Comparison Between Integer Order and Fractional ... - IEEE Xplore

17th IEEE Mediterranean Electrotechnical Conference, Beirut, Lebanon, 13-16 April 2014.

Comparison between integer order and fractional order controllers Roy ABI ZEID DAOU

Xavier MOREAU IMS Laboratory, CRONE Group, University Bordeaux I, Bordeaux, France email: [email protected]

Biomedical Technologies Department, Public Health Faculty, Lebanese German University, Jounieh, Lebanon email: [email protected]

methods were and still are proposed for this purpose in order to get the optimal values of the integration and differentiation orders [5]. Most recently, the CRONE controller, also based on the fractional integration and differentiation, was introduced in three generations [6]. However, a main difference, other than the way used to synthesize these controllers, is encountered. The number of parameters to be defined differs between these controllers. In fact, the PID needs to define three parameters, the generalized PID needs 5 parameters and the CRONE controller needs four parameters. Nowadays, the use of the fractional controllers is almost necessary in almost all engineering domains. The reasons behind this use are diverse; among them we list the most important: - The identification of several physical and natural properties showed that a fractional order differentiation in implemented when modelling them using transfer function. Some of the examples are the thermal diffusive interfaces [7], the muscles activities [8] and much more… - The analogue [9] and digital [10] implementation of the fractional order controllers is easy; - Once the user specifications and/or the open loop shape are defined, the synthesis of the fractional order controller is not complicated. As for the fractional calculus, its idea was born in the last decade of the XVII century after letters exchange between Leibniz and L’Hospital [11]. Several definitions of the fractional integration and differentiation were proposed [12]. The applications in this domain started almost three centuries later with applications in almost all the engineering domains [13-15]. Concerning the control systems, the well known PID was the mostly used controller till 1961 when Manabe introduced the fractional order concept to such regulators [16]. In 1975, Oustaloup developed a regulator of order 3/2 in order to control a laser beam [17] and proposed, some years after, the CRONE control system [15]. As well, new methods were proposed in order to search the best values of the generalized PIλDμ or the CRONE controllers [18-19]. To sum up, this paper aims to do a double comparison. It presents a comparison between fractional order and integer order controllers, on one hand, and between a priori and a posteriori controllers on the other hand.

Abstract—This paper presents a comparison between fractional order controllers and integer order controllers. The well-known PID is the integer order controller chosen whereas the generalized PID and the CRONE (“Commande Robuste d’Ordre Non Entier” which stands for fractional order robust controller) are the fractional order controllers studied. Another big difference between these controllers is that the structure of the PID and the generalized PIλDμ is a priori fixed whereas the structure of the CRONE is a posteriori deduced from the loop shaping. Results show that the fractional order controllers are more robust when small variations appear in the plant and their behavior is more suitable regarding the user specifications requirements. Keywords—fractional order controllers; PID and generalized PID controllers; CRONE controller; robustness; behavior study;

I. INTRODUCTION This paper presents a double comparison between: 1. integer order controllers and fractional order controllers; 2. controllers synthesized using a priori method and controllers synthesized using a posteriori method. For the interger order controllers, the PID is used whereas the generalized PID and the CRONE represent the fractional order controllers. As for these controllers, the PID and the generalized PID controllers are a priori fixed, which means that the computation of these controllers transfer functions are done directly according to the user specifications (stability degree, bandwidth, rejection level of the measured noise, rejection level of the output disturbance,…). Concerning the CRONE controller, the posterior synthesis method is used. In this case, the definition of the controller is made with respect to the openloop constraints (robust loop shaping). Going back to the controller’s debut, their synthesis and their realization has started before almost a century. One of the earliest forms of a PID controller was developed by Elmer Sperry in 1911 [1]. However, the first published work presenting a PID controller was proposed by Russian American engineer Nicolas Minorsky in 1922 [2]. Many years later, the generalized PIλDμ controller, where the integration and differentiation order can be any positive and real number less than the unit (e.g., 0 < λ ,η < 1 ∀λ ,η ∈ \ +* ), was proposed while respecting the tuning rules proposed by Ziegler and Nichols for integer PIDs [3-4]. Several tuning

