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FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR IN PATH TRACKING DESIGN
Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19-21, 2006
FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR IN PATH TRACKING DESIGN Alexandre Poty, Pierre Melchior and Alain Oustaloup LAPS - UMR 5131 CNRS Université Bordeaux I - ENSEIRB 351 cours de la Libération - F 33405 Talence cedex - France Tél: +33 (0) 540 006 607 - Fax: +33 (0) 540 006 644 Email: [email protected]
Abstract: A new approach to path tracking using fractional differentiation is proposed in this paper. It is an extension of a previous method considering a Davidson-Cole transfer function as I/O transfer function. A methodology for synthesized a prefilter is defined using I/O Frequency Band-Limited Fractional Differentiator (FBLFD) transfer function whose two main properties are having no overshoot on the plant output and by adding numerator to have maximum control value for short time. These properties are available whatever the values of its constitutive parameters. This permits synthesis from only three parameters. The optimum settling time of the I/O transfer function is thus obtained. Transfer function synthesis requires the maximum value of the control signal by taking into account a frequential constraint. The prefilter can be implanted in the same way as a classical digital filter. Copyright © 2006 IFAC Keywords: Davidson-Cole filter, Fractional prefilter, Motion control, Path tracking, Control, Fractional systems. 1. INTRODUCTION To increase the speed of machine tools, lighter materials are used increasing their flexibility. Execution times must be optimized without exciting resonance. A prefilter is used in industrial path tracking designs, as it easy to implement and adapt for reducing overshoots. This reduces the high frequency energy of the path planning signal using a low-pass filter with trial-and-error determined parameters. Nevertheless, for classic linear prefilter approaches, when overshoots are reduced, dynamic performances are also reduced. This type of path tracking, based on position step filtering, does not permit separate control over maximal values of velocity and acceleration, which stay proportional to the amplitude of the step applied. Using a time-domain bound in the frequency domain is difficult (Horowitz, 1992). It is not therefore easy to take into account actuator
saturations in the design of prefilters. So, the prefilter often only narrows the frequency band of the control loop reference input. For the polynomial approach (Dombre, 1988)(Khalil, 1999), maximal values of velocity and acceleration can not be kept. The path completion time is thus under optimal. The Bang Bang approach (Dombre, 1988)(Khalil, 1999) takes into account the same physical constraints but does provide a minimal path completion time. However, as for the polynomial approach, the dynamics (bandwidth) of the control loop are not taken into account, so overshoots can appear on the end actuator. Cubic spline functions (order 3 piecewise polynomials) are now widely used in robotics. They are minimal curvature curves (Dombre, 1988) and the optimization proposed by Lin (Lin, 1983), or De Luca (De Luca, 1991), based on the non-linear simplex optimization algorithm (Nedler, 1965) offers a complete-path reference solution. However, as in the polynomial or Bang Bang approach, the dynamics of the control loop are not
taken into account: overshoots on the end actuator appear for small displacements. Thus, to limit actuator saturation during transitions, the actuator dynamics must be taken into account, so the above techniques are often combined with a prefilter. When the mathematical expression of the trajectory is known, and the control loop is defined perfectly, algorithms of Shin and McKay (Shin, 1985)(Shin, 1987) or Bobrow (Bobrow, 1985) allow synthesis of the optimal control which takes into account constraints on the control inputs and the details of the dynamics manipulator. The dynamic model of the process must be designed by applying Lagrange formalism. The use of curvilinear abscissa allows reduction of the number of variables without loss of information. The minimal path time is determined from the phase curve using the Pontryagyn maximum principle. However, this is fastidious and must be done for each trajectory. Furthermore, for such tasks as painting or cloth cutting, the trajectories are very complex and numerous. So, the algebraic calculus takes too long without providing tracking accuracy (Kieffer, 1997). Moreover, during this time the task will not take place. In the aerospace industry, flexible mode frequencies are well defined, but weakly damped. Here, the input shaper technique reduces vibration in path tracking design. Input shaping is obtained by convolving desired input with an impulse sequence. This generates vibration-reducing shaped command, which is more effective than conventional filters (Singhose, 1995). When the target is unknown, nonlinearity, such as saturation, causes the integral of the error to accumulate to a much larger value than in the linear case. This large integrated error, known as integral windup, causes a large percentage overshoot and a long settling time. The aim of antiwindup compensation is thus to modify the dynamics of a control loop when control signals saturate (Ohr, 1998). This technique uses a fast control loop but no prefilter: the control loop reference input signal is equal to a real time target position. The antiwindup compensation does not take into account the reference input when the trajectory is known in advance. Here, the actuator must not be saturated, so there is no need for antiwindup compensation. A recent approach to path tracking using this fractional (or non-integer order) derivative (Miller, 1993)(Oustaloup, 1995)(Samko, 1993) have been developed by Melchior (Melchior, 2000)(Melchior, 2000). With a Davidson-Cole (Davidson, 1951)(Le Mehaute, 1991) prefilter, the reference-input results from its step response. It is thus possible to limit the resonance of the feedback control loop, by a continuous variation on τ , but also on n. It permits the generation of optimal movement referenceinput, leading to a minimum path completion time, taking into account both the physical constraints of the actuators (maximum velocity, acceleration and torque) and the bandwidth of the closed-loop system. The filter can be implemented as a classical digital filter. It is synthesized in the frequency
domain, thus the power spectral density of the position permits absolute control of the high frequency energy. To separate speed and acceleration control, a Davidson-Cole speed filter has been developed allowing intermediate speed control for path tracking (Melchior, 2000). As the spline function, made up of one jerk step per point, is a reference in robotics, a Davidson-Cole jerk filter has also been developed (Melchior, 2000). However, only the control loop reference input is optimized, and not the plant output. In this paper, a method based on an I/O transfer function FBLFD type is proposed to optimize plant output. The FBLFD type I/O transfer function (figure 1) properties are having no overshoot on the plant output and by adding numerator to have maximum control value for starting time. These properties are available whatever the values of its constitutive parameters (n,τ 1,τ 2 ) which are optimized by minimizing the output settling time of the plant, by maximizing the bandwidth energy transfer between the input and the output, and including the time-domain bound on the control signal. The transmission of energy from input to control is maximized. Overshoots are avoided on the control signal by including a frequency bound on the transfer function. Finally, once the I/O transfer function and the controller have been determined the prefilter can be deduced. The remainder of this paper is divided as follow. Section 2, defines the time-domain bound in the frequency domain. Section 3 summarizes the fractional derivative filter used as a prefilter. Section 4 presents the FBLFD I/O transfer function for a filtered feedback control loop and the synthesis methodology of the prefilter. Section 5 gives simulation performances obtained using this method on a Parvex RX 120 DC motor. 2. FREQUENCY-DOMAIN SYNTHESIS With step type input references, high frequencies are not transmitted to the output, due to the low-pass character of the control loops. Thus tracking accuracy is not efficient. Also, these frequencies solicit unacceptably high velocities and accelerations from the actuators and other tools. So, to keep the control signal below its maximum value, the static reference input must be bounded. Also, during transitions, control signals overshoots must be avoided. 2.1 Definition For a Single Input - Single Output (SISO) path tracking design (figure 1), the filter, F (s ) decouples the dynamics behaviors in position control and regulation. However, the accuracy on the output position depends on the controller efficiency to reject noise and disturbances. Also, to allow the controller to reduce effects of these unexpected signals, the power spectral density of the reference input must be within the sensitivity bandwidth.
