Robustness Margin in Linear Time Invariant Fractional Order Systems

4th IFAC Symposium on System, Structure and Control Università Politecnica delle Marche Ancona, Italy, Sept 15-17, 2010

Robustness Margin in Linear Time Invariant Fractional Order Systems Kamran Akbari Moornani*. Mohammad Haeri** * Advanced Control System Lab, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran (Tel: +98-21-66165971; e-mail: [email protected]). ** Advanced Control System Lab, Electrical Engineering Department, Sharif University of Technology, Tehran, Iran (Tel: +98-21-66165964; e-mail: [email protected]) Abstract: In this paper, the computation of robustness margin for linear time invariant fractional order systems is studied. For the definition of robustness margin, we employ the one introduced for polynomials (i.e. integer order) and extend it to fractional order functions. Using the well known concept of the value set and knowing its shape for the intended functions, this paper presents an easy way to obtain the robustness margin for fractional order systems. To illustrate the results, a numerical example is provided. Keywords: Fractional order systems; LTI systems; BIBO-stability; Stability robustness margin; Value set; Real parametric uncertainties. 1. INTRODUCTION

floor( x) is considered as the largest integer less than or equal

to x ∈

In recent years, an increasing number of studies have been focussed on a class of infinite-dimensional systems, namely fractional order systems, to model the real world phenomena and processes, and to design efficient control structures (for example refer to Kilbas et al. (2006), Hilfer (2000), Das (2008), Sabatier et al. (2007), and references therein).

k2

∑f

i = k1

G( s) =

i

. Assume

= ∅ when k2 < k1 ( fi is a scalar but Fi

i = k1

□

′ ′ ′ v( s) bn′ sα n′ + bn′−1 sα n′−1 + + b1 sα1 + b0 = , p( s) an sα n + an −1 sα n−1 + + a1 sα1 + a0

(1)

where a0 , a1 ,… , an , b0 , b1 ,… , bn′ ∈ , 0 < α1 < α 2 < < α n , 0 < α1′ < α 2′ < < α n′′ , an ≠ 0 , and α n ≥ α n′′ . p( s ) and v( s ) can be multi-valued functions when some of their powers are non-integer. Define the single-valued functions:

The remainder of the paper is organized as follows. Section 2 contains some definitions and background material. Main results are given in Section 3. The theorem presented in this section can be employed for computing the robustness margin of LTI fractional order systems. The given example in Section 4 illustrates the results. Finally, conclusions are presented in Section 5.

⎧n α jα Arg( s ) + a0 if s ≠ 0, ⎪∑ ah | s | h e h p pb ( s) = ⎨ h =1 ⎪a if s = 0, ⎩ 0

(2)

⎧ n′ α′ jα ′ Arg( s ) + b0 if s ≠ 0, ⎪∑ bh | s | h e h v pb ( s) = ⎨ h =1 ⎪b if s = 0, ⎩ 0

(3)

where Arg( s ) ∈] − π , π ] denotes the principal value of the argument of s assuming s ≠ 0 . The defined functions p pb ( s) and v pb ( s ) are respectively the principal branches of p( s ) and v( s ) . (Indeed, p( s ) and v( s ) are considered on

the main Riemann sheet.) Assume p pb ( s ) and v pb ( s ) have

2. DEFINITIONS AND PRELIMINARIES

no common zeros in CRHP. According to Theorem 3.1 of Bonnet & Partington (2000), it follows that G is BIBOstable if and only if p pb ( s ) has no zero in CRHP.

Notation. Closed right half plane (CRHP) is defined as {s ∈ : Re( s ) ≥ 0} . # D denotes the number of distinct

978-3-902661-83-8/10/$20.00 © 2010 IFAC

∪F

x − floor( x) for all x ∈

Consider a fractional order system G with the following transfer function:

In this paper, we consider the problem of computing “robustness margin” for LTI fractional order systems. The intended system should be BIBO-stable itself (otherwise, the robustness margin would be meaningless). The employed definition for “robustness margin” is a straightforward extension of the one mentioned in page 96 of Barmish (1994) (which is a special case of Tsypkin & Polyak, 1991) to fractional order systems.

