Least Square and Instrumental Variable System Identification of AC Servo Position Control System with Fractional Gaussian Noise Saptarshi Das, Abhishek Kumar, Indranil Pan, Anish Acharya, Shantanu Das, and Amitava Gupta
Abstract--In this paper, the classical Least Square Estimator (LSE) and its improved version the Instrumental Variable (IV) estimator have been used for the identification of an ac servo motor position control system. The data for system identification has been collected from a practical test set-up for fixed command on the final angular position of the servo motor with varying level of velocity and acceleration. The measured data is corrupted then with externally induced random noise having a Gaussian distribution, commonly known as white Gaussian noise (wGn). Performance of the LSE and IV estimators are also compared for fractional Gaussian noise (fGn) which have heavy tails in its statistical distribution and are capable of modeling real world signals having spiky nature. Index Terms--1/fα noise; ARX; fractional Gaussian noise; instrumental variable; servomotor position control; system identification.
S
I. INTRODUCTION
YSTEM identification System identification is a widely used technique in control system engineering for model based controller design. Practical measurement of process variables often gets corrupted with noise. In conventional statistical estimation theory, the random disturbances that get induced externally in the measurement are generally considered to have a zero-mean and Gaussian distribution. The LSE based linear estimators are well equipped to handle these kinds of uncorrelated random variables for parametric estimation of physical systems [1]-[2]. But the actual noise dynamics, which often occur in many physical processes, are non-Gaussian and well mimicked with a 1/fα type drooping power spectral density known as fractional Gaussian noise [3][5]. The fGn can be considered as the output of a fractional This work has been supported by the Department of Science & Technology (DST), Govt. of India under the PURSE programme. S. Das and A. Gupta are with School of Nuclear Studies and Applications, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700098, India. (e-mail:
[email protected]). A. Kumar and I. Pan are with Department of Power Engineering, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700098, India. A. Acharya is with Department of Instrumentation and Electronics Engineering, Jadavpur University, Salt Lake Campus, LB-8, Sector 3, Kolkata-700098, India. Sh. Das is with Reactor Control Division, Bhabha Atomic Research Centre, Mumbai-400085, India.
978-1-4673-0136-7/11/$26.00 ©2011 IEEE
order operator, triggered with white Gaussian noise [6]. The classical estimation based system identification theory is based on the fact that the measured input and output data has to be uncorrelated with the externally induced noise. Least square based estimators, developed with the consideration that measurement noise is white Gaussian and memory-less, will give wrong estimate for the system parameters in the presence of colored noise and even more for fGn, which has a lingering memory. In this paper, a practical test data for the position control loop of an ac servo motor is measured and corrupted with externally induced noise to compare the performance of the classical least square and its improved version namely instrumental variable estimator in the presence of fGn with long memory behavior. Estimation in the presence of non-Gaussian noise has been extensively studied in [7]-[8]. Estimation of the order of the fractional noise has been studied by Pilgram and Kaplan [9]. Other specific signal processing applications like signal detection [10], maximum likelihood estimation [11]-[12] have been carried out in the presence of 1/f family of noise. Recent applications like system identification with linear and nonlinear estimators [13], filtering [14] and denoising [15] have been attempted with the consideration of fGn in measurement and consequently extended for instrumental variable estimators in the present paper. AC servomotors are widely used in industries owing to its precise position and speed control property. It requires no maintenance because of brushless structure and control of ac servomotor has now become very easy due to advancements in power electronics. Servomotor uses feedback and works on the principle of servomechanism to precisely control position of the motor. AC