978-1-4799-2337-3/14/$31.00 ©2014 IEEE

292


The stability degree is specified with respect to the phase margin MФ. However, it can also be specified using the module margin MM or the gain margin MG. (8) M Φ = π + arg β ( jωu ) ,

In more details, this paper will be composed as follow: in section 2, the system components and the performance specifications, in frequency domain, will be show. In section 3, the PID controller will be presented. Both the integer order and the fractional order PIDs are shown. Section 4 presents the CRONE controller whereas section 5 shows an application example to illustrate the advantages and drawbacks of each controller. Section 6 ends up with a conclusion and some future works.

where β(s) represent the open-loop transfer function (relation (7)) and ωu is the crossover frequency in the open-loop defined as follow: β ( jωu ) = 1 . (9) The stability degree specification can be presented as the following constraint: M Φ ≥ M Φmin , (10)

II. SYSTEM COMPONENTS AND USER SPECIFICATIONS In this section, the representation of the system containing the plant and the controller will be introduced as well as the performance specifications that the user would specify in order to get the best performance from the designed controller.

where M Φ min represents the minimal acceptable value of the phase margin. The above leads to the following relation,

A. System components Figure 1 presents the block diagram of the feedback control system. Input disturbance Du (s) Yref(s) Reference signal

Error signal U(s) ε(s) + C(s) +

-

Controller

arg β ( jωu ) ≥ − π + MΦmin ,

that can be reduced and becomes:

arg C ( jωu ) ≥ − π + MΦmin − arg G ( jωu ) .

Sensor noise Bm(s) G(s)

+

Measured output

where ωu min represents the minimal acceptable value of ωu.

Fig. 1 – Block diagram of the feedback control system

Referring to equation (9) and knowing that: β ( jω ) = C ( jω ) G ( jω ) ,

From this block diagram, the closed-loop relations are established: - for the measured output: - for the error signal:

C ( jωu ) ≥ G ( jωu min )

ε (s ) = − S (s ) Bm (s ) − G (s )S (s ) Du (s ) + S (s ) Yref (s ) , (2) U (s ) = − R (s ) Bm (s ) − T (s ) Du (s ) + R (s ) Yref (s ) , (3)

where 1 + β (s) - the complementary sensitivity function is: β ( s) T (s) = = 1− S ( s) , 1 + β ( s) - the plant input sensitivity function is: R ( s) = C (s) S ( s)

- the open-loop transfer function is: β (s) = C (s)G (s) .

−1

(15) . Concerning the rejection level of the measured noise, it is calculated using a specification applied to the complementary sensitivity function module as follow:

- for the control signal:

- the sensitivity function is: S ( s ) =

(14)

a controller gain constraint at the frequency ωu can be deduced from relation (13) as follow:

Y (s ) = S (s ) Bm (s ) + G (s )S (s ) Du (s ) + T (s ) Yref (s ) ,(1)

1

(12)

The bandwidth specification is also computed at the crossover frequency ωu. The main goal of this constraint is to fix the closed-loop dynamics speed. Hence, the frequency range that is concerned by this condition is ωu ≥ ωu min , (13)

Y(s) = Ur(s)

Plant model

(11)

−1

,

∀ ω ≥ ωT , T ( jω ) = β ( jω ) 1 + β ( jω ) ≤ AT (16) where AT shows the desired rejected noise level for the given frequency ωT: T ( jωT ) = AT . (17)

(4)

(5) (6)

If the value of ωT is chosen to be much bigger than ωu, relation (17) can be rewritten as: ∀ ω ≥ ωT , T ( jω ) ≈ β ( jω ) ≤ AT . (18)

(7)

Hence, a new controller gain constraint around the frequency ωT is deduced from relation (18) as follow: ∀ ω ≥ ωT ,

B. Performance specifications The performance specifications concern different aspects in frequency domain, from among we list the following: • the stability degree; • the bandwidth; • the rejection level of the measured noise; • the rejection level of the output perturbation; • the plant input sensitivity.

