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. (38) 2 One can check easily that for rational systems, ν = 1, (38) − reduces to: Ω3 + 2ζ2 1 Ω=0, (39) − Fig. 3. closed-loop transfer function equivalence which strictly positive solution is given, as expected, by π 2 if 1 < ν 2 and cos ν 2 <ζ < the stable Ωr = 1 2ζ , (40) • ≤ − ∞ ⇒ − system is always resonant, provided that the following known condition is satisfied: if ν < ν 2 and ζ <ζ < where ν and ζ p • 0 ≤ 0 ∞ 0 0 √2 are computed numerically and plotted as the lower- 1 2ζ2 > 0 ζ < . (41) − ⇒ 2 left limits of the green region of Fig2 the stable system has two resonant frequencies. ⇒ The third order equation in Ων (38), can have positive real- valued, negative real-valued or complex-valued solutions. 4. OPEN-LOOP – CLOSED-LOOP ANALYSIS The number of resonant frequencies of the studied system, zero one or two, depends on the number of positive real- The transfer function (4) can be viewed as the closed-loop valued solutions corresponding to maxima of (jΩ). Care F system of Fig.1, or by considering only the outer loop, as must be taken, because some of the strictly positive so- the closed-loop system of Fig.3: lutions correspond to minima (anti-resonance), especially β(s) when a double resonance is present, then the gain presents F (s)= , (42) a minimum between the two maxima. 1+ β(s) where the open-loop transfer function β(s) is given by: Solving (38) analytically is not easy. A numerical solution K K is obtained for all combinations of ν and ζ and plotted in β(s)= 1 2 , (43) 2ν K2 Fig.2 (yellow and green regions represent combinations of s (1 + sν ) ν and ζ which produce resonant systems). The main result K and K being defined in (5) and (6). of this paper is plotted in Fig.2 and summarized below. 1 2 The open-loop transfer function β(s) is studied for differ- if 0 < ν 0.5 and cos ν π <ζ < 0 the stable • ≤ − 2 ⇒ ent values of ν and ζ and its frequency response plotted system is always resonant, for ζ = 0.7, ζ = +0.7, and ζ = 2 in the Nichols charts of if 0 < ν 0.5 and 0 < ζ the stable system is never Figs 4, 5,− and 6. • resonant,≤ ⇒ π ζ ζ . if 0.5 < ν 1 and cos ν 2 <ζ < the For negative , see for instance Fig.4 with = 0 7, the • stable system≤ is resonant− if an additional∞ condition ⇒ steady state gain of the open-loop transfer function,− β(s), is satisfied ζ < ζ0, where ζ0 is computed numerically is negative. Hence, in low frequencies, the Nichols chart of and plotted in Fig.2 as the upper bound of the β(s) is inside the Nichols magnitude contours. When sta- π resonant region in the interval ζ ]0.5, 1]. In the bility condition (28) is satisfied, here ζ > cos 0.7 2 = √2 ∈ 0.45, β(s) passes on the right of the critical− point; other- particular case when ν = 1, ζ0 = , 2 −wise it passes on its left. Consequently, for negative ζ, the ν 1 The multi-valued function s becomes holomorphic in the comple- system can either be stable and resonant or unstable. ment of its branch cut line as soon as a branch cut line, i.e. R− is specified. The following restrictions on arguments of s are imposed: For 0 < ζ 1, see for instance Fig.5 with ζ = +0.7, π ≤ −π < arg(s) <π. Hence, the only possible argument of j is 2 . the steady state gain of β(s) is positive. When ν < ν0, 1 (jΩ) = (33) π ν 2ν π ν 2ν |F | 1+2ζ cos ν 2 Ω + cos(νπ)Ω + j 2ζ sin ν 2 Ω + sin (νπ)Ω The gain in dB is now given by: 2 2 π ν 2ν π ν 2ν |F(jΩ)|dB = −10 log 1 + 2ζ cos ν Ω + cos (νπ)Ω + 2ζ sin ν Ω + sin(νπ)Ω (34) 2 2 π π (jΩ) = 10 log 1+4ζ2 cos2 ν Ω2ν + cos2 (νπ)Ω4ν +4ζ cos ν Ων +2cos(νπ)Ω2ν + |F |dB − 2 2 h π π π 4ζ cos ν cos(νπ)Ω3ν + 4ζ2 sin2 ν Ω2ν + sin2 (νπ)Ω4ν +4ζ sin ν sin (νπ)Ω3ν (35) 2 2 2 π π i (jΩ) = 10 log 1+4ζ2Ω2ν +Ω4ν +4ζ cos ν Ων +2cos(νπ)Ω2ν + 4ζ cos ν Ω3ν (36) |F |dB − 2 2 i

