EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 844-868
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Numerical computation of Lower bounds of Structured
Singular Values

M. Fazeel Anwar1, Mutti-Ur Rehman1,∗

1 Department of Mathematics, Sukkur IBA University, 65200 Sukkur, Pakistan

Abstract. In this article we have considered numerical approximation of lower bounds of Struc-
tured Singular Values, SSV. The SSV is a well-known mathematical quantity which is widely used
to analyse and syntesize the robust stability and instability analysis of linear feedback systems in
control theory. The SSV establishes a link between numerical linear algebra and system theory.
The computation of lower bounds of SSV by means of ordinary differential equations based tech-
nique is presented. The obtained numerical results for the lower bounds of SSV are compared with
the well-known MATLAB function mussv available in MATLAB control toolbox.

1. Introduction

The Structured Singular Values known as µ-value is a well-known mathematical tool in
control, introduced by J. C. Doyle around 1980’s [12]. This tool can be used to discuss both
stability and instability analysis of linear systems when subject to a certain perturbations.
For more applications on SSV, the interested reader can consult [13] which describe the
engineering motivation for µ-values. Due it’s computational complexity, the approximation
of an exact value of SSV appears to be NP-hard see [2]. In fact, the computation of bounds
of SSV, especially the computation of upper bounds when certain properties are under
consideration appers to be a NP-hard problem [14].

There has been done an extensive amount of research in order to develop new numeri-
cal algorithms which are very efficient and provides tighter bounds for µ-value. The power
method [9] approximate the lower bound SSV while taking pure complex perturbations
into an account. But unfortunately power method fail to converge; this happens when
pure real uncertainties are under consideration, for more detail see [15]. The structured
perturbations addressed by µ-tool is very generic and it allows to cover all types of para-
metric perturbations that can be incorporated into the linear control system theory by
help of both real and complex Linear Fractional Transformations (LFT’s). For more de-
tail please see [1, 3, 5–8, 10] and the references therein for the applications of SSV. The

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3301

Email addresses: mutti.rehman@iba-suk.edu.pk (M. Rehman),
fazeel.anwar@iba-suk.edu.pk (M. F. Anwar)

http://www.ejpam.com 844 c© 2018 EJPAM All rights reserved.



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 845

message from the approximation of an upper bound of µ-tool provides conditions which
guarantee the stability of feedback linear systems. The well-known Matlab function mussv
available in MATLAB controlboox approximates an upper bounds for SSV by means of
diagonal balancing technique and Linear Matrix Inequlaity (LMI) techniques [4].

1.1. Overview of the article

Section 2 provides the basic framework. In particular, it explain how the computation
of the SSV can be addressed by an inner-outer algorithm, where the outer algorithm de-
termines the perturbation level ε and the inner algorithm determines a (local) extremizer
of the structured spectral value set. In Section 3 it is explain that how the inner algorithm
works for the case of pure complex structured perturbations. An important character-
ization of extremizers shows that one can restrict himself to a manifold of structured
perturbations with normalized and low-rank blocks. In Section 4, we construct a gradient
system of ordinary differential equations in order to solve the local optimization problem.
Finally, Section 5 presents a range of numerical experiments to compare the quality of the
lower bounds to those obtained with mussv.

2. Framework

Consider A ∈ Cr,r or A ∈ Rr,r and let ΘB′ , a perturbation set defined as

ΘB′ = {diag(αiIri , Γs) : αi ∈ C(R), Γs ∈ Cms,ms(Rms,ms)}.

Here, Ii is an identity matrix with dimension i.
Definition 2.1. [8]. The structured singular value for an operator A ∈ Cr,r or A ∈ Rr,r

w.r.t ΘB′ is defined as follows:

µΘB′ (A) :=
1

min {‖℘‖2 : ℘ ∈ ΘB′ , det(I −A℘) = 0}
. (1)

In above given Definition 2.1, the quantity det(·) denotes determinants of operator (I −
A℘). The above Definition 2.1 for the case of pure complex perturbation takes the form:

µΘB(A) =
1

min {‖℘‖2 : ℘ ∈ ΘB, ρ(A℘) = 1}
. (2)

In Equ. (2.2), the quantity ρ(·) is known as the spectral radius that is ρ(A℘) = max|λi|
where λi is the spectrum of an operator A℘.