C ( jω ) ≤ AT G ( jω )

−1

.

(19)

The rejection level of the output disturbance is used for low frequencies (ωS < ωu) and is computed using a specification of the sensitivity module function as follow: ∀ ω ≤ ωS ,

S ( jω ) = 1 + β ( jω )

−1

(20) ≤ BS , where BS represents the desired rejected output disturbance level for the frequency ωS:

293


S ( jωS ) = BS . (21) When choosing ωS > 1 (minimization of the noise effect caused by the integration due to the phase delay of ωu), system (33) becomes: ⎧Cu = CPID ( jωu ) = C0 a ⎪ ⎨ , (34) ⎪ϕ = arg C ( jω ) = (arctan (b ) − π / 2 ) + ϕ u PID u m ⎩

follow:

(22) C ( jω ) ≥ (BS G ( jωS ) ) . The plant input sensitivity is computed using a specification of the form: −1

∀ ω ≤ ωS ,

(

R( jω ) = C ( jω ) (1 + β ( jω ))−1 ≤ DR , (23)

∀ ω ≥ ωR ,

where DR represents the maximal value at the frequency ωR: R( jωR ) = DR . (24) If the frequency ωR is much bigger than ωu, the relation (23) can be rewritten as: ∀ ω >> ωR , C ( jω ) ≤ DR . (25) The constraints (19) and (25) represent almost the same aspect which is the controller gain at high frequencies. Hence, they can be reduced to one constraint by choosing the lowest value of these two relations. Thus,

∀ ω >> ωu , C ( jω ) ≤ Min ⎡ AT G ( jωT ) ⎢⎣

−1

(

, DR ⎤ . (26) ⎥⎦

⎛⎛ ⎜⎜ ⎜ ⎜1 + ⎜ ⎜⎝ ⎝

⎞ ⎛ ⎟ ⎜ s ⎟/⎜ ωi ⎟⎠ ⎜⎝ ωi

⎞⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎠⎠

(28)

and CD ( s ) =

⎛⎛ ⎜⎜ ⎜ ⎜1 + ⎜ ⎜⎝ ⎝

⎞ ⎛ ⎟ ⎜ ⎟ / ⎜1 + ωb ⎟⎠ ⎜⎝

⎞⎞ ⎟⎟ ⎟⎟ ωh ⎟⎠ ⎟⎠ ,

(29)

s

s

s

C PI λ D μ (s ) =

C PID (s ) =

⎛ ⎜ 1+ a ⎜ ⎜ ⎛ ⎜ 1 + s /⎜ ⎝ ⎝

⎞ ⎟ ⎟ ⎞ ωu ⎟ ⎟⎟ ⎠⎠

s / ωu a

( )

(

)))

))

(35)

⎛1 + b s /ω u C0 ⎜⎜ / b s ω u ⎝

⎞ ⎟ ⎟ ⎠

λ ⎛⎜

⎞ ⎟ ⎟ ωu ⎞⎟ ⎟⎟ ⎠⎠

1 + a s / ωu

⎜ ⎜ ⎜1 + ⎝

s / ⎛⎜ a ⎝

μ

,(36)

where λ and μ are the integration and the differentiation real order and vary strictly between 0 and 1. The module and the phase of the generalized PIλDμ controller are as follow: λ/2 ⎧ μ /2 ⎛⎜1 + (bω / ω ) 2 ⎞⎟ ⎛ ⎞ ⎪ u ⎜ 1 + a (ω / ωu ) 2 ⎟ ⎠ ⎝ ⎪ CPI λ D μ ( jω ) =C0 ⎜ ⎟ ⎪ ⎜ 1 + a − 1 (ω / ω ) 2 ⎟ (bω / ωu )λ ⎨ u ⎠ ⎝ ⎪arg CPI λ D μ ( jω )= λ (arctan (bω / ωu ) − π / 2 ) ⎪ ⎪⎩ + μ arctan a ω / ωu − arctan ω / a ωu (37) At the open-loop crossover frequency ωu, relation (37) can be rewritten as follow: ⎧ 2 λ /2 ⎪Cu* = C λ μ ( jωu ) = C0 1 + b aμ /2 PI D ⎪ bλ ⎪ ⎨ . (38) ⎪ϕ * = arg C λ μ ( jω ) = λ (arctan(b ) − π / 2 ) u u PI D ⎪ ⎪ + μ arctan a − arctan 1 / a ⎩

