Computing µ -Values for Real and Mixed µ Problems

: In various modern linear control systems, a common practice is to make use of control in the feedback loops which act as an important tool for linear feedback systems. Stability and instability analysis of a linear feedback system give the measure of perturbed system to be singular and non-singular. The main objective of this article is to discuss numerical computation of the µ -values bounds by using low ranked ordinary differential equations based technique. Numerical computations illustrate the behavior of the method and the spectrum of operators are then numerically analyzed.


Introduction
In system theory, µ-value known as structured singular value (SSV) is a well-defined tool, first introduced by Doyle [1].It is used to analyze the robustness of linear feedback systems when they are subject to uncertainties, particularly structured uncertainties.The µ-value tool has vast applications in control engineering, for instance, for the analysis of controller of an unstable system, the MIMO margins by using µ and for the space shuttle robustness analysis [2].For more applications, we refer [3,4] and the references therein.
The numerical approximation of SSV appears as NP-hard problem [5,6].The computation of the upper bounds of µ-values is a hard problem; in fact, the NP-hard is in nature when only real uncertainties appear [7].
The power method [3,8] is used to approximate the lower bounds of µ-values when only pure complex uncertainties are under consideration.This is a seemingly robust numerical method due to the fact that it depends solely on matrix-vector products.The power method is easy to apply only when mixed real perturbations and complex perturbations are taken into account [9][10][11].This method resembles a mixture of power techniques used for the computation of eigenvalues and singular values of the constant matrices.This method also prove that µ-values act as an equilibrium point of the power technique.The power method fails to numerically compute the lower bounds of µ-values only when pure real uncertainties are under discussion.We refer to [12][13][14][15][16][17][18][19][20] and references therein for a comprehensive study of the numerical algorithms to compute numerically the bounds of µ-values from below.
An approximation of bounds of µ-values involve fundamental difficulties when one considers pure real uncertainties because the real µ-values might be discontinuous function of the given problem data [21,22].Thus the available algorithms [13][14][15]19] to deal such problems have a computational cost which has exponential growth in nature.The idea of regularization to such a problem helps to reduce computational cost by making use of the mixed real/complex power iteration.
In this article, we use an iterative method [23] in order to numerically compute the µ-values bounds from below for a set of constant matrices generated randomly.The idea is based on a two level algorithm described in a great detail in [23].The inner algorithm provides basis to develop a gradient system of ordinary differential equations (ODEs) and eventually solves this system by making use of extremizers of structured spectral-value sets.On the other hand, The outer algorithm helps to adjust the small positive parameter which acts as valid perturbation > 0, when making use of fast Newton's iteration.
This paper is organized as follows: In Section 2 we provide the basic definitions of pseudo-spectra and structured singular values.In Section 3 we give the reformulation of the definition of structured singular values.Section 4 of this article contains the computation of pseudo-spectra, where in Section 5 we explain how the computation of structured singular values can be addressed by making use of an inner-outer algorithm.In the outer algorithm, we determine a small perturbation level while in inner algorithm we determines a local extremizer of the structured spectral value set.Section 6 refers to numerical experiments to compare the lower bounds of SSV obtained with algorithm [23] to those obtained with MATLAB function mussv.Section 7 summarizes our conclusions.

Preliminaries
Definition 1.The uncertainties in terms of mixed real and complex scalar blocks are denoted by B rc and defined by Definition 2. The set of block diagonal matrices in terms of pure real scalar blocks is denoted by B r and defined by Definition 3. The set of uncertainties in terms of pure full complex blocks are denoted by B f , defined as Definition 4. The uncertainties in terms of blocks consist of r repeated real scalar blocks and F square full complex blocks are denoted by B R and defined as follows Definition 5.The block diagonal operators entirely consist of S repeated complex scalar blocks and F square full complex blocks are denoted by B R and defined as Definition 6.The spectrum of a square complex valued matrix M ∈ C n,n is represented by Λ(M) and mathematically defined as Definition 7. The pseudo-spectrum of a n-dimensional complex valued operator is represented by Λ (M) and defined as: Definition 8.The µ-value of M ∈ C n×n underlying block diagonal operator B is mathematically denoted by µ B R (M) and defined as follows

Reformulation of µ-Values
In this section we give the reformulation of µ-values based on structured spectral value sets [24].The reformulation of µ-values allows us to construct an optimization problem in order to maximize the spectrum of the perturbed system.Definition 9.The -spectral valued set of an n-dimensional complex operator underlying a small positive real parameter > 0 is mathematically denoted by ∆ B R (M) and defined as: where the quantity Λ( M∆) represent the spectrum of an operator ( M∆) while ||∆|| 2 represents the greatest singular value of an operator ∆ ∈ B R .
Definition 10.For a given n-dimensional complex operator M and an underlying perturbation in the form of B, the spectral value set is represented by Σ B R (M) and is defined as the following Definitions 9 and 10 make it possible to formulate again µ-value as a maximization problem.
Definition 11.For M ∈ C n×n and a set of pure complex uncertainties B R , µ-value is defined as the following