≤ k2 ≥ k1

= 0 and

k2

is a set, and let them as some functions of index i ).

Bounded-input bounded-output (BIBO) stability of linear time invariant (LTI) fractional order systems has been investigated in Matignon (1998), and Bonnet & Partington (2000 & 2002). The robust BIBO-stability of fractional order systems with interval real parametric uncertainties have been studied in Petras et al. (2004a & 2004b), Chen et al. (2006), Ahn & Chen (2008), and Tan et al. (2009).

members of the set D . Define

i

. frac( x)

{x ∈ : k1 ≤ x ≤ k2 } .

For the sake of simplicity, the Hurwitz-stability of a 198

10.3182/20100915-3-IT-2017.00001

SSSC 2010 Ancona, Italy, Sept 15-17, 2010

fractional order function can be defined as follows.

where qhk can be computed for h = 0,1,…, n − 1 as follows:

Definition 1. The fractional order function p( s ) = an sα n + an −1 sα n−1 +

+ a1 sα1 + a0

(where n ∈ , a0 , a1 ,… , an ∈ , and 0 < α1 < α 2 < < α n ) is said to be Hurwitz-stable if and only if the function p pb ( s) defined in (2) has no zero in CRHP.

qhk

+ a1 sα1 + a0 ,

rmargin (q0+ , q1+ ,… , qn+−1 ) as in Definition 2. It follows that:

a) If H 0 = ∅ , then rmargin (q0+ , q1+ ,… , qn+−1 ) = +∞ .

(4)

b) Assume H 0 ≠ ∅ . Let m , β1 , β 2 ,… , β m , and q1 , q 2 ,…, q m be as in Definition 3. Then inf T (ω ) where

(5)

T (0)

⎧⎪a0 q0+ ⎨ ⎪⎩+∞

T (ω )

⎧ Im[ p0 pb ( jω )e − jπβk ] if ω > 0 & m ≥ 2, ⎪ max − jπβ k ⎪ k =1, 2,…, m Im[e pb ( jω , q )e k ] ⎪ − jπβ ⎪ Re[ p0 pb ( jω )e 1 ] if ω > 0 & m = 1 & (13) ⎨ 1 Im[ p0 pb ( jω )e − jπβ1 ] = 0, e pb ( jω , q ) ⎪ ⎪ ⎪+∞ if ω > 0 & m = 1 & ⎪ Im[ p0 pb ( jω )e − jπβ1 ] ≠ 0, ⎩

(6)

The “robustness margin” of p0 ( s ) is defined as: rmargin (q0+ , q1+ ,… , qn+−1 )

sup{r ≥ 0 : Pr is robustly Hurwitz-stable}.

+ 1

Definition 3. Consider q , q ,…, q

+ n −1

(7) □

p0 pb ( jω )

e pb ( jω , q k )

and frac(0.5α h ) = β i }, i = 1, 2,…, m

H i′ {h ∈ H 0 : floor(0.5α h ) is an odd number and frac(0.5α h ) = β i }, i = 1, 2,…, m. 1

2

n −1

∑ q ωα e h =1

k h

h

j 0.5πα h

n −1

+ a1ω α1 e j 0.5πα1 + a0 ,

+ q0k , k = 1, 2,… , m.

(14) (15) □

q1 , q 2 ,… , q m are only performed once (since their values are independent from ω ).

(9)

Indeed, to obtain rmargin (q0+ , q1+ ,… , qn+−1 ) , it is not essential to depict T (ω ) for all ω ∈ [0, +∞[ . Actually, we only need to consider T (ω ) for ω ∈ [0, ωmax ] (i.e. rmargin (q0+ , q1+ ,… , qn+−1 ) =

m

[q0k q1k … qnk−1 ]T , k = 1, 2,… , m

n −1

n

(8)

The vectors q , q ,… , q are defined as: qk

n

When one would like to depict the function T (ω ) with respect to the variable ω , it should be noted that a0 , a1 ,… , an −1 , α1 , α 2 ,… , α n , and q0+ , q1+ ,… , qn+−1 are known fixed values and the computation of m , β1 , β 2 ,… , β m , and

m (# B0 ) . Define the following sets: {h ∈ H 0 : floor(0.5α h ) is an even number

ω α e j 0.5πα + an −1ω α e j 0.5πα +

Proof: The proof is given in Appendix A.

