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Resonance of Fractional Transfer Functions of the Second Kind Rachid Malti, Xavier Moreau, Firas Khemane

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Resonance of Fractional Transfer Functions of the Second Kind Rachid Malti, Xavier Moreau, Firas Khemane Resonance of fractional transfer functions of the second kind Rachid Malti, Xavier Moreau, Firas Khemane To cite this version: Rachid Malti, Xavier Moreau, Firas Khemane. Resonance of fractional transfer functions of the second kind. The 3th IFAC Workshop on Fractional Differentiation and its Applications, FDA08, Nov 2008, Ankara, Turkey. pp.1-6. hal-00326418 HAL Id: hal-00326418 https://hal.archives-ouvertes.fr/hal-00326418 Submitted on 26 Jan 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Resonance of fractional transfer functions of the second kind Rachid MALTI, Xavier MOREAU, and Firas KHEMANE ∗ Bordeaux University – IMS, 351, cours de la Lib´eration, 33405 Talence Cedex, France. firstname.lastname @ims-bordeaux.fr { } Abstract: Canonical fractional transfer function of the second kind is studied in this paper. Stability and resonance conditions are determined in terms of pseudo-damping factor and commensurable order. Keywords: Resonance, fractional system, second order transfer function, canonical form. 1. INTRODUCTION F (s) + + 1 1 Σ K1 ν Σ K2 Commensurable fractional systems can be represented in s sν a transfer function form as: mB − − b sνj ν j T (s ) j=0 H (s)= = , (1) ν PmA R(s ) νi 1+ ais Fig. 1. closed-loop transfer function equivalence i=1 2 where (ai,bj) R , ν is the commensurableP differentiation ∈ rational systems: it is not a damping factor, unless ν = 1.It order, mB and mA are respectively numerator and denom- will be referred to as pseudo-damping factor in this paper. inator degrees, with mA >mB for strictly causal systems. ν Stability of fractional differentiation systems is addressed When two complex conjugate s -poles are present, the in the following theorem. representation (4) is generally preferred to (3), because Theorem 1.1. (Stability Matignon (1998)). A commensu- all parameters in (4) are real-valued. rable transfer function with a commensurable order ν, as The transfer function of the second kind (4), can also be in (4), with T and R two coprime polynomials, is stable viewed as two nested closed-loop transfer functions of Fig. if and only if (iff) 0 <ν< 2 and p C such as R(p) = 1 with gains, π ∀ ∈ 0, arg(p) > ν 2 . ν | | ω0 K1 = , (5) The commensurable transfer function (1) can always be 2ζ ν decomposed in a modal form: K2 =2ζω0 , (6) N vk Ak,q and two integrators of order ν. H(s)= ν q , (2) (s + pk) k=1 q=1 The closed-loop representation of Fig. 1 will be used in X X ν the open-loop transfer function analysis of 4, based on where ( pk), with k =1,...,N, represent the s -poles of − Nichols charts. § integer multiplicity vk. The representation (2) is constituted of elementary trans- Plotting the asymptotic frequency response of a transfer fer functions of the first kind: function such as (2) requires usually to decompose (2) into elementary transfer functions of the first and the second K˜ F˜(s)= , (3) kind and the contribution of each elementary function sν + b is plotted as it appears depending on the transitional studied in Hartley and Lorenzo (1998). frequencies. The elementary transfer function of the first Combining two elementary functions of the first kind (3) kind has already been studied in Hartley and Lorenzo yields an elementary function of the second kind written (1998). On the other hand, the properties of rational ν in a canonical form as: second order systems, with ( = 1), are well known: K ζ > F s . the system is stable, if the damping factor 0, ( )= ν 2ν (4) • √2 s s <ζ< 1+2ζ + the system is resonant, if 0 2 , ω0 ω0 • the system has two complex conjugate poles, if 0 < • As in rational systems, K and ω0 in (4) represent respec- ζ < 1, tively steady-state gain and cut-off frequency. However, the system has a real double pole if ζ = 1, the parameter ζ does not have the same meaning as in • the system is overdamped, if ζ > 1. • The main concern of this paper is to study elementary Substituting (12) in (16) yields properties of fractional transfer functions of the second π 1 ζ2 π kind written in the canonical form (4). First, stability tan ν < − tan (17) 2 ζ ≤ 2 conditions are established in terms of the pseudo-damping p − factor ζ and the commensurable differentiation order ν. π 1 ζ2 tan2 ν < − (18) Then, resonance conditions are established. 