A numerical investigation for robust stability of fractional-order uncertain systems

ISA Transactions 53 (2014) 189–198

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

A numerical investigation for robust stability of fractional-order uncertain systems Bilal Senol a,n, Abdullah Ates a, B. Baykant Alagoz b, Celaleddin Yeroglu a a b

Computer Engineering Department, Inonu University, 44280 Malatya, Turkey Electrical and Electronics Engineering Department, Inonu University, 44280 Malatya, Turkey

art ic l e i nf o

a b s t r a c t

Article history: Received 3 April 2013 Received in revised form 26 June 2013 Accepted 3 September 2013 Available online 6 October 2013 This paper was recommended for publication by Prof. Y. Chen

This study presents numerical methods for robust stability analysis of closed loop control systems with parameter uncertainty. Methods are based on scan sampling of interval characteristic polynomials from the hypercube of parameter space. Exposed-edge polynomial sampling is used to reduce the computational complexity of robust stability analysis. Computer experiments are used for demonstration of the proposed robust stability test procedures. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Robust stability Fractional-order systems Uncertain coefficients Uncertain orders

1. Introduction In control theory, increasing trend for fractional-order control system implementation promises superior control performance comparing to integer-order ones. Fractional order calculus (FOC) has been widely studied since discovery of Newton and Leibniz. They proposed an alternative formulation instead of integer-order calculus in the 17th century [1,2]. FOC can be utilized in wide application areas [3] such as physics [4], electrical circuit theory and fractances [5], mechatronic systems [6], signal processing [7], chemical mixing [8], chaos theory [9] etc. Recently FOC has gained a wider attention in control system applications. Robust control of real systems is one of the substantial problems in control practice due to the fact that parameter uncertainty or model perturbation may cause unstable responses of control systems in real applications, even though the systems were theoretically designed as stable systems in computer simulations. In order to avoid the risk of unstable responses of control systems in applications, stability of systems should be guaranteed within possible ranges of parametric uncertainties and therefore the system should be designed to remain stable against parameter perturbations of uncertain system. Robust control has been studied in many works for integer [10] and fractional-order systems (FOS) [11,12]. Several works revealed n

Corresponding author. Tel.: þ 90 5052984717. E-mail addresses: [email protected], [email protected] (B. Senol), [email protected] (A. Ates), [email protected] (B. Baykant Alagoz), [email protected] (C. Yeroglu).

the advantages of FOS [13–15]. Recently, there is a growing effort for stability analysis of linear time invariant (LTI) systems in the literature: Robust stability test method for FOS was presented for fractional-order linear systems with interval uncertainties [16,17]. Petras et al. presented an experimental approach for interval uncertainties for FO-LTI systems [18]. An interval boundary box method for stability testing of the FO-LTI systems with interval uncertainties was presented in [19]. Stabilization analysis of FO-LTI systems based on Lyapunov inequality and linear matrix inequality (LMI) was proposed in [20–22]. Robust stability check based on four Kharitonov's polynomials was presented for commensurate order LTI fractional-order system in [16,17,23]. A sufficient and necessary condition for the robust asymptotical stability of fractional-order interval systems with the order satisfying 0 o α o 1 was demonstrated in [24]. Another study presented a solution for robust stability of time-varying systems by using linear fractional representation [25]. Lim et al. proposed a method for asymptotical stabilization of fractional order linear systems subject to input saturation provided by using Gronwall– Bellman lemma [26]. In addition, bounded-input bounded-output (BIBO) stability of a large class of neutral type fractional delay systems is investigated in [27]. Stability analysis mostly depends on calculating eigenvalues of state equations of LTI systems in the literature [28,29]. Stability check via root locus is a useful technique for closed loop control system design practice. Hence, extending the root locus based stability check procedures for the fractional order systems involving interval uncertainty of polynomial orders and coefficients may yield new horizons in fractional order control study.

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.09.004

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This study is devoted for numerical investigation of robust stability of fractional-order closed-loop control systems with parameter uncertainty. The study also considers both coefficient and order uncertainty cases of FOS. The study benefits from s ¼ vm mapping in order to turn fractional-order characteristic polynomials of control systems into expanded integer-order polynomials [12]. Two numerical methods based on Radwan et al. procedure [12] are discussed for the stability analysis of interval polynomial families. The first method uses scan sampling of test polynomials from intervals of uncertain parameters, and consequently, computational complexity of the method exponentially grows with respect to the spatial sampling density of parameter space. The second method based on Edge Theorem samples edge polynomials from parameter space, and hence considers the boundary line of root plane and therefore it considerably decreases the computational complexity. The Edge Theorem has been successfully utilized in the robust stability analysis of integer-order systems. This theorem establishes the fundamental property that the root space boundary of a polytopic family of polynomials is contained in the root locus evaluated along the exposed edges [10]. In a previous work, Kang et al. briefly showed a diligent use of Edge Theorem for robust stability analysis of commensurate fractional-order interval polynomials [27]. Thanks to Edge Theorem that effectively and constructively reduce the problem of determining the root space under multiple parameter uncertainty to a set of one-parameter root locus problem [10]. Motivation of this study comes from root region investigation of sampled interval polynomials according to Radwan procedure for the computer aided robust stability analysis of fractional-order control systems with interval uncertainty. This study contributes to numerical investigation of robust stability of fractional-order closed-loop control systems.