e(t) input
F (s ) prefilter
+
ε(t)
-
C(s)
controller
I/O transfer function
u(t)
G(s)
y(t) output
plant feedback
Fig. 1. Filtered unity-feedback control loop. The transfer function, A(s ) , of the filtered unityfeedback control is given by: S (s ) C (s )G (s ) A(s ) = = F (s ) , (1) 1 + C (s )G (s ) E (s ) where F (s ) , C (s ) and G (s ) are the transfer functions of the filter, the controller and the plant. Good tracking performances require that Se (s ) , the sensitivity transfer function 1 (2) S e (s ) = 1 + C (s )G (s ) be small in magnitude for small frequencies, so that effect of disturbances is attenuated. It is also required that T (s ) , the complementary sensitivity transfer function: C (s )G (s ) , (3) T (s ) = 1 + C (s )G (s ) be small in magnitude for large frequencies, so that effect of the sensor noise is attenuated, and be unity for small frequencies to follow asymptotically the reference input. The transfer function between control and input is called reference sensibility transfer function: U (s ) C (s ) A(s ) . (4) S1 (s ) = = F (s ) = E (s ) 1 + C (s )G (s ) G (s ) Final value theorem leads to a condition for having a maximum static constant value, umax , on the control signal in response to a constant signal emax applied on the prefilter input: u lim S1 (s ) = max . (5) s →0 emax Otherwise, comparison of expressions (3) and (4) gives: T (s ) . (6) S1 (s ) = F (s ) G (s ) The transfer function of the filter could also be expressed: G (s )S1 (s ) . (7) F (s ) = T (s ) As the complementary sensivity transfer function verifies: (8) lim T (s ) = 1 , s →0
the static behaviour of the filter transfer function can also be deduced: G (s ) S1 (s ) lim F (s ) = lim , (9) s →0 s →0 T (s ) and using expressions (5) and (8): u lim F (s ) = lim G (s ) max . (10) s →0 s →0 emax
It is now convenient to break down the plant transfer function into: G (s ) = G0 (s ) G1 (s ) , (11) where (12) G0 (s ) =ˆ lim G (s ) , s →0
which correspond to the static behaviour of the plant. It is also deduced from (10), that if the low frequency behavior of the plant is: K G0 (s ) = m0 , (13) s the low frequency behavior of the prefilter must be: K u lim F (s ) = m0 max . (14) s →0 s emax In addition, using relation (8), the I/O transfer function also verify: K u lim A(s ) = m0 max ; (15) s →0 s emax Thus, the low frequency integration number of the prefilter and the plant must be the same. This result is used in section 3 to fix the structure of the FBLFD I/O transfer function for the path tracking design. 2.2 Time-domain bound into frequency-domain Translation of time-domain bounds into frequencydomain bounds is a difficult problem (Horowitz, 1992). However, to keep the control signal below its maximum value during transitions, frequential criteria needs to be found on the reference sensitivity transfer function S1 (s ) . Evenif the frequency response of S1(s) is always inferior to the static value S1(0), overshoots can appear on the step response. However, with order 2 transfer function: 1 (16) H (s ) = s s² + 1+ 2 z ωn ωn ² and Cole-Cole transfer function (Le Mehaute, 1991): 1 , (17) H (s ) = 1 + (τ 1 s )n the maximum value of the frequency response magnitude is analytically linked to the first overshoot of the unit step response (Oustaloup, 1995). In these particular cases, the infinity-norm of time-domain and frequency-domain of these systems are linked: H1 ( jω ) ∞ H 2 ( jω ) ∞ s1 (t ) ∞ s2 (t ) ∞ ≥ ⇔ ≥ , (18) H1 (0 ) H 2 (0) s1 (∞ ) s2 (∞ ) where s1(t), s2(t), H1(jω) and H2(jω) are step responses and I/O frequency responses, both being order 2 or Cole-Cole systems. In these two particular cases, the reduction of the frequency response magnitude resonance is equivalent to a reduction of the unity step response first overshoot (Oustaloup, 1995). Although relation (18) is not proved for all cases, it is often true. So, to respect (14) and to avoid overshoots, we impose that S1(jω) has no resonance. Also, using relation (5), which fixes the low frequency behavior, the following frequential constraint is deduced:
u max , (19) emax is the maximum admissible value for
∀ω > 0 , S1 ( j ω ) ≤
where u max the control signal, and e max is the corresponding value of the input. Thus, overshoot on the control signal should be avoided for a step input as there is no resonance on S1 ( jω ) . This approach is similar to QFT where a magnitude frequency-domain bound is constructed for a set of transfer functions. A time-domain bound corresponding to this set of transfer functions is deduced. Finally, as in our method, the frequencydomain bound is considered to ensure the timedomain bound (Horowitz, 1992).