(

where ωm represents the frequency for which the controller phase is maximal (e.g., ϕm = Max[argC(jω)] = argC(jωm)). Inserting ωu in equation (27) leads to the following PID transfer function: ⎞ ⎟ ⎟ ⎠

( (

B. Generalized PID The computation of the generalized PID (or PIλDμ) is made based on the values found when calculating the PID controller. In fact, the generalized PID requires 5 parameters. Its form is as follows:

where ωi, ωb et ωh are the transitional frequencies and C0 is a constant. In order to calculate these parameters, the user constraints must be used. In the following, the method to calculate the optimal values of the PID parameters is shown. Let’s start by introducing some new constants as follow: a = ωh / ωb , b = ωu / ωi and ωm = ωb ωh , (30)

⎛1 + b s /ω u C0 ⎜⎜ b s / ω u ⎝

)

where ϕ m = arcsin((a − 1) / (a + 1)) .

III. PID CONTROLLER A. Integer order PID First, the well known PID controller is described. It is the eldest controller. It consists on a gain, an integration of order 1 and a derivation of order 1. Two different arrangement of the PID controller exist: the parallel arrangement and the cascade arrangement. The cascade arrangement will be treated in this paper. Its transfer function can be presented as follow: CPID (s ) = C0 C I (s ) CD (s ) , (27) with CI ( s ) =

(

(

)

( (

(

. (31)

(

Hence, the PID module and phase are given by:

294

( )

)

(

))

)))


values of the vector θ once all five relations ((42) to (47)) are valid. So, different simulations may lead to different values of the five parameters as the simulation stops once these five equations are satisfied. The initial vector and the lower and upper bound vectors have a big impact on the output values. Note that, in order to emphasize the fractional behaviour of the generalized PIλDμ controller, the value of the integrator and differentiator orders, λ and μ, should vary between 0.2 and 0.8.

If b >> 1 (minimization of the noise effect caused by the integration due to the phase delay of ωu), system (38) becomes: ⎧Cu* = C λ μ ( jωu ) = C0 a μ / 2 PI D ⎪ ⎨ , (39) ⎪ * ⎩ϕu = arg CPI λ D μ ( jωu ) = λ (arctan(b ) − π / 2 ) + ϕ m

where ϕ m = μ arcsin ((a − 1) / (a + 1)) . (40) After recalculating the controller module and phase at central frequency ωu, we obtain for low frequencies (ω > ωu) ⎧⎪ C λ μ ( jω ) ≈ C0 a μ PI D ⎨ . (42) arg ⎪⎩ CPI λ D μ ( jω ) ≈ 0 Considering the system (31), the constraints listed above can be rewritten to suit best the generalized PIλDμ controller as follow: λ (arctan(b) − π / 2) + ϕ m ≥ − π + M Φ min − arg G ( jωu ) ,(43)

(

)

C0 1 + b2

λ /2

b−λ a μ /2 ≥ G ( jωu min ) −1

−1

IV. CRONE CONTROLLER After presenting both PID controllers, this paragraph presents the CRONE controller. The CRONE controller is designed using the open-loop constraints (robust loop shaping), which means it is based on the a posterior synthesis method. Three generations of the CRONE controller have been developed [20]. • In the first generation, the constant phase nπ/2 characterizes this controller around frequency ωu. When the frequency ωu varies, the constant phase controller doesn’t contribute to the phase margin variations; • In the second generation, a phase change is observed in the plant when varying the frequency ωu; • In the third generation, the plant’s phase and margin change when varying the frequency ωu. The first step consists on defining the necessary specifications for the synthesis of the nominal plant transfer function. All calculations are made in the frequency domain. The second step consists on reassigning the frequency closed-loop specifications into open-loop frequency specifications for the nominal plant. These new conditions take into account the plant behaviour at: • low frequencies in order to have good accuracy in the steady state; • middle frequencies, especially around the frequency ωu, to get the stability degree robustness; • high frequencies to have good input plant sensitivity.

