Tuning of Fractional Complex Order PID Controller ⋆ Ayadi Guefrachi ∗ Slaheddine Najar ∗ Messaoud Amairi ∗ Mohamed Aoun ∗ ∗
University of Gabes, National Engineering School of Gabes (ENIG), Research Laboratory Modeling, Analysis and Control of Systems (MACS) LR16ES22, Omar Ibn el Khattab street, 6029 Gabes, Tunisia. (
[email protected],
[email protected],
[email protected],
[email protected]). Abstract: This paper deals with a new structure of Fractional Complex Order Controller (FCOC) with the form P IDx+iy , in which x and y are the real and imaginary parts of the derivative complex order, respectively. A tuning method for the Controller based on numerical optimization is presented to ensure the controlled system robustness toward gain variations and noise. This can be obtained by fulfilling five design requirements. The proposed design method is applied for the control of a Second Order Plus Time Delay resonant system. The effectiveness of the FCOC design method is checked through frequency and time domain analysis. Keywords: Fractional calculus, Complex order PID controller, robust control, gain variations, numeric optimization. 1. INTRODUCTION Fractional calculus is more and more improving in system identification as well as in controller design. In fact, integer differentiations revealed limitations, in front of describing some physical phenomena. Such physical phenomena can only be described by a non-integer differential equation leading to a fractional order transfer function; see Chetoui et al. (2012), Heymans (2004), and Jalloul et al. (2013). In addition, many researches have shown the efficiency of fractional controllers in maintaining system robustness towards different forms of parametric variations; see Podlubny (1999), TenreiroMachado (2010), and Xue et al. (2006). Fractional order transfer functions FOTF are irrational transfer function with real or complex power of the Laplace variables. The majority of FTOF, are characterized with real orders; see Lanusse et al. (2014), Li et al. (2010), and Saidi et al. (2015a,b). The first and second generations of the CRONE controller, based on real fractional differentiation, were introduced by Oustaloup; see Oustaloup et al. (1995, 2013), and Oustaloup et al. (1999). The CRONE controller has shown robustness to gain changes. The first generation CRONE controllers can be used for plants with magnitude variation and constant phase. When the plant phase varies with respect to the frequency, the second generation of CRONE controllers can be used. Monje et al. presented a method for tuning and auto-tuning a Fractional Order PID controller (FOPID) based on an optimization method of a nonlinear function under nonlinear constraints; see Monje et al. (2008). The Monje’s method ensures a local robustness of the controlled system towards ⋆ This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia.
gain changes thanks to the local flatness of the controlled open-loop phase curve. Saidi et al. developed a numerical optimization algorithm to extended the flatness area of the open-loop phase curve, which consequently improved the robustness of the controlled system with respect to gain variation; see Saidi et al. (2015a). Despite that complex order differentiation was mathematically defined a long time ago, the use of complex order transfer functions is not as widely used as the real order ones. Fractional Complex Order Controllers have more parameters than real order ones. Thus, more requirements can be achieved. The researches of the CRONE team are one of the most developed works in this context; see Sabatier et al. (2015).They proposed the third generation CRONE controller to handle more uncertainties. They used the complex order to define the open-loop transfer function as follows: −sign(y) ωcg x+iy β (s) = C0 Re/i , (1) s where s is the Laplace complex variable, ωcg is the gaincrossover frequency, Re/i [.] is the real part with respect to i, C0 to ensure a unity-gain of the open-loop at the gain-crossover frequency ωcg , x, and y are the real and imaginary parts of the fractional complex order, respectively. An optimization algorithm is performed to obtain the optimal open-loop transfer function (1) that minimize certain criteria. The frequency response of the controller is then calculated based on the frequency response of both the plant and the optimal open-loop. The rational controller results of a frequency-domain approximation. Inspired from the third generation CRONE open-loop definition, Shahiri et al. proposed a structure of Fractional Complex Order PI controller (FCO-PI) defined by