Structured spectral value sets. Consider the given input arguments A ∈ Cr×r and ε,
the desired perturbation level. The structured spectral value set is the set containing all
the eigenvalues of an operator (εA℘) and is defined as:

Λ
ΘB′
ε∗ (A) = {λ ∈ Λ(ε∗A℘) : ℘ ∈ ΘB′}. (3)



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 846

Here, Λ(·) is the set of all eigenvalues of an operator and ‖℘‖2 = 1. If we consider both
mixed real and complex perturbations, then structured spectral value set is of the form:

Σ
ΘB′
ε∗ (A) = {η = 1− λ1 : λ1 ∈ Λ

ΘB′
ε∗ (A)}. (4)

The above formulation in Equ. (2.4) helps us to write down the alternative definition of
structured singular value as given in Equ. (1.2) as follows:

µΘB′ (A) =
1

arg min{0 ∈ Σ
ΘB′
ε∗ (A)}

. (5)

While for the case when we have only pure complex perturbations, then Equ. (2.3) allows
us to alternatively express structured singular values as

µΘB(A) =
1

arg min{max |λ1| = 1}
. (6)

Problem under consideration. Our goal is to solve following optimization problem,

ξ(ε∗) = arg min |η|. (7)

In above Equ. (2.7), η ∈ Σ
ΘB′
ε∗ (A) for a fixed parameter ε > 0. In order to solve the

optimization problem addressed in Equ. (2.7), we suggests a two-level algorithm. In the
inner algorithm, we give a solution of Equ. (2.7) by constructing and then solving a
gradient system of ordinary differential equations. In the outer algorithm, with the help
of an iterative method we first vary the perturbation level ε.

First we address the case of a purely complex perturbations when ΘB by taking the
inner algorithm in order to compute a local extremizer for

λ(ε) = arg max |λ1|. (8)

In above Equ. (2.8), λ1 ∈ ΛΘB
ε (M) which gives us a lower bound for structured singular

values in case of pure complex uncertainties that is µ∆B(M). First we consider the case
of pure complex perturbations that is by taking into account the perturbations set ΘB
instead of ΘB′ .

3. Pure Complex perturbations

In this section, we compute the solution to optimization problems as discussed in Equ.
(2.8). For this we consider the estimation of µΘB(A) for the given operator A ∈ Cr,r. Also,
in this case we consider the pure complex perturbations that is

ΘB = {diag(α1I1 , ..., αnIn;℘1, ..., ℘F ) : αi ∈ C, ℘j ∈ Cmj ,mj}. (9)

The following Lemma 3.1 gives the behavior of the spectrum of a matrix valued function.



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 847

Lemma 3.1. Consider the matrix valued function Υ : R → Cn,n. Also consider the
fact that λ(t) is an eigenvalue of matrix valued function Υ(t) which approaches to simple
eigenvalue that is λ∗ of Υ0 = Υ(0) as t→ 0. Then λ(t) is analytic near t = 0 with

dλ

dt
=
w∗

0Υ1v0

w∗
0v0

,

where Υ1 = Υ̇(0) and v0, w0 are right and left eigenvectors of Υ0 associated to λ∗.
As now our goal is to deal with the an optimization problem as mentioned in Equ.

(2.8). This needs the computation of an uncertainty ℘local so that ρ(εA℘local) achieves
maximum growth along the perturbation ℘ ∈ ΘB′ with ‖℘‖2 ≤ 1. In below we consider
that λ be the greatest eigenvalue when |λ| equals to the spectral radius.

Definition 3.2. A matrix valued function ℘ ∈ ΘB such that ‖℘‖2 = 1 and (εA℘)
possesses the maximum eigenvalue which increases the modulus for structured spectral
vale set ΛΘB

ε (A), known as a local maximizer. In theorem 3.3, we give the characterization
of local maximizer of the gradient system of ordinary differential equations.

Theorem 3.3 [11]. Consider that

℘local = diag(α1I1, ..., αnIn;℘1, ..., ℘F ).