2. Methodology

[19,20]. This mapping simply transforms fractional-order polynomials to expanded degree integer-order polynomials, and thus it reduces a fractional-order system stability analysis problem to stability analysis of an integer-order system in the Riemann sheets. This allows the assessment of the stability of fractional-order polynomials according to root locus analysis in their root spaces. Assuming α ¼ 1=m; m number of Riemann sheets are defined in complex root plain (s ¼ jsjejφ ) as ð2k þ 1Þπ o ϕ oð2k þ 3Þπ , where k ¼ 1; 0; :::; m  2 [30]. Applying transformation s ¼ vm for a positive integer m 4 1, expanded integer-order form of the characteristic polynomial in Eq. (1) can be written as, n

Δm ðvÞ ¼ ∑ ai vβi ;

ð3Þ

i¼1

where, Δ ðvÞ represents m  th order expanded characteristic polynomial of the fractional-order control system and βi ¼ mαi are the expanded integer orders. In order to expand a fractionalorder characteristic polynomial to a minimal degree integer-order characteristic polynomial, m expansion order can be chosen as the least common multiplier (LCM) of d1 ; d2 ; d3 ; :::; dn that transforms the fractional orders into minimal integer orders [31]. In this case, the degree of the polynomial can be given as ξ ¼ max fβ1 ; β2 ; :::; βn g [17] and this integer-order polynomial with degree of ξ is referred to expanded integer-order representation of the fractional-order system. An expanded integer order polynomial of a fractional-order control system has ξ roots on the Riemann surface [20]. Let denote these roots as v1 ; v2 ; v3 ; :::; vξ . Roots residing in the phase range of φ A ð  π =m; π =mÞ, which coincides to the first Riemann sheet, are physically meaningful and these roots of the expanded integer order system are used for the stability analysis of control systems [32]. Root location of expanded integer-order characteristic polynom mial (Δ ðvÞ) in the root space was effectively used to decide stability of fractional order LTI systems [12,16,32,33]. The systematic approach in the robust stability analysis is provided by Radwan et al. [12]. Radwan procedure can be summarized as, m

2.1. Theoretical background Fig. 1 shows general representation of a closed loop control structure. The characteristic polynomial of this system can be given as,

ΔðsÞ ¼ 1 þ CðsÞGðsÞHðsÞ

ð1Þ

Let us define a fractional-order characteristic polynomial of this control system in the form of n

ΔðsÞ ¼ ∑ ai sαi ;

ð2Þ

i¼1

where, the parameters ai A R are the polynomial coefficients for i ¼ 1; 2; 3; :::; n and αi ¼ ci =di are the fractional orders of polynomial [3]. Here, ci and di are the positive integer numbers. The parameter αi is assumed to satisfy the condition of α1 ¼ 0 for a constant term of polynomials and αi þ 1 4 αi . One of the common ways to decide the stability of a FO-LTI system is to apply s ¼ vm mapping to the system transfer function

n Calculate absolute values of root phases (jϕv j ¼ jargðvr Þj),   r A 1; ξ ) of expanded integer order system polynomial in the first Riemann sheet (ϕ A ð  π =m; π =mÞ). n If the values of root phases are in the range of π =2m o jϕv j o π =m in the first Riemann sheet, this system is stable. n If the values of root phases are equal to π =2m, the system oscillates. n Otherwise, the system is unstable.

As a special case of s ¼ vm mapping for m ¼ 1, the principle Riemann surface refers to the open left half of the complex plane. This half plane is stability region of integer order characteristic polynomials. The stability condition for integer-order systems can be written as π =2 ojϕv j o π for m ¼ 1. Fig. 2 illustrates stability regions according to absolute root phase values (jϕv j ¼ jargðvr Þj) in the root spaces for m ¼ 1 and m 4 1 cases. 2.2. Robust stability analysis for fractional-order systems with uncertain coefficients For the robust stability analysis of the closed loop control system containing interval uncertainty in the predefined finite ranges, it is convenient to define uncertain characteristic polynomials in the form of an interval polynomial as follows, n

Fig. 1. Block diagram of a closed loop control system.

Δu ðsÞ ¼ ∑ ½a i ai sαi ; i¼1

ð4Þ

B. Senol et al. / ISA Transactions 53 (2014) 189–198

191

Fig. 2. (a) Stability region of integer order characteristic polynomials. (b) Stability region of fractional order characteristic polynomials in the first Riemann surface for s ¼ vm mapping with m 41.