(44)

,

∀ ω ≥ ωT ,

C0 a μ ≤ AT G( jωT )

∀ ω ≤ ωS ,

C0 (ωu / (b ω ))λ ≥ (BS G ( jωS ) )

,

(45) −1

,(46)

μ (47) and ∀ ω >> ωR , C0 a ≤ DR . In order to determine the optimal values of the parameters vector θ = [C0, b, a, λ, μ], two phases are required. - The first one depends on the initial values of the vector θ depending on some values found when computing the PID controller; - The second phase involves the search for the optimal values of this vector. In order to accomplish the first phase, the five following steps must be achieved. 1 – We consider λ = 1, μ = 1, b = 10, ωu = ωu min and

Once the behaviour of the system in the open-loop is defined, a decision should be made concerning the CRONE generation that will be used. This decision depends on the plant uncertainties. In this paper, only the first or the second generations will be applied as the third one presents complex differentiation and integration orders, and the use of special programs or toolboxes as the one designed by the CRONE group [21]. Hence, the open-loop behaviour due to the plant gain uncertainties (e.g., taking into consideration the second CRONE generation) leads to the following transfer function:

MΦ = M Φmin ; 2

–

We

ρ 0 = C I ( jω u ) G ( jω u )

calculate

and

Φ 0 = arg C I ( jωu ) + arg G ( jωu ) ;

3 – We deduce ϕ m = M Φ − π − Φ 0 ; 4 – We calculate the value of a with respect to the relation a = (1 + sin ϕ m ) / (1 − sin ϕ m ) ; 5 – Knowing that β ( jωu ) = 1 , we deduce the value of C0 as

follow:

(

C0 = a

μ /2

ρ0

)

−1

n

n

⎛ 1+ s / ω b ⎞ b ⎛ 1+ s / ω h ⎞ − nh β ( s ) = β0 ⎜ ,(48) ⎟ ⎜ ⎟ (1 + s / ω h ) ⎝ s / ω b ⎠ ⎝ 1+ s / ω b ⎠ where ωb and ωh represent the low and high transitional frequencies, n is the fractional order varying between 1 and 2, nb and nh are the asymptotic order behaviours for low and high frequencies and β0 is a constant that assures a unit gain at the frequency ωu. This constant is calculated as follow:

.

The second phase consists on determining the optimal values of the vector θ. For this purpose, the optimization toolbox of Matlab and its function fmincon is used or the genetic algorithms can be also used. In this paper, the optimisation function will be used. This function requires a large number of inputs and returns the five

295


(

β0 = (ωu / ω b ) 1 + (ωu / ω b ) 2

2

( n −nb ) /2

)

( nh − n) /2

(1 + (ω / ω ) ) 2

u

h

.

(49) Knowing that β ( s ) = Ccrone ( s ) G ( s ) , (50) the CRONE transfer function Ccrone(s) is deduced for the nominal plant value, which is to say: (51) Ccrone ( s ) = β ( s ) / G ( s ) . V. APPLICATION EXAMPLE After presenting the three controllers, an application will be shown in this part. The plant is a hydro electromechanical system constituted of a double direction pump, a perturbation pump, a water level sensor and two tanks as shown in figure 2.