n
⎛τ ⎞ q(t ) = ⎜⎜ 2 ⎟⎟ [δ (t ) + ⎝ τ1 ⎠
(− 1)k Γ(n + k ) ⎛⎜ 1 − 1 ⎞⎟ ∑ k!(k − 1)! Γ(n ) ⎜⎝ τ1 τ 2 ⎟⎠ k =1 ∞
bandwidth filter described by the transmittance: n
⎛1+τ2 s ⎞ ⎟⎟ F (s ) = ⎜⎜ (20) ⎝ 1 + τ1 s ⎠ with n ∈ R et τ 2 < τ1 . The use of real poles prevents frequency resonance. The choice of identical poles allows the greatest possible energy on a given bandwidth (figure 2). F ( jω ) dB
ω [rad/s]
t
k −1
e
−
t
τ1
(21)
⎤ u (t )⎥ ⎥ ⎦
n
⎛τ ⎞ q& (t ) = ⎜⎜ 2 ⎟⎟ [u (t ) + ⎝ τ1 ⎠
(− 1)k Γ(n + k )Γ(k ) ⎛⎜ τ 2 − 1⎞⎟ ∑ ⎜τ ⎟ Γ(n ) k!(k − 1)! ⎝ 1 ⎠ k =1 10 5
⎞⎤ Γ⎛⎜ k , t ⎞⎟ ⎟⎥ τ 2⎠⎟ ⎜1 − ⎝ ⎥ ⎜ Γ(k ) ⎟⎥ ⎜ ⎟⎥ ⎝ ⎠⎦ .(22) ⎛
k⎜
1
τ1
0
Modulus [dB]
Polynomial interpolation and Bang-Bang laws have a bandwidth that varies with the length of the displacements are due to these variations. Numerical filters have a fixed bandwidth, allowing optimization in the frequency domain once and for all, and for all displacements, to limit end actuator vibration. For motion control, the FBLFD position filter generates a reference-input from the step reply of a low
k
and the velocity by
∞
3. FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR
0 dB
The FBLFD position filter methodology defines analytic profile expressions of position, speed, acceleration and their maxima, using only three parameters (n,τ 1,τ 2 ) . The position is expressed by
-5 -10
n
-15 -20
1
-25
τ2
-30 -35 -40 -2 10
closed-loop FBLFD filter
-1
0
10 10 Frequency [rad/s]
1
10
Fig. 3. Power spectral density assignment of the FBLFD filter compared to resonance frequency placement of the control loop to which it is applied. 4. FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR I/O TRANSFER FUNCTION
order 1 pole order 4 pole
Fig. 2. Pole assignment for a maximum energy in a given pass-band. The filter given by expression (20), where parameters n is real and no longer restricted to be an integer, is an FBLFD filter (Le Mehaute, 1991)(Orsoni, 2001)(Oustaloup, 1995). Thus the filter given by (20) reduces energy of the signal at high frequencies by defining bandwidth (time constants τ 1 and τ 2 ) and, through the continuous nature of the selectivity (real order n) as can be seen in figure 3. The optimization of parameters (n,τ1,τ 2 ) considers the static constraints
(Vmax , Amax , J max ) and the dynamic (ωc < ωr ) to reduce resonance.
constraints
An FBLFD transfer function step response has no overshoot for any value of (n, τ 1, τ 2 ) such as n ∈ R+ and 0 < τ 2 < τ1 . A method is now proposed using an FBLFD transfer function to optimize output plant. This method is an extension of Davidson-Cole transfer function synthesis (Orsoni, 2001). It requires an FBLFD type I/O transfer function (figure 1) to have: - no overshoot on the plant output, - maximum bandwidth energy - and maximum control at starting time. The I/O transfer function A(s ) desired can be written as: n
⎛ 1 + τ 2s ⎞ ⎟⎟ . A(s ) = ⎜⎜ ⎝ 1 + τ1s ⎠
(23)
4.1 Frequency constraint From equation (19) and (23), the frequency constraint (19), which keeps the control signals below its maximum value, becomes: ∀ω > 0,
1 1 + jτ 2ω G ( jω ) 1 + jτ1ω
n
<
umax . (24) emax
This expression is remarkable: the I/O transfer function can be design without knowing which controller is to be used; the first corner frequency of the plant limits the I/O transfer function bandwidth. The high frequency ω2 = 1 / τ 2 can be expressed
in function of (n,τ1 ) thanks to the initial value theorem: (25) lim h(t ) = lim sH (s ) . t →0
s →∞
So, applying (25) to (24), it leads to 1/ n
⎡ u ⎤ τ 2 = ⎢G ( p → ∞ ) max ⎥ τ1 . (26) emax ⎦ ⎣ Thanks to (26), only the two parameters (n,τ1 ) have to be found.
4.2 Integral gap optimization The fastest FBLFD transfer function is now to be determined. Using the frequency constraint (23) saturation of the control input signal are avoided. Integral Gap is often used to determine the dynamic performance of a step response without overshoot. The Integral Gap analytic expression for the FBLFD step response is (Oustaloup, 1995): I e = n (τ 1 − τ 2 ) . (27) Remark: if τ 2 = 0 , the Integral Gap analytical expression is for function: I e = nτ 1 .
a
Davidson-Cole
transfer
The optimal values of parameters (n, τ1, τ 2 ) are finding by the Matlab Toolbox Optimization (fmincon for example). 4.3 Prefilter design The controller and I/O transfer function being known and using relation (1), the prefilter transfer function is deduced: ⎛ 1 + τ 2 s ⎞ 1 + C (s )G (s ) ⎟⎟ F (s ) = ⎜⎜ . (28) ⎝ 1 + τ 1 s ⎠ C (s )G (s ) When the plant and /or the controller are complex transfer functions, using relation (28) leads to a very complex prefilter transfer function. Numeric problems can be avoided in the implementation of such a solution, through a graphic design using the Identification unit of CRONE software (Melchior 1999). This permits synthesis of the rational filter from its desired frequency response (Oustaloup, 2000). The ideal Bode diagram of the complex prefilter is drawn. The approximation prefilter is thus designed with n
few poles and zeros to fit the frequency magnitude and phase responses. 5. SIMULATION RESULTS The DC motor PARVEX RX 120 characteristics are given in table 1. Motor characteristic Value Inertia moment (J) 5 10-5 Kg.m2 Viscous friction (f) 4,2 10-5 m.s/rd 0,11 V.m/A Electromagnetic torque ratio (Kc) Induced inductance (L) 7,5 10-3 H Induced resistance (r) 2,5 Ω Amp/volt ratio (Ki) 1,93 A/V Maximal control (usat) 3V Table 1. PARVEX RX 120 DC motor characteristics The plant modelization and the identification of the various parameters lead to the following transfer function: K0 , (29) G (s ) = s s2 1 + 2z + 2
ωn
where K 0 = 750 rd / s / V , z = 0.09 .