servo motor is highly reliable for high speed and large torque as compared to dc servo motor. This is due to the fact that dc servo motor encounters problems at high speed and torque. Also AC servo motor responds very quickly so it can be used in analog form and it gives precise motion control due to the ease of digitization. AC servomotor has found applications in diverse fields like plastic industry, food and packaging industry, textile industry machine tool, conveying technology, printing, wood processing etc. Rest of the paper is organized as follows. Section II describes the theory of basic LSE and need for Instrumental Variable estimator. A brief description of the hardware set-up used for measurement of the position control system is presented in Section III. Identification and model reduction results for the ac servo motor position control system is
outlined in Section IV. The paper ends with the conclusion as Section V, followed by the references. II. THEORETICAL FOUNDATION FOR SYSTEM IDENTIFICATION A. Nominal Least Square Estimator and Its Limitations Let, at time event t , the input and output of an unknown system be u ( t ) and y ( t ) respectively. Then the system can simply be described by the linear difference equation (1) y ( t ) + a1 y (t − 1) + " + a n y ( t − n ) (1) = b1u ( t − 1) + " + bm u ( t − m ) The above equation can be re-written in the following form if the values of input-output data at each time step are known: y ( t ) = − a1 y (t − 1) − " − a n y ( t − n ) (2)
+ b1u ( t − 1) + " + bm u ( t − m ) The calculated value of the output can thus be represented by y ( t ) = ϕ T ( t ) ⋅ θ (3)
where, system parameters θ = [ a1 " a n b1 " bm ]T and measured input-output data T ϕ ( t ) = [ − y ( t − 1) " − y ( t − n ) u ( t − 1) " u ( t − m ) ] Hence, the prediction error becomes ε (t , θ ) = y ( t ) − ϕ T (t ) ⋅ θ
(4) (5) (6)
N
Now, from the input-output data ( Z ) over a time interval (1 ≤ t ≤ N ) the co-efficient vector θ can be calculated (7) satisfying the condition θˆ = min V (θ , Z N ) θ
N
where, Z N = {u (1), y (1), " , u ( N ), y ( N )}
(8)
1 N 2 (9) ( y (t ) − ϕ (t ) ⋅ θ ) ∑ N t =1 To find out the minima (7), the derivative of V N with respect to θ can be set to zero. i.e. d 2 N (10) V N (θ , Z N ) = ∑ ϕ ( t ) ( y ( t ) − ϕ T (t ) ⋅ θ ) = 0 dθ N t =1 V N (θ , Z N ) =
⇒
N
N
t =1
t =1
∑ ϕ (t ) y (t ) = ∑ ϕ (t )ϕ
T
( t )θ
−1
⎡ N ⎤ N (11) = ⎢ ∑ ϕ ( t )ϕ T ( t ) ⎥ ∑ ϕ ( t ) y ( t ) ⎣ t =1 ⎦ t =1 Since the sum of the squared residuals (9) is minimized using equation (10), the method is known as the Least Square algorithm for system identification [1]-[2]. Also with the known value of input and output data at each instant i.e. ϕ ( t ) vector, using relation (11) the coefficients of the discrete transfer function model i.e. θˆ can be calculated, provided the inverse in (11) exists. In most of the analysis the observed data is considered to be generated by (12) with v ( t ) being a stochastic disturbance. (12) y ( t ) = ϕ T (t ) ⋅ θ 0 + v ( t ) ⇒
θ
LS
For v ( t ) being zero mean white noise, (2) and (12) can be cast into the most common equation error model, known as the AutoRegressive eXogenous (ARX) model, (13) A ( q −1 ) y ( t ) = B ( q −1 ) u ( t ) + v ( t )
where, q − 1 is the backward shift operator. Also from (11) the difference between the estimate θˆ and the true value of θ 0 can be written as [2]: ⎡1
⎤
N
−1
⎡1
N
⎤
θ − θ 0 = ⎢ ∑ φ (t )φ T ( t ) ⎥ ⎢ ∑ φ ( t ) y (t ) ⎥ ⎣ N t =1 ⎦ ⎣ N t =1 ⎦ when N tends to infinity (14) results in (15)
(14)
(15) θ − θ 0 = ⎡⎣ E (φ ( t )φ T (t ) ) ⎤⎦ ⎡⎣ E (φ ( t ) v ( t ) ) ⎤⎦ Thus θˆ shows asymptotic bias (inconsistent estimate) unless (16) E [φ (t ) v (t ) ] = 0 −1
It is clear that (16) is satisfied if and only if v ( t ) is white noise. The LSE can be used for colored noise also if the property of the noise model is known which is unrealistic in many practical cases [1]. Thus nominal LSE and its variant ARX are applicable in case of white noise only and to remove this disadvantage of LS method, IV method was introduced. B. Instrumental Variable Technique In many cases, due to the correlation between output and disturbance, the LSE in (15) will not tend to zero. Let us consider a general correlation vector ζ ( t ) , known as the instrument such that the instrument must be correlated with the regression variables but uncorrelated with the noise. i.e. (17) a) E ⎡ζ ( t )φ T ( t ) ⎤ be nonsingular ⎣ ⎦ b) E [ζ ( t ) v ( t ) ] = 0
(18)