Nichols plot for ζ = −0.7 and different values of ν Nichols plot for ζ = +0.7 and different values of ν 40 40 ν = 0.10 dB 0 dB ν = 0.4 30 30 ν = 0.7 (unstable) ν = 1 (rational unstable system) 20 1 dB 20 1 dB 2 dB 2 dB 10 10 6 dB 6 dB

0 0

−10 −10 Open−Loop Gain (dB) Open−Loop Gain (dB)

−20 −20 ν = 0.5 ν = 0.95 ν −30 −30 = 1.05 ν = 1.7

−40 −40 −250 −200 −150 −100 −50 0 −250 −200 −150 −100 −50 0 Open−Loop Phase (deg) Open−Loop Phase (deg)

Fig. 4. Nichols charts for ζ = 0.7 and different values of Fig. 5. Nichols charts for ζ = +0.7 and different values of ν. − ν. in this example ν 1, the Nichols chart remains outside 0 ≈ the 0dB Nichols magnitude contour. For ν0 <ν< 2, the Nichols chart crosses the 0dB Nichols magnitude contour which makes the closed-loop system resonant. The system Nichols plot for ζ = 2 and different values of ν is unstable if the stability condition (28), expressed in 40 ν = 0.5 terms of ν, is not satisfied, here ν > arccos( ζ) 2 =1.50. 0 dB π ν = 0.95 − 30 For ζ > 1, see for instance Fig.6 with ζ = 2, the steady ν = 1.05 ν = 1.7 state gain of β(s) is positive. When ν 1, the Nichols 20 ≤ 1 dB chart remains outside the 0dB Nichols magnitude contour. 2 dB For 1 <ν< 2, the Nichols chart crosses the 0dB Nichols 10 6 dB magnitude contour which makes the closed-loop system resonant. The system is unstable if ν 2 as specified by 0 ≥ theorem 1.1. −10 Open−Loop Gain (dB)

5. TIME-DOMAIN SIMULATIONS OF FRACTIONAL −20 MODELS −30

Due to the consideration that real physical systems gener- −40 −250 −200 −150 −100 −50 0 ally have bandlimited fractional behavior and due to the Open−Loop Phase (deg) practical limitations of input and output signals (Shan- non’s cut-oﬀ frequency for the upper band and the spec- Fig. 6. Nichols charts for ζ = 2 and diﬀerent values of ν. trum of the input signal for the lower band), fractional operators are usually approximated by high order rational models. As a result, a fractional model and its rational approximation have the same dynamics within a limited frequency band. The most commonly used approximation Oustaloup (1995). Trigeassou et al. Trigeassou et al. (1999) ν of s in the frequency band [ωA,ωB] is the recursive suggested to use an integrator outside the frequency range distribution of zeros and poles proposed by Oustaloup [ωA,ωB] instead of a gain: s−ν 10 s s s 1+ ω′ 1+ ω′ 1+ ω′ 1 1 2 N 0 s 1+ s 1+ s s ω1 ω2 1+ ωN Ωr1

Gain in dB −10

−20 Fig. 7. Approximation of a fractional integrator using a −1 0 1 rational model 10 10 10

log(η) log(α) log(η) log(α) log(η) 0

−ν 20dB −50 Phase in degrees Magnitude (dB) −100 −1 0 1 10 10 10 Frequency in rad/sec