ωn
ωn = 0.476 rd / s
and
5.1 Static parameters The PARVEX DC motor maximum control value is usat = 3V . (30) The controller is design so that 20% of the control signal may be used for the regulation function. The maximum value of the control signal available for the positioning function is thus: u max = 0.8u sat , (31) and the maximum desired is set to emax = 1800 rd .s −1 . (32) 5.2 Dynamic optimization The optimization according criteria (24) and constraint (27) and (umax , emax ) leads to: n = 3.4 τ1 = 1.09 s. and τ 2 = 4.35.10−1 s. .
(33)
From expression (28), the prefilter is deduced after reduction, using the Crone software Identification module (Oustaloup, 2000): F0 (s + b1 ) 2 (s + b 2 )(s2 + b3s + b 4 ) . (34) (s + a1 ) (s + a 2 ) (s + a 3 ) (s 2 + a 4s + a 5 ) where numerator and denominator coefficients are in table 2. Numerator denominator F0 = 152.8687 a1 = 10 7 FFBLFD (s )
b1 = 6.87
a2 = 1.183
b2 = 118
a 3 = 0.4871
b3 = 5.574 b4 = 16.89
a 4 = 2.844 a 5 = 2.496
Table 2. Numerator and denominator coefficients of the FBLFD prefilter
To valid the synthesis methodology, a emax = 1800 rd/s is applied. A PID controller is
(b)
150 speed [rd/s]
ω f = 100 rd/s, and phase margin Φ m = 45° . The
speed [rd/s]
designed with crossover frequency ω u = 6.57 rd/s, corner frequency for the integral action ωi = 1.77 rd/s, for high frequency filter
(a)
3000 2000
100
1000
50
following controller is obtained: 0
(35)
0
10 time [s]
control [% ]
100
50
10 5
(36)
The prefilter is also deduced after reduction, using the Crone software Identification module (Oustaloup, 2000): FDC (s ) =
0
10
15
A filtered noise is added on the feedback. The simulation is also done for Davidson-Cole I/O transfer function. The optimal parameters (n, τ ) are obtained as (Orsoni, 2001) and they are: n = 2.78 τ = 1.30 .
0
time [s]
control [% ]
(s + 1.825)(s + 1.774) . C (s ) = 87.6204 s (s + 100 )(s + 23.65)
2
F0 (s + b1 ) (s + b 2 )(s + b3s + b 4 ) . (37) (s + a1 ) (s + a 2 ) (s + a 3 ) (s + a 4 )
(s 2 + b5s + b6 ) (s 2 + b7s + b8 ) (s 2 + a 5s + a 7 )(s 2 + a 7s + a 8 ) where numerator and denominator coefficients are in table 3.
0
0
10
0
0
time [s]
10 time [s]
without prefilter with DC prefilter with FBLFD prefilter
Fig. 4. Output speed and control (a): maximum speed (V=1800 rd/s) (b): short speed (V=100 rd/s)
a 3 = 248
6. CONCLUSION The methodology of the Davidson-Cole prefilter synthesis is applied to the FBLFD prefilter. Its filteredfeedback control loop is considered as an FBLFD transfer function, which allows synthesis from three parameters. Yet, only two (n, τ1 ) are optimize because
b3 = 4.45 b4 = 19.8
a 4 = 17.6
τ 2 depends of the two others. The optimum settling
b5 = 19.6 b6 = 196
a5 = 1.039 a6 = 0.3329
time of the I/O FBLFD transfer function is thus obtained. Thanks to the frequency constraint, the maximum control value is set at the initial instant without exceeding it. The prefilter can be implemented in the same way as a classical digital filter. A simulation on a Parvex DC RX 120 motor model validates the methodology and its use for continuous optimization of fractional derivatives orders. The use of the FBLFD prefilter is also highlighted: outputs have no overshoot, without saturation on the control signal, and the maximum available control signal value is reached for short time and kept for long displacements. Considering little displacements, control value is also set to maximum for short time as long time as possible. The extra settling-time is just enough to ensure improved output and control.
Numerator
denominator
F0 = 4.90e −4
a1 = 8.13e 4
b1 = 2790
a 2 = 5000
b2 = 4.2e
b7 = 225
4
a 7 = 4.374 a 8 = 5.905
b8 = 1.562e 4
Table 3. Numerator and denominator coefficients of the DC prefilter Simulation results for a maximum speed (V=1800 rd/s) and short speed (V=100 rd/s) are respectively given in figures (4a) and (4b). The prefilter increase the settling time ( t90% ≈ 6.2s. ) but the control signal stays below its maximum value. However, the maximum value of the control signal is reached for the short time and kept. Without a prefilter, u max is not respected and greater than the maximum admissible value. Also, the two main properties of the FBLFD transfer function are respected: - no overshoot, - and at starting time, the control signal is as important as possible.
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Dombre E., Khalil W. (1988), Modélisation et commande des robots, Editions Hermès. Horowitz I. (1992), Quantitative Feedback design Theory (QFT), QFT publications. Khalil W., Dombre E. (1999), Modélisation, identification et commande des robots, Editions Hermès. Kieffer J. (1997), Cahill A.J., James M.R., Robust and Accurate Time-Optimal Path-Tracking Control for Robot Manipulators, IEEE Trans. on Robotics and Automation, vol. 13, n°6, pp. 880-890. Lin C.S., Chang P.R., Luh J.Y.S. (1983), Formulation and Optimisation of Cubic Polynomial Joint Trajectories for Industrial Robots, IEEE Transactions on Automatic Control, vol. 28, n°12, pp. 1066-1073. De Luca A., Lanari L., Oriolo G. (1991), A sensivity Approach to Optimal Spline Robot Trajectories, IEEE Transactions on Automatic Control, vol. 27,n°3, pp.535-539. Le Mehaute A., Heliodore F., Oustaloup A. (1991), Cole-Cole relaxation and CRONE relaxation, IMACS-IFAC symposium MCTS 91, Lille, France, May. Melchior P., Lanusse P., Dancla F., Cois O. (1999), Valorisation de l'approche non entière par le logiciel CRONE, CETSIS-EEA'99, Montpellier, France. Melchior P., Orsoni B., Badie Th., Robin G. (2000), Génération de consigne optimale par filtre à dérivée généralisée implicite : Application au véhicule électrique, IEEE CIFA'2000, Lille, France. Melchior P., Orsoni B., Badie Th., Robin G. (2000), Oustaloup. A., Non-Integer Motion Control: Application to an XY Cutting Table, 1st IFAC Conference on Mechatronic Systems, Darmstadt, Germany, Sept.
