Documentation

mussv

Compute bounds on structured singular value (µ)

Syntax

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')

Description

bounds = mussv(M,BlockStructure) calculates upper and lower bounds on the structured singular value, or µ, for a given block structure. M is a double array, an frd model, or a state-space (ss) model.

• If M is an N-D array (with N ≥ 3), then the computation is performed pointwise along the third and higher array dimensions.

• If M is a frd model, then the computations are performed pointwise in frequency (as well as any array dimensions).

• If M is a ss model, the computations are performed using state-space algorithms. Frequencies are adaptively selected, and upper bounds are guaranteed to hold over each interval between frequencies. M must be a single system, without array dimensions.

BlockStructure is a matrix specifying the perturbation block structure. BlockStructure has 2 columns, and as many rows as uncertainty blocks in the perturbation structure. The i-th row of BlockStructure defines the dimensions of the i'th perturbation block.

• If BlockStructure(i,:) = [-r 0], then the i-th block is an r-by-r repeated, diagonal real scalar perturbation;

• if BlockStructure(i,:) = [r 0], then the i-th block is an r-by-r repeated, diagonal complex scalar perturbation;

• if BlockStructure(i,:) = [r c], then the i-th block is an r-by-c complex full-block perturbation.

• If BlockStructure is 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.

If M is a two-dimensional matrix, then bounds is a 1-by-2 array containing an upper (first column) and lower (second column) bound of the structured singular value of M. For all matrices Delta with block-diagonal structure defined by BlockStructure and with norm less than 1/bounds(1) (upper 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 equal to 1/bounds(2) (lower bound), for which the matrix I - M*DeltaS is singular.

The format used in the 3rd output argument from lftdata is also acceptable for describing the block structure.

If M is an frd, the computations are always performed pointwise in frequency. The output argument bounds is a 1-by-2 frd of upper and lower bounds at each frequency. Note that bounds.Frequency equals M.Frequency.

If M is an N-D array (either double or frd), the upper and lower bounds are computed pointwise along the 3rd and higher array dimensions (as well as pointwise in frequency, for frd). For example, suppose that size(M) is r×c×d1×...×dF. Then 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 M(:,:,i), and bounds(1,2,i) is the lower bound for the structured singular value of M(:,:,i). Here, any i between 1 and d1·d2...dF (the product of the dk) would be valid.

If M is a ss model, bounds is returned as an frd model.

bounds = mussv(M,BlockStructure,Options) specifies computation options. Options is a character vector, containing any combination of the following characters:

Option

Meaning

'a'

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.

'f'

Force fast upper bound (typically not as tight as the default)

'G'

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.

'U'

Upper-bound “only” (lower bound uses a fast/cheap algorithm).

'gN'

Use gain-based lower bound method multiple times. The value of N sets the number of times, according to 10+N*10. For example, 'g6' uses gain-based lower bound 70 times. Larger numbers typically give better lower bounds.

If all uncertainty blocks described by blk are real, then the default is 'g1'. If at least one uncertainty block is complex, then mussv uses power iteration lower bound by default.

'i'

Reinitialize lower bound computation at each new matrix (only relevant if M is ND array or frd).

'mN'

Randomly reinitialize lower bound iteration multiple times. N is an integer between 1 and 9. For example, 'm7' randomly reinitializes the lower bound iteration 7 times. Larger numbers are typically more computationally expensive, but often give better lower bounds.

'p'

Use power iteration method to compute lower bound. When at least one of the uncertainty blocks described by BlockStructure is complex, then 'p' is the default lower bound method.

's'

Suppress progress information (silent).

'd'

Display warnings.

'x'

Decrease iterations in lower bound computation (faster but not as tight as default). Use 'U' for an even faster lower bound.

'an'

Same as 'a', but without automatic prescaling.

'o'

Run “old” algorithms, from version 3.1.1 and before. Included to allow exact replication of earlier calculations.

[bounds,muinfo] = mussv(M,BlockStructure) returns muinfo, a structure containing more detailed information. The information within muinfo must be extracted using mussvextract.

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. M is a double or frd object. M and BlockStructure are as before. F is an additional (double or frd).

ubound = mussv(M,F,BlockStructure,'s') adds an option to run silently. Other options are ignored for generalized µ problems.

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 (M,F), with respect to the block-diagonal uncertainty described by BlockStructure. Consequently ubound is 1-by-1 (with additional array dependence, depending on M and F). For all matrices Delta with block-diagonal structure defined by BlockStructure and norm<1/ubound, the matrix [I-Delta*M;F] is guaranteed not to lose column rank. This is verified by the matrix Q, which satisfies mussv(M+Q*F,BlockStructure,'a')<=ubound.

Examples

See mussvextract for 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 F is a complex 2-by-5 matrix. The block structure BlockStructure is 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);

The quantities optbounds(1) and ubound should be extremely close, and significantly lower than bounds(1) and bounds(2).

[optbounds(1) ubound]
ans =

2.2070    2.1749
[bounds(1)  bounds(2)]
ans =

4.4049    4.1960

Algorithms

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 MATLAB implementation, 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 al., 1991).

References

 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. 