Here, ℘local is such that ‖℘local‖2 = 1 and is a local maximizer for ΛΘB
ε (A). Additionally,

we consider that an operator (εA℘local) having a simple maximum eigenvalue which is
λ = |λ|eiθ, having right and left eigenvectors v and w and are scaled so that s = eiθw∗v > 0.
Upon partitioning, we get

v = (vT
1 , . . . , v

T
n , v

T
n+1, . . . , v

T
n+F )T; (10)

u = (uT
1 , . . . , u

T
n , u

T
n+1, . . . , u

T
n+F )T. (11)

Where u = A∗w. Additionally we assume that

u∗kvk 6= 0 ∀ k = 1, . . . , n, (12)

‖un+h‖2 · ‖vn+h‖2 6= 0 ∀ h = 1, . . . , F. (13)

Then
|sk| = 1 ∀ k = 1, . . . , n and ‖∆h‖2 = 1 ∀h = 1, . . . , F,

4. A system of ODEs to compute extremal points of Λ℘B
ε (A).

In order to compute the local extremizer to |λ| such that |λ| ∈ ΛΘB
ε (A). To do so first

we compute matrix valued function ℘(t) so that the maximum eigenvalue that is λ(t) of
an operator (εA℘(t)) attains the maximum value. We then construct and give an optimal
solution to a gradient system of ordinary differential equation’s. This system of ordinary
differential equations satisfies the choice of ℘(t).



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 848

4.1. Local optimization problem

Let λ = |λ1|eiθ is the a simple eigenvalue of (εA℘(t)). Further consider that the
corresponding eigenvectors v, w are normalized as

‖w‖ = ‖v‖ = 1, w∗v = |w∗v|e−iθ. (14)

By the help of Lemma 3.1, we get

d

dt
|λ1|2 = 2|λ1|Re

( u∗℘̇v

eiθw∗v

)
=

2|λ1|
|w∗v|

Re(u∗℘̇v), (15)

where u = A∗w.
Suppose that ℘ ∈ ΘB and we search the direction ℘̇ = τ that given maximum local

growth of the modulus of λ1. This gives us

τ = diag(ω1Ir1 , . . . , ωsIrN ,Ω1, . . . ,ΩF ). (16)

This acts as a solution to the maximization problem

τ∗ = arg max{Re(u∗τx)}

subject to Re(δiωi) = 0, i = 1 : N,

and Re〈℘j ,Ωj〉 = 0, j = 1 : F. (17)

In Lemma 4.1, we give the solution τ∗ to the maximization problem as discussed in the
Equ. (3.4).

Lemma 4.1 [11]. The solution τ∗ with

τ∗ = diag(ω1Ir1 , . . . , ωNIrN ,Ω1, . . . ,ΩF ), (18)

with

ωi = νi (v∗i ui −Re (v∗i uisi) si) , i = 1, . . . , N (19)

Ωj = ζj
(
uN+jv

∗
N+j −Re〈℘j , uN+jv

∗
N+j〉℘j

)
, j = 1, . . . , F. (20)

The coefficient νi > 0 is the reciprocal of the absolute value of the expression that appears
in the right-hand side in Equ. (4.6) when it’s different from zero and the coefficient νi = 1
else. While on the other hand the coefficient ζj > 0 is the reciprocal of the Frobenius
norm of an operator that appear in the right hand side of Equ. (4.7) if it’s different from
zero and the coefficient ζj = 1 else.

Now we express the result as obtained in the previous Lemma 3.1 as:

τ∗ = S1PΘB (u(t)v(t)∗)− S2℘. (21)

In above Equ. (4.8), PΘB(·) is the orthogonal projection while S1, S2 ∈ ΘB are diagonal
operators.



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 849

4.2. Gradient system of ordinary differential equations

The result in the previous Lemma 4.1 allows us to have the following differential
equation on the manifold ΘB:

℘̇(t) = S1PΘB(u(t)v(t)∗)− S2℘(t). (22)

Here, v(t) is eigenvector with ‖v(t)‖2 = 1 and is associated with the simple eigenvalue
λ(t) for an operator (εA℘(t)) for fixed perturbation level that is ε > 0. The differential
equation (4.9) is a gradient system of ODE’s because of the fact that it’s right-hand side
is nothing but projected gradient of τ 7→ Re(u∗τv).

4.3. Choice of initial value matrix and ε

[11]. For the computation of the admissible perturbation level ε, we take the pertur-
bation ℘ which is obtained for the previous value that is ε1 as the initial value matrix.
In order to produce the maximal growth for the eigenvalue |λ(t)|, we take the initial value
matrix as:

℘0 = S P℘B(w(t)v(t)∗). (23)

The operator S is chosen such that ℘0 ∈ ΘB. For a very natural choice of the initialization
of the perturbation level, we consider ε as:

ε =
1

µ̂ΘB(A)
. (24)

In the above equation, µ̂ΘB(A) is the upper bound of µ-value approximated by MATLAB
function mussv, which approximates both upper and lower bounds of structured singular
values.

