Robust FOPID stabilization of retarded type fractional order plants with interval uncertainties and interval time delay

Robust FOPID Stabilization of Retarded Type Fractional Order Plants with Interval Uncertainties and Interval Time Delay

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Robust FOPID Stabilization of Retarded Type Fractional Order Plants with Interval Uncertainties and Interval Time Delay Majid Ghorbani, Mahsan Tavakoli-Kakhki, Ali Akbar Estarami PII: DOI: Reference:

S0016-0032(19)30609-X https://doi.org/10.1016/j.jfranklin.2019.08.035 FI 4119

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

30 December 2018 31 May 2019 27 August 2019

Please cite this article as: Majid Ghorbani, Mahsan Tavakoli-Kakhki, Ali Akbar Estarami, Robust FOPID Stabilization of Retarded Type Fractional Order Plants with Interval Uncertainties and Interval Time Delay, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.08.035

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Robust FOPID Stabilization of Retarded Type Fractional Order Plants with Interval Uncertainties and Interval Time Delay Majid Ghorbania , Mahsan Tavakoli-Kakhkia,∗ , Ali Akbar Estaramia a Faculty

of Electerical Engineering, K. N. Toosi University of Technology, Tehran, Iran.

Abstract This study investigates the robust stability of the retarded type of interval fractional order plants with an interval time delay. To this end, the characteristic quasi-polynomial is divided into two terms. The first term is simply the denominator interval polynomial of the open loop system and the second term is the multiplication of the interval delay term in the numerator of the open loop system which is an interval polynomial. Each of these two terms of the characteristic quasi-polynomial makes their own value sets in the complex plane for a given frequency. In this paper, based on these two value sets and by using the zero exclusion principle, the robust stability of the closed loop system by applying a FOPID controller is analyzed. Finally, two numerical examples and an experimental verification are provided to demonstrate the effectiveness of the proposed method in the robust stabilization of fractional order plants with interval uncertainties and interval time delay. Keywords: robust stabilization, interval time delay, interval uncertainties, fractional order controller.

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Preprint submitted to Journal of the Franklin Institute

September 3, 2019

1. Introduction 1.1. Background 5

In recent years, fractional order calculus has been widely used in industrial applications [1]. Since the stability analysis of the closed loop system is an essential step for control system design, this issue has been studied in many research works in the field of fractional order systems. For example in [2], the concept of BIBO stability for linear time invariant fractional order systems has

10

been presented and analyzed. The stability analysis of linear fractional order system with time delays of the retarded type has been given in[3]. Also, several analytical and graphical methods have been presented in literature for the stability analysis of fractional order time delay systems [4, 5, 6]. The stability regions of different kinds of fractional order controllers in control of

15

various plants have been obtained in literature. For instance, the boundaries of stability regions of fractional order PI and PID controllers have been specified in [7]. The stabilizing region of fractional order PD controller for integrating time delay integer order systems and time delay fractional order systems have been presented in [8] and [9], respectively. Moreover, in [10] investigate the stabiliz-

20

ing region of FOPID controllers for the fractional order counterpart of a First Order Plus Dead Time (FOPDT) system. The stabilizing region of fractional order PD controller for fractional order system with interval uncertainties and an interval delay has been obtained in [11]. In practical applications, which different kinds of uncertainty may exist in the

25

plant model, the robust stability analysis of the designed control system would be a main stage in designing any kind of robust controllers. In literature, this research topic has been studied in both time and frequency domains. In the time domain, the robust stability of fractional order linear systems has been studied in [12, 13, 14]. In these works, the stabilization conditions are derived

30

in the form of linear matrix inequalities. In the frequency domain, a significant theorem for the robust stability analysis of integer order systems is the Kharitonov’s theorem proposed in [15]. This theorem defines four polynomials

2

for an interval integer order system. If these polynomials are Hurwitz-stable, it can be concluded that for any set of parameters chosen from the uncertainty 35

space the interval system would be stable [16, 17, 18, 19]. It is shown that the Kharitonov’s theorem is not applicable for interval fractional order polynomials [20]. As stated in [20], the value set of any interval fractional order polynomial is a polygon in the complex plane for any frequency and the polygon edges are determined by defining the set of vertices [20]. Based on the

40

zero exclusion principle, robust stability criteria for an interval fractional order polynomial has been presented in [20]. Furthermore, the vertices presented for the value set of the interval fractional order polynomial in [20] are not the consecutive vertices of the polygon which leads to redundant computational cost. In [21], firstly the consecutive vertices of the value set of an interval fractional

45

order polynomial have been suggested. Then, the robust stability of interval fractional order polynomial has been studied based on the obtained vertices. In [22], by an analytical method, the stabilization of the fractional order counterpart of an interval first order system has been done by using Fractional Order PID (FOPID) controllers.

50

In [23], the consecutive vertices of the value set of interval fractional order polynomials have been obtained and a design procedure for a fractional order PD controller with a fractional order filter has been proposed. In [24, 25], using the concept of the value sets the robust stability criterion has been presented for fractional order time delay systems of the retarded and neutral types.

55

It has been shown that the characteristic polynomial of a closed loop system consists of a fractional order controller and an interval fractional order plant with time delay (without uncertainty) has a polygon shape [26]. In [26, 27, 28], by the Minkowski sum the consecutive vertices of the value set of the characteristic quasi-polynomial has been obtained. Furthermore, in [26, 27, 28], some

60

auxiliary functions have been proposed to be used in the robust stability analysis of interval fractional order systems with time delay (without uncertainty). It is worth mentioning that none of the results presented in the mentioned research works can be employed for robust stability analysis of interval fractional 3

order plants with interval time delay. This issue will be completely discussed in 65

Subsection 2.2 of this paper. 1.2. Motivations and Challenges In [29], the robust stability theorem proposed in [20] has been extended for robust stabilization of fractional order interval systems with an interval time delay using a FOPID controller. The presented theorem in [29] (see Theorem

70

3.1 in [29]) which has been proposed to evaluate the robust stability of the compensated system has a crucial defect which will be discussed in the rest of this paper. Furthermore, in [30] for a classic second order transfer function with time delay a robust FOPID controller has been designed. To this end, at first 8 numbers of vertices have been obtained for the characteristic quasi-polynomial.

75

Then, it has been claimed that if Theorem 3 in [20] is satisfied for the set of the exposed edges, then the FOPID controller is robust stable. The counterexample presented in Subsection 2.2 (Problem Statement) of this paper reveals the significant drawback of [29, 30]. For the robust stability analysis of an uncertain system based on the zero ex-

80

clusion principle it is necessary to have an exact determination of the value set of the characteristic quasi-polynomial. In [31], it has been mentioned that existing an uncertain time delay in the characteristic quasi-polynomial leads to a nonconvex shape of the value set. In such cases, i.e. when the value set of the characteristic quasi-polynomial does not have a polygon shape, an im-

85

portant challenge is to determine whether the characteristic quasi-polynomial value set contains the origin of the complex plane or not. To solve this problem, in Example 1 of [24] the value set of the characteristic quasi-polynomial has been specified plotting one hundred thousand randomly generated values in the uncertainty space at any frequency which needs too much computational cost.