Miller K.S., Ross B. (1993), An introduction to the fractional calculus and fractional differential equations, Wiley-Interscience Publication. Nedler J.A., Mead R. (1965), A simplex method for function minimization, Compter Journal, vol. 7, pp. 308-313. Öhr J., Sternad M., Rönnbäck S. (1998), H2-optimal Anti-Windup Performance in SISO Control Systems, 4th SIAM Conf. on Control and its Applications, Jacksonville, USA, May. Orsoni B., Melchior P., Oustaloup A. (2001), Davidson-Cole transfer function in path tracking, 6th IEEE European Control Conference ECC'2001, pp. 1174-1179, Porto, Portugal, September 4-7. Oustaloup A. (1995), La dérivation non entière : théorie, synthèse et applications, Editions Hermès. Oustaloup A., Melchior P., Lanusse P., Cois O., Dancla F. (2000), The CRONE toolbox for Matlab, IEEE Int. Symp. on Computer-Aided Control-System Design, Anchorage, USA. Samko S.G., Kilbas A.A., Marichev O.I. (1993), Fractional Integrals and Derivatives, Gordon and Breach Science Publishers. Shin K.G., McKay N.D. (1985), Minimum-time control of robotic manipulators with geometric path constraints, IEEE Trans. on Automatic Control, vol. 30, n°6, pp. 531-541. Shin K.G., McKay N.D. (1987), Robust trajectory planning for robotic manipulators under payload uncertainties, IEEE Trans. on Automatic Control, vol. 32, n°12, pp.1044-1054. Singhose W., Singer N., Seering W. (1995), Comparison of Command Shaping Methods for Reducing Residual Vibration, 3rd European Control Conference, Rome, Italy, Sept.
FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR IN PATH TRACKING DESIGN Alexandre Poty, Pierre Melchior and Alain Oustaloup LAPS - UMR 5131 CNRS Université Bordeaux I - ENSEIRB 351 cours de la Libération - F 33405 Talence cedex - France Tél: +33 (0) 540 006 607 - Fax: +33 (0) 540 006 644 Email: [email protected]
Abstract: A new approach to path tracking using fractional differentiation is proposed in this paper. It is an extension of a previous method considering a Davidson-Cole transfer function as I/O transfer function. A methodology for synthesized a prefilter is defined using I/O Frequency Band-Limited Fractional Differentiator (FBLFD) transfer function whose two main properties are having no overshoot on the plant output and by adding numerator to have maximum control value for short time. These properties are available whatever the values of its constitutive parameters. This permits synthesis from only three parameters. The optimum settling time of the I/O transfer function is thus obtained. Transfer function synthesis requires the maximum value of the control signal by taking into account a frequential constraint. The prefilter can be implanted in the same way as a classical digital filter. Copyright © 2006 IFAC Keywords: Davidson-Cole filter, Fractional prefilter, Motion control, Path tracking, Control, Fractional systems. 1. INTRODUCTION To increase the speed of machine tools, lighter materials are used increasing their flexibility. Execution times must be optimized without exciting resonance. A prefilter is used in industrial path tracking designs, as it easy to implement and adapt for reducing overshoots. This reduces the high frequency energy of the path planning signal using a low-pass filter with trial-and-error determined parameters. Nevertheless, for classic linear prefilter approaches, when overshoots are reduced, dynamic performances are also reduced. This type of path tracking, based on position step filtering, does not permit separate control over maximal values of velocity and acceleration, which stay proportional to the amplitude of the step applied. Using a time-domain bound in the frequency domain is difficult (Horowitz, 1992). It is not therefore easy to take into account actuator
saturations in the design of prefilters. So, the prefilter often only narrows the frequency band of the control loop reference input. For the polynomial approach (Dombre, 1988)(Khalil, 1999), maximal values of velocity and acceleration can not be kept. The path completion time is thus under optimal. The Bang Bang approach (Dombre, 1988)(Khalil, 1999) takes into account the same physical constraints but does provide a minimal path completion time. However, as for the polynomial approach, the dynamics (bandwidth) of the control loop are not taken into account, so overshoots can appear on the end actuator. Cubic spline functions (order 3 piecewise polynomials) are now widely used in robotics. They are minimal curvature curves (Dombre, 1988) and the optimization proposed by Lin (Lin, 1983), or De Luca (De Luca, 1991), based on the non-linear simplex optimization algorithm (Nedler, 1965) offers a complete-path reference solution. However, as in the polynomial or Bang Bang approach, the dynamics of the control loop are not
taken into account: overshoots on the end actuator appear for small displacements. Thus, to limit actuator saturation during transitions, the actuator dynamics must be taken into account, so the above techniques are often combined with a prefilter. When the mathematical expression of the trajectory is known, and the control loop is defined perfectly, algorithms of Shin and McKay (Shin, 1985)(Shin, 1987) or Bobrow (Bobrow, 1985) allow synthesis of the optimal control which takes into account constraints on the control inputs and the details of the dynamics manipulator. The dynamic model of the process must be designed by applying Lagrange formalism. The use of curvilinear abscissa allows reduction of the number of variables without loss of information. The minimal path time is determined from the phase curve using the Pontryagyn maximum principle. However, this is fastidious and must be done for each trajectory. Furthermore, for such tasks as painting or cloth cutting, the trajectories are very complex and numerous. So, the algebraic calculus takes too long without providing tracking accuracy (Kieffer, 1997). Moreover, during this time the task will not take place. In the aerospace industry, flexible mode frequencies are well defined, but weakly damped. Here, the input shaper technique reduces vibration in path tracking design. Input shaping is obtained by convolving desired input with an impulse sequence. This generates vibration-reducing shaped command, which is more effective than conventional filters (Singhose, 1995). When the target is unknown, nonlinearity, such as saturation, causes the integral of the error to accumulate to a much larger value than in the linear case. This large integrated error, known as integral windup, causes a large percentage overshoot and a long settling time. The aim of antiwindup compensation is thus to modify the dynamics of a control loop when control signals saturate (Ohr, 1998). This technique uses a fast control loop but no prefilter: the control loop reference input signal is equal to a real time target position. The antiwindup compensation does not take into account the reference input when the trajectory is known in advance. Here, the actuator must not be saturated, so there is no need for antiwindup compensation. A recent approach to path tracking using this fractional (or non-integer order) derivative (Miller, 1993)(Oustaloup, 1995)(Samko, 1993) have been developed by Melchior (Melchior, 2000)(Melchior, 2000). With a Davidson-Cole (Davidson, 1951)(Le Mehaute, 1991) prefilter, the reference-input results from its step response. It is thus possible to limit the resonance of the feedback control loop, by a continuous variation on τ , but also on n. It permits the generation of optimal movement referenceinput, leading to a minimum path completion time, taking into account both the physical constraints of the actuators (maximum velocity, acceleration and torque) and the bandwidth of the closed-loop system. The filter can be implemented as a classical digital filter. It is synthesized in the frequency