The above method is known as the Instrumental Variable estimate of θ and is given by (19): −1
⎡ N ⎤ ⎡ N ⎤ (19) = ⎢ ∑ ζ (t )φ T ( t ) ⎥ ⎢ ∑ ζ ( t ) y ( t ) ⎥ ⎣ t =1 ⎦ ⎣ t =1 ⎦ The most common choice of the instruments is done by linear filtering of the input. i.e. ζ ( t ) = ζ ( t , u t −1 ) (20)
θ
IV
Optimum choice of the instruments and other theoretical analysis is done in [2], [16]. However, for the present problem, the four step IV estimator (IV4) is used as recommended by Ljung [1]. The focus of the present study is to compare the performance of the ARX and IV4 estimator if the data is corrupted with fractional Gaussian noise. C. Concept of fractional Gaussian noise (fGn) Many natural processes exhibit Long Range Dependence (LRD) [3]-[4] and are more likely to deviate from the common notion of wGn which is adequately handled by the above mentioned estimators. The fGn can be viewed as the output of a fractional integrator ( 1 s α , 0 < α < 1 ), driven by continuous time stationary wGn w ( t ) with variance σ 2 . The output fGn v ( t ) can be written in the convolution form as [17]: vα ( t ) =
1
t
α −1
w (τ )( t − τ ) Γ (α ) ∫
dτ
(21)
−∞
The autocorrelation of the fGn ( vα ( t ) ) is given as: r (τ ) = σ 2
2 α −1
τ 2 Γ ( 2α ) cos απ
(22)
The autocovariance function of fGn can be represented by the associated Hurst parameters also as follows: 1 2H 2H (23) Γ (τ ) = ⎡ (τ + 1) + (τ − 1) ⎤ − τ 2 H , τ > 0 ⎦ 2⎣ where, Hurst parameter is the degree of self-similarity in the data and is related to the order of the fractional integrator as: (24) α = 2H −1 The frequency domain representation of the autocorrelation function (22) known as Power Spectral Density (PSD) falls as a power law unlike the white Gaussian noise [13]-[15]. Performance of the above mentioned estimators in the presence of fGn has been studied in the present paper. III. DESCRIPTION OF THE EXPERIMENTAL SET-UP
signal is then fed to the inverter to give controlled ac signal to drive the motor. Normally the encoder of servo motor provides feedback signal (i.e. position, speed and torque) to the position control, speed control and current control section. These control sections are monitored by MCU and results in the gating signal. This gating signal is used to drive the IGBT switch of the inverter and thus precise motion control of ac servo motor is attained. Advantech PCI-1220 Common Motion Driver has been used to control the servo system. Visual C++ code has been used for interfacing hardware with the control card to control the position of AC servo motor. The unit used for angular position, speed and acceleration is pulse per unit (ppu), ppu/s and ppu/s2 respectively. B. Data Acquisition System
A. Description of the AC Servo Motor Drive We have used microcomputer controlled AC servo system that consists of VersaMotion SERVO MOTOR (Cat. No. IC800VMM01LNKSE25A) and Versa Motion SERVO DRIVE (Cat. No. IC800VMA012-AA) [18], manufactured by GE Fanuc (Fig. 1). The servo drive can itself control the motor (as the gain of controller can be set manually in the drive) or can be connected to external controller to control the motor. The servo motor has an inbuilt incremental encoder that gives 2500 pulse per revolution (10,000 quadrature count per revolution) resulting in a resolution of 0.036 degree angular rotation. Figure 2. “S” shaped identification data for low acceleration.
Figure 1. Experimental set-up of the servo motor position control system.
In the servo drive there are three connectors viz. CN1, CN2 and CN3. CN1 is the input/output connector and is used to connect the external controller to the drive. This provides interfacing for analog speed and torque command signal, input pulse and reference voltage signal. CN2 is encoder connector and is used to connect integrated servomotor incremental encoder to drive input/output. CN3 is communication connector and is used to connect host controller via a serial communication cable. The drive used for experiment is comprised of Analog to Digital Converter (ADC), Digital to Analog Converter (DAC), control power, regenerative resistors, protection circuit and display unit. The Servo drive also has rectifier, dc link, inverter and Master Control Unit (MCU) that controls the speed, position and torque (current). In servo drive, rectification of ac signal is performed by converter and dc link. The dc link is used to remove the ripple from the converted signal. This rectified
Figure 3. Truncated ramp type identification data for high acceleration.