−sign(y) ωcg x+iy C (s) = Kp + Kn Re/i , (2) s where only Kp and Kn were computed via a M constraint integral gain optimization (FC-MIGO) algorithm. In fact, real and imaginary parts of the fractional order x and y have been taken as known values parameters. Moreover, the controller output has high values for low set-point which raise questions about its real-time implementation; see Shahiri et al. (2016). In Guefrachi et al. work, an explicit expression of the frequency response of a generalized transfer function with a complex order, was introduced for the first time. It has been proven that the frequency response of a transfer function defined by Z(s) = H(s)x+iy , (3) can be computed knowing only the magnitude and the phase of the rational transfer function H(s); see Guefrachi et al. (2012). In Machado (2013), Machado used an evolutionary optimization algorithm for tuning a fractional complex order controller with conjugated order derivatives. In this paper, we consider a new structure of FCOC P IDx+iy given by kn (4) + kd Re/i sx+iy , s kp , kn and kd are the proportional, integral and derivative real constants, respectively, x and y are the real and imaginary parts of the derivative complex order. The idea is to compute parameters of the FCOC defined in (4) (x, y, kp , kn , and kd ), to fulfil the aimed performances defined by five design specifications, ensuring the controlled system robustness toward gain variation (through a desired phase margin and an open-loop flat phase curve at ωcg ) and noise (through constraints on sensibility functions). The paper is organized as follows. Section 2 presents the notations used in the following sections and some generalities about fractional differentiation. The controller design problem is stated in Section 3. The proposed design method is detailed in 4. Section 5 presents simulation results on a SOPTD resonant system. Section 6 summarizes and concludes. C (s) = kp +
2. NOTATIONS AND MATHEMATICAL BACKGROUNDS 2.1 Notations
2.2 Fractional differentiation Fractional differentiation is a generalization of differentiation to non-integer (fractional) order fundamental operators represented by a Dtα . Several definitions of this operator have been proposed in the literature, one of the most used ones is the Riemann- Louiville (RL) definition. Definition 1. The Riemann-Liouville integral of a function f (t) for an order α ∈ C, is defined by Ross et al. (1978) Zt 1 f (τ ) −α α (5) a It f (t) = a Dt f (t) = 1−α dτ, Γ (α) (t − τ ) a
with Re (α) > 0 to ensure convergence of the integral and the function f is such that on [a, t] the integral converges; see Davis (1936). Γ (.) is given by Z∞ Γ (α) = e−k k α−1 dk. (6) 0
The Riemann’s definition is obtained when a = 0, otherwise if a = −∞ Liouville’s definition is obtained. The Laplace transform of the fractional derivative of an order α of any function f (t) relaxed at t = 0 (i.e. all derivatives of f (t) are equal to zero when t 6 0) is given by (7) L {a Dtα f (t); s} = sα F (s) ; see Podlubny (1998). Consider a transfer function Z(s) = sx+iy , (8) where s can be replaced by jω and ω is the angular frequency; i ∈ Ci and j ∈ Cj . Equation (8) can be written as follows x+iy Z (jω) = (jω) . (9) Note that Cj and Ci are such that: Ci = Cj + iCj where i is the couple (0,1) of Cj × Cj . Two operations are defined here; if z = a + ib and z ′ = a′ + ib′ where a, b, a′ , b′ ∈ Cj , then z + z ′ = (a + a′ ) + i (b + b′ ) and z · z ′ = (aa′ − bb′ ) + i (a′ b + ab′ ); see Aoun (2005). Then (9) can be written as follows i h i h Z (jω) = Re/i (jω)
stands for the operator of Throughout this paper differentiation of arbitrary order , a and t are the limits (t is an arbitrary but fixed based point) and α ∈ C is the differentiation order. a Itα denotes the operator of integration of arbitrary order. Γ (.) is the Euler’s gamma function. F (s) is the Laplace transform of f (t) and s is the Laplace operator. SOPTD is the abbreviation of ”Second Order Plus Time Delay”, FCOC is the abbreviation of ”Fractional Complex Order Controller”. Cj and Ci stands for two different complex planes. Re/i [.] and Im/i [.] are the real and imaginary parts w.r.t i (in the complex plane Ci ), respectively.