Fig. 3. An illustration of corner (g 1 ; g2 ; g3 ; :::; g 8 ) and exposed edges (e1 ; e2 ; e3 ; :::; e12 ) of hypercube A in three dimensional parameter space in (a) and corresponding the boundary points of the root region in the root space in (b).

and correspondingly, one can write the expanded integer-order polynomial of the fractional-order interval polynomial family as, n

βi Δm u ðvÞ ¼ ∑ ½a i ai v ;

ð5Þ

i¼1

where, a i and ai are the lower and upper bounds of the uncertain coefficient ai . The finite limits of coefficient perturbation defines a hypercube A in the polynomial coefficient space as [10] A ¼ fa : a i r ai rai ; i ¼ 1; 2; :::; ng;

ð6Þ

where a ¼ ½a1 a2 a3 ::: an  denotes coefficient vectors in the polynomial coefficient space [10]. All polynomials represented in A forms a family of characteristic polynomials denoted by Ω A Rn . The set Ω is a closed bounded set and therefore compact [10]. The complex roots of the interval polynomial family Ω forms a set of root points defined as, RðΩÞ ¼ fvr : Δu ða; vr Þ ¼ 0; m

8 a A A; r A 1; 2; 3; :::; ξg:

ð7Þ

Authors refer RðΩÞ set as the root region in complex root space. The corner characteristic polynomials of hypercube A are formed by all possible upper and lower bounds combinations of the polynomial coefficient (ai A fa i ; ai g). Number of corner polynomials is 2n . Corner characteristic polynomials can be expressed as,

Δgk ¼ Δm ðgk ; vÞ;

ð8Þ

j ¼ 1; 2; 3; :::; n and j ak). Let define edge polynomial families in A as, Sðak ; hÞ ¼ ha k þ ð1  hÞak for h A ð0; 1Þ

ð10Þ

The exposed-edge characteristic polynomials of Ω can be expressed as,

Δel ¼ Δm ðel ; vÞ; where n2

n1

ð11Þ

edge vectors (ek ) of hypercube A can be composed as

ek ¼ fa 1 ; a1 g  fa 2 ; a2 g  ::::  Sðak ; hÞ  ::::  fa n ; an g h A ð0; 1Þ and l ¼ 1; 2; 3; :::; n2n  1

ð12Þ

Fig. 3 shows an illustration of exposed edges of hypercube A in three dimensional coefficient space and corresponding the boundary of the root region RðΩÞ in the root space. Corner polynomial set for three dimensional coefficient space can be written as,

Δg1 ¼ Δm ðg 1 ; vÞ; Δg2 ¼ Δm ðg 2 ; vÞ; Δg3 ¼ Δm ðg 3 ; vÞ; Δg4 ¼ Δm ðg 4 ; vÞ; Δg5 ¼ Δm ðg 5 ; vÞ; Δg6 ¼ Δm ðg 6 ; vÞ; Δg7 ¼ Δm ðg 7 ; vÞ; Δg8 ¼ Δm ðg 8 ; vÞ ð13Þ where, the corner polynomial vectors are, g 1 ¼ ½a 1 a 2 a 3 ; g 2 ¼ ½a 1 a 2 a3 ; g 3 ¼ ½a 1 a2 a 3 ; g 4 ¼ ½a 1 a2 a3 ; g 5 ¼ ½a1 a 2 a 3 ; g 6 ¼ ½a1 a 2 a3 ; g 7 ¼ ½a1 a2 a 3 ; g 8 ¼ ½a1 a2 a3 

ð14Þ

where corner vectors g k of hypercube A can be composed as

Edge polynomial set for this parameter space can be written as,

g k ¼ fa 1 ; a1 g  fa 2 ; a2 g  fa 3 ; a3 g

Δe1 ¼ Δm ðe1 ; vÞ; Δe2 ¼ Δm ðe2 ; vÞ; Δe3 ¼ Δm ðe3 ; vÞ; Δe4 ¼ Δm ðe4 ; vÞ; Δe5 ¼ Δm ðe5 ; vÞ; Δe6 ¼ Δm ðe6 ; vÞ; Δe7 ¼ Δm ðe7 ; vÞ; Δe8 ¼ Δm ðe8 ; vÞ; Δe9 ¼ Δm ðe9 ; vÞ; Δe10 ¼ Δm ðe10 ; vÞ; Δe11 ¼ Δm ðe11 ; vÞ; Δe12 ¼ Δm ðe12 ; vÞ

 ::::  fa n ; an g

n

k ¼ 1; 2; 3; :::; 2 ;

ð9Þ

where () refers to Cartesian product operator. Exposed edges are all line segments connecting corners of the hypercube A and they can be obtained by varying only one polynomial coefficient in its uncertainty interval (ak ¼ ha k þð1 hÞak for h A ð0; 1Þ) while the rest is chosen to be either upper or lower bounds (aj A fa j ; aj g

ð15Þ where edge polynomial vectors, e1 ¼ ½a 1 a 2 Sða3 ; hÞ; e2 ¼ ½a 1 Sða2 ; hÞ a 3 ; e3 ¼ ½Sða1 ; hÞ a 2 a 3 ;