0 and

B0 {frac(0.5α h ) : h ∈ H 0 } . Let β1 , β 2 ,… , β m be all distinct members of B0 sorted as 0 ≤ β1 < β 2 < < β m < 1 where Hi

(12)

an − 2ω α n−2 e j 0.5πα n−2 +

, α1 , α 2 ,… , α n , and set

H 0 as in Definition 2. Assume H 0 ≠ ∅ . Define α 0

if q0+ > 0, if q0+ = 0,

where

To compute the robustness margin, we need to obtain some values and vectors given in the following definition. + 0

rmargin (q0+ , q1+ ,… , qn+−1 ) =

ω∈[0, +∞[

where q0+ , q1+ ,…, qn+−1 ≥ 0 (the fixed values q0+ , q1+ ,… , qn+−1 can be interpreted as scale factors similar to page 96 of Barmish, 1994). Let H 0 {h ∈ ≤≥n0−1 : qh+ > 0} . The following fractional order interval function family can be defined for each arbitrary r ∈ ≥ 0 : { p0 ( s ) + r ⋅ e( s ) : e( s ) ∈ E }.

p0 ( s ) , q0+ , q1+ ,…, qn+−1 , H 0 , and

Theorem 1. Consider

where n ∈ , a0 , a1 ,… , an−1 ∈ , and 0 < α1 < α 2 < < α n . Assume that p0 ( s ) is Hurwitz-stable. Define the fractional order interval function family ⎧ n −1 α h ⎫ + + ⎨∑ qh s + q0 : qh ∈ [− qh , qh ] for h = 0,1,… , n − 1⎬ ⎩ h =1 ⎭

(11) □

The following theorem can be employed to compute the robustness margin of Hurwitz-stable fractional order functions. It should be noted that we do not define the robustness margin when the function is not Hurwitz-stable, since this concept is meaningless for such a case.

Definition 2. Consider the following fractional order function: p0 ( s ) = sα n + an −1 sα n−1 + an − 2 sα n−2 +

and

3. MAIN RESULTS

The definition of “robustness margin” given in page 96 of Barmish (1994) (which is a special case of Tsypkin & Polyak, 1991) can be extended to fractional order functions as follows.

Pr

⎛ k −1 ⎞ ⎛ m ⎞ if h ∈ ⎜ ∪ H i ⎟ ∪ ⎜ ∪ H i′ ⎟ , ⎝ i =1 ⎠ ⎝ i = k ⎠ if otherwise.

□

The concept of “robust Hurwitz-stability” can be considered as usual (i.e. the Hurwitz-stability of all the functions in the given family of fractional order functions).

E

⎧ + ⎪qh ⎨ ⎪−q + ⎩ h

k = 1, 2,…, m

inf

(10)

ω∈[0,ωmax ]

199

T (ω ) ) where ωmax can be computed as in the

SSSC 2010 Ancona, Italy, Sept 15-17, 2010

following four Cases 1, 2, 3, and 4 (This is a consequence of part “a” of Lemma 1 of Akbari Moornani & Haeri (2010a) and Definition 2).

of Argument” (Th. 1.2 of Marden, 1966), one can conclude that the function p0 pb ( s ) has no zero interior to the curve Γ . Then, p0 pb ( s ) has no zero in CRHP and consequently,

Case 1 ( H 0 = ∅ ): There is no need to define ωmax in this

p0 ( s ) is Hurwitz-stable.

case and we have rmargin (q0+ , q1+ ,… , qn+−1 ) = +∞ .

imaginary

Case 2 ( H 0 ≠ ∅ and m ≥ 2 ): ωmax can be obtained as ⎧⎪ ⎡⎛ n −1 max ⎨1, ⎢⎜ ∑ | ah ⎪⎩ ⎣⎝ h = 0