5. Numerical Testing

In the very final section of this artile, we present numerical experimentations for both lower
and upper bounds of structured singular values. The numerical results are computed by
well-known MATLAB function mussv and the algorithm [11].
Example 1. Consider two dimensional real matrix A1.

A1 =


−1 0 0 0 0
0 0 0 1 0
0 0 −1 0 0
0 1 0 0 0
0 0 0 0 −1

 .
The set of block diagonal matrices is taken as:

ΘB′ = {diag(δ1I1, δ2I1, ℘1) : δ1, δ2 ∈ R, ℘1 ∈ C3,3}.



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 850

The admissible perturbation structure ℘̂ obtained by using MATLAB routine mussv is:

℘̂ =


0 0 0 0 0
0 0 0 0 0
0 0 −1 0 0
0 0 0 0 0
0 0 0 0 0

 .
The ‖℘̂‖2 = 1. The computed upper bound is µupperPD = 2.3224. The same lower bound
is obtained, that is, µlowerPD = 2.3224. By using algorithm [11], the perturbation structure
ε∗℘∗ is obtained as:

℘∗ =


0 0 0 0 0
0 0.5 0 0.5 0
0 0 0 0 0
0 0.5 0 0.5 0
0 0 0 0 0

 .
In this case, ε∗ = 1.0000 and ‖℘∗‖2 = 1. The obtained lower bound is as µlowerNew = 2.3224.
In the following Figure 1, we give the comparison of lower bounds of structured singular
values approximated by our New algorithm with the Lower and Upper bounds approxi-
mated by MATLAB function mussv for matrix valued function B1(w) for w=1:9, where
w ∈ Ω and Ω represents the frequency range in R+. Frequency response w is measure of
output of (B1 − ℘) system.
Example 2. Consider two dimensional real matrix A2.

A2 =



0 1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 −3 1 3 2 −3 −2 −4 1
−1 2 −2 −4 −2 3 2 4 −1
1 −3 2 4 2 −3 −2 −5 1
1 −1 0 2 2 −1 0 −2 0
−1 −1 0 0 0 0 −1 0 0
−1 0 0 −1 −1 0 0 0 0
1 −2 2 3 2 −3 −2 −4 0


.

The set of block diagonal matrices is taken as:

ΘB′ = {diag(δ1I1, ℘1) : δ1 ∈ R, ℘1 ∈ C8,8}.

The admissible perturbation structure ℘̂ obtained by using MATLAB routine mussv is:



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 851

℘̂ =



−0.0641 0 0 0 0 0 0 0 0
0 −0.0000 −0.0100 0.0105 −0.0118 −0.0047 −0.0008 0.0011 −0.0097
0 0.0000 0.0065 −0.0069 0.0077 0.0030 0.0005 −0.0007 0.0063
0 0.0000 0.0140 −0.0147 0.0164 0.0065 0.0011 −0.0015 0.0136
0 0.0000 0.0083 −0.0087 0.0098 0.0039 0.0007 −0.0009 0.0081
0 −0.0000 −0.0115 0.0121 −0.0135 −0.0054 −0.0009 0.0012 −0.0112
0 −0.0000 −0.0075 0.0079 −0.0088 −0.0035 −0.0006 0.0008 −0.0073
0 −0.0000 −0.0166 0.0175 −0.0196 −0.0078 −0.0013 0.0018 −0.0162
0 0.0000 0.0028 −0.0030 0.0033 0.0013 0.0002 −0.0003 0.0028


.

The ‖℘̂‖2 = 0.1258. The computed upper bound is µupperPD = 15.6004. The lower bound is
obtained, that is, µlowerPD = 15.5921. By using algorithm [11], the perturbation structure
ε∗℘∗ is obtained as:

℘∗ =



−1.0000 0 0 0 0 0 0 0 0
0 −0.0005 −0.1559 0.1640 −0.1834 −0.0729 −0.0123 0.0167 −0.1518
0 0.0003 0.1016 −0.1069 0.1195 0.0475 0.0080 −0.0109 0.0989
0 0.0007 0.2178 −0.2291 0.2563 0.1018 0.0171 −0.0233 0.2120
0 0.0004 0.1296 −0.1363 0.1525 0.0606 0.0102 −0.0139 0.1262
0 −0.0005 −0.1794 0.1887 −0.2110 −0.0838 −0.0141 0.0192 −0.1746
0 −0.0003 −0.1164 0.1224 −0.1369 −0.0544 −0.0092 0.0125 −0.1133
0 −0.0008 −0.2596 0.2731 −0.3054 −0.1213 −0.0204 0.0278 −0.2527
0 0.0001 0.0443 −0.0466 0.0521 0.0207 0.0035 −0.0047 0.0431


.