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e4 ¼ ½a 1 Sða2 ; hÞ a3 ; e5 ¼ ½Sða1 ; hÞ a 2 a3 ; e6 ¼ ½a 1 a2 Sða3 ; hÞ; e7 ¼ ½Sða1 ; hÞ a2 a 3 ; e8 ¼ ½Sða1 ; hÞ a2 a3 ; e9 ¼ ½a1 a 2 Sða3 ; hÞ; e10 ¼ ½a1 Sða2 ; hÞ a 3 ; e11 ¼ ½a1 Sða2 ; hÞ2 a3 ; e12 ¼ ½a1 a2 Sða3 ; hÞ ð16Þ Eq. (16) can be used to investigate the region based stability of the uncertain system. Region based robust stability analysis is an extension of Radwan procedure for interval polynomials. Theorem 1. Let the complex roots of the interval family of polynomials Ω be RðΩÞ, then the control system defined by interval family of characteristic polynomials are robustly stable, if and only if π =2m o jargfRðΩÞgj o π =m condition is satisfied. Proof. If root phases of all polynomials in the interval polynomial family in a hybercube A satisfy π =2m ojϕv j o π =m in the first Riemann sheet, all control systems represented by these characteristic polynomials are stable according to Radwan procedure. Since the root region RðΩÞ contains all roots of characteristic polynomial family in A, the condition of π =2m o jargfRðΩÞgj o π =m can be satisfied, if only if all characteristic polynomial are stable. □ Region-based robust stability analysis was widely implemented in different aspects in the literature. For instance, a random sampling of interval polynomial family is used to detect a member of the interval polynomial family [17,18]. Smallest rectangular box containing all random sampled polynomial family is discussed to improve the reliability of this approach [17]. However, convex or concave forms of root region boundaries may mislead robust stability checking process. Random sampling of polynomials from Ω may not truly expose the boundary of RðΩÞ. For this reason, scan sampling containing exposed edges and corners of hypercube A can give more reliable results so that it is the most likely to contain sampled polynomials from boundaries of root region RðΩÞ. Procedure of region based robust stability test with scan sampling can be summarized as follows: n Apply s ¼ vm transformation and obtain expanded integer order polynomials of the fractional-order interval polynomial family according to Eq. (5). n Scan sample the hypercube A spatially by unit sampling lengths; Δa1 ; Δa2 ; Δa3 ; :::; Δan . n Computes roots in the first Riemann sheet for each sampled characteristic polynomial obtained from sampled hypercube A. n Construct root region RðΩÞ and if π =2m o jargfRðΩÞgj o π =m is satisfied for all members of RðΩÞ, the control system with interval polynomial is robust stable.

This method has disadvantage of high computational complexity depending on sampling density of the hypercube A. In the following section, computational complexity of robust stability analysis is considerably reduced for the sake of Edge Theorem [12]. The s ¼ vm mapping of fractional-order characteristic polynomials yields integer-order polynomials. This allows to employ celebrated Kharitonov's polynomials [16,17,23,24] and Edge Theorem [34] in the robust stability analysis of fractional-order control systems. Edge Theorem [10]: Let Ω A Rn be a polytope of polynomials which satisfies assumption that the sign of ai is constant over Ω, either always positive or always negative. Then the boundary of RðΩÞ is contained in the root space of the exposed edges of Ω. The Edge Theorem states us that under assumption of a Ω polynomial family with parameters either all ai o 0 or all ai 4 0, the boundary of root region RðΩÞ is contained in the root space of exposed edges of Ω [10]. In other word, the boundary of RðΩÞ is formed by roots coming from exposed-edge polynomials of

interval characteristic polynomial family. This theorem considerably reduces the number of polynomials to be tested for robust stability analysis of interval polynomials family. Procedure of exposed-edge based robust stability test can be summarized as follows: n Apply s ¼ vm transformation and obtain expanded integer order polynomial of the fractional-order interval polynomial family according to Eq. (5). n Sample exposed-edge and corner polynomials from hypercube A by unit sampling of parameters as Δa1 ; Δa2 ; Δa3 ; :::; Δan . n Computes roots in the first Riemann sheet for each sampled characteristic polynomials from the exposed-edges and corners. Compose set H  RðΩÞ from this roots. the outline n If π =2m o argfRðΩÞg o π =m is satisfied for all H, the control system with interval polynomials is robustly stable.

2.3. Robust stability analysis for fractional-order systems with uncertain orders Robust stability analysis procedures are adopted for the case of uncertain orders in this section. For the robust stability analysis of the closed loop control system containing uncertain orders in the predefined finite ranges, it is convenient to define characteristic polynomials in the form of, n

Δuo ðsÞ ¼ ∑ ai s½α i ;αi  ;

ð17Þ

i¼1

where, α i and αi are the lower and upper bounds of the fractional order αi and correspondingly, one can write the expanded integerorder polynomial of the uncertain fractional-order polynomial family as, n

½β ;β i  i Δm ; uo ðvÞ ¼ ∑ ai v

ð18Þ

i¼1

where, β and β i are the lower and upper bounds of the uncertain i parameter βi . The finite limits of order perturbation defines a hypercube Ao in the parameter space as, Ao ¼ fβ : β r β i r βi ; i ¼ 1; 2; ; ; ng; i

ð19Þ

where β ¼ ½β 1 β2 β 3 ::: β n  denotes parameter vectors in the polynomial parameter space. Accordingly, the complex roots of the polynomial family Ωo forms a set of root points, RðΩo Þ ¼ fvr : Δu ðβ ; vr Þ ¼ 0; 8 β A Ao ; r A 1; 2; 3; ; ; ξ g: m

ð20Þ

Corner characteristic polynomials can be expressed as

Δgk ¼ Δm ðgok ; vÞ;