ωmax

n −1 ⎤ ⎞ | ⎟ + T0 ⋅ ∑ qh+ ⎥ h=0 ⎠ ⎦

1 (α n −α n −1 )

⎫⎪ ⎬ ⎪⎭

R0

(16)

Γ

R1

where T0 is an arbitrary non-negative real number greater

− R1

than or equal to rmargin (q0+ , q1+ ,… , qn+−1 ) . For instance, one can select T0 = T (1) (i.e. the value of T (ω ) for ω = 1 ). Indeed,

R0 R1

real

− R0

one can assign T0 = T (ω ) for each arbitrary ω ∈ ≥ 0 where T (ω ) ≠ +∞ as well, and it is evident that the smaller T0 results in smaller ωmax (which is more desirable from a computational point of view).

Fig. 1. Jordan curve of Γ .

Case 3 ( H 0 ≠ ∅ , m = 1 , and q0+ > 0 ): ωmax can be computed as in (16) where T0 = a0 q0+ . Case 4 ( H 0 ≠ ∅ , m = 1 , and q0+ = 0 ): Depicting the function Im[ p0 pb ( jω )e − jπβ1 ]

for

ω∈

>0

,

we

encounter

situations. When we have Im[ p0 pb ( jω )e

− jπβ1

ω∈

+ 1

>0

+ 0

two

] ≠ 0 for all

, it can be written that rmargin (q , q ,… , qn+−1 ) = +∞ .

Another situation is when there exists a value ω1 ∈ that Im[ p0 pb ( jω1 )e

− jπβ1

>0

such

] = 0 . Then, ωmax can be computed as

Fig. 2. The path of p0 pb ( s ) .

in (16) where T0 = T (ω1 ) . 4. ILUSTRATIVE EXAMPLE In Podlubny (1999), the fractional order PD μ controller C ( s ) = k p + k D s μ = 20.5 + 3.73s1.15 has been applied to fractional order system H ( s ) = 1 (0.8s 2.2 + 0.5s 0.9 + 1) . The transfer function of the closed loop system can be written as: G ( s) =

4.6625s1.15 + 25.625 . s 2.2 + 4.6625s1.15 + 0.625s 0.9 + 26.875

(17)

Consider the fractional order function p0 ( s ) = s 2.2 + 4.6625s1.15 + 0.625s 0.9 + 26.875 .

(18)

We show that p0 ( s ) is Hurwitz-stable. According to Lemma 1 of Akbari Moornani & Haeri (2010a), all the zeros of the function p0 pb ( s ) are in the set {s ∈ : | s |∈ [ R1 , R0 ]} where

Fig. 3. A better view around the point 0 . We verify the robustness margin of p0 ( s ) in the following two Cases.

R0 = 27.263 and R1 = 1 . Assume the Jordan curve Γ as in Fig. 1. By considering that s moves on the curve Γ once over, the path of p0 pb ( s ) can be plotted as in Fig. 2. In Fig.

Case A: Let the scaling factors be q0+ = q1+ = q2+ = 1 (i.e.

{ p0 ( s ) + r ⋅ (q2 s1.15 + q1 s 0.9 + q0 ) : q0 , q1 , q2 ∈ [−1,1]} ). We have:

Pr

3, a better view around the origin of the complex plane is provided. As it is seen in Figs. 2 and 3, the path of p0 pb ( s )

H 0 = {0,1, 2}, B0 = {0, 0.45, 0.575},

does not encircle the point 0 . Then, based on the “Principle 200

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SSSC 2010 Ancona, Italy, Sept 15-17, 2010

m = 3, β1 = 0, β 2 = 0.45, β 3 = 0.575, H1 = {0}, H 2 = {1}, H 3 = {2}, H1′ = H 2′ = H 3′ = ∅,

(21)

T

q = [q q q ] = [−1 − 1 − 1] ,

(22)

q = [q

q ] = [1 − 1 − 1] ,

(23)

1

2

1 0

2 0

1 1

2 1

q

1 T 2

2 T 2

(20)

T

q = [q q q ] = [1 1 − 1] . 3

3 0

3 1

3 T 2

T

(Fig. 5). The value set V (4.4052, 2.1121, Q) (i.e. the value set V (ω , r , Q) for ω = 4.4052 and r = 2.1121 ) is highlighted by Gray colour in Fig. 5 to better demonstrate this fact.