90

Also as it was mentioned before, the defined value sets in [29, 30] are not reliable generally. Therefore, determining the exact shape of the value set of a fractional order quasi-polynomial with interval delay, which is not a convex shape, is a significant challenge in robust stability analysis of interval fractional order plant 4

with interval time delay. The other challenge would be decreasing the computa95

tional cost in the robust stability analysis of controlled interval fractional order plants with interval time delay. Such an issue will be also elaborately discussed in Section 3 (Main result) of this paper. Consequently, the above mentioned issues motivate the authors of this paper to give an efficient and exact method for robust stability analysis of interval fractional order plants with interval time

100

delay based on the zero exclusion principle. 1.3. Main Works and Contributions In this paper, the robust stability analysis for the retarded type of time delayed fractional order plants with interval uncertainties and an interval time delay is investigated. The characteristic quasi- polynomial is divided into two

105

terms. One of these terms is an interval fractional order polynomial and the other term is a quasi-interval fractional order polynomial due to the existence of the delay term. In Section 2, the value sets of these two terms are introduced. Based on the zero exclusion principle, three theorems are given which help to check the robust stability of a designed control system. According to the

110

presented theorems, an algorithm is given for robust stability analysis of the closed loop system. Therefore, the main contributions of this paper can be summarized as follows. 1. Presenting an accurate graphical procedure for checking whether the corresponding value set of the characteristic quasi- polynomial of a fractional

115

order plant with interval uncertainties and interval time delay controlled by a FOPID controller contains the origin or not. 2. Effectively reducing of the computational cost for the robust stability analysis an interval fractional order plant with an interval time delay controlled by a FOPID controller in three ways noted as follows.

120

(a) Presenting a finite frequency test interval instead of the infinite frequency interval to investigate the robust stability of the closed loop system.

5

(b) Introducing an auxiliary function whose sign check simply determines whether the value set of the characteristic quasi- polynomial contains 125

the origin or not. (c) Presenting some exact value sets whose plots indicate whether the value set of the characteristic quasi- polynomial contains the origin or not. This method is much simpler and more accurate than plotting too many randomly generated points in the uncertainty space of the

130

characteristic quasi- polynomial. 1.4. Organization The remainder of this paper is structured as follows. In Section 2, some definitions are stated in Subsection 2.1 and the problem statement is described in Subsection 2.2. Section 3 presents the main results of this paper. Theorems

135

given in this section can be used to verify the robust stability of controlled fractional order plants with interval uncertainties and an interval time delay. In Section 4, two numerical examples and an experimental verification are given to illustrate the applicability of the presented theorems in the robust stability analysis of controlled interval fractional order plants with interval time delay.

140

Section 5 concludes the paper.

2. Preliminaries 2.1. Definitions Define A\B , {x | x ∈ A , x ∈ / B} and A∪B , {x : x ∈ A or x ∈ B} for any Pk2 two sets A and B. Assume i=k1 fi = 0 for k2 < k1 where fi , (i = k1 , · · · , k2 ) k2 [ are scalar functions and Fi = ∅ for k2 < k1 where Fi , (i = k1 , · · · , k2 ) are i=k1

numerical sets. The number of elements of the set A is denoted by n(A). Also, ∂(U ) denotes the boundary of the value set U in the complex plane. In this paper, an interval fractional order plant with interval time delay as k e−ls , αi q s i i=0

P (s) = Pn

6

(1)

is supposed, where the fractional orders αi (i = 0, 1, · · · , n) satisfy 0 = α0 < α1 < · · · < αn , qi (i = 0, 1, · · · , n) are the real uncertain coefficients lying in the intervals [qi , qi ], and qn 6= 0. Also, l is a positive constant time delay lying in the interval [l, l] and k is a positive open loop gain lying in the interval [k, k]. Moreover k, l and qi , (i = 0, 1, · · · , n) are independent real uncertain parameters. Also, a FOPID controller with fractional orders with λ , µ ∈ (0, 2) and µ < αn is considered as follows. C(s) = kp +

ki + kd sµ . sλ

(2)

The corresponding characteristic quasi-polynomial for P (s) in (1) and C(s) in (2) is obtained as ∆(s) in (3). ∆(s) =

n X

qi sαi +λ + k(kp sλ + ki + kd sµ+λ )e−ls .

(3)

i=0

According to [3], the fractional order characteristic quasi-polynomial (3) is of the retarded type if αn > µ, and it is of the neutral type if αn = µ. In this paper, it is supposed that αn > µ which indicates that the resulted closed loop system is a fractional order time delay system of the retarded type. The characteristic quasi-polynomial ∆(s) in (3) can be rewritten as ∆(s) = D(s) + ∆N d (s), where, D(s) =

n X

qi sαi +λ , ∆N d (s) = N (s)e−ls and N (s) = k(kp sλ +ki +kd sµ+λ ). (4)

i=0

Zero Exclusion Principle [21] : Assume that ∆(s) has at least one Hurwitz145

stable member. It follows that ∆(s) is robustly Hurwitz-stable if and only if 0∈ / ∆(jω) for all ω ≥ 0. Some required definitions for the rest of the paper are given as follows. Definition 1. For any two quasi-polynomials (or polynomials) F (s) and G(s) the convex combination e(F (s), G(s)) is defined as follows. e(F (s), G(s)) = ηF (s) + (1 − η)G(s),

η ∈ [0, 1].

(5)

Definition 2. Two vertices v1N (s) and v2N (s), and the exposed edge PEN (s) corresponding to the value set of the interval fractional order polynomial N (s) 7

in (4) are defined as (6) and (7), respectively.   v (s) = k(k sλ + k + k sµ+λ ), 1N p i d  v (s) = k(k sλ + k + k sµ+λ ). 2N

p

i

(6)

d

PEN (s) = e(v1N (s), v2N (s)).

(7)

Moreover, the polygon PED (s) in (8) is defined as the set of the exposed edges corresponding to the value set of the interval fractional order polynomial D(s) in (4).

PED (s) =

 2m [    {e(viD (s), v(i+1)D (s))},   

m > 1,

i=1

     

e(v1D (s), v2D (s)),

(8) m = 1.

In (8), viD (s) (i = 1, · · · , 2m) are the vertices of D(s) and v(2m+1)D (s) = v1D (s) [21]. Definition 3. Substituting s = jω, the largest absolute magnitudes of the interval fractional order polynomials N (jω) and D(jω) in (4) are respectively defined as RN max (ω) in (9) and RDmax (ω) in (10). RN max (ω) = k|(kp (jω)λ + ki + kd (jω)µ+λ )|.