ð21Þ

g ok ¼ fβ ; β 1 g  fβ ; β 2 g  fβ β 3 g 1

2

3

 ::::  fβ ; β n g k ¼ 1; 2; 3::2n n

ð22Þ

If a line in Ao was defined as Sðβk ; hÞ ¼ hβ þ ð1  hÞβk for k hA ð0; 1Þ. The exposed-edge characteristic polynomials can be expressed as,

Δel ¼ Δm ðeol ; vÞ;

ð23Þ

eok ¼ fβ ; β 1 g  fβ ; β 2 g  ::::  Sðβk ; hÞ  :::: 1

2

fβ ; βn g h A ð0; 1Þ and l ¼ 1; 2; 3; :::; n2n  1 n

ð24Þ

Corner polynomial set for three dimensional parameter space can be written as,

Δg1 ¼ Δm ðgo1 ; vÞ; Δg2 ¼ Δm ðgo2 ; vÞ; Δg3 ¼ Δm ðg o3 ; vÞ; Δg4 ¼ Δm ðg o4 ; vÞ; Δg5 ¼ Δm ðgo5 ; vÞ; Δg6 ¼ Δm ðgo6 ; vÞ; Δg7 ¼ Δm ðg o7 ; vÞ; Δg8 ¼ Δm ðg o8 ; vÞ

ð25Þ

B. Senol et al. / ISA Transactions 53 (2014) 189–198

where, the corner polynomial vectors are, g o1 ¼ ½β β β ; g o2 ¼ ½β β β3 ; g o3 ¼ ½β β 2 β ; g o4 ¼ ½β β 2 β3 ; 1

2

3

1

2

1

3

3

2

3.1. Example 1: integer-order control system with uncertain coefficients

1

g o5 ¼ ½β 1 β β ; g o6 ¼ ½β 1 β β3 ; g o7 ¼ ½β 1 β2 β ; g o8 ¼ ½β 1 β2 β 3  2

193

3

ð26Þ Edge polynomial set for this parameter space can be written as,

Δe1 ¼ Δm ðeo1 ; vÞ; Δe2 ¼ Δm ðeo2 ; vÞ; Δe3 ¼ Δm ðeo3 ; vÞ; Δe4 ¼ Δm ðeo4 ; vÞ;

The proposed test procedure provides a general methodology, which covers integer-order characteristic polynomials stability analysis as a special case for m ¼ 1. This example validates relevance and generality of the procedures. Consider the following integer order plant transfer function with uncertain coefficients provided from [35]. 0:5 a4 s3 þ a3 s2 þa2 s þ a1

Δe5 ¼ Δm ðeo5 ; vÞ; Δe6 ¼ Δm ðeo6 ; vÞ; Δe7 ¼ Δm ðeo7 ; vÞ; Δe8 ¼ Δm ðeo8 ; vÞ;

G1 ðsÞ ¼

Δe9 ¼ Δm ðeo9 ; vÞ; Δe10 ¼ Δm ðeo10 ; vÞ; Δe11 ¼ Δm ðeo11 ; vÞ;

where the uncertain parameters are reorganized as, a4 A ½0:5 ; 1:5; a3 A ½3 ; 4; a2 A ½1 ; 2 and a1 A ½0:45 ; 0:5. Applying s ¼ vm mapping for m ¼ 1, one can express the characteristic polynomial family of the closed loop system as,

Δe12 ¼ Δ ðeo12 ; vÞ m

ð27Þ

Where, edge polynomial vectors are,

ð29Þ

eo1 ¼ ½β β Sðβ3 ; hÞ; eo2 ¼ ½β Sðβ2 ; hÞ β ; eo3 ¼ ½Sðβ1 ; hÞ β β ;

Δ1u ðvÞ ¼ ½0:5; 1:5v3 þ ½3; 4v2 þ ½1; 2v1 þ ½0:45; 0:5 þ 0:5

eo4 ¼ ½β Sðβ2 ; hÞ β 3 ; eo5 ¼ ½Sðβ1 ; hÞ β β 3 ; eo6 ¼ ½ β β2 Sðβ 3 ; hÞ;

The robust stability test procedure is applied to the polynomial family in Eq. (30) by unit sampling lengths of Δa1 ¼ 0:01, Δa2 ¼ 0:01, Δa3 ¼ 0:01 and Δa4 ¼ 0:0005. Fig. 4(a) shows the roots region RðΩÞ of this system. Phases of all sampled roots complies with the stability condition π =2 o jargfRðΩÞgj o π for m ¼ 1, and therefore the system is robustly stable for the given interval ranges. Fig. 4(b) presents the roots of corner and edge polynomials in the root spaces and it confirms robust stability of the system. Fig. 4(b) shows the sampled roots of corner and edge polynomials in the root spaces and it confirms robust stability of the system by considering less polynomials. Because exposed edge and corner polynomials contains the boundary roots of root region and the number of sampled polynomials from parameter hypercube considerably decreases. Fig. 4(c) demonstrates the step-responses of 16 corner polynomials shown in Fig. 4(b). Step responses of corner polynomials clearly indicate the robust stability of the characteristic equation of the interval system given in Eq. (29).