(24)

We obtain T (0) = 26.875 1 = 26.875 and ⎧⎪ Im[ p0 pb ( jω )] Im[ p0 pb ( jω )e − j 0.45π ] T (ω ) = max ⎨ , , 1 2 − j 0.45π ] ⎪⎩ Im[e pb ( jω , q )] Im[e pb ( jω , q )e (25) Im[ p0 pb ( jω )e− j 0.575π ] ⎪⎫ ⎬ Im[e pb ( jω , q 3 )e− j 0.575π ] ⎪⎭

for

ω >0.

The

functions

p0 pb ( jω ) ,

e pb ( jω , q1 ) ,

e pb ( jω , q 2 ) , and e pb ( jω , q 3 ) are as follows: p0 pb ( jω ) = ω 2.2 e j1.1π + 4.6625ω1.15 e j 0.575π + 0.625ω 0.9 e j 0.45π + 26.875, e pb ( jω , q1 ) = −ω 1.15 e j 0.575π − ω 0.9 e j 0.45π − 1,

e pb ( jω , q ) = −ω 2

e pb ( jω , q ) = −ω 3

1.15

1.15

e

j 0.575π

e

j 0.575π

Fig. 5. The value sets V (ω , 2.1121, Q) for ω ≥ 0 in Case A.

(26)

Case B: Consider the scaling factors as q0+ = q1+ = 0 and

(27)

−ω e

j 0.45π

+ 1,

(28)

q2+ = 1 (i.e. Pr write:

+ω e

j 0.45π

+ 1.

(29)

H 0 = {2}, B0 = {0.575}, m = 1, β1 = 0.575,

(32)

H1 = {2}, H1′ = ∅, q = [0 0 − 1] .

(33)

0.9

0.9

The graph of T (ω ) is depicted in Fig. 4. As is seen, it can be written that: rmargin (q , q , q ) = inf T (ω ) = 2.1121 . + 0

+ 1

+ 2

ω∈[0, +∞[

{ p0 ( s ) + r ⋅ (q2 s1.15 ) : q2 ∈ [−1,1]} ). We can

1

T

Then, one obtains T (0) = +∞ and

(30)

⎧ Re[ p0 pb ( jω )e− j 0.575π ] if ω > 0 & ⎪ Im[ p0 pb ( jω )e − j 0.575π ] = 0, (34) T (ω ) = ⎨ −ω1.15 e j 0.575π ⎪ − j 0.575π ] ≠ 0. ⎩+∞ if ω > 0 & Im[ p0 pb ( jω )e

Hence, it can be written that: ⎧3.5629 if ω = 4.4837 T (ω ) = ⎨ if ω > 0 & ω ≠ 4.4837 ⎩+∞ Therefore, we have: rmargin (q0+ , q1+ , q2+ ) = inf T (ω ) = 3.5629 . ω ∈[0, +∞[

(35)

(36)

Fig. 4. The graph of the function T (ω ) in Case A. As a test for correctness of the obtained robustness margin, we depict the value sets V (ω , r , Q) { p0 pb ( jω ) + r ⋅ (q2ω1.15 e j 0.575π + q1ω 0.9 e j 0.45π + q0 ) : q0 , q1 , q2 ∈ [−1,1]}

(31)

in the complex plane for r = 2.1121 and ω ∈ [0, +∞[ (Note that there is no need to depict the value sets V (ω , r , Q) to obtain the robustness margin and it is done only as a test). It is observed that the point 0 (the origin of the complex plane) lies on the edges of the depicted value sets when ω = 4.4052

Fig. 6. The value sets V (ω ,3.5629, Q) for ω ≥ 0 in Case B. To check the correctness of the computed robustness margin, 201