x+iy
+ iIm/i (jω)
,
(10)
where
and α a Dt
x+iy
i h x x+iy = (jω) cos (y ln (jω)) , Re/i (jω)
(11)
i h x x+iy = (jω) sin (y ln (jω)) . Im/i (jω)
(12)
Note that (11) will be used to compute the frequency response of the controller defined in (4). 3. PROBLEM STATEMENT Consider a system modeled by its transfer function G (s) controlled by a controller presented by its transfer function C (s) as presented in Fig. 1.
β (s) Setpoint
+
-
with S0 the desired value of the sensitivity function for frequencies ω = ωs rad/s . • Steady-state error cancellation: Since the proposed FCOC has an integer integrator, a zero steady-state error is guaranteed.
G (s)
C (s)
Fig. 1. Unity-feedback closed-loop schema. The main purpose of this work is to design a controller that maintains a certain degree of robustness, towards gain variation, high frequency noise and output disturbance, of the system to be controlled. Note that the proposed method for the controller design is based on frequencydomain analysis. Robustness towards gain variation can be obtained by making the open-loop phase slope equal to zero at a certain angular frequency ωcg (local flatness of the phase curve). The high frequency noise rejection and the output disturbance rejection can be ensured by making certain condition on the sensibility and complementary sensibility functions. An additional design requirement can be added as the cancellation of the steady-state error. Note that the controlled open-loop is noted β (s) and defined by β (s) = C(s)G (s). Since frequency-domain analysis are used in the design method, the controller’s real and the imaginary parts need to be explicitly calculated. In this manuscript, a new structure of FCOC is presented and checked. The structure of the proposed FCOC (P IDx+iy ) is novel and, to our best knowledge, was not introduced in previous work. The designed controller has to satisfy five design specifications. Hence, the design problem is formulated as follows: • phase margin and gain-crossover frequency: The phase margin is related to the damping of the system and can be considered as an important measurement of the system stability. The equations that define the phase margin and the gain-crossover frequency are |β (jωcg )|dB = 0 dB,
(13)
ϕm = π + arg (β (jωcg )) . (14) • Robustness to variations in the gain of the plant : This can be ensured with an open-loop flat phase at ωcg that can be mathematically expressed by d arg ((β (jω))) = 0. (15) dω ω=ωcg
• High frequency noise rejection: The constraint on the complementary sensitivity function T can be as follows: T (jω) =
β (jω) 1 + β (jω)
6 T0 dB, ∀ω > ωt rad/s. dB
(16) where T0 is the desired noise attenuation for ω = ωt rad/s . • A good output disturbance rejection: We can define a constraint on the sensitivity function S as follows: 1 6 S0 dB, ∀ω 6 ωs rad/s. S (jω) = 1 + β (jω) dB (17)
where ϕm , |.| and arg (.) are the magnitude and the phase, respectively As the controller defined in (4) has five parameters, the desired specifications listed above can be satisfied. Yet, an analytical solution for these nonlinear equation is not simple to be found. In fact, a nonlinear optimization problem has to be solved: a numerical optimization algorithm must find out the optimal solution of FCOC parameters ensuring the five design requirements. For this purpose, the Matlab function fmincon, that allow to minimize a multi-variables function under constraints, is used. Note that the fmincon algorithm solution depends on the initial values of the parameters to be optimized, and may be trapped in local minima. Evolutionary algorithms (genetic algorithms) can be a solution for this problem and will be treated in future works. In our case, specification (13) is considered as the main cost function to be minimized, the remaining specifications (from (14) to (17)) are taken as constraints for the minimization problem. 