RDmax (ω) = max

2m [

(9)

{|viD (jω)|}.

(10)

i=1

Definition 4. Substituting s = jω, the smallest absolute magnitudes of the interval fractional order polynomials N (jω) and D(jω) in (4) are respectively defined as RN min (ω) in (11) and RDmin (ω) in (12) as follows. RN min (ω) = k|(kp (jω)λ + ki + kd (jω)µ+λ )|.

RDmin (ω) =

   

min

2m [

{|e(viD (jω), v(i+1)D (jω))|},

(11)

m > 1,

i=1

   min |e(v (jω), v (jω))|, 1D 2D 8

m = 1.

(12)

Figure 1 and Figure 2 depict the boundary of the value set ∆N d (jω) , i.e. 2π 2π ∂(∆N d (jω)) for ω ∈ (0, l−l ) and ω ∈ [ l−l , ∞), respectively. Therefore, for a

fixed frequency ω ∈ (0, ∞), ∂(∆N d (jω)) ⊆ PE∆N d (jω),

(13)

where PE∆N d (jω) is defined as    {e(v1N (jω)e−jωl , v2N (jω)e−jωl ),       e(v1N (jω)e−jωl , v2N (jω)e−jωl ),   PE∆N d (jω) = v1N (jω)e−jω (e(l,l)) , v2N (jω)e−jω (e(l,l)) }, 0 < ω < ωci ,           {v (jω)e−jω (e(0,2π)) , v (jω)e−jω (e(0,2π)) }, ω ≥ ω , 1N 2N ci (14)

with ωci =

2π . l−l

Figure 1: The boundary ∂(∆N d (jω)) and its four vertices for a fixed frequency ω ∈ (0 , ωci ). 150

2.2. Problem Statement In this subsection, at first a counterexample is presented to investigate the drawback of the Theorem 3.1 in [29] and the applied technique in [30]. Counterexample: Consider the following fractional order uncertain plant P (s) =

k e−ls , q0 = 1, q1 s1.6 + q0 9

(15)

Figure 2: The boundary ∂(∆N d (jω)) for a fixed frequency ω ∈ [ωci , ∞) (solid line) and two radiuses RN max (ω) in (9) and RN min (ω) in (11).

where the range of the parameters k, q1 and l are considered as 0.66 ≤ k ≤ 0.68 , 0.5 ≤ q1 ≤ 0.6 , 1 ≤ l ≤ 2.5.

(16)

Applying the FOPID controller (2), the characteristic quasi-polynomial of the interval fractional order plant (15) is achieved as (17). ∆(s) = sλ (q1 s1.6 + 1) + k(kp sλ + ki + kd sµ+λ )e−ls .

(17)

As it can be seen, the characteristic quasi-polynomial (17) has three interval parameters. Thus, it has eight number of vertices and twelve exposed edges [29]. Based on [29], vertices of ∆(s) are obtained as P1 (s, q) = sλ (0.5s1.6 + 1) + 0.66(kp sλ + ki + kd sµ+λ )e−s , P2 (s, q) = sλ (0.5s1.6 + 1) + 0.66(kp sλ + ki + kd sµ+λ )e−2.5s , P3 (s, q) = sλ (0.5s1.6 + 1) + 0.68(kp sλ + ki + kd sµ+λ )e−s , P4 (s, q) = sλ (0.5s1.6 + 1) + 0.68(kp sλ + ki + kd sµ+λ )e−2.5s , P5 (s, q) = sλ (0.6s1.6 + 1) + 0.66(kp sλ + ki + kd sµ+λ )e−s , P6 (s, q) = sλ (0.6s1.6 + 1) + 0.66(kp sλ + ki + kd sµ+λ )e−2.5s , P7 (s, q) = sλ (0.6s1.6 + 1) + 0.68(kp sλ + ki + kd sµ+λ )e−s , P8 (s, q) = sλ (0.6s1.6 + 1) + 0.68(kp sλ + ki + kd sµ+λ )e−2.5s , 10

(18)

where, q = [kp , ki , kd , λ, µ]. Finally, the set of exposed edges is derived as follows:



PE (s, q) = e(P1 (s, q), P2 (s, q)), e(P1 (s, q), P3 (s, q)), e(P1 (s, q), P5 (s, q)), e(P2 (s, q), P4 (s, q)),

e(P2 (s, q), P6 (s, q)), e(P3 (s, q), P4 (s, q)), e(P3 (s, q), P7 (s, q)), e(P4 (s, q), P8 (s, q))  , e(P5 (s, q), P6 (s, q)), e(P5 (s, q), P7 (s, q)), e(P6 (s, q), P8 (s, q)), e(P7 (s, q), P8 (s, q)) . (19)

Let C(s) be a FOPID controller which is given by C(s) =

0.5s0.2 + 0.42 + 0.001s0.4 . s0.2

(20)

Now, Theorem 3 in [3] is used to test the stability of each vertex Pi (s, q)(i = 1, 2, · · · , 8) in (18) by using the FOPID controller (20). To this end, the generalised modified Mikhalov’s plot corresponding to eight characteristic quasi-polynomials (18) have been depicted in Figure 3. As this figure shows, none of the generalised modified Mikhailov’s plot of vertices (18) encircles the origin of the complex plane which means that all the vertices (18) are Hurwitz-stable by applying the FOPID controller (20). Therefore, based on Theorem 3.1 in [29], it is concluded that the interval fractional order plant (15) is robust stable by applying the FOPID controller (20). In the rest, the method used in [30] is also applied for this example. The robust controller design procedure proposed in [30] is based on Theorem 3 in [20]. According to [30], the value set of PE (s, q) in (19) has been drawn in Figure 4. As Figure 4 demonstrates, the origin is not included in the value sets of PE (s, q) in (19). Therefore it is concluded that the closed loop system is robust stable by the designed FOPID controller (20). As it has been discussed above, based on [29, 30] the robust stability of the closed loop system for P (s) in (15) and C(s) in (20) is concluded, while the following considered sample case shows that the proposed methods in [29, 30] is not reliable for robust stability analysis of an interval fractional order plant with an interval

11

time delay. (In fact, the proposed robust stability procedures in both of these papers are based on [20] whose theorems presented for robust stability analysis of interval fractional order plants.) As a sample case, consider the characteristic

Figure 3:

The generalized modified Mikhailov plot for the vertex quasi-polynomials

Pi (s, q)(i = 0, 1, · · · , 8) in (18) and C(s) in (20) and its better view around origin.

Figure 4: The set of the exposed edges PE (s, q) in (19) and C(s) in (20) and its better view around origin for ω = 0 : 0.01 : 2 (rad/sec).

quasi-polynomial ∆∗ (s) in (21) obtained based on (17) for k = 0.67, q1 = 0.5, and l = 1.8. ∆∗ (s) = s0.2 (0.5s1.6 + 1) + 0.67(0.5s0.2 + 0.42 + 0.001s0.4 )e−1.8s .