Fig. 9. Bode plot for ζ = 0.5 and ν =0.5 0 −

−ν π/2 Step Response

2 Phase (rad) 1.5 −π/2 1 ω’ ω ω’ ω ω ω’ ω 1 1 2 2 Frequency N−1 N N Amplitude Fig. 8. Bode diagrams of a fractional integrator and the 0.5

asymptotic behavior of its rational approximation 0 0 5 10 15 20 25 30 35 40 45 50 Time (sec) 1 ν 1+ s − Impulse Response ν ν C0 ωA s− s− = 2 [ωA,ωB ] s 1+ s → ωB ! N s 1 1+ ′ C0 ωk

, (44) Amplitude ≈ s 1+ s 0 k ωk Y=1 −1 where ωk = αωk′ , ωk′ +1 = ηωk′ and 0 2 4 6 8 10 12 Time (sec) log α ν =1 , (45) − log αη Fig. 10. Step and Impulse responses for ζ = 0.5 and − α and η are real parameters which depend on the diﬀer- ν =0.5 entiation order ν (see Fig 8). The bigger N the better the 6.2 Example 2 ν approximation of the integrator s− . The approximation (44) will be used while plotting the impulse and the step Consider now the case ν = 1.9, and ζ = 2, which responses of systems studied in the following section. presents the particularity of having a double resonance as established in Fig.2. Eq. (38), now written as: 6. EXAMPLE Ω5.7 5.93Ω3.8 +8.95Ω1.9 1.98=0, (48) − − has three real-valued solutions: 6.1 Example 1 Ωr1 =0.50, Ωr2 =1.47, Ωr3 =1.96. (49) As shown in Fig.11, Ωr corresponds to a resonance, Ωr to In this example, the particular case ν =0.5 and ζ = 0.5 1 2 − an anti-resonance, and Ωr3 to a second resonance. When a is studied. Eq. (38), which reduces to: system presents two resonant frequencies, it always has a 3√2 Ω0.5 √2 minimum between these two maxima. Moreover, the step Ω1.5 Ω+ =0, (46) − 4 2 − 4 and impulse responses, plotted in Fig 12, are oscillatory as expected due to the resonant frequencies. has a single strictly positive real-valued solution and two complex conjugate ones: 7. CONCLUSION Ωr =0.87, Ωr ,r = 0.37 0.08j. (47) 1 1 2 − ± Only the positive solution is acceptable and corresponds The fractional transfer functions of the second kind (4) to the resonant frequency, as shown in the Bode diagram is studied in this paper. First, stability conditions are of Fig. 9. Moreover, the step and impulse responses, determined in terms of the pseudo-damping factor ζ (it plotted in Fig.10, are underdamped as expected due to is known that the commensurable order, ν, must satisfy the resonant frequency. 0 <ν< 2) and is summarized in (28). Then, conditions 50

Ωr1

Gain in dB −50 Ωr2 Ωr3

−100 −1 0 1 10 10 10

−100

−200

−300 Phase in degrees

−400 −1 0 1 10 10 10 Frequency in rad/sec

Fig. 11. Bode plot for ζ = 2 and ν =1.9

Step Response

1.5

1 Amplitude 0.5

0 0 10 20 30 40 50 60 70 80 90 100 Time (sec) Impulse Response

0.5

Amplitude 0

−0.5 0 20 40 60 80 100 120 Time (sec)

Fig. 12. Step and Impulse responses for ζ = 2 and ν =1.9 on ζ and ν are determined so that the system is resonant. The latter conditions are diﬃcult to express analytically. They are computed numerically and plotted in Fig.2. It is also shown, in Fig.2, that some combinations of ζ and ν yield two resonant frequencies.

REFERENCES T.T. Hartley and C.F. Lorenzo. A solution to the funda- mental linear fractional order differential equation. In NASA/TP–1998-208693 report, Lewis Research Center, 1998. D. Matignon. Stability properties for generalized fractional differential systems. ESAIM proceedings - Systèmes Différentiels Fractionnaires - Modèles, Méthodes et Ap- plications, 5, 1998. A. Oustaloup. La dérivation non-entière. Hermès - Paris, 1995. J.-C. Trigeassou, T. Poinot, J. Lin, A. Oustaloup, and F. Levron. Modeling and identification of a non integer order system. In ECC, Karlsruhe, Germany, 1999.