4. FCOC PROPOSED DESIGN METHOD To formulate the optimization constraints, we need to compute the real and imaginary parts of the suggested FCOC (defined in (4)). By replacing s by jω in (4), we get i h kn x+iy C (jω) = kp + . (18) + kd Re/i (jω) jω Back to (11), we can write π i h π + j sin x , (19) (jω)x = ω x cos x 2 2 and cos (y ln (jω)) = RE1 + jIM1 . (20) With some trigonometric manipulations, we can obtain π RE1 = cos (y ln ω) cosh y , (21) 2 and π IM1 = − sin (y ln ω) sinh y . (22) 2 Therefore, (11) can be rewritten as i h x+iy = REd + jIMd , (23) Re/i (jω) where
REd = ω x [REd1 + REd2 ] ,
(24)
in which
π π REd1 = cos x cos (y ln ω) cosh y , 2 2 and π π REd2 = sin x sin (y ln ω) sinh y . 2 2 By the same way we have IMd = ω x [IMd1 − IMd2 ] , in which π π IMd1 = sin x cos (y ln ω) cosh y , 2 2 and π π sin (y ln ω) sinh y . IMd2 = cos x 2 2
(25) (26) (27) (28) (29)
5. SIMULATION RESULTS In order to validate the FCOC proposed design method, a simulation on a SOPTD resonant system was performed. We remind here the transfer function of such a system. kωn 2 G (s) = e−Ls 2 , (33) ωn + 2ξωn s + s2 where k, ξ and ωn denote the static gain, the damping coefficient and the undamped natural frequency of the system, respectively. The SOPTD simulation model (33) is chosen with L = 1s, k = 1, ξ = 0.2 and ωn = 0.5rad/s. For this example, whose Bode diagrams are plotted in Fig. 2, the following design specifications are required: • gain-crossover frequency, ωcg = 0.65 rad/s; • phase margin, ϕm = 45o ; • robustness to variations in the gain of the plant must be fulfilled; • noise rejection: |T (jω)|dB 6 −15 dB, ∀ω > ωt = 10 rad/s ; • sensitivity function: |S (jω)|dB 6 −15 dB, ∀ω 6 ωs = 0.042 rad/s . 0
Magnitude (dB) Phase (deg)
−1
0
10
1
10
10
2
10
0 −90
−2
−1
10
10
0
1
10 Frequency (rad/s)
10
2
10
Fig. 3. Ideal and approximated FCOC Bode diagrams. The Bode diagrams of the ideal and approximated openloop are drawn in Fig. 4. 10 Ideal open−loop Approximated open−loop
0 −10 −20 0,1
wcg=0.65
1
10
0 −100 −135 −180 −270 0,1
wcg=0.65 1 Frequency (rad/s)
10
Fig. 4. Ideal and approximated compensated open-loop Bode diagrams for the SOPTD system. As can be observed in Fig. 4, specifications of gaincrossover frequency and phase margin are respected. Moreover, the phase of the compensated system is forced to be flat at ωcg and its neighborhood. Which means system robustness with respect to to gain changes is enhanced. The magnitudes of the functions T (jω) and S (jω) are shown in Fig. 5 and Fig. 6, respectively, fulfilling the corresponding specifications.
−20 −40 −60 0,01
Phase (deg)
0
−200
0 −100 −180 −300 −400 −500 −600 −700 −800 0,01
0,1
wcg=0.65 1
10
Magnitude of ideal T(jw) Magnitude of approximated T(jw)
0 −15
0,1
wcg=0.65 1 Frequency (rad/s)
10
Magnitude (dB)
Magnitude (dB)
20
Ideal FCOC Approximated FCOC
50
200
Magintude (dB)
The expression of the controller obtained in (30) permits to compute any needed transfer function throughout the optimization problem.