SSSC 2010 Ancona, Italy, Sept 15-17, 2010

Matignon, D. (1998). Stability properties for generalized fractional differential systems. ESAIM Proceedings, vol. 5, Paris, France (pp. 145-158). Petras I., Chen, Y.Q., and Vinagre, B.M. (2004a). Robust stability test for interval fractional order linear systems. In Blondel, V.D. and Megretski, A. (ed.), Unsolved problems in mathematical systems and control theory, Problem 6.5, pp. 208-211. Princeton, NJ: Princeton University Press. Petras I., Chen, Y.Q., Vinagre, B.M., and Podlubny, I. (2004b). Stability of linear time invariant systems with interval fractional orders and interval coefficients. In Proceedings of the International Conference on Computational Cybernetics (ICCC04), Vienna Technical University, Vienna, Austria, Aug. 30-Sep. 1, pp.341-346. Podlubny, I. (1999). Fractional-order systems and PI λ Dμ controllers. IEEE Transactions on Automatic Control, 44(1), 208-214. Sabatier, J. (Ed.), Agrawal, O.P. (Ed.), and Machado, J.A.T. (Ed.) (2007). Advances in fractional calculus. The Netherlands: Springer. Tan, N., Ozguven, O.F., and Ozyetkin, M.M. (2009). Robust stability analysis of fractional order interval polynomials. ISA Transactions, 48, 166-172. Thomas, G.B. and Finney, R.L. (1996). Calculus and analytic geometry (9th Ed.). Addison-Wesley Publishing Company. Tsypkin, Y.Z. and Polyak, B.T. (1991). Frequency domain criteria for l p -robust stability for continuous linear systems. IEEE Transactions on Automatic Control, 36(12), 1464-1469.

we depict the value sets V (ω , r , Q) { p01 ( jω ) + r ⋅ (q2ω1.15 e j 0.575π ) : q2 ∈ [−1,1]}

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for r = 3.5629 and ω ∈ [0, +∞[ . It is seen in Fig. 6 that the point 0 lies on one of the vertices of the depicted value sets when ω = 4.4837 . Indeed, one of the vertices of value set V (4.4837,3.5629, Q) (i.e. the value set V (ω , r , Q) for ω = 4.4837 and r = 3.5629 ) locates on the origin of the complex plane (the point 0 ). Each of the depicted value sets is a straight line segment when m = 1 . 5. CONCLUSIONS In this paper, a theorem is provided to compute the robustness margin for an arbitrary BIBO-stable fractional order system. To compute the robustness margin, we need to consider some scale factors q0+ , q1+ ,… , qn+−1 ≥ 0 . Though the theorem is held for any scale factors, the determination of the scale factors that best fitted to the probably other constraints and conditions imposed on the actual system is not an easy task in general and can be an interesting problem for the future works. REFERENCES Ahn, H.S. and Chen, Y.Q. (2008). Necessary and sufficient condition of fractional-order interval linear systems. Automatica, 44, 2985-2988. Akbari Moornani, K and Haeri, M. (2009). On robust stability of linear time invariant fractional-order systems with real parametric uncertainties. ISA Transactions, 48(4), 484-490. Akbari Moornani, K and Haeri, M. (2010a). On robust stability of LTI fractional-order delay systems of retarded and neutral type. Automatica, 46(2), 362-368. Akbari Moornani, K. and Haeri, M. (2010b). Robust stability testing function and Kharitonov-like theorem for fractional order interval systems. IET Control Theory & Applications, Accepted for publication. Barmish, B.R. (1994). New tools for robustness of linear systems. NY: Macmillan Publishing Company. Bonnet, C. and Partington, J.R. (2000). Coprime factorizations and stability of fractional differential systems. Systems & Control Letters, 41(3), 167-174. Bonnet, C. and Partington, J.R. (2002). Analysis of fractional delay systems of retarded and neutral type. Automatica, 38, 1133-1138. Chen, Y.Q., Ahn, H.S., and Podlubny, I. (2006). Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Processing, 86, 2611-2618. Das, S. (2008). Functional fractional calculus for system identification and controls. Berlin: Springer-Verlag. Hilfer, R. (Ed.) (2000). Applications of fractional calculus in physics. Singapore: World Scientific. Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and applications of fractional differential equations. The Netherlands: Elsevier B.V. Marden, M. (1966). Geometry of polynomials (2nd Ed.). Providence, RI: American Mathematical Society.