1

2

1

1

3

2

2

3

1

eo7 ¼ ½Sðβ 1 ; hÞ β 2 β ; eo8 ¼ ½Sðβ 1 ; hÞ β2 β 3 ; eo9 ¼ ½ β 1 β Sðβ 3 ; hÞ; 3

2

eo10 ¼ ½β1 Sðβ 2 ; hÞ β ; eo11 ¼ ½β1 Sðβ 2 ; hÞ β 3 ; eo12 ¼ ½ β 1 β2 Sðβ 3 ; hÞ 3

ð28Þ Region based robust stability test procedure can be written for the case of uncertain orders as n Apply s ¼ vm transformation and obtain expanded integer order polynomials of the fractional-order polynomial family according to Eq. (18). n Scan sample the hypercube Ao spatially by unit sampling lengths; Δβ 1 ,Δβ2 ,Δβ 3 ,,,Δβn . n Compute roots in the first Riemann sheet for each sampled characteristic polynomial obtained from sampled hypercube Ao. n Construct root region RðΩÞ and if π =2m o jargfRðΩÞgj o π =m is satisfied for all members ofRðΩÞ, the fractional order control system with uncertain orders is robust stable.

Edge based robust stability test procedure can be written for the case of uncertain orders as:

3.2. Example 2: fractional-order control systems with uncertain coefficients Consider the fractional order uncertain plant transfer function provided from [36], G2 ðsÞ ¼

n Apply s ¼ v transformation and obtain expanded integer order polynomial of the fractional-order polynomial family according to Eq. (18). n Sample exposed-edge and corner polynomials from hypercube Ao by unit sampling of parameters as Δβ1 ; Δβ 2 ; Δβ2 ; :::; Δβ n . n Computes roots in the first Riemann sheet for each sampled characteristic polynomials from the exposed-edges and corners. Compose set H  RðΩÞ from this roots. the outline n If π =2m o argfRðΩÞg o π =m is satisfied for all H, the fractional order control system with order uncertainty are robustly stable. m

3. Computer experiments This section includes four illustrative examples for the application of the proposed procedures. First example illustrates robust stability test of an integer order system with uncertain coefficients to present the reliability of the proposed method. Second and third examples illustrate robust stability test of fractional order systems with uncertain coefficients. Finally, the fourth example examines the stability of a fractional order system containing uncertain orders.

ð30Þ

1 1 ¼ P 2 ðsÞ a4 s1:8 þ a3 s1:3 þ a2 s0:5 þ a1

ð31Þ

where the uncertain parameters are arranged as, a4 A ½1 ; 1:2; a3 A ½0:7 ; 0:9; a2 A ½1:8 ; 2 and a1 A ½0:9 ; 1. Applying s ¼ vm mapping for m ¼ 10, one can express the characteristic polynomial family of this closed loop control system as, 18 13 5 Δ10 u ðvÞ ¼ ½1; 1:2v þ ½0:7; 0:9v þ ½1:8; 2v þ ½0:9; 1 þ 1

ð32Þ

In this example, region-sampled stability test procedure was applied to the polynomial family by unit sampling lengths of Δa4 ¼ 0:002, Δa3 ¼ 0:002, Δa2 ¼ 0:002 and Δa1 ¼ 0:001. Corner polynomials of the fractional order interval system with four coefficients can be written as

Δg1 ¼ Δm ð½a 1 a 2 a 3 a 4 ; vÞ; Δg2 ¼ Δm ð½a 1 a 2 a 3 a4 ; vÞ; Δg3 ¼ Δm ð½a 1 a 2 a3 a 4 ; vÞ Δg4 ¼ Δm ð½a 1 a 2 a3 a4 ; vÞ; Δg5 ¼ Δm ð½a 1 a2 a 3 a 4 ; vÞ; Δg6 ¼ Δm ð½a 1 a2 a 3 a4 ; vÞ Δg7 ¼ Δm ð½a 1 a2 a3 a 4 ; vÞ; Δg8 ¼ Δm ð½a 1 a2 a3 a4 ; vÞ; Δg9 ¼ Δm ð½a1 a 2 a 3 a 4 ; vÞ Δg10 ¼ Δm ð½a1 a 2 a 3 a4 ; vÞ; Δg11 ¼ Δm ð½a1 a 2 a3 a 4 ; vÞ; Δg12 ¼ Δm ð½a1 a 2 a3 a4 ; vÞ Δg13 ¼ Δm ð½a1 a2 a 3 a 4 ; vÞ; Δg14 ¼ Δm ð½a1 a2 a 3 a4 ; vÞ; Δg15 ¼ Δm ð½a1 a2 a3 a 4 ; vÞ Δg16 ¼ Δm ð½a1 a2 a3 a4 ; vÞ:

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Fig. 4. Stability test results for the uncertain integer order control system given by Eq. (29). (a) Root space obtained for region-sampled robust stability analysis, (b) root space obtained for edge-sampled robust stability analysis, and (c) step responses of corner polynomials.