12

(21)

Figure 5 shows the generalized Mikhailov’s plot corresponding to the characteristic quasi-polynomial ∆∗ (s). As this figure indicates, the generalized modified Mikhailov’s plot encircles the origin of the complex plane twice in the clockwise direction. Consequently, the designed controller C(s) in (20) cannot stabilized P (s) in (15) for k = 0.67, q1 = 0.5, and l = 1.8. This issue means that the designed control system would not be robust stable. This result is in contrast with the result obtained based on [29, 30]. Figure 6 confirms that the closed loop system is not robust stable, because the origin is included in the value set of ∆(s) in (17). In this figure, the images of 45000 randomly generated point in the uncertainty space of the characteristic quasi-polynomial (17) have been plotted for ω = 1.2(rad/sec). Also in Figure 6, the exposed edges PE (s, q) in (19) have been depicted for ω = 1.2(rad/sec). As it is seen in Figure 6, these exposed edges do not enclose the origin. Therefore, the robust stability analysis in [29, 30] is not generally correct for an interval fractional order plant with an interval time delay. It is because that in both of these research works the relation (22) is not satisfied. ∂(∆(jω)) ⊆ PE (jω, q).

(22)

According to the above discussed deficiencies, this paper focuses on presenting a reliable graphical procedure to verify whether the value set of ∆(s) in (3) for s = jω contains the origin of the complex plane or not. Also, in this paper a 155

finite frequency test interval is presented instead of the infinite frequency test interval used in literature for checking the value set of the characteristic quasipolynomial (3). To this end, in Theorem 1 proved in this paper the necessary and sufficient conditions are presented for robust stability of a closed loop system consisting an interval fractional order plant and a FOPID controller. Also, in

160

Theorem 2 a finite frequency test interval is offered which considerably reduces the computational cost. Furthermore, for special cases, Lemma 1 and Lemma 2 are presented to facilitate the robust stability analysis of the characteristic quasi-polynomial (3).

13

Figure 5: The generalised modified Mikhailov’s plot of ∆∗ (s) in (21).

Figure 6: The value set of the characteristic quasi-polynomial (17) by plotting the images of 45000 randomly generated points in the uncertainty space (blue color), and the exposed edges PE (s, q) in (19) corresponding to this characteristic polynomial (black color) for the frequency ω = 1.2(rad/sec).

3. Main results 165

In this section, three theorems and two lemmas are presented for the robust stability analysis of fractional order interval plants with interval time delay. As it was mentioned before, the characteristic quasi-polynomial ∆(s) is the sum of D(s) and ∆N d (s) in (4), that D(s) is an interval fractional order polynomial lying in a polygon for a given frequency. On the other hand, ∆N d (s) has its 14

170

own specific shape in the complex plane for a fixed frequency that was depicted in Figure 1 and Figure 2. According to these value sets, Theorem 1 is presented to investigate the robust stability of the fractional order closed loop system in the presence of interval uncertainties of all the plant parameters. Theorem 1. Assume ∆(jω) in (3) has at least one Hurwitz-stable member by

175

applying the fractional order PID controller (2). The fractional order closed loop system with the characteristic quasi-polynomial defined (3) is robust stable if and only if: 1. The value sets of PE∆N d (jω) in (14) and PΞ (jω) in (23) do not have any overlap in the complex plane for any frequency ω ∈ (0, ωci ).  2m [    e(ejπ viD (jω), ejπ v(i+1)D (jω)), m > 1,    i=1 PΞ (jω) =       e(ejπ v (jω), ejπ v (jω)), m = 1. 1D 2D

(23)

2. D(ω) > 0 holds for ω ∈ [ωci , ∞), where

D(ω) = RDmin (ω) − RN max (ω).

(24)

Proof. Based on the zero exclusion principle, the closed loop system is stable if and only if the characteristic quasi-polynomial (3) contains at least one stable quasi-polynomial and the origin is not included in the value set of ∆(jω) for ω ∈ [0, ∞). According to (3) and (4), the value set of ∆(jω) may contain the origin for specific frequencies ω 6= 0, if the equality ∆N d (jω) = −D(jω) holds. Therefore, for the case 0 < ω < ωci , if ∂(∆N d (jω)) depicted in Figure 1 and the value set PΞ (jω) do not have any overlap in the complex plane, the origin is not included in the value set of ∆(jω). Also for the case ω ≥ ωci , i.e. when ∂(∆N d (jω)) is as depicted in Figure 2, ∆(jω) does not include the origin if the inequality (25) holds. RDmin (ω) > RN max (ω).

(25)

It is because that the closed loop system is of the retarded type and for ω = ∞ the inequality lim RDmin (ω) > lim RN max (ω) holds. Therefore, two value ω→∞

ω→∞

15

180

sets PE∆N d (jω) and PΞ (jω) would not have any overlap for ω ∈ [ωci , ∞), if the inequality (25) is satisfied. For more clarification, Figure 7 schematically shows a circumstance that the inequality (25) does not hold for the specific frequency ω. Therefore, in such a case ∆(jω) includes the origin and the closed loop system would not be robust stable. Finally, for ω = 0 the characteristic

185

quasi-polynomial ∆(jω) (3) equals kki which is a nonzero value. Remark 1. Although the boundary of the value set of the characteristic quasipolynomial (3) is not a polygon, Theorem 1 can be used to determine whether this value set contains the origin or not. This investigation can be done by simply plotting the value set presented in Theorem 1. Also, for ω ∈ [ωci , ∞) the

190

auxiliary function D(ω) is introduced whose sign check indicated whether the value set of the characteristic quasi- polynomials contains the origin or not.

Figure 7: The boundary∂(∆N d (jω)) (black color) and the polygon of PED (jω) (red color) for a fixed frequency ω ∈ [ 2π , ∞). l−l

As it was mentioned before, to check the robust stability of a closed loop system based on the zero exclusion principle it is necessary to check whether the value set of ∆(jω) for the infinite frequency interval [0 , ∞) includes the 195

origin or not. In the following, this frequency test interval is reduced to a finite frequency test interval [ωmin , ωmax ] as stated in Theorem 2. Theorem 2. The origin is not included in the value set of ∆(jω) in (3), if

16

ω ∈ [0 , ∞)\[ωmin , ωmax ], where ωmax , min{ωmax 1 , ωmax 2 },

ωmin , max{ωmin 1 , ωmin 2 }.

(26)

In (26), ωmax 1 is defined as (27). ωmax1 , max{1, σ1 ε1 },

σ1 ,

Pn−1 i=0

ε1 ,

1 , αn − max{αn−1 , µ} (27)

max{|qi |, |qi |} + k(|kp | + |ki | + |kd |) , min{|qn |, |qn |}

Also, ωmax 2 is defined as (28).

