an
ccomplex full-block perturbation.
BlockStructureis omitted, its default is
ones(size(M,1),2), which implies a perturbation structure of all 1-by-1 complex blocks. In this case, if
size(M,1)does not equal
size(M,2), an error results.
M is a two-dimensional matrix, then
1-by-2 array containing an upper (first column)
and lower (second column) bound of the structured singular value of
For all matrices
Delta with block-diagonal structure
BlockStructure and with norm less than
bound), the matrix
I - M*Delta is not singular.
Moreover, there is a matrix
DeltaS with block-diagonal
structure defined by
BlockStructure and with norm
1/bounds(2) (lower bound), for which the
I - M*DeltaS is singular.
The format used in the 3rd output argument from
also acceptable for describing the block structure.
M is an
frd, the computations
are always performed pointwise in frequency. The output argument
1-by-2 frd of upper and lower bounds at each
frequency. Note that
M is an N-D array (either
the upper and lower bounds are computed pointwise along the 3rd and
higher array dimensions (as well as pointwise in frequency, for
For example, suppose that
size(M) is r×c×d1×...×dF.
size(bounds) is 1×2×d1×...×dF.
Using single index notation,
bounds(1,1,i) is the
upper bound for the structured singular value of
bounds(1,2,i) is the lower bound for the structured
singular value of
M(:,:,i). Here, any
1 and d1·d2...dF (the
product of the dk) would
M is a
returned as an
bounds = mussv(M,BlockStructure,Options)
specifies computation options.
Options is a character
vector, containing any combination of the following characters:
Upper bound to greatest accuracy, using LMI solver. This is the default behavior when the number of decision variables within the D/G scalings is less than 45.
Force fast upper bound (typically not as tight as the default)
Force upper bound to use gradient method. This is the default behavior when the number of decision variables within the D/G scalings is greater than or equal to 45.
Upper-bound “only” (lower bound uses a fast/cheap algorithm).
Use gain-based lower bound method multiple times. The
all uncertainty blocks described by
Reinitialize lower bound computation at each new matrix
(only relevant if
Randomly reinitialize lower bound iteration multiple
Use power iteration method to compute lower bound. When
at least one of the uncertainty blocks described by
Suppress progress information (silent).
Decrease iterations in lower bound computation (faster
but not as tight as default). Use
Run “old” algorithms, from version 3.1.1 and before. Included to allow exact replication of earlier calculations.
[bounds,muinfo] = mussv(M,BlockStructure) returns
a structure containing more detailed information. The information
muinfo must be extracted using
Generalized Structured Singular Value
ubound = mussv(M,F,BlockStructure)
calculates an upper bound on the generalized structured singular value
(generalized µ) for a given block structure.
F is an additional (
ubound = mussv(M,F,BlockStructure,'s')
adds an option to run silently. Other options are ignored for generalized
Note that in generalized structured singular value computations,
only an upper bound is calculated.
ubound is an
upper bound of the generalized structured singular value of the pair
with respect to the block-diagonal uncertainty described by
ubound is 1-by-1 (with additional
array dependence, depending on
For all matrices
Delta with block-diagonal structure
[I-Delta*M;F] is guaranteed not to lose
column rank. This is verified by the matrix
a detailed example of the structured singular value.
A simple example for generalized structured singular value can be done with random complex matrices, illustrating the relationship between the upper bound for µ and generalized µ, as well as the fact that the upper bound for generalized µ comes from an optimized µ upper bound.
M is a complex 5-by-5 matrix and
a complex 2-by-5 matrix. The block structure
an uncertain real parameter δ1, an uncertain
real parameter δ2, an uncertain complex
parameter δ3 and a twice-repeated uncertain
complex parameter δ4.
rng(929,'twister') M = randn(5,5) + sqrt(-1)*randn(5,5); F = randn(2,5) + sqrt(-1)*randn(2,5); BlockStructure = [-1 0;-1 0;1 1;2 0]; [ubound,Q] = mussv(M,F,BlockStructure); bounds = mussv(M,BlockStructure); optbounds = mussv(M+Q*F,BlockStructure);
be extremely close, and significantly lower than
ans = 2.2070 2.1749
ans = 4.4049 4.1960
The lower bound is computed using a power method, Young and Doyle, 1990, and Packard et al. 1988, and the upper bound is computed using the balanced/AMI technique, Young et al., 1992, for computing the upper bound from Fan et al., 1991.
Peter Young and Matt Newlin wrote the original function.
The lower-bound power algorithm is from Young and Doyle, 1990, and Packard et al. 1988.
The upper-bound is an implementation of the bound from Fan et
al., 1991, and is described in detail in Young et
al., 1992. In the upper bound computation, the matrix is
first balanced using either a variation of Osborne's method (Osborne,
1960) generalized to handle repeated scalar and full blocks,
or a Perron approach. This generates the standard upper bound for
the associated complex µ problem. The Perron eigenvector method
is based on an idea of Safonov, (Safonov, 1982). It gives the exact
computation of µ for positive matrices with scalar blocks, but
is comparable to Osborne on general matrices. Both the Perron and
Osborne methods have been modified to handle repeated scalar and full blocks.
Perron is faster for small matrices but has a growth rate of n3,
compared with less than n2 for
Osborne. This is partly due to the
which greatly favors Perron. The default is to use Perron for simple
block structures and Osborne for more complicated block structures.
A sequence of improvements to the upper bound is then made based on
various equivalent forms of the upper bound. A number of descent techniques
are used that exploit the structure of the problem, concluding with
general purpose LMI optimization (Boyd et al.),
1993, to obtain the final answer.
The optimal choice of
Q (to minimize the
upper bound) in the generalized µ problem is solved by reformulating
the optimization into a semidefinite program (Packard et
 Boyd, S. and L. El Ghaoui, “Methods of centers for minimizing generalized eigenvalues,” Linear Algebra and Its Applications, Vol. 188–189, 1993, pp. 63–111.
 Fan, M., A. Tits, and J. Doyle, “Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics,” IEEE Transactions on Automatic Control, Vol. AC–36, 1991, pp. 25–38.
 Osborne, E., “On preconditioning of matrices,” Journal of Associated Computer Machines, Vol. 7, 1960, pp. 338–345.
 Packard, A.K., M. Fan and J. Doyle, “A power method for the structured singular value,” Proc. of 1988 IEEE Conference on Control and Decision, December 1988, pp. 2132–2137.
 Safonov, M., “Stability margins for diagonally perturbed multivariable feedback systems,” IEEE Proc., Vol. 129, Part D, 1992, pp. 251–256.
 Young, P. and J. Doyle, “Computation of with real and complex uncertainties,” Proceedings of the 29th IEEE Conference on Decision and Control, 1990, pp. 1230–1235.
 Young, P., M. Newlin, and J. Doyle, “Practical computation of the mixed problem,” Proceedings of the American Control Conference, 1992, pp. 2190–2194.