In this case, ε∗ = 1.0000 and ‖℘∗‖2 = 0.0641. The obtained lower bound is as µlowerNew =
15.5995.
In the following Figure 2, we give the comparison of lower bounds of structured singular
values approximated by our New algorithm with the Lower and Upper bounds approxi-
mated by MATLAB function mussv for matrix valued function B2(w) for w=1:9, where
w ∈ Ω and Ω represents the frequency range in R+. Frequency response w is measure of
output of (B2 − ℘) system.
Example 3. Consider two dimensional real matrix A3.

A3 =


0 0 1 0 0
0 0 0 1 0
1 0 0 0 0
0 1 0 0 0
−1 −1 −1 −1 −1

 .
The set of block diagonal matrices is taken as:

ΘB′ = {diag(δ1I1, δ2I1, ℘1) : δ1, δ2 ∈ R, ℘1 ∈ C3,3}.



M. F. Anwar, M. Rehman / Eur. J. Pure Appl. Math, 11 (3) (2018), 844-868 852

The admissible perturbation structure ℘̂ obtained by using MATLAB routine mussv is:

℘̂ =


0.4354 0 0 0 0

0 0.4354 0 0 0
0 0 0.0337 0.0337 −0.2740
0 0 0.0337 0.0337 −0.2740
0 0 0.0226 0.0226 −0.1840

 .
The ‖℘̂‖2 = 0.4354. The computed upper bound is µupperPD = 2.2966. The same lower bound
is obtained, that is, µlowerPD = 2.2966. By using algorithm [11], the perturbation structure
ε∗℘∗ is obtained as:

℘∗ =


1.0000 0 0 0 0

0 1.0000 0 0 0
0 0 0.0773 0.0774 −0.6293
0 0 0.0773 0.0774 −0.6293
0 0 0.0519 0.0520 −0.4227

 .
In this case, ε∗ = 1.0000 and ‖℘∗‖2 = 2.2966. The obtained lower bound is as µlowerNew =
0.4354.
In the following Figure 3, we give the comparison of lower bounds of structured singular
values approximated by our New algorithm with the Lower and Upper bounds approxi-
mated by MATLAB function mussv for matrix valued function B3(w) for w=1:6, where
w ∈ Ω and Ω represents the frequency range in R+. Frequency response w is measure of
output of (B3 − ℘) system.
Example 4. Consider two dimensional real matrix A4.

A4 =


0 0 1 0 0
0 0 0 1 0
1 0 0 0 0
0 1 0 0 0
−1 −1 −1 −1 −1

 .
The set of block diagonal matrices is taken as:

ΘB′ = {diag(δ1I1, δ2I1, ℘1) : δ1, δ2 ∈ R, ℘1 ∈ C3,3}.

The admissible perturbation structure ℘̂ obtained by using MATLAB routine mussv is:

℘̂ =


0.4354 0 0 0 0

0 0.4354 0 0 0
0 0 0.0337 0.0337 −0.2740
0 0 0.0337 0.0337 −0.2740
0 0 0.0226 0.0226 −0.1840

 .



REFERENCES 853

The ‖℘̂‖2 = 0.4354. The computed upper bound is µupperPD = 2.2966. The same lower bound
is obtained, that is, µlowerPD = 2.2966. By using algorithm [11], the perturbation structure
ε∗℘∗ is obtained as:

℘∗ =


1.0000 0 0 0 0

0 1.0000 0 0 0
0 0 0.0773 0.0774 −0.6293
0 0 0.0773 0.0774 −0.6293
0 0 0.0519 0.0520 −0.4227

 .
In this case, ε∗ = 1.0000 and ‖℘∗‖2 = 2.2966. The obtained lower bound is as µlowerNew =
0.4354.
In the following Figure 4, we give the comparison of lower bounds of structured singular
values approximated by our New algorithm with the Lower and Upper bounds approxi-
mated by MATLAB function mussv for matrix valued function B4(w) for w=1:2, where
w ∈ Ω and Ω represents the frequency range in R+. Frequency response w is measure of
output of (B4 − ℘) system.
In the following Figures [5-14], we give the comparison of lower bounds of structured
singular values approximated by our New algorithm with the Lower and Upper bounds
approximated by MATLAB function mussv for the various matrix valued functions.