ωmax 2 , (1 +

σ2 ) ε2 , min{|q n |, |qn |}

σ2 , max{M1 ∪ M2 }, M1 ,

n−1 [ i=1

{max{|qi |, |qi |}} ∪ {max{|q0 |, |q0 |} + k|kp | , k|kd | , k|ki |},

  {|k||k | + max{|q |, |q |} | α = µ , i = 1, 2, · · · , n − 1 }, d i i i M2 ,  ∅, ε2 ,

1

min{γ1 , γ22 , · · ·

γ

n+2−n(M2 ) , n+2−n(M } 2)

n ≥ 2, n = 1.

,

   x = 1,  min{αn − αn−1 , αn − µ}, x−1 [ γx ,  x = 2, 3, · · · , n + 2 − n(M2 ),   min{A1 \ {γh }}, h=1  n−1 [    {αn − αi } ∪ {αn − µ, αn + λ}, n(M2 ) = 0,   i=0 A1 , n−1 [     {αn − αi } ∪ {αn + λ}, n(M2 ) = 1.  i=0

Likewise ωmin 1 and ωmin 2 are defined as (29) and (30), respectively.

17

(28)

ωmin 1 , min{1, σ3 ε3 },

ε3 ,

1 , λ (29)

k|ki | . max{|q |, |q i i |} + k(|kp | + |kd |) i=0

σ3 , Pn

ωmin 2 , (

1 1+

σ4 k|ki |

)ε4 ,

ˆ1 ∪ M2 }, σ4 , max{M

ˆ1 , M

i=n [ i=1

{max{|qi |, |qi |}} ∪ {max{|q0 |, |q0 |} + k|kp | , k|kd | },

ε4 ,

βx ,

1 min{β1 , β22 , · · ·

β

n+2−n(M2 ) , n+2−n(M } 2)

   λ,    

(30)

x = 1,

x−1  [    min{A \ {βh }},  2 

x = 2, 3, · · · , n + 2 − n(M2 ),

h=1

A2 ,

,

 [ n   {αi + λ} ∪ {µ + λ},      

i=0 n [

{αi + λ},

n(M2 ) = 0, n(M2 ) = 1.

i=0

Proof. The proof is given in Appendix A. Remark 2. It is important to note that the presented frequency test interval in Theorem 2 i.e. [ωmin , ωmax ] is less than or equal to the presented frequency test 200

interval obtained based on Lemma 2 and Lemma 3 in [24]. 18

Theorem 3. If the characteristic quasi-polynomial ∆(jω) in (3), has at least one Hurwitz-stable member, the corresponding closed loop fractional order system is robust stable if and only if the value sets of PE∆N d (jω) in (14) and PΞ (jω) in (23) do not have any overlap in the complex plane for any frequency 205

ω ∈ [ωmin , ωmax ] introduced in Theorem 2. Proof. The proof of this theorem is straightforward according to Theorem 1 and Theorem 2. Lemma 1. Assume the characteristic quasi- polynomial ∆(jω) in (3), has at least one Hurwitz-stable member. If ωmin < ωci < ωmax , the corresponding

210

closed loop fractional order system is robust stable if and only if 1. The value sets of PE∆N d (jω) in (14) and PΞ (jω) in (23) do not have any overlap in the complex plane for any frequency ω ∈ [ωmin , ωci ). 2. The inequality D(ω) > 0 is satisfied for ω ∈ [ωci , ωmax ]. Proof. Considering Theorem 2 and the shape of the boundary ∂(∆N d (jω)) for

215

ω ∈ (0, ωci ) and ω ∈ [ωci , ∞) (Figure 1 and Figure 2), the frequency interval introduced in Theorem 2 can be divided to two intervals [ωmin , ωci ) and [ωci , ωmax ). Based on Theorem 3, if the value set of PE∆N d (jω) and PΞ (jω) do not have any overlap for ω ∈ [ωmin , ωci ), then the origin is not included in the value set of ∆(jω). Also, according to Theorem 1, if the inequality D(ω) > 0

220

holds for ω ∈ [ωci , ωmax ], then it is also concluded that the origin is not included in the value set of ∆(jω) for ω ∈ [ωci , ωmax ]. In this way, the proof is completed according to the zero exclusion principle. Lemma 2. Assume the characteristic quasi-polynomial ∆(jω) in (3) has at least one Hurwitz-stable member. If ωmin ≥ ωci , the corresponding closed loop

225

fractional order system is robust stable if and only if D(ω) > 0 is satisfied for ω ∈ [ωmin , ωmax ]. Proof. According to Theorem 2, the origin is not included in the value set of ∆(jω) in (3) for ω ∈ (0, ωmin ) and ω ∈ (ωmax , ∞). Also, the inequality 19

ωmin ≥ ωci implies that the shape of ∂(∆N d (jω)) is same as Figure 2 for ω ∈ 230

[ωmin , ωmax ]. Based on Theorem 1, if D(ω) > 0 holds, then the origin is not included in the value set of ∆(jω) for ω ∈ [ωmin , ωmax ]. Now, the proof is completed by benefiting from the zero exclusion principle. Based on Thereom 2, Theorem 3, Lemma 1 and Lemma 2 the robust stability of an interval fractional order plant with an interval time delay controlled

235

by a FOPID controller can be analyzed by benefiting from Algorithm 1.

Algorithm 1 (Robust Stability Analysis): Step 1. The stability of the characteristic quasi-polynomial (3) for an arbitrary values of the uncertainty space is checked by the method proposed in [3]. If the 240

characteristic quasi-polynomial can not be stabilized for the selected uncertainty values, then the interval plant cannot be stabilized. Otherwise, the algorithm is continued in the next step. Step 2. By calculating ωmin and ωmax according to Theorem 2, three conditions may occur:

245

1. If ωmax < ωci , then the robust stability is checked by Theorem 3. 2. If ωmin < ωci < ωmax , then the robust stability is checked by Lemma 1. (It is suggested that to check the inequality D(ω) > 0, firstly. If this inequality does not hold, the characteristic quasi-polynomial (3) is not robust stable. Otherwise, the overlap between the value sets PE∆N d (jω)

250

in (14) and PΞ (jω) in (23) in the complex plane should be checked for any frequency ω ∈ [ωmin , ωci )). 3. If ωci ≤ ωmin , then the robust stability is checked by Lemma 2. 4. Illustrative example Example 1. In this example by applying Algorithm 1 described in Section 3, the robust stability of a closed loop system is investigated. Consider the fractional order plant and the designed FOPID controller as (31) and (32),

20

respectively [24]. P (s) =

C(s) = 0.6152 +

ke−ls , q1 s + q0

0.01 + 4.3867s0.4773 , s0.8968

(31)

(32)

50 433.33 where k ∈ [ 3.13 1.5 , 3.13×1.5] , l ∈ [ 1.5 , 50×1.5] , q0 = 1 and q1 ∈ [ 1.5 , 433.33× 255