100
−50 −2 10
Phase (deg)
Finally, the FCOC defined in (18) can be expressed by C (jω) = RE + jIM, (30) with RE = kp + kd REd , (31) and kn IM = − + kd IMd . (32) ω
−50
Fig. 2. Bode diagrams of the SOPTD system. We applied the proposed design method, with zero initial values of the controller parameters, and we get the following FCOC 0.2157 C (s) = 1.4228 + + 1.5042 Re/i s1.4794+i0.6484 . s (34)
0,01
0,1
1
wt=10
100
Frequency (rad/s)
Fig. 5. Magnitude of ideal and approximated T (jω).
Magnitude of ideal S(jw) Magnitude of approximated S(jw)
We used the Matlab function invfreqs to approximate only the proportional and complex derivative of the i h branches x+iy . designed controller CP D (jω) = kp +kd Re/i (jω) The Bode diagrams of the ideal and approximated FCOC are plotted in Fig. 3. Note that the approximation is performed between ωs and ωt frequencies. In addition, the approximated FCOC have to behave like a low-pass filter for ω > ωt so that the high frequency noise will not affect the control signal.
Magnitude (dB)
10
0
−10 −15
−30
ws=0.042
1 Frequency (rad/s)
10
100
Fig. 6. Magnitude of ideal and approximated S (jω).
In many experimental systems, the noise affecting the feedback is a 50Hz frequency noise (and its harmonics). For this reason, a noise signal with an amplitude equal to 0.1 and a frequency of 50Hz was added to the feedback signal. A disturbance with an amplitude equal to 0.1 was positively added to the system output. The noise rejection on the system output and on the control signal, the disturbance rejection on the signal output will be checked. The unitary step responses of the unity-feedback closedloop for three parametric states of SOPTD system are plotted in Fig. 7.
5.1 Back-calculation anti-windup method The anti-windup structure used in Fig. 9 is inspired from a P ID back-calculation anti-windup scheme; see Markaroglu et al. (2006) for more details. A feed-back loop is used to discharge the controller’s integrator if the controller output is out of the saturation limits (±10). The advantage of the back-calculation anti-windup method is that the integrator is reset dynamically and not instantaneously. Setpoint +
-
k=1 k=0.83 k=1.2
Amplitude
+
G (s)
+ sat.model
kn
1,4
1.1 1 0.95
Cpd (s) +
1 s
+
plant with input saturation
+ FCOC with anti-windup controller
0,6
Fig. 9. FCOC with anti-windup control schema.
0,4 0,2 0 0
20
30
40
60
80
100
120
140
160
Time (seconds)
The resulting step response and control signals are plotted, for the same three parametric states, in Fig. 10 and Fig. 11, respectively. 1.1 1 0.95
Through Fig. 7, we can see that the stability of the controlled system is significant for SOPTD system static gain variation (k = 1, k = 0.83, k = 1.2). In fact, zero overshoot is maintained for the three parametric states of the system, which proves the robustness of the designed controller towards gain perturbations. Knowing that the settling time of the studied system is around 28 s, Fig. 7 show that the closed-loop settling time is reduced to 21.2 s, 22.3 s and 23.6 s, for k = 1.2, k = 1 and k = 0.83, respectively.
0
−0.5 0
20
30
40
60
80
100
120
140
Fig. 10. Step responses of the controlled SOPTD system with static gain variation (k = 1, k = 0.83, k = 1.2) using antiwindup. 10 k=1 k=0.83 k=1.2
0 5
−50 −100 0
20
40
60
80
100
120
140
160
Time (seconds) Zoomed Amplitude
160
Time (seconds)
k=1 k=0.83 k=1.2
50
k=1 k=0.83 k=1.2
0.5
k=1 k=0.83 k=1.2
4
Amplitude
Amplitude
100
Amplitude
Fig. 7. Step responses of the controlled SOPTD system with static gain variation (k = 1, k = 0.83, k = 1.2).