Pseudospectra of Square Matrices
In this section, the pseudospectra for matrices are under consideration.For this purpose we make use of the software package EigTool [25].EigTool is routinely used to plot the unstructured pseudospectra of the matrices under consideration.Figures 1 and 2 show the graphical representation of the pseudospectrum of 6-dimensional and 4-dimensional real valued matrices.

Proposed Methodology
In this section, the main objective is to discuss the optimization problem as mentioned in Definition 11, we make use of a numerical approach which is based on low-ranked ordinary differential equations.The numerical method is composed of a two-level algorithm, the inner algorithm and an outer algorithm.In this Section, we first discuss the inner-algorithm which shows how to give a theoretical construction to the system of ordinary differential equations.Secondly, we discuss how to deal with such systems mathematically.Next, we discuss the outer algorithm, which helps to modify the small positive real parameter > 0. The outer algorithm is based on Newton's iteration for approximating the roots of non-linear equations.The outer-algorithm computes an exact derivative of an extremizer say, ∆( ) for ∆ ∈ B and > 0. Complete details of the numerical method under consideration are given in [23,24].Finally, we discuss the computation of an extremizer.For this purpose, we approximate the derivative of an eigenvalue matrix Λ(p) of a smooth matrix family say A(p) for some fixed parameter p.

The Basic Theory
Consider some smooth matrix family A(p) ∈ C n,n for a small parameter p.Let Λ(p) ∈ C n,n and X(p) ∈ C n,n be the eigenvalue and eigenvector matrices corresponding to A(p).In matrix form In Equation ( 12), computation of eigenvector matrix X(p) is not unique unless all of the eigenvectors remain fixed.For distinct eigenvalues all of the eigenvectors are computed up to a constant multiplier.
The theory of computing X(p) is extended if eigenvalue derivatives are repeated while the second order derivatives of eigenvalues remain distinct.This theory is further generalized to complex-valued and Hermitian matrix.
Here and hereafter the symbol ( ) denotes the derivative while we write A = A(p), that is, we omit the dependency on p.By differentiating Equation ( 12) we get, Our goal is to determine Λ , the derivative of eigenvalues matrix.For this purpose, we multiply Equation ( 13) with X −1 throughout to have, Since X −1 X = I, this implies: let L = X −1 X , we find X = XL , thus Equation ( 14) gives: Finally, this gives the required result for the computation of the derivative of spectrum of matrix valued function A(p) as follows

Approximation of Extremizers
A matrix valued function ∆ ∈ B having the largest eigenvalue bounded above by 1 and the matrix valued function (I − M∆) having a smallest eigenvalue which minimizes the modulus of structured spectral value set, ∑ B (M) is known as an extremizer.The following theorem computes an extremizer for a chosen smallest complex number belonging to the set ∑ B (M).

Gradient System of ODEs
The gradient system of ODEs for an admissible perturbation ∆ ∈ B to approximate a local extremizer of smallest eigenvalue λ = |λ| iθ , is obtained as ) is a characteristic function.For a further discussion on the construction of gradient system of ODEs in above equations, we refer to [23].

Outer-Algorithm
In an outer-algorithm the main aim is to vary > 0, the perturbation level, by means of a fast Newton's iteration.In turn, 1/ will provide the approximation of lower bound of µ-value.
We make use of fast Newton's iteration in order to solve a problem In order to solve Equation ( 16), we need to compute ( Proof.For proof we refer to [23].
Remark 1. Choice of suitable initial value matrix and initial perturbation level: For a suitable choice of the initial value matrix ∆ 0 and an initial perturbation level 0 , we refer to [23].