Appendix A. PROOF OF THEOREM 1 Part a) We have qh+ = 0 for h = 0,1,… , n − 1 when H 0 = ∅ . Then, E (see Definition 2) contains only the zero function. Hence, the fractional order interval function family Pr (see Definition 2) has only one member p0 ( s ) for each r ≥ 0 .

Consequently,

rmargin (q0+ , q1+ ,… , qn+−1 ) = +∞

regarding the

presumption of the Hurwitz-stability of p0 ( s ) . Part b) Indeed, the function T (ω ) has been considered as T (ω ) = sup{r ≥ 0 : 0 ∉ V (ω , r , Q)},

(38)

where the value set V (ω , r , Q) is defined for each ω , r ≥ 0 as n −1

V (ω , r , Q) { p0 pb ( jω ) + r (∑ qhω α h e j 0.5πα h + q0 ) : h =1

+ h

(39)

+ h

qh ∈ [− q , q ] for h = 0,1,… , n − 1}.

Since p0 ( s ) is Hurwitz-stable, then a0 > 0 . (In the case of a0 ≤ 0 , based on the well-known “intermediate value theorem” (Th. 9 of Thomas & Finney, 1996), p0 pb ( s ) has at

least one non-negative real zero since p0 pb (0) = a0 ≤ 0 and lim

s∈ , s →+∞

p0 pb ( s ) = +∞ . This contradicts the assumption of the

Hurwitz-stability of p0 ( s ) .) 202

SSSC 2010 Ancona, Italy, Sept 15-17, 2010

First, consider the case when ω = 0 . We have 0 ∉ V (0, r , Q)

segment with vertices A1 (ω )

if and only if 0 ∉ [a0 − rq0+ , a0 + rq0+ ] . According to a0 > 0

A2 (ω )

+ 0

p0 pb ( jω ) + re pb ( jω , −q1 ) = p0 pb ( jω ) − re pb ( jω , q1 )

and q ≥ 0 , one obtains T (0) as in (12).

(in

Now, consider the case when ω > 0 . We obtain T (ω ) in two Cases I and II.

t ∈ [−1,1]} ).

− | e pb ( jω , q1 ) | e jπβ1 ).

(40)

Then, we can write Im[e pb ( jω , q1 )e − jπβ1 ] = 0 and

Regarding r > 0 , it follows based on Theorem 3.1 of Akbari Moornani & Haeri (2010b) that 0 ∈ V (ω , r , Q) if and only if: Im ([ p0 pb ( jω ) + re pb ( jω , q )]e k

− jπβ k

) ≤ 0,

V (ω , r ,Q ) ={[ p0 pb ( jω )e − jπβ1 − tr | e pb ( jω , q1 )|]e jπβ1 : t ∈[−1,1]}⇒

{

V (ω , r , Q) = ⎡⎣( Re[ p0 pb ( jω )e− jπβ1 ] − tr | e pb ( jω , q1 ) |) +

(41)

k = 1, 2,… , m

}

j Im[ p0 pb ( jω )e− jπβ1 ]⎤⎦ e jπβ1 : t ∈ [−1,1] .

(46)

Hence, when r > 0 , it follows that 0 ∈ V (ω , r , Q) if and only

and Bk′ (ω )

if

− Im ([ p0 pb ( jω ) + re pb ( jω , q m + k )]e − jπβk ) ≤ 0,

(42)

k = 1, 2,… , m

Ak′ (ω ) < − Bk′ (ω ) & q r Im[e pb ( jω , q )e k

m+ k

− jπβ k

k

⎧ Re[ p0 pb ( jω )e − jπβ1 ] ⎪⎪ e pb ( jω , q1 ) T (ω ) = ⎨ ⎪ ⎪⎩+∞

k

] < 0 ⎯⎯→

Hence, defining r0

(43)