Data was collected for different velocity profiles with given set point i.e. with predefined position command. A customized C++ function was used to interface the PC based data acquisition system with the servo motor and record 5000 samples in 5 seconds. Data was gathered for two different sets of acceleration each associated with nine different velocities. As shown in Fig. 2-3 we get truncated ramp type position curve for an acceleration of 5×106 ppu/s2, while for acceleration 1×106 ppu/s2 we have a “S”-shaped structure. This change in the structure of position is due to fact that high acceleration the rate of increase in velocity is 5 times as compared to that at low acceleration. So the motor achieved its final velocity in much less time in case of operating at
acceleration and since integrating speed gives position so we get a truncated ramp in this case. IV. IDENTIFICATION RESULTS AND DISCUSSIONS A. Model Validation and Identification Accuracy Now, the quality of the identified models and model orders can be selected from an appropriate statistical criterion like Akaike’s Information Criterion (AIC), Final Prediction Error (FPE) etc. The statistical measure of the quality of the identified model can be judged using the AIC [1]: ⎧ ⎡1 N ⎤ ⎫ 2d (25) AIC = log ⎨ det ⎢ ∑ ε ( t , θ N )ε T (t , θ N ) ⎥ ⎬ + N N ⎣ i =1 ⎦⎭ ⎩ where, N is the number of measurement points, θ is the identified system parameters and d is the number of parameters to be identified. It is obvious that the better model should have a smaller AIC value. The position data of the ac servo motor in Fig. 2-3 is corrupted with a random variable (wGn) within the band ± 1000 ppu and the corresponding fGn with α = 0.75 . The estimation accuracies of the ARX and IV4 estimators for the identification of the position control loop at different velocity and acceleration are shown in Table I. TABLE I.
Figure 4. ARX models from corrupted data with wGn for low acceleration.
IDENTIFICATION ACCURACIES AT DIFFERENT SPEED AND ACCELERATION FOR DATA CORRUPTED WITH FGN AIC Values with Different Estimators At Low Acceleration (1×106 ppu/s2) ARX IV4
At High Acceleration (5×106 ppu/s2) ARX IV4
5×104
10.802
11.1738
12.1518
12.2073
6×104
10.801
11.0907
12.308
12.5071
4
7×10
10.7341
11.4149
12.3847
12.6219
8×104
11.0684
11.1803
12.4816
12.7986
9×104
11.003
11.6206
12.7744
13.0015
Velocity (ppu/s)
5
1×10
10.6549
11.5927
12.5175
13.1994
1.1×105
10.9403
11.8199
12.7621
13.152
1.2×105
11.3712
11.7181
12.8116
13.476
5
11.0796
11.7298
12.6754
13.4252
1.3×10
The AIC values in Table I corresponds to the identified 5th order transfer function models as studied in Chao et al. [19]. Also, from Table I, it is clear that IV4 estimates are relatively less accurate than the ARX in the presence of fGn. The identified discrete time models are converted into the continuous time models with the known value of the sampling time ( Ts = 0.1sec ) and with the Bilinear Transform or Tustin’s method, that does one to one mapping between the s ↔ z plane, i.e. (26) s = ( 2 Ts ) (1 − z − 1 ) (1 + z − 1 )
(
)
Figure 5. ARX models from corrupted data with wGn for high acceleration.
Figure 6. IV4 models from corrupted data with wGn for low acceleration.
increasing velocity is not always monotonic, especially in the presence of fGn in the data.
Figure 7. IV4 models from corrupted data with wGn for high acceleration. Figure 10. IV4 models from corrupted data with fGn for low acceleration.
Figure 8. ARX models from corrupted data with fGn for low acceleration. Figure 11. IV4 models from corrupted data with fGn for high acceleration.
B. Reduction of the Identified Models The higher order models are reduced to the First Order Plus Time Delay (FOPTD) template by minimizing the discrepancy between the H 2 -norm of the identified and reduced FOPTD models as also studied in Das et al. [21]. The sufficiency of the FOPTD template is justified from the fact that the time responses of the position control loop in Fig. 2-3 are sluggish and not oscillatory. The reduced FOPTD template is given by (27), with { K , L , T } denoting the process dc-gain, time delay and time constant respectively. i = Ke − Ls ( sT + 1) G
(27)
Figure 9. ARX models from corrupted data with fGn for high acceleration.
Minimization of the deviation in H 2 -norm is justified since it
Nyquist plots of the identified models are shown in Fig. 4-7 for wGn and Fig. 8-11 for fGn, which show that with higher velocity as if the gain of the position control system increases with no increase in overshoot [20], as the frequency response move towards the ( − 1, 0 ) point in the complex Nyquist plane,
is a measure of the energy of the output of a system, subjected to an impulse excitation which persistently excites the model to capture its delicate dynamic behaviors. i (s) (28) Here, the objective function J = G ident ( s ) − G
which is in perfect agreement with the test data shown in Fig. 2-3. Also, for identified models at high acceleration the shifting of the Nyquist diagrams towards instability with
2
Since there is no offset in tracking the position command in Fig. 2-3, only the delay ( L ) and time constants ( T ) are searched within the optimization process for the FOPTD template (27) which shows that T decreases with increased velocity for the ARX and IV identified models (Table II-III).