domain, thus the power spectral density of the position permits absolute control of the high frequency energy. To separate speed and acceleration control, a Davidson-Cole speed filter has been developed allowing intermediate speed control for path tracking (Melchior, 2000). As the spline function, made up of one jerk step per point, is a reference in robotics, a Davidson-Cole jerk filter has also been developed (Melchior, 2000). However, only the control loop reference input is optimized, and not the plant output. In this paper, a method based on an I/O transfer function FBLFD type is proposed to optimize plant output. The FBLFD type I/O transfer function (figure 1) properties are having no overshoot on the plant output and by adding numerator to have maximum control value for starting time. These properties are available whatever the values of its constitutive parameters (n,τ 1,τ 2 ) which are optimized by minimizing the output settling time of the plant, by maximizing the bandwidth energy transfer between the input and the output, and including the time-domain bound on the control signal. The transmission of energy from input to control is maximized. Overshoots are avoided on the control signal by including a frequency bound on the transfer function. Finally, once the I/O transfer function and the controller have been determined the prefilter can be deduced. The remainder of this paper is divided as follow. Section 2, defines the time-domain bound in the frequency domain. Section 3 summarizes the fractional derivative filter used as a prefilter. Section 4 presents the FBLFD I/O transfer function for a filtered feedback control loop and the synthesis methodology of the prefilter. Section 5 gives simulation performances obtained using this method on a Parvex RX 120 DC motor. 2. FREQUENCY-DOMAIN SYNTHESIS With step type input references, high frequencies are not transmitted to the output, due to the low-pass character of the control loops. Thus tracking accuracy is not efficient. Also, these frequencies solicit unacceptably high velocities and accelerations from the actuators and other tools. So, to keep the control signal below its maximum value, the static reference input must be bounded. Also, during transitions, control signals overshoots must be avoided. 2.1 Definition For a Single Input - Single Output (SISO) path tracking design (figure 1), the filter, F (s ) decouples the dynamics behaviors in position control and regulation. However, the accuracy on the output position depends on the controller efficiency to reject noise and disturbances. Also, to allow the controller to reduce effects of these unexpected signals, the power spectral density of the reference input must be within the sensitivity bandwidth.
e(t) input
F (s ) prefilter
+
ε(t)
-
C(s)
controller
I/O transfer function
u(t)
G(s)
y(t) output
plant feedback
Fig. 1. Filtered unity-feedback control loop. The transfer function, A(s ) , of the filtered unityfeedback control is given by: S (s ) C (s )G (s ) A(s ) = = F (s ) , (1) 1 + C (s )G (s ) E (s ) where F (s ) , C (s ) and G (s ) are the transfer functions of the filter, the controller and the plant. Good tracking performances require that Se (s ) , the sensitivity transfer function 1 (2) S e (s ) = 1 + C (s )G (s ) be small in magnitude for small frequencies, so that effect of disturbances is attenuated. It is also required that T (s ) , the complementary sensitivity transfer function: C (s )G (s ) , (3) T (s ) = 1 + C (s )G (s ) be small in magnitude for large frequencies, so that effect of the sensor noise is attenuated, and be unity for small frequencies to follow asymptotically the reference input. The transfer function between control and input is called reference sensibility transfer function: U (s ) C (s ) A(s ) . (4) S1 (s ) = = F (s ) = E (s ) 1 + C (s )G (s ) G (s ) Final value theorem leads to a condition for having a maximum static constant value, umax , on the control signal in response to a constant signal emax applied on the prefilter input: u lim S1 (s ) = max . (5) s →0 emax Otherwise, comparison of expressions (3) and (4) gives: T (s ) . (6) S1 (s ) = F (s ) G (s ) The transfer function of the filter could also be expressed: G (s )S1 (s ) . (7) F (s ) = T (s ) As the complementary sensivity transfer function verifies: (8) lim T (s ) = 1 , s →0
the static behaviour of the filter transfer function can also be deduced: G (s ) S1 (s ) lim F (s ) = lim , (9) s →0 s →0 T (s ) and using expressions (5) and (8): u lim F (s ) = lim G (s ) max . (10) s →0 s →0 emax
It is now convenient to break down the plant transfer function into: G (s ) = G0 (s ) G1 (s ) , (11) where (12) G0 (s ) =ˆ lim G (s ) , s →0
which correspond to the static behaviour of the plant. It is also deduced from (10), that if the low frequency behavior of the plant is: K G0 (s ) = m0 , (13) s the low frequency behavior of the prefilter must be: K u lim F (s ) = m0 max . (14) s →0 s emax In addition, using relation (8), the I/O transfer function also verify: K u lim A(s ) = m0 max ; (15) s →0 s emax Thus, the low frequency integration number of the prefilter and the plant must be the same. This result is used in section 3 to fix the structure of the FBLFD I/O transfer function for the path tracking design. 2.2 Time-domain bound into frequency-domain Translation of time-domain bounds into frequencydomain bounds is a difficult problem (Horowitz, 1992). However, to keep the control signal below its maximum value during transitions, frequential criteria needs to be found on the reference sensitivity transfer function S1 (s ) . Evenif the frequency response of S1(s) is always inferior to the static value S1(0), overshoots can appear on the step response. However, with order 2 transfer function: 1 (16) H (s ) = s s² + 1+ 2 z ωn ωn ² and Cole-Cole transfer function (Le Mehaute, 1991): 1 , (17) H (s ) = 1 + (τ 1 s )n the maximum value of the frequency response magnitude is analytically linked to the first overshoot of the unit step response (Oustaloup, 1995). In these particular cases, the infinity-norm of time-domain and frequency-domain of these systems are linked: H1 ( jω ) ∞ H 2 ( jω ) ∞ s1 (t ) ∞ s2 (t ) ∞ ≥ ⇔ ≥ , (18) H1 (0 ) H 2 (0) s1 (∞ ) s2 (∞ ) where s1(t), s2(t), H1(jω) and H2(jω) are step responses and I/O frequency responses, both being order 2 or Cole-Cole systems. In these two particular cases, the reduction of the frequency response magnitude resonance is equivalent to a reduction of the unity step response first overshoot (Oustaloup, 1995). Although relation (18) is not proved for all cases, it is often true. So, to respect (14) and to avoid overshoots, we impose that S1(jω) has no resonance. Also, using relation (5), which fixes the low frequency behavior, the following frequential constraint is deduced:
u max , (19) emax is the maximum admissible value for
∀ω > 0 , S1 ( j ω ) ≤
where u max the control signal, and e max is the corresponding value of the input. Thus, overshoot on the control signal should be avoided for a step input as there is no resonance on S1 ( jω ) . This approach is similar to QFT where a magnitude frequency-domain bound is constructed for a set of transfer functions. A time-domain bound corresponding to this set of transfer functions is deduced. Finally, as in our method, the frequencydomain bound is considered to ensure the timedomain bound (Horowitz, 1992).