1.5]. The generalised modified Mikhailov’s plot of the characteristic quasipolynomial for kˆ = 3.13, ˆl = 50 and qˆ = 433.33 has been presented in Figure 8. Since the generalised modified Mikhalov’s plot depicted in Figure 8 does not encircle the origin of the complex plane, the considered characteristic quasipolynomial would be Hurwitz-stable. According to Theorem 2, the frequencies

260

ωmin1 , ωmin2 , ωmax 1 and ωmax 2 are obtained as 0 rad/sec, 0 rad/sec , 1 rad/sec and 1.1392 rad/sec respectively. Therefore, ωmin and ωmax equal 0 rad/sec and 1 rad/sec, respectively. Also, for this interval plant it is obtained that ωci = 0.1508 rad/sec. The values of D(ω) for ω ∈ [0.1508, 1] have been shown in Figure 9. According to this figure, the second condition of Lemma

265

1 is satisfied. Based on viD (jω), (i = 1, 2) in (33) and viN (jω), (i = 1, 2) in (34), the value sets of PΞ (jω) and PE∆N d (jω) have been drawn in Figure 10 for ω = 0.01 : 0.03 : 0.15 rad/sec.   v (jω) = (jω)0.8968 ( 433.33 jω + 1), 1D 1.5  v (jω) = (jω)0.8968 (433.33 × 1.5jω + 1).

(33)

2D

  v (jω) = 3.13 (0.6152(jω)0.8968 + 0.01 + 4.3867(jω)0.4773+0.8968 ), 1N 1.5  v (jω) = 3.13 × 1.5(0.6152(jω)0.8968 + 0.01 + 4.3867(jω)0.4773+0.8968 ). 2N

(34)

As this figure depicts, these value sets do not have any overlap at these

frequency test points. In other words, the same color shapes in this figure do not have any overlap. Hence, the closed loop system is robust stable based on Lemma 1. In [24], the border of the corresponding characteristic quasipolynomial to P (s) in (31) and C(s) in (32) has been specified using one hundred thousand randomly generated values in the uncertainty space at each frequency 21

interval ω ∈ [0 , 1] to investigate whether the origin belongs to the value sets of the characteristic quasi-polynomial or not. Nevertheless, according to Lemma 1 presented in this paper, it is simply sufficient to check the overlap of PΞ (jω) and PE∆N d (jω) for ω ∈ [0, 0.1508) and the inequality D(ω) > 0 for ω ∈ [0.1508, 1].

Figure 8: The generalised modified Mikhailov’s plot of the characteristic quasi-polynomial for ˆ = 3.13 , ˆ the interval (31) and controller (32) for k l = 50 , qˆ1 = 433.33.

Figure 9: D(ω) in (24) for the interval plant (31) and controller (32) in the frequency range 0.1508 ≤ ω ≤ 1.

Example 2. Consider the interval fractional order plant with an interval time delay as (35) P (s) =

ke−ls , q2 s1.2 + q1 s0.5 + q0

(35)

where k ∈ [1.2 , 1.5] , l ∈ [1 , 4] , q0 ∈ [1.6 , 2] , q1 ∈ [1.4 , 1.5] and q2 ∈ [1.3 , 1.5]. 22

Figure 10: The value sets of PΞ (jω) in (23) and PE∆N d (jω) in (14) for ω = 0.01 : 0.03 : 0.15 rad/sec where PΞ (jω) and PE∆N d (jω) are dash line and solid line, respectively.

Based on [32], the stabilization region of the FOPID controller (2) for kˆ = 1.2, ˆl = 1.5, qˆ0 = 2 , qˆ1 = 1.5, qˆ2 = 1.5, λ = 0.5, µ = 0.5 and kd = 0.1 has been drawn in Figure 11. From Figure 11, two controllers C1 (s) in (36) and C2 (s) in (37) are chosen in the stability region as kp = 1 and ki = 1.4 for C1 (s) and kp = 0.5 and ki = 0.4 for C2 (s). C1 (s) = 1 +

1.4 + 0.1s0.5 . s0.5

C2 (s) = 0.5 +

0.4 + 0.1s0.5 . s0.5

(36)

(37)

In the following, the robust stability of the corresponding characteristic quasipolynomial for P (s) in (35) and C1 (s) in (36) is analyzed. According to Theorem 2, the frequencies ωmin1 , ωmin2 , ωmax 1 and ωmax 2 equal to 0.0638 rad/sec , 0.1052 rad/sec , 11.6487 rad/sec and 10.0257 rad/sec, respectively. Therefore, ωmin and ωmax are obtained as 0.1052 rad/sec and 10.0257 rad/sec, respectively. Also, for the considered plant (35), it is obtained that ωci = 2.0944 rad/sec. The values of D(ω) for ω ∈ [2.0944 , 10.0257] have been shown in Figure 12. According to this figure, the second condition of Lemma 1 is satisfied. Based on viD (jω), (i = 1, 2, · · · , 6) in (38) and viN (jω), (i = 1, 2) in (39), checking the first step of Lemma 1 shows that the value sets of PΞ (jω) and PE∆N d (jω) have

23

overlap in ω = 1 rad/sec (see Figure 13).    v1D (jω) = 1.3(jω)1.7 + 1.4(jω) + 1.6(jω)0.5 ,      v (jω) = 1.3(jω)1.7 + 1.4(jω) + 2(jω)0.5 ,  2D      v (jω) = 1.3(jω)1.7 + 1.5(jω) + 2(jω)0.5 , 3D   v4D (jω) = 1.5(jω)1.7 + 1.5(jω) + 2(jω)0.5 ,       v5D (jω) = 1.5(jω)1.7 + 1.5(jω) + 1.6(jω)0.5 ,      v (jω) = 1.5(jω)1.7 + 1.4(jω) + 1.6(jω)0.5 .

(38)

6D

  v (jω) = 1.2((jω)0.5 + 1.4 + 0.1(jω)), 1N  v (jω) = 1.5((jω)0.5 + 1.4 + 0.1(jω)).

(39)

2N

Consequently, the closed loop system would not be robust stable using the proposed controller. Now, the robust stability of the corresponding characteristic quasi-polynomial for P (s) in (35) and C2 (s) in (37) is analyzed. According to Theorem 2, ωmin1 , ωmin2 , ωmax 1 and ωmax 2 equal 0.0066 rad/sec , 0.0221 rad/sec , 6.851 rad/sec and 7.4285 rad/sec, respectively. Therefore, ωmin and ωmax are obtained as 0.0221 rad/sec and 6.851 rad/sec, respectively. The values of D(ω) for ω ∈ [2.0944, 6.851] have been shown in Figure 14. Also, based on viD (jω), (i = 1, 2, · · · , 6) in (38) and viN (jω), (i = 1, 2) in (40), the value sets of PΞ (jω) and PE∆N d (jω) have been drawn in Figure 15 for ω = 0.1 : 0.3 : 2 rad/sec.   v (jω) = 1.2(0.5(jω)0.5 + 0.4 + 0.1(jω)), 1N  v (jω) = 1.5(0.5(jω)0.5 + 0.4 + 0.1(jω)).