6. Conclusion

In this article we have considered the numerical approximation of µ-values for the matrix
representations of finite symmetric groups Sn over the filed of complex numbers by using
well-known MATLAB function mussv and our algorithm [11]. The experimental results
indicates the different behaviors of lower bounds of µ-values with once computed by mussv
and our algorithm.

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[3] Chen, Jie and Fan, Michael KH and Nett, Carl N. Structured singular values with
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Appendix



Appendix 855

0 1 2 3 4 5 6 7 8
0.5

1

1.5

2

2.5

3

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bound by musv
SSV Loweer bound by musv
SSV Lower bound by NewAlgo

Figure 1: Comparison of Lower and Upper bounds of SSV



Appendix 856

0 1 2 3 4 5 6 7 8
0

2

4

6

8

10

12

14

16

18

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bound by mussv
SSV Lower bound by mussv
SSV Lower bound by NewAlgo

Figure 2: Comparison of Lower and Upper bounds of SSV



Appendix 857

0 1 2 3 4 5
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bound by mussv
SSV Lower bound by mussv
SSV Lower bound by NewAlgo

Figure 3: Comparison of Lower and Upper bounds of SSV



Appendix 858

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
4

4.5

5

5.5

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bounds by mussv
SSV Lower bounds by mussv
SSV Lower bounds by NewAlgo

Figure 4: Comparison of Lower and Upper bounds of SSV



Appendix 859

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0

2

4

6

8

10

12

14

16

18

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bound by mussv
SSV Lower bound by mussv
SSV Lower bound by NewAlgo

Figure 5: Comparison of Lower and Upper bounds of SSV



Appendix 860

0 0.5 1 1.5 2 2.5 3 3.5 4
0

1

2

3

4

5

6

7

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
Upper bounds by mussv
Lower bounds by mussv
Lower bounds by NewAlgo

Figure 6: Comparison of Lower and Upper bounds of SSV



Appendix 861

0 1 2 3 4 5
0

0.5

1

1.5

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bound by mussv
SSV Lower bound by mussv
SSV Lower bound by NewAlgo

Figure 7: Comparison of Lower and Upper bounds of SSV



Appendix 862

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2

2.1

2.2

2.3

2.4

2.5

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
Upper bounds by mussv
Lower bounds by mussv
Lower bounds by NewAlgo

Figure 8: Comparison of Lower and Upper bounds of SSV



Appendix 863

0 1 2 3 4 5
0

1

2

3

4

5

6

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bounds by mussv
SSV Lower bounds by mussv
SSV Lower bounds by NewAlgo

Figure 9: Comparison of Lower and Upper bounds of SSV



Appendix 864

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0

1

2

3

4

5

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
Upper bounds by mussv
Lower bounds by mussv
Lower bounds by NewAlgo

Figure 10: Comparison of Lower and Upper bounds of SSV



Appendix 865

0 1 2 3 4 5
2.5

3

3.5

4

4.5

5

5.5

6

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bound by mussv
SSV Lower bounds by mussv
SSV Lower bounds by NewAlgo

Figure 11: Comparison of Lower and Upper bounds of SSV



Appendix 866

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
4.5

5

5.5

6

6.5

7

7.5

8

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
Upper bounds by mussv
Lower bounds by mussv
Upper bounds by NewAlgo

Figure 12: Comparison of Lower and Upper bounds of SSV



Appendix 867

0 0.5 1 1.5 2 2.5 3 3.5 4
1

1.5

2

2.5

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
SSV Upper bounds by mussv
SSV Lower bounds by mussv
SSV Lower bounds by NewAlgo

Figure 13: Comparison of Lower and Upper bounds of SSV



Appendix 868

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0

0.5

1

1.5

2

2.5

Frequency(rad/sec)

U
pp

er
/L

ow
er

 b
ou

nd
s

 

 
Upper bounds by mussv
Lower bounds by mussv
Lower bounds by NewAlgo

Figure 14: Comparison of Lower and Upper bounds of SSV