Numerical Computation
In this section, we present numerical results for both lower and upper bounds of µ-values for randomly generated matrices in MATLAB (R2018a, MathWorks, Natick, MA, USA).The numerical computation of the lower bounds of structured singular values with the help of Nalgo, show the tighter results for the µ-values lower bounds computed by means of well-known MATLAB routine mussv.In each experiment the admissible uncertainties ∇ and E are computed with the help of mussv and Nalgo [23] 0.0615 0.0421 0.0607 0.0427 0.0479 0.0453 0.0692 0.0473 0.0683 0.0480 0.0539 0.0509 0.0545 0.0373 0.0538 0.0378 0.0425 0.0401 0.0628 0.0430 0.0619 0.0436 0.0489 0.0462 0.0609 0.0417 0.0601 0.0423 0.0475 0.0448 0.0727 0.0497 0.0717 0.0504 0.0566 0.0535 , with ∇ 2 = 0.3157.The main drawback with mussv is that it does not compute the unit 2-norm of each block of admissible perturbation ∆ and by this way it loose the optimality condition.The optimality condition requires that: not only the admissible perturbation ∆ should possess unit 2-norm but each of its block also possesses a unit 2-norm.
The approximated lower bound of µ-value is 3.1677.Moreover, by using the algorithm [23], the admissible perturbation E is obtained as The algorithm [23] computes the unit 2-norm of admissible perturbation E. It does not violate the optimality condition and hence computes the unit 2-norm against each block of E. The lower bound approximation is 3.5693 which is a much better approximation than the one computed numerically.
If instead of full complex blocks, one chose the uncertainties to be repeated as complex blocks, then the mussv routine will not compute the desired admissible perturbations such that each repeated complex block possesses a unit 2-norm.On the hand, Nalgo computes each repeated complex block in such a way that not only the admissible perturbation possesses a unit 2-norm but each individual block also has a largest singular value to be exactly equal to 1.
In Table 1, we present the comparison for the lower and upper bounds of µ-value approximated numerically with the help of mussv and an efficient numerical algorithm [23] for matrix M in Example 1.The first column presents the size of an operator M, where the second, third and fourth columns represent the approximations towards both upper and lower bounds which are computed numerically and the lower bounds µ Nalgo l computed with the help of [23] respectively.
The lower bound approximation is 2.6983.If instead of full complex blocks, one choose the uncertainties to be repeated complex blocks then the mussv routine will not compute the desired admissible perturbations such that each repeated complex block possesses a unit 2-norm.On the other hand, Nalgo computes each repeated complex block in a such a way that not only the admissible perturbation possesses a unit 2-norm but each individual block also have a largest singular value to be exactly equal to 1.
Table 2 represents the comparison of the µ-values approximated numerically with the mussv routine and an iterative method [23] for matrix M taken in Example 2. In the very first column, we present the size of the matrix M. In the second column, the upper bounds of µ-values are presented while lower bounds are presented in the third and fourth columns, respectively.In Figure 3, we show numerical approximation of bounds of µ-values for matrix valued function M(ω i ) at various frequencies ω i , ∀i = 1 : 5.In Cases 1-6, the matrix valued functions M(ω i ) are obtained in MATLAB by using the "rand" command.The geometrical interpretation of µ-values bounds show the comparison of µ-values bounds computed numerically from below with the help of Nalgo and the one computed with the help of mussv.The frequency response ω i is the quantitative measure of output of M − ∆ system.

Conclusions
In this article, we presented numerical computation of lower and upper bounds of µ-values a well-known mathematical quantity in control.The µ-values discuss robustness and performance of linear feedback systems in control and their bounds examine stability analysis of such systems.The numerical computations show that the attained results for µ-values bounds from below are more appropriate than those obtained by MATLAB function mussv.The EigTool discussed the geometrical interpretation of the pseudo-spectrum of the family of matrices.

with ∆ 2 = 1 ,
acts as a local extremizer of structured spectral value set.For a simple smallest eigenvalue λ = |λ|e ιθ , θ ∈ R of matrix valued function (I − M∆) having right and left eigenvetors x and y scaled as S = e ιθ y * x and let z = M * y.The non-degeneracy conditions z * k x k = 0, ∀ = 1 : S Re(z * k x k ) = 0, ∀ = 1 : S + 1 : S and ||z s+h ||.||x s+h || = 0, ∀h = 1 : F, hold.Then magnitude of each complex scalar δ i ∀i = 1 : s appears to be exactly equal to 1 while each full block possesses a unit 2-norm.
bounds Upper bounds by mussv Lower bounds by mussv Lower bounds by Nalgo (a) Case 1. M = C 8bounds Upper bounds my mussv Lower bounds by mussv Lower bounds my Nalgo (b) Case 2. M = R 9bounds Upper bounds by mussv Lower bounds by mussv Lower bounds by Nalgo (f) Case 6. M = C 13,13

Figure 3 .
Figure 3. Numerical approximation of bounds of µ-values at different frequency response. respectively.

Table 1 .
Computation of bounds of µ-values.We choose the perturbation set B = Diag{∆ 1 : ∆ 1 ∈ C 4,4 }.The perturbation set contains only pure complex full blocks.The admissible perturbation ∇ is obtained as

Table 2 .
Computation of bounds of µ-values.