(40)

2N

According to this figure, these value sets do not have any overlap at these frequency test points. Thus according to Algorithm 1, the closed loop system is 270

robust stable by applying the fractional order controller C2 (s) in (37).

Example 3. In this example, to show the effectiveness of the paper results, a FOPID controller has been designed based on [32] for the laboratory scale level control process GUNT- RT512 (Figure 16) [33] and the designed controller has been practically applied. 24

ˆ = 1.2, ˆ Figure 11: The stabilization region of the controller parameters kp and ki for k l = 1.5, qˆ0 = 2 , qˆ1 = 1.5 , qˆ2 = 1.5 in (35) and λ = 0.5, µ = 0.5 and kd = 0.1 in (2).

Figure 12: D(ω) in (24) for the interval plant (35) and the controller (36) in the frequency range 2.0944 ≤ ω ≤ 10.0257.

The approximation method proposed in [34] is used for implementation of FOPID controller. The order and the frequency range of the approximation filter are considered as N = 3 and ω ∈ [0.001, 100] rad/sec respectively. Also, Ts = 0.4 Sec is selected as the sampling time. The process is identified by applying different step inputs and the identified model is as follows. P (s) =

ke−ls , q1 s + q0

25

(41)

where k ∈ [2.94, 4.59] , l ∈ [4.8, 14.4] , q0 = 1 and q1 ∈ [26.4, 35.6]. Based on

[32], the stabilization region of the FOPID controller (2) for kˆ = 4, ˆl = 10, qˆ0 = 1 , qˆ1 = 34, λ = 0.65, µ = 0.5 and kd = 2 has been drawn in Figure 17. From Figure 17, the FOPID controller is selected as (42) whose parameters locate in the obtained stability region. C(s) = 0.01 +

0.1 + 2s0.5 . s0.65

(42)

According to Theorem 2, the frequencies ωmin1 , ωmin2 , ωmax 1 and ωmax 2 are obtained as 4.2317 × 10−4 rad/sec , 1.6313 × 10−4 rad/sec , 1 rad/sec and 1.8164

rad/sec, respectively. Therefore, ωmin and ωmax are obtained as 4.2317 × 10−4

rad/sec and 1 rad/sec, respectively. Also, for this interval plant it is obtained that ωci = 0.6545 rad/sec. The values of D(ω) for ω ∈ [0.6545, 1] have been shown in Figure 18. According to this figure, the second condition of Lemma 1 is satisfied. Based on viD (jω), (i = 1, 2) in (43) and viN (jω), (i = 1, 2) in (44), the value sets of PΞ (jω) and PE∆N d (jω) have been drawn in Figure 19 for ω = 0.1 : 0.1 : 0.6 rad/sec.   v (jω) = (jω)0.65 (26.4jω + 1), 1D  v (jω) = (jω)0.65 (35.6jω + 1).

(43)

2D

  v (jω) = 2.94(0.01(jω)0.65 + 0.1 + 2(jω)1.15 ), 1N  v (jω) = 4.59(0.01(jω)0.65 + 0.1 + 2(jω)1.15 ).

(44)

2N

As this figure depicts, these value sets do not have any overlap at these frequency test points. Thus, according to Lemma 1, the closed loop system is robust stable by applying the designed fractional order controller C(s) in (42). The obtained 275

experimental results by applying the FOPID controller (42) are shown in Figure 20.

5. Conclusions In this paper, the general form of an interval retarded type fractional order plant with an interval delay term was considered and the robust stability of 26

Figure 13: Overlap between two value sets PΞ (jω) in (23) (dashedline) and PE∆N d (jω) in (14) (solid line) for ω = 1 rad/sec.

Figure 14: D(ω) in (24) for the interval plant (35) and the controller (37) in the frequency range 2.0944 ≤ ω ≤ 6.851.

280

such a plant by applying a FOPID controller was investigated. In Theorem 1 of the paper, the necessary and sufficient conditions were presented for robust stability of FOPID controlled interval fractional order plants with interval time delay. By proving Theorem 2, it was shown that the origin of the complex plane would not be included in the value set of the characteristic quasi-polynomial of

285

the closed loop system in the frequency intervals (0, ωmin ) and (ωmax , ∞) where the values of ωmin and ωmax have been given in this theorem. In fact, Theorem 2 helps to reduce the computational cost in verifying the robust stability of the closed loop system. In Theorem 3, which is based on the zero exclusion princi27

Figure 15: The value set of PΞ (jω) in (23) (dashed line) and PE∆N d (jω) in (14) (solid line) for ω = 0.1 : 0.3 : 2 rad/sec.

ple, it was presented that the robust stability of the closed loop system can be 290

investigated by drawing two defined value sets in the finite frequency interval [ωmin , ωmax ] introduced in Theorem 2. According to the mentioned theorems in this paper, an algorithm was proposed which helps to investigate the robust stability of an interval fractional order plant with interval time delay. Finally, the assessment of the paper results was offered by two numerical examples and also

295

an experimental example. In the continuation of this work, the robust stability analysis of interval fractional order plant with interval time delay possessing interval fractional order terms in both the numerator and the denominator of the plant transfer function controlled by a class of fractional order controllers can be a noticeable future research. Also, such analysis can be done in the presence

300

of simultaneous uncertainties in the coefficients and also in the fractional orders of the interval plant with interval time delay.

Appendix A. (Proof of Theorem 2) To prove Theorem 2, it is sufficient to show that |∆(jω)| > 0 for ω ∈ (0, ωmin ) and ω ∈ (ωmax , ∞). In the other words, investigating the inequality |∆(jω)| > 0 for these two frequency ranges ensures that the origin is located outside of the value set of ∆(jω).

28

Figure 16: Laboratory scale level control process in the Process Control Laboratory at K. N. Toosi University of Technology.

ˆ = 4, ˆ Figure 17: The stabilization region of the controller parameters kp and ki for k l = 10, qˆ0 = 1 , qˆ1 = 34 in (41) and λ = 0.65, µ = 0.5 and kd = 2 in (2).

Suppose that ω > 1: Based on ∆(jω) in (3) and the triangle inequality ||A|−|B|| ≤ |A+B| ≤ |A|+|B|, it is concluded that |∆(jω)| ≥ min{|qn |, |qn |}|ω|αn +λ − λ

n−1 X i=0

max{|qi |, |qi |}|ω|αi +λ µ+λ

−k(|kp ||ω| + |ki | + |kd ||ω|

29

).

(45)

Figure 18: D(ω) in (24) for the interval plant (41) and the controller (42) in the frequency range 0.6545 ≤ ω ≤ 1.