2 ζ2 ≤ ∞ π The authors came to this study when they wanted to ζ2 < cos2 ν . (19) simulate a resonant and stable transfer function of the 2 second kind as in (4). They noticed then that the above- Since ζ is negative, it must satisfy the following inequali- mentioned properties of second order rational systems do ties: π not apply for fractional systems. 1 < cos ν < ζ 0. (20) − − 2 ≤ 2. STABILITY OF FRACTIONAL TRANSFER Subcase 0 ζ < 1: In this case, FUNCTIONS OF THE SECOND KIND ≤ 1 ν < 2 and (21) ≤ The application of theorem 1.1 on (4), requires the com- 1 ζ2 putation of both sν -poles: − < 0. (22) ζ ! ν ν 2 p − s1,2 = ω0 ζ ζ 1 , (7) π − ± − Hence, θ is in the second quadrant: θ 2 , π . According which can either be real if ζ p1, or complex conjugate to theorem 1.1, the system is stable iff:∈ if ζ < 1. Hence, two cases| are| ≥ distinguished. π π | | ν <θ<π. (23) 2 ≤ 2 2.1 Case ζ 1 Consequently, substituting (12) in (23) yields: | |≥ 2 ν π 1 ζ Two real s -poles are present. According to theorem 1.1, tan ν π < − < tan(0) (24) sν 2 − ζ the transfer function (4) is stable if both -poles are p − negative: π 1 ζ2 2 ν > > tan −2 0 (25) sν < 0 ων ζ ζ2 1 < 0 (8) 2 ζ 1,2 ⇒ 0 − ± − π ζ2 > cos2 ν . (26) ζ ζ2p 1 < 0. (9) 2 ⇒− ± − Since ζ is positive, it must satisfy the following inequali- In this case, this conditionp is always true and hence the system is always stable. ties: π 0 < cos ν <ζ< 1. (27) 2.2 Case ζ < 1 − 2 | | Two complex conjugate sν -poles are present: 2.3 Summary ν ν 2 s1,2 = ω0 ζ j 1 ζ (10) The transfer function (4) is stable iff: − ± − ν jθ π = ω0 e± , p (11) cos ν <ζ< and 0 <ν< 2. (28) − 2 ∞ where 0 <θ<π and θ is given by: 2 The stable transfer function (4) has two complex conjugate 1 − ζ ν arctan if − 1 < ζ ≤ 0 s -poles iff: −ζ π θ = p ! (12) 2 cos ν <ζ< 1. (29) 1 − ζ − 2 arctan + π if 0 ≤ ζ < 1. −ζ Condition (29) is not necessary for the transfer function p ! of the second kind (4) to be resonant. According to theorem 1.1, the system is stable iff: π 0 < ν <θ<π. (13) 3. RESONANCE OF FRACTIONAL TRANSFER 2 FUNCTIONS OF THE SECOND KIND Both conditions expressed in (12) are treated below. Subcase 1 < ζ 0: In this case, The frequency response of (4) is given by: − ≤ 0 < ν 1 and (14) K F (jω)= ν 2ν . (30) ≤ jω jω 2 ζ 1 ζ 1+2 ω0 + ω0 − > 0. (15) ζ Define Ω = ω as the normalized frequency. Since the p − ! ω0 π resonance does not depend on the gain K, it will be set to Consequently, θ is in the first quadrant: θ 0, 2 . According to theorem 1.1, the system is stable iff:∈ one in the following. Hence, define: π π 1 0 < ν <θ . (16) (jΩ) = . (31) 2 ≤ 2 F 1+2ζ (jΩ)ν + (jΩ)2ν The gain of (jΩ) 1 , F 3 1 No resonant frequency (jΩ) = π , (32) jν ν jνπ 2ν |F | 1+2ζe 2 Ω + e Ω 2.5 A single resonant frequency Two resonant frequencies is further detailed in (33)-(36). Unstable system 2 In case F (s) is resonant, (jΩ) has one or multiple ζ Stability limit maxima at positive frequencies.|F Since| log is a strictly 1.5 increasing function, finding the maximum of (jΩ) cor- |F | 1 responds to finding the maximum of (jΩ) dB. Hence, d (jΩ) dB |F | F (s) is resonant if |F dΩ | = 0 has at least one real 0.5 and strictly positive solution. Based on (36), all real and Pseudo−damping factor strictly positive solutions of the following equation need to 0 be evaluated: −0.5 d|F(jΩ)|dB 4ν−1 π 3ν−1 = 0 ⇒ Ω + 3ζ cos ν Ω + d Ω 2 −1 2 2ν−1 π ν−1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2ζ + cos (νπ) Ω + ζ cos ν Ω = 0. (37) Commensurable order ν 2 Since Ω = 0, is not an acceptable solution, the common ν 1 Fig. 2. Resonance and stability regions of the fractional factor Ω − in (37) can be simplified, so as to obtain: system (4) in the ζ versus ν plane d (jΩ) dB 3ν π 2ν |F | =0 Ω +3ζ cos ν Ω + F (s) dΩ ⇒ 2 + π Σ β(s) 2ζ2 + cos(νπ) Ων + ζ cos ν =0.
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