G (s) =

2.24 ×10−4

. (53) ⎛ ⎞ s s 1 + ⎜ ⎟ 3.6 × 10−3 ⎠ ⎝ As for the user specifications, they can be resumed by the following: - the stability degree: M Φ min = 45° ; (54) - the bandwidth : ωu min = 0.0108 rad / s ; - the rejection level of the measured noise:

(

)

∀ ω ≥ ωT = 0.108 rad / s, T ( jω ) ≤ 0.1 ;

(55)

- the rejection level of the output perturbation: ∀ ω ≤ ωS = 0.00108 rad / s, S ( jω ) ≤ 0.1 ;

(56)

- the plant input sensitivity: ∀ ω ≥ ωR = 10.8 rad / s, R ( jω ) ≤ 100 .

(57)

Depending on the user specifications and the methods used to calculate the controllers defined previously, the transfer functions of the PID, the generalized PID and the CRONE controller of second generation are as follow: ⎛ 1 + s / 1.1 10−3 ⎞ ⎛ 1 + s / 6 10−3 ⎞ ⎟ , (58) ⎟⎜ CPID (s ) = 83.62 ⎜ ⎜ s / 1.1 10− 3 ⎟ ⎜ 1 + s / 19.6 10− 3 ⎟ ⎠ ⎠⎝ ⎝ ⎛ 1 + s / 8.3 10− 4 ⎞ ⎟ C PI λ D μ (s ) = 46 ⎜ ⎜ s / 8.3 10− 4 ⎟ ⎝ ⎠

⎛ 1 + s / 7.95 10− 3 ⎞ ⎟ ⎜ ⎜ 1 + s / 1.47 10 − 2 ⎟ ⎠ ⎝

Figure 2 – hydro-electro mechanical system The identification of this system shows that it can be presented by the following transfer function form: k , (52) G ( s) = ( s (1 + s / ω0 ) )

,

(59)

0.5435

⎛ 1 + s / 6.98 10− 4 ⎞ ⎜ ⎟ . (60) ⎜ 1 + s / 0.5014 ⎟ ⎝ ⎠ 1 1 + s / 0.5014 In the following, some frequency and time domain output show the behaviour and the robustness of the three controllers when varying the values of the plant’s variables. As we are limited in space, figure 3 shows the step responses for the system when taking into consideration the first and the third operational points. As for the frequency response, almost the same is observed. So, the first overshoot is maintained for the CRONE controller when varying the water level sensor, which is not the case for the other controllers.

Table 1 – Values of k and ω0 for each operational point

k (V/V) 8.30 10-4 5.90 10-4 2.24 10-4 3.7

0.5

⎛ 1 + s / 3.6 10−3 ⎞ ⎟ Ccrone (s ) = 32.62 ⎜ ⎜ s / 3.6 10− 3 ⎟ ⎝ ⎠

where k represents the constant of the water level sensor and ω0 the response time of the system. A complete modelling of the system can be found in [22]. As the plant’s transfer function is nonlinear, three different water levels were chosen and a linear study around these points was made. For each level, the values of k and ω0 were maintained. One of these three levels was identified to be the nominal level and the two others were used to study the robustness of the controllers. Table 1 summarizes these values.

Water Level (cm) 2 4 6 Max/Min

0.2

ω0 (rad/s) 4.0 10-3 3.8 10-3 3.6 10-3 1.11

VI. CONCLUSION A comparison between three controllers was proposed in this work. Two of the controllers are of fractional order (the generalized PIλDμ and the CRONE) whereas the third one has an integer order integration and derivation (the PID).

The third operational point, which refers to the 6cm water level, was chosen to be the nominal transfer function. Hence, this transfer function can be written as follow:

296


Also, two controllers are synthesized using the a priori method (the PID and generalized PIλDμ) whereas the CRONE is calculated using a posteriori synthesis method. The methods used to calculate the three controllers were shown and these controllers were implemented to control a hydro electromechanical system. The obtained results showed that the CRONE controller is robust to plant small variations, which is not the case for the two other controllers. As a future work, we will apply this comparative study to other systems such as diffusive thermal systems.