⎫ , k = 1, 2,… , m ⎪ − jπβ k k Im[e pb ( jω , q )e ] ⎪ ⎬⇔ − jπβ k Im[ p0 pb ( jω )e ] ⎪ (43) →r ≥ (42) ←⎯⎯ , k = 1, 2,… , m ⎪ Im[e pb ( jω , q k )e− jπβk ] ⎭

Im[e pb ( jω , q k )e − jπβk ] k =1, 2,… , m

rmargin (q0+ , q1+ ,… , qn+−1 )

, k = 1, 2,… , m ⇔

r ≥ max Im[ p0 pb ( jω )e − jπβk ] Im[e pb ( jω , q k )e − jπβk ]

and

if Im[ p0 pb ( jω )e − jπβ1 ] = 0, if Im[ p0 pb ( jω )e

− jπβ1

(47)

] ≠ 0.

inf T (ω ) , one obtains:

ω ∈[0, +∞[

(48)

According to Hurwitz-stability of p0 ( s ) (i.e. there exists at least one Hurwitz-stable member of the family Pr ) and the “zero exclusion condition theorem” (here, we need only a special case of Theorem 3.2 of Akbari Moornani & Haeri, 2009), one concludes that:

− Im[ p0 pb ( jω )e − jπβk ]

Im[ p0 pb ( jω )e − jπβk ]

e pb ( jω , q1 ) ≠ 0

r0 = sup{r ≥ 0 : 0 ∉ V (ω , r , Q) for all ω ∈ [0, +∞[}.

Then, it can be written for r > 0 that: (43) (41) ←⎯⎯ →r ≥

to

| Re[ p0 pb ( jω )e− jπβ1 ] |≤

Summing up the mentioned materials in Cases I and II, T (ω ) can be computed as in (13) for all ω > 0 when H 0 ≠ ∅ .

r >0

Im[e pb ( jω , q k )e − jπβk ] < 0, k = 1, 2,… , m.

According

and

0 ∉ V (ω , 0, Q) , it follows that we can write:

= −q & e pb ( jω , −q ) = −e pb ( jω , q ) ⇒ k

Im[ p0 pb ( jω )e − jπβ1 ] = 0

r | e pb ( jω , q1 ) | .

where q m + k = −q k for k = 1, 2,… , m . Considering r > 0 and attention to the proof of Theorem 3.1 of Akbari Moornani & Haeri (2010b), one concludes Ak′ (ω ) < − Bk′ (ω ) for ω > 0 , m ≥ 2 , and k = 1, 2,… , m . Hence, we can write:

r≥

V (ω , r , Q ) = { p0 pb ( jω ) + tre pb ( jω , q1 ) :

words

3) A2 (ω ) − A1 (ω ) =| A2 (ω ) − A1 (ω ) | e jπβ1 (i.e. e pb ( jω , q1 ) =

for all ω ≥ 0 ), it can be written that:

Ak′ (ω )

other

2) | A2 (ω ) − A1 (ω ) |≠ 0 (i.e. e pb ( jω , q1 ) ≠ 0 ).

Case I ( ω > 0 & m ≥ 2 ): When r = 0 , according to Hurwitz-stability of p0 ( s ) (and consequently p0 pb ( jω ) ≠ 0 0 ∉ V (ω , 0, Q).

p0 pb ( jω ) + re pb ( jω , q1 ) and

sup{r ≥ 0 : Pr is robustly Hurwitz-stable} = r0 .

(44)

Hence, regarding r > 0 , it follows that 0 ∈ V (ω , r , Q) if and only if (44) holds. According to this note and (40), one concludes:

T (ω ) = max Im[ p0 pb ( jω )e− jπβk ] Im[e pb ( jω , q k )e− jπβk ] . (45) k =1, 2,…, m

Case II ( ω > 0 & m = 1 ): When r = 0 , we have 0 ∉ V (ω , 0, Q) with the same reason as in Case I. Considering r > 0 , one obtains the three following notes based on Procedure 3.1 of Akbari Moornani & Haeri (2010b):

1) V (ω , r , Q) is the set of all points of the straight line 203

(49) ■