n
⎛τ ⎞ q(t ) = ⎜⎜ 2 ⎟⎟ [δ (t ) + ⎝ τ1 ⎠
(− 1)k Γ(n + k ) ⎛⎜ 1 − 1 ⎞⎟ ∑ k!(k − 1)! Γ(n ) ⎜⎝ τ1 τ 2 ⎟⎠ k =1 ∞
bandwidth filter described by the transmittance: n
⎛1+τ2 s ⎞ ⎟⎟ F (s ) = ⎜⎜ (20) ⎝ 1 + τ1 s ⎠ with n ∈ R et τ 2 < τ1 . The use of real poles prevents frequency resonance. The choice of identical poles allows the greatest possible energy on a given bandwidth (figure 2). F ( jω ) dB
ω [rad/s]
t
k −1
e
−
t
τ1
(21)
⎤ u (t )⎥ ⎥ ⎦
n
⎛τ ⎞ q& (t ) = ⎜⎜ 2 ⎟⎟ [u (t ) + ⎝ τ1 ⎠
(− 1)k Γ(n + k )Γ(k ) ⎛⎜ τ 2 − 1⎞⎟ ∑ ⎜τ ⎟ Γ(n ) k!(k − 1)! ⎝ 1 ⎠ k =1 10 5
⎞⎤ Γ⎛⎜ k , t ⎞⎟ ⎟⎥ τ 2⎠⎟ ⎜1 − ⎝ ⎥ ⎜ Γ(k ) ⎟⎥ ⎜ ⎟⎥ ⎝ ⎠⎦ .(22) ⎛
k⎜
1
τ1
0
Modulus [dB]
Polynomial interpolation and Bang-Bang laws have a bandwidth that varies with the length of the displacements are due to these variations. Numerical filters have a fixed bandwidth, allowing optimization in the frequency domain once and for all, and for all displacements, to limit end actuator vibration. For motion control, the FBLFD position filter generates a reference-input from the step reply of a low
k
and the velocity by
∞
3. FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR
0 dB
The FBLFD position filter methodology defines analytic profile expressions of position, speed, acceleration and their maxima, using only three parameters (n,τ 1,τ 2 ) . The position is expressed by
-5 -10
n
-15 -20
1
-25
τ2
-30 -35 -40 -2 10
closed-loop FBLFD filter
-1
0
10 10 Frequency [rad/s]
1
10
Fig. 3. Power spectral density assignment of the FBLFD filter compared to resonance frequency placement of the control loop to which it is applied. 4. FREQUENCY BAND-LIMITED FRACTIONAL DIFFERENTIATOR I/O TRANSFER FUNCTION
order 1 pole order 4 pole
Fig. 2. Pole assignment for a maximum energy in a given pass-band. The filter given by expression (20), where parameters n is real and no longer restricted to be an integer, is an FBLFD filter (Le Mehaute, 1991)(Orsoni, 2001)(Oustaloup, 1995). Thus the filter given by (20) reduces energy of the signal at high frequencies by defining bandwidth (time constants τ 1 and τ 2 ) and, through the continuous nature of the selectivity (real order n) as can be seen in figure 3. The optimization of parameters (n,τ1,τ 2 ) considers the static constraints
(Vmax , Amax , J max ) and the dynamic (ωc < ωr ) to reduce resonance.
constraints
An FBLFD transfer function step response has no overshoot for any value of (n, τ 1, τ 2 ) such as n ∈ R+ and 0 < τ 2 < τ1 . A method is now proposed using an FBLFD transfer function to optimize output plant. This method is an extension of Davidson-Cole transfer function synthesis (Orsoni, 2001). It requires an FBLFD type I/O transfer function (figure 1) to have: - no overshoot on the plant output, - maximum bandwidth energy - and maximum control at starting time. The I/O transfer function A(s ) desired can be written as: n
⎛ 1 + τ 2s ⎞ ⎟⎟ . A(s ) = ⎜⎜ ⎝ 1 + τ1s ⎠
(23)
4.1 Frequency constraint From equation (19) and (23), the frequency constraint (19), which keeps the control signals below its maximum value, becomes: ∀ω > 0,
1 1 + jτ 2ω G ( jω ) 1 + jτ1ω
n
<
umax . (24) emax
This expression is remarkable: the I/O transfer function can be design without knowing which controller is to be used; the first corner frequency of the plant limits the I/O transfer function bandwidth. The high frequency ω2 = 1 / τ 2 can be expressed
in function of (n,τ1 ) thanks to the initial value theorem: (25) lim h(t ) = lim sH (s ) . t →0
s →∞
So, applying (25) to (24), it leads to 1/ n
⎡ u ⎤ τ 2 = ⎢G ( p → ∞ ) max ⎥ τ1 . (26) emax ⎦ ⎣ Thanks to (26), only the two parameters (n,τ1 ) have to be found.