Fig. 4 – Output responses in tracking mode for a step input of 10 mV λ μ with: (a) the PID controller, (b) the generalized PI D controller and (c) the CRONE controller

REFERENCES B. Stuart, “A Brief History of Automatic Control”, June 1996. N. Minorsky, “Directional stability of automatically steered bodies”, J. Amer. Soc. Naval Eng., vol. 34, issue 2, pp. 280–309, 1922 [3] V. Duarte, J. Sa da Costa, “Tuning rules for fractional PID controllers”, in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its applications, Porto, Portugal, July 19-21, 2006. [4] J. G. Ziegler and N. B. Nichols, “Optimum settings for automatic controllers”, Transactions of ASME, vol. 64, pp. 759-768, 1942. [5] I. Podlubny, “Fractional-order systems and PID-controllers”, IEEE Transaction on Automatic Control, Vol.44, pp.208-214, 1999. [6] M. Moze, J. Sabatier, A. Oustaloup, “Synthesis of a Third Generation crone Controller using μ-Analysis Tools”, IEEE Industrial Electronics, IECON 2006, November 2006, Paris, France [7] FADI – ASCC 2013 [8] A. Sommacal, P. Melchior, J.M. Cabelguen, A. Oustaloup, A.J. Ijspeert, “Fractional Multi-Models of the Gastrocnemius Frog Muscle, Journal of Vibration and Control. Sage Publishing”, Vol. 14, No. 9-10, pp. 14151430, 2008. [9] R. Abi Zeid Daou, C. Francis and X. Moreau, “Fractional order systems applied to electrical domain – part 2: implementation results and Uncertainties influence”, 2nd International Conference on Advances in Computational Tools for Engineering Applications (ACTEA), Lebanon, December 2012. [10] S. Das, “Functional Fractional Calculus for System Identification and Controls”, Springer, ISBN: 978-3540727026, 2007. [11] K.S. Miller and B. Ross, “An introduction to the fractional calculus and fractional differential equations”, A Wiley-Interscience Publication, 1993. [12] K.B. Oldham and J. Spanier, “The fractional calculus”, Academic Press, New-York and London, 1974. [13] O. Cois, « Systèmes linéaires non entiers et identification par modèle non entier : application en thermique », Thèse de Doctorat de l’Université Bordeaux 1, 2002. [14] J. Lin, « Modélisation et identification de systèmes d'ordre non entier », Thèse de Doctorat, Université de Poitiers, 2001. [15] A. Oustaloup, « La dérivation non entière : théorie, synthèse et applications », Edition Hermès, Paris, 1995. [16] S. Manabe, “The non-integer integral and its application to control systems”, ETJ of Japan, Vol. 6, pp.83-87, 1961. [17] A. Oustaloup, « Etude et réalisation d'un système d'asservissement d'ordre 3/2 de la fréquence d'un laser à colorant continu », Thèse de Docteur-Ingénieur, Université Bordeaux 1, 1975. [18] S. G.Samko, A. A. Kilbas and O. I. Marichev, “Fractional integrals and derivatives: theory and applications”, Gordon and Breach Science Publishers, 1993. [19] B. M. Vinagre, I. Podlubny, L. Dorcak and V. Feliu, “On fractional PID controllers: a frequency domain approach”, IFAC Workshop on Digital Control, Past, Present and Future of PID Control, pp. 53-58, Terressa, Spain, 2000. [20] J. Sabatier, A. Garcia Iturricha, A. Oustaloup, F. Levron, "Third generation CRONE control of continuous linear time periodic systems", Proceedings of IFAC Conference on System Structure and Control, CSSC'98, Nantes, France, July 8-10, 1998 [21] http://www.imsbordeaux.fr/CRONE/toolbox/pages/accueilSITE.php?guidPage=home_p age [22]R. Abi Zeid Daou, X. Moreau, “A comparison between integer order and fractional order controllers applied to a hydro-electromechanical system”, Transaction on Control and Mechanical Systems, V.2, No. 3, pp. 131-143, 2013 [1] [2]

(a)

(b)

(c)

297