Figure 19: The value sets of PΞ (jω) in (23) (dashed line) and PE∆N d (jω) in (14) (solid line) for ω = 0.1 : 0.1 : 0.6 rad/sec.

The inequalities (46) hold for ω > 1. ω αn +λ > ω max{αn−1 +λ

, µ+λ}

> · · · > ωλ > ω0 .

(46)

From (45) and (46), the inequality (47) is resulted. αn +λ

|∆(jω)| ≥ min{|qn |, |qn |}|ω|

−(

n−1 X i=0

k(|kp | + |ki | + |kd |))|ω|max{αn−1 +λ

30

max{|qi |, |qi |}+

, µ+λ}

.

(47)

Figure 20: Level output and control signal by applying the robust stable FOPID controller (42).

The right hand side of (47) is equal to α −max{α

n−1 min{|qn |, |qn |}|ω|max{αn−1 +λ , µ+λ} (|ω| n Pn−1 max{|qi |, |qi |} + k(|kp | + |ki | + |kd |) )). ( i=0 min{|qn |, |qn |}

, µ}

−

From (48) and according to the inequality min{|qn |, |qn |}|ω|max{αn−1 +λ

(48)

, µ+λ}

>

0, if the inequality ω > σ1 ε1 holds, then |∆(jω)| > 0.

Also, by using the triangle inequality, the following inequality is derived from (45).  |∆(jω)| ≥ min{|qn |, |qn |}|ω|αn +λ 1 −

n−1 X 1 max{|qi |, |qi |} min{|qn |, |qn |} i=1

|ω −1 |αn −αi + (max{|q0 |, |q0 |} + k|kp |)|ω −1 |αn + |ki ||ω −1 |αn +λ 
+|kd ||ω −1 |αn −µ .

The right hand side of (49) equals    min{|qn |, |qn |}|ω|αn +λ 1 −   P  −1 αn −αi  σ2 ( n−1 | +|ω −1 |αn +|ω −1 |αn +λ +|ω −1 |αn −µ )
i=1 |ω  ,  min{|qn |,|qn |}        min{|qn |, |qn |}|ω|αn +λ 1 −   Pn−1    σ2 ( i=1 |ω−1 |αn −αi +|ω−1 |αn +|ω−1 |αn +λ )
min{|qn |,|qn |}

31

,

(49)

n(M2 ) = 0, (50)

n(M2 ) = 1.

Also, (50) can be rewritten regarding (28) as follows.  min{|qn |, |qn |}|ω|αn +λ 1 −

 σ2 (|ω −1 |γ1 + |ω −1 |γ2 + · · · + |ω −1 |γn+2−n(M2 ) ) . min{|qn |, |qn |}

(51)

On the other hands, the inequalities (52) hold for ω > 1. 2

1

(ω −1 ) ε2 ≥ (ω −1 )γ1 , (ω −1 ) ε2 ≥ (ω −1 )γ2 , · · · , (ω −1 )

n+2−n(M2 ) ε2

(52)

≥ (ω −1 )γn+2−n(M2 ) .

From (49) , (51) and (52), the inequality (53) is concluded. |∆(jω)| ≥ min{|qn |, |qn |}|ω|αn +λ (1 − 1

2

σ2 (|ω −1 | ε2 + |ω −1 | ε2 + · · · + |ω −1 | min{|qn |, |qn |}

n+2−n(M2 ) ε2

)

(53) ).

The right hand side of (53) equals n+2−n(M2 )

1

min{|qn |, |qn |}|ω|

αn +λ

ε2 σ2 |ω −1 | ε2 1 − |ω −1 | (1 − ( )( 1 min{|qn |, |qn |} 1 − |ω −1 | ε2

)).

(54)

Relations (53) and (54) result in inequality (55) as follows. 1

|∆(jω)| ≥ min{|qn |, |qn |}|ω|αn +λ (1 − (

σ2 |ω −1 | ε2

1

(min{|qn |, |qn |})(1 − |ω −1 | ε2 )

)). (55)

Due to the inequality min{|qn |, |qn |}|ω|αn +λ > 0, satisfying the inequality (56) yields to |∆(jω)| > 0. ω > (1 +

σ2 )ε2 . min{|an |, |an |}

(56)

Suppose 0 < ω < 1: Based on the triangle inequality, it is resulted that |∆(jω)| ≥ k|ki | −

n X i=0

max{|qi |, |qi |}|ω|αi +λ − k(|kp ||ω|λ + |ki ||ω|µ+λ ).

(57)

Also, the inequalities (58) hold for 0 < ω < 1. 1 > ω λ > · · · > ω αn +λ . 32

(58)

From (57) and (58), the inequality (59) is concluded. n X |∆(jω)| ≥ k|ki | − ( max{|qi |, |qi |} + k(|kp | + |ki |))|ω|λ .

(59)

i=0

Based on (59) and satisfying the inequality (60), it is concluded that |∆(jω)| > 0. ω < (σ3 )ε3 .

(60)

Also, (57) results in |∆(jω)| ≥ k|ki | −

n X i=0

max{|qi |, |qi |}|ω|αi +λ − k(|kp ||ω|λ + |ki ||ω|µ+λ ) ≥

   Pn  αi +λ λ µ+λ  + |ω| + |ω| , n(M2 ) = 0,  σ4 i=1 |ω|   k|ki | − Pn  αi +λ  + |ω|λ , n(M2 ) = 1.  σ4 i=1 |ω|

(61)

The right hand side of (61) can be rewritten regarding (30) as follows. k|ki | − σ4 (|ω|β1 + |ω|β2 + · · · + |ω|βn+2−n(M2 ) ).

(62)

On the other hand, the inequalities (63) hold for 0 < ω < 1. 1

2

ω ε4 ≥ ω β1 , ω ε4 ≥ ω β2 , · · · , ω

n+2−n(M2 ) ε4

≥ ω βn+2−n(M2 ) .

(63)

From (61) and (63), the inequality (64) is concluded. 1

2

|∆(jω)| ≥ k|ki | − σ4 (|ω| ε4 + |ω| ε4 + · · · + |ω|

n+2−n(M2 ) ε4

(64)

).

The right hand side of (64) is equal to 1

k|ki | − σ4 |ω| ε4

(1 − |ω|

n+2−n(M2 ) ε4

1 − |ω|

1 ε4

)

(65)

.

From (64) and (65), the inequality (66) is resulted. |∆(jω)| ≥ k|ki | − σ4 |ω|

1 ε4

(1 − |ω|

n+2−n(M2 ) ε4

)

1

1 − |ω| ε4

If the inequality (67) holds, then |∆(jω)| > 0. ω<(

1

1+



33

σ4 k|ki |

)ε4 .

1

≥ k|ki | −

σ4 |ω| ε4

1

1 − |ω| ε4

.

(66)

(67)

305

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