0
−5
2 0 20
40
60
80
100
120
140
Time (seconds)
Fig. 8. Control signals of the designed FCOC with static gain variation (k = 1, k = 0.83, k = 1.2). As seen in Fig. 8,for the SOPTD system three parametric states, The controller output is too high at the transitorystate (for a very short duration comparing to the system time response). To solve this problem we inserted an antiwindup mechanism with the synthesised FCOC. Assuming that the system to be controlled has an input saturation related to an actuator, which is the case in many experimental systems, we added to the simulated SOPTD system, a saturation model with upper and lower limits equals to 10 and −10, respectively. The control schema of the FCOC with anti-windup mechanism is drawn in Fig. 9.
−10 0
20
40
60
80
100
120
140
160
Time (seconds)
Fig. 11. Control signals of the designed FCOC with static gain variation (k = 1, k = 0.83, k = 1.2) using antiwindup. Fig. 10 shows that the controlled system robustness is maintained, for system gain variation, even with the existence of control input saturation. And the control signal is within the saturation range as demonstrated in Fig. 11. The system output and the control signal were remarkably improved with the anti-windup mechanism comparing to those without saturation model. We remark a complete noise rejection on the system output, with or without control input saturation. The output disturbance was also rejected and the noise affecting the control signal was attenuated by almost 50%.
6. CONCLUSION A new Structure of Fractional Complex Order Controller P IDx+iy was introduced in this paper. With its five parameters to be calculated, up to five design specifications can be met. An optimization method that ensure the controlled system robustness towards gain variations, high frequency noise and system output disturbance, was presented. The proposed design method was validated through simulated Second Order Plus Time Delay resonant system. In order to solve the problem of the high control signal values in the transitory-state, an anti-windup mechanism was inserted with the designed FCOC, taking into account a control input saturation. The results on controlled system step response and control signal were remarkably improved. All design requirements were fulfilled and an eventual application on an experimental system was checked. ACKNOWLEDGEMENTS This work was supported by the Ministry of the Higher Education and scientific research in Tunisia. REFERENCES Aoun, M. (2005). Systemes lin´eaires non entiers et identification par bases orthogonales non entieres. Ph.D. thesis, Bordeaux 1. Chetoui, M., Malti, R., Thomassin, M., Aoun, M., Najar, S., Oustaloup, A., and Abdelkrim, M.N. (2012). EIV methods for system identification with fractional models. IFAC Proceedings Volumes, 45(16), 1641–1646. Davis, H.T. (1936). The theory of linear operators from the standpoint of differential equations of infinite order. Bloomington, Ind. Guefrachi, A., Najar, S., Amairi, M., and Abdelkrim, M. (2012). Frequency response of a fractional complex order transfer function. In 13th International conference on Sciences and Techniques of Automatic control & computer engineering, 765–773. Heymans, N. (2004). Fractional calculus description of non-linear viscoelastic behaviour of polymers. Nonlinear Dynamics, 38(1), 221– 231. doi:10.1007/s11071-004-3757-5. URL http://dx.doi.org/10.1007/s11071-004-3757-5. Jalloul, A., Trigeassou, J.C., Jelassi, K., and Melchior, P. (2013). Fractional order modeling of rotor skin effect in induction machines. Nonlinear Dynamics, 73(1), 801–813. doi:10.1007/s11071-013-0833-8. URL http://dx.doi.org/10.1007/s11071-013-0833-8. Lanusse, P., Oustaloup, A., and Sabatier, J. (2014). Robust factional order PID controllers: The first generation crone csd approach. In ICFDA’14 International Conference on Fractional Differentiation and Its Applications 2014. Li, H., Luo, Y., and Chen, Y. (2010). A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Transactions on control systems technology, 18(2), 516–520. Machado, J.T. (2013). Optimal controllers with complex order derivatives. Journal of Optimization Theory and Applications, 156(1), 2–12.
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