Similarly, the edge polynomials can be obtained for h A ½0; 1 as follows,

Δe1 ¼ Δm ð½a 1 a 2 a 3 Sða4 ; hÞ; vÞ; Δe2 ¼ Δm ð½a 1 a 2 Sða3 ; hÞ a 4 ; vÞ; Δe3 ¼ Δm ð½a 1 Sða2 ; hÞ a 3 a 4 ; vÞ; Δe4 ¼ Δm ð½Sða1 ; hÞ a 2 a 3 a 4 ; vÞ; Δe5 ¼ Δm ð½a 1 a 2 Sða3 ; hÞ a4 ; vÞ; Δe6 ¼ Δm ð½a 1 Sða2 ; hÞ a 3 a4 ; vÞ Δe7 ¼ Δm ð½Sða1 ; hÞ a 2 a 3 a4 ; vÞ; Δe8 ¼ Δm ð½a 1 a 2 a3 Sða4 ; hÞ; vÞ; Δe9 ¼ Δm ð½a 1 Sða2 ; hÞ a3 a 4 ; vÞ; Δe10 ¼ Δm ð½Sða1 ; hÞ a 2 a3 a 4 ; vÞ; Δe11 ¼ Δm ð½a 1 Sða2 ; hÞ a3 a4 ; vÞ; Δe12 ¼ Δm ð½Sða1 ; hÞ a 2 a3 a4 ; vÞ Δe13 ¼ Δm ð½a 1 a2 a 3 Sða4 ; hÞ; vÞ; Δe14 ¼ Δm ð½a 1 a2 Sða3 ; hÞ a 4 ; vÞ; Δe15 ¼ Δm ð½Sða1 ; hÞ a2 a 3 a 4 ; vÞ; Δe16 ¼ Δm ð½a 1 a2 Sða3 ; hÞ a4 ; vÞ; Δe17 ¼ Δm ð½Sða1 ; hÞ a2 a 3 a4 ; vÞ; Δe18 ¼ Δm ð½a 1 a2 a3 Sða4 ; hÞ; vÞ Δe19 ¼ Δm ð½Sða1 ; hÞ a2 a3 a 4 ; vÞ; Δe20 ¼ Δm ð½Sða1 ; hÞ a2 a3 a4 ; vÞ; Δe21 ¼ Δm ð½a1 a 2 a 3 Sða4 ; hÞ; vÞ; Δe22 ¼ Δm ð½a1 a 2 Sða3 ; hÞ a 4 ; vÞ; Δe23 ¼ Δm ð½a1 Sða2 ; hÞ a 3 a 4 ; vÞ; Δe24 ¼ Δm ð½a1 a 2 Sða3 ; hÞ a4 ; vÞ Δe25 ¼ Δm ð½a1 Sða2 ; hÞ a 3 a4 ; vÞ; Δe26 ¼ Δm ð½a1 a 2 a3 Sða4 ; hÞ; vÞ; Δe27 ¼ Δm ð½a1 Sða2 ; hÞ a3 a 4 ; vÞ; Δe28 ¼ Δm ð½a1 Sða2 ; hÞ a3 a4 ; vÞ; Δe29 ¼ Δm ð½a1 a2 a 3 Sða4 ; hÞ; vÞ; Δe30 ¼ Δm ð½a1 a2 Sða3 ; hÞ a 4 ; vÞ Δe31 ¼ Δm ð½a1 a2 Sða3 ; hÞ a4 ; vÞ; Δe32 ¼ Δm ð½a1 a2 a3 Sða4 ; hÞ; vÞ: Fig. 5(a) shows the roots region RðΩÞ of this uncertain system. The set RðΩÞ complies with the stability condition of π =20 o jargfRðΩÞgj o π =10, and therefore the system is robustly stable for

the given uncertainty ranges of the coefficients. Fig. 5(b) shows the roots of corner and edge polynomials in the root spaces and indicates the robust stability of the polynomial family given by Eq. (32). Fig. 5(c) demonstrates the step-responses of corner polynomials and confirms robust stability of the system. Similar to Example 1, 32 edge polynomials contain all roots of the uncertain system in Eq. (31) for all parameters perturbations. Four of corner polynomials {Δg4 , Δg13 , Δg10 , Δg7 } are Kharitonov polynomials. The Kharitonov polynomials validate the robust stability of this system [16]. 3.3. Example 3: fractional-order control systems with uncertain coefficients including unstable poles Consider the fractional order uncertain plant transfer function provided from [34], G3 ðsÞ ¼

1 1 ¼ P 3 ðsÞ s4 þ a4 s2:9 þ a3 s2:1 þ a2 s1:1 þa1 s0:1

ð33Þ

where the uncertain coefficients are given as, a4 A ½2:56 ; 6:56, a3 A ½2:871 ; 12:614, a2 A ½3:164 ; 15:868 and a1 A ½1:853 ; 23:677. Applying s ¼ vm mapping for m ¼ 10, one can express the characteristic polynomial family of the closed loop control system as 40 29 21 Δ10 u ðvÞ ¼ v þ ½2:56 ; 6:56v þ ½2:871 ; 12:614v

þ ½3:164 ; 15:868v11 þ ½1:853 ; 23:677 þ 1

ð34Þ

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195

Fig. 5. Stability test results for the uncertain fractional order control system given by Eq. (31). (a) Root space obtained for region- sampled robust stability analysis, (b) root space obtained for edge-sampled robust stability analysis, and (c) step responses of corner polynomials.