4.2 Integral gap optimization The fastest FBLFD transfer function is now to be determined. Using the frequency constraint (23) saturation of the control input signal are avoided. Integral Gap is often used to determine the dynamic performance of a step response without overshoot. The Integral Gap analytic expression for the FBLFD step response is (Oustaloup, 1995): I e = n (τ 1 − τ 2 ) . (27) Remark: if τ 2 = 0 , the Integral Gap analytical expression is for function: I e = nτ 1 .
a
Davidson-Cole
transfer
The optimal values of parameters (n, τ1, τ 2 ) are finding by the Matlab Toolbox Optimization (fmincon for example). 4.3 Prefilter design The controller and I/O transfer function being known and using relation (1), the prefilter transfer function is deduced: ⎛ 1 + τ 2 s ⎞ 1 + C (s )G (s ) ⎟⎟ F (s ) = ⎜⎜ . (28) ⎝ 1 + τ 1 s ⎠ C (s )G (s ) When the plant and /or the controller are complex transfer functions, using relation (28) leads to a very complex prefilter transfer function. Numeric problems can be avoided in the implementation of such a solution, through a graphic design using the Identification unit of CRONE software (Melchior 1999). This permits synthesis of the rational filter from its desired frequency response (Oustaloup, 2000). The ideal Bode diagram of the complex prefilter is drawn. The approximation prefilter is thus designed with n
few poles and zeros to fit the frequency magnitude and phase responses. 5. SIMULATION RESULTS The DC motor PARVEX RX 120 characteristics are given in table 1. Motor characteristic Value Inertia moment (J) 5 10-5 Kg.m2 Viscous friction (f) 4,2 10-5 m.s/rd 0,11 V.m/A Electromagnetic torque ratio (Kc) Induced inductance (L) 7,5 10-3 H Induced resistance (r) 2,5 Ω Amp/volt ratio (Ki) 1,93 A/V Maximal control (usat) 3V Table 1. PARVEX RX 120 DC motor characteristics The plant modelization and the identification of the various parameters lead to the following transfer function: K0 , (29) G (s ) = s s2 1 + 2z + 2
ωn
where K 0 = 750 rd / s / V , z = 0.09 .
ωn
ωn = 0.476 rd / s
and
5.1 Static parameters The PARVEX DC motor maximum control value is usat = 3V . (30) The controller is design so that 20% of the control signal may be used for the regulation function. The maximum value of the control signal available for the positioning function is thus: u max = 0.8u sat , (31) and the maximum desired is set to emax = 1800 rd .s −1 . (32) 5.2 Dynamic optimization The optimization according criteria (24) and constraint (27) and (umax , emax ) leads to: n = 3.4 τ1 = 1.09 s. and τ 2 = 4.35.10−1 s. .
(33)
From expression (28), the prefilter is deduced after reduction, using the Crone software Identification module (Oustaloup, 2000): F0 (s + b1 ) 2 (s + b 2 )(s2 + b3s + b 4 ) . (34) (s + a1 ) (s + a 2 ) (s + a 3 ) (s 2 + a 4s + a 5 ) where numerator and denominator coefficients are in table 2. Numerator denominator F0 = 152.8687 a1 = 10 7 FFBLFD (s )
b1 = 6.87
a2 = 1.183
b2 = 118
a 3 = 0.4871
b3 = 5.574 b4 = 16.89
a 4 = 2.844 a 5 = 2.496
Table 2. Numerator and denominator coefficients of the FBLFD prefilter
To valid the synthesis methodology, a emax = 1800 rd/s is applied. A PID controller is
(b)
150 speed [rd/s]
ω f = 100 rd/s, and phase margin Φ m = 45° . The
speed [rd/s]
designed with crossover frequency ω u = 6.57 rd/s, corner frequency for the integral action ωi = 1.77 rd/s, for high frequency filter
(a)
3000 2000
100
1000
50
following controller is obtained: 0
(35)
0
10 time [s]
control [% ]
100
50
10 5
(36)
The prefilter is also deduced after reduction, using the Crone software Identification module (Oustaloup, 2000): FDC (s ) =
0
10
15
A filtered noise is added on the feedback. The simulation is also done for Davidson-Cole I/O transfer function. The optimal parameters (n, τ ) are obtained as (Orsoni, 2001) and they are: n = 2.78 τ = 1.30 .
0
time [s]
control [% ]
(s + 1.825)(s + 1.774) . C (s ) = 87.6204 s (s + 100 )(s + 23.65)
2
F0 (s + b1 ) (s + b 2 )(s + b3s + b 4 ) . (37) (s + a1 ) (s + a 2 ) (s + a 3 ) (s + a 4 )
(s 2 + b5s + b6 ) (s 2 + b7s + b8 ) (s 2 + a 5s + a 7 )(s 2 + a 7s + a 8 ) where numerator and denominator coefficients are in table 3.
0
0
10
0
0
time [s]
10 time [s]
without prefilter with DC prefilter with FBLFD prefilter
Fig. 4. Output speed and control (a): maximum speed (V=1800 rd/s) (b): short speed (V=100 rd/s)
a 3 = 248
6. CONCLUSION The methodology of the Davidson-Cole prefilter synthesis is applied to the FBLFD prefilter. Its filteredfeedback control loop is considered as an FBLFD transfer function, which allows synthesis from three parameters. Yet, only two (n, τ1 ) are optimize because
b3 = 4.45 b4 = 19.8
a 4 = 17.6
τ 2 depends of the two others. The optimum settling
b5 = 19.6 b6 = 196
a5 = 1.039 a6 = 0.3329
time of the I/O FBLFD transfer function is thus obtained. Thanks to the frequency constraint, the maximum control value is set at the initial instant without exceeding it. The prefilter can be implemented in the same way as a classical digital filter. A simulation on a Parvex DC RX 120 motor model validates the methodology and its use for continuous optimization of fractional derivatives orders. The use of the FBLFD prefilter is also highlighted: outputs have no overshoot, without saturation on the control signal, and the maximum available control signal value is reached for short time and kept for long displacements. Considering little displacements, control value is also set to maximum for short time as long time as possible. The extra settling-time is just enough to ensure improved output and control.
Numerator
denominator
F0 = 4.90e −4
a1 = 8.13e 4
b1 = 2790
a 2 = 5000
b2 = 4.2e
b7 = 225
4
a 7 = 4.374 a 8 = 5.905
b8 = 1.562e 4
Table 3. Numerator and denominator coefficients of the DC prefilter Simulation results for a maximum speed (V=1800 rd/s) and short speed (V=100 rd/s) are respectively given in figures (4a) and (4b). The prefilter increase the settling time ( t90% ≈ 6.2s. ) but the control signal stays below its maximum value. However, the maximum value of the control signal is reached for the short time and kept. Without a prefilter, u max is not respected and greater than the maximum admissible value. Also, the two main properties of the FBLFD transfer function are respected: - no overshoot, - and at starting time, the control signal is as important as possible.
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