Unit sampling lengths of the uncertain parameters were set to Δa4 ¼ 0:04, Δa3 ¼ 0:097, Δa2 ¼ 0:127 and Δa1 ¼ 0:218. Fig. 6(a) shows the sampled roots region RðΩÞ obtained for the system. Since all root phases does not comply with the stability condition of π =20 o argfRðΩÞg o π =10, the system is not robustly stable in this uncertainty ranges. For instance, root phase of corner polym nomial Δg1 ¼ Δ ð½a 1 a 2 a 3 a4 ; vÞ is out of phase boundaries. Fig. 6(b) shows roots of corner and edge polynomials in the root spaces. As expected, some corner and edge polynomials do not fall into the region π =20 o jargfRðΩÞgj o π =10, therefore, one can deduce that this control system is not robustly stable for the given intervals by investigating only corner and edge polynomials from the hypercube. Fig. 6(c)–(e) show step responses for three corner polynomials sampled from root region: The one with label 1 exhibits damping oscillation because it is close to line with the angle of π =20, the second with label 2 behaves stable due to accommodating in the stability region of π =20o jargfRðΩÞgj o π =10 and the third with label 3 behaves unstable because of residing at the out of the stability region. Since all roots of the system do not lie in the stability region, one can deduce that the system is not robust stable in the given uncertainty intervals. Fig. 6(b) also illustrates root location of four Kharitonov polynomials by labels K1, K2, K3, K4. Indeed, four Kharitonov polynomials coincide with 4 corner polynomial marked with circles. The robust stability check based on Kharitonov polynomials [16] demonstrates that the interval system

is not robust stable. Kharitonov polynomials yield very consistent results for stability check, however we do not yet guaranties that one of convex boundaries of root region may spread over instability region while four Kharitonov polynomials accommodate in the stability region. 3.4. Example 4: fractional-order control system with uncertain orders This section illustrates an example for a closed loop fractional order system with orders uncertainty. Example 4. Consider the following fractional order plant transfer function, 1 G1 ðsÞ ¼ a ð35Þ s 3 þ sa2 þ sa1 where the uncertain orders are taken as, a3 A ½2:1 ; 2:6; a2 A ½1:2 ; 1:7; a1 A ½0:4 ; 0:9. Applying s ¼ vm mapping for m ¼ 100, one can express the characteristic polynomial family of the closed loop system as, ½210 ; 260 Δ100 þ v½120 ; 170 þ v½40 ; 90 þ 1 u ðvÞ ¼ v

ð36Þ

The robust stability test procedure was applied to the polynomial family given by Eq. (36) by using unit sampling lengths of Δa1 ¼

196

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Fig. 6. Stability test results for the uncertain fractional order control system in Eq. (33). (a) Root space obtained for region- sampled robust stability analysis, (b) root space obtained for edge-sampled robust stability analysis, (c) step responses of a sample oscillative polynomial labeled by 1 in 6(b), (d) step response of a sample stable polynomial labeled by 2 in 6(b), and (e) step response of a sample unstable polynomial labeled by 3 in 6(b).

Δa2 ¼ Δa3 ¼ 1. Fig. 7(a) shows the sampled roots region RðΩÞ of this system. Phases of all sampled roots complies with the stability condition π =200 o argfRðΩÞg o π =100 for m ¼ 100, and therefore the system is robustly stable for the given interval uncertainty ranges. Fig. 4(b) shows that the roots of corner

and edge polynomials surround root region of the system. These roots also prove robust stability of the system. Fig. 4(c) demonstrates the step-responses of all corner polynomials, which clearly validate the robust stability of the system.

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197

Fig. 7. Stability test results for the uncertain fractional order control system given by Eq. (35). (a) Root space obtained for region-sampled robust stability analysis, (b) root space obtained for edge-sampled robust stability analysis, and (c) step responses of corner polynomials.

In the four illustrative examples, computer experiments have shown the advantage of the proposed procedures for the both integer and fractional order systems with coefficient and order uncertainty.

geometrical box methods (rectangular, circular, etc.) so that it allows precise trace of root region boundary. References

4. Conclusions This study presents numerical methods for the robust stability investigation of fractional-order closed loop control system with interval parameter uncertainty. These methods are based on s ¼ vm mapping, which reduces the stability analysis problem of fractionalorder system to stability analysis of expanded integer-order fractional system in the first Riemann sheet. Two procedures as an extension of Radwan stability check procedure were used for analysis of fractional-order characteristic polynomials with interval uncertainty. The first one applies the scan sampling of polynomial family from hypercube in the parameter space and checks if all root locus fall into the stability region defined as π =2 m o jargfRðΩÞgj o π =m. The other method uses only edge polynomials sampling from parameter space and mainly consider boundaries of root region for checking root locus in the stability region. Thus, it reduces computational complexity of the numerical stability analysis. Scan sampling method can yield more reliable results than random sampling of interval polynomial family so that it can uniformly spread over the root region RðΩÞ. Edge-sampled robust stability procedure is more accurate comparing to any smallest

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