TABLE II. REDUCED ORDER FOPTD MODEL PARAMETERS FOR CORRUPTED DATA WITH WHITE GAUSSIAN NOISE Low Acceleration (1×106 ppu/s2) Velocity (ppu/s) 4
5×10
[7]
High Acceleration (5×106 ppu/s2)
ARX Estimator IV4 Estimator ARX Estimator IV4 Estimator T
L
T
L
T
L
T
L
1.3978
0.745
1.4939 0.918 1.6644
0.923
1.4661
1.1
6×104 1.1399
0.954
1.2112 0.766 1.3596
0.69
1.245
0.444
7×104 0.9901
0.766
1.014
0.587
1.0116 0.459
0.8658
0.801
0.8621 0.508 1.0306
0.609
0.8052 0.295
9×104 0.8396
0.71
0.6143
0.8718
0.596
0.7974 0.475
4
8×10
5
1×10
0.817 1.1169 0.33
0.7794
0.728
0.9017 0.657 0.7605
0.524
0.6693 0.602
1.1×105 0.7782
0.745
0.7968 0.486 0.7479
0.591
0.7107 0.755
1.2×105 0.7837
0.788
0.6887 0.649 0.6309
0.39
0.5875 0.432
1.3×105 0.8772
0.824
0.7446 0.353 0.6024
0.476
0.5757 0.408
TABLE III. REDUCED ORDER FOPTD MODEL PARAMETERS FOR CORRUPTED DATA WITH FRACTIONAL GAUSSIAN NOISE High Acceleration (5×106 ppu/s2)
Low Acceleration (1×106 ppu/s2) Velocity (ppu/s) 4
5×10
T
L
T
L
T
L
T
L
1.57
0.982
1.3198
1.5627
0.871
1.4211
1.18
0.843
1.0975 0.695 1.3191
0.802
1.0349
1.43
7×104 1.1138
1.05
0.8711 0.807 1.2222
0.782
0.9775
0.88
0.7819 0.987
8×10
0.8749
0.993
0.9116 0.658 1.0646
0.699
9×104
0.821
0.992
0.8503 0.676 0.8453
0.761
0.6378
1.01
1×105 0.8921
1.03
0.9294 0.638 0.7215
0.616
0.5079
1.03
1.1×105 0.9425
1.04
0.817
0.862 0.8258
0.638
0.5744 0.765
1.2×105 0.7868
0.86
0.8651 0.712 0.6349
0.631
0.4798 0.735
1.3×105 0.8873
0.927
0.8569 0.691 0.5777
0.592
0.427
Performance of the ARX and IV4 estimators are compared for the identification of the position control loop of an ac servo motor in the presence of fGn. Simple LSE based ARX estimator shows better modeling accuracy and the identified models are reduced to FOPTD template which shows consistent decrease in time constant with increased velocity. Future works may be directed towards analysis and controller design [22] for such position control systems. REFERENCES
[3]
[4]
[5]
[6]
[11]
[12]
[14]
[15]
0.763
V. CONCLUSION
[2]
[10]
ARX Estimator IV4 Estimator ARX Estimator IV4 Estimator 1.6418
[1]
[9]
[13]
6×104 1.2649 4
[8]
Lennart Ljung, “System identification: theory for the user,” PrenticeHall, Upper Saddle River, 1999. Torsten Soderstrom and Petre Stoica, “System identification”, Prentice Hall, 1989. Benoit B. Mandelbrot and John W. Vann Ness, “Fractional Brownian motion, fractional noises and applications”, SIAM Review, vol. 10, no. 4, pp. 422-437, Oct. 1968. Benoit Mandelbrot, “Some noises with 1/f spectrum, a bridge between direct current and white noise”, IEEE Transactions on Information Theory, vol. 13, no. 2, pp. 289-, April 1967. Min Shao and Chrysostomos L. Nikias, “Signal processing with fractional lower order moments: stable processes and their applications”, Proceedings of the IEEE, vol. 81, no. 7, pp.986-1010, July 1993. Chien-Cheng Tseng, Soo-Chang Pei, and Shih-Chang Hsia, “Computation of fractional derivatives using Fourier transform and
[16]
[17]
[18] [19]
[20]
[21]
[22]
digital FIR differentiator”, Signal Processing, vol. 80, no. 1, pp. 151159, Jan. 2000. Panayiotis G. Georgiou, Panagiotis Tsakalides, and Chris Kyriakakis, “Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise”, IEEE Transactions on Multimedia, vol. 1, no. 3, pp. 291-301, Sept. 1999. George A. Tsihrintzis and Chysostomos L. Nikias, “Fast estimation of the parameters of Alpha-stable impulsive interference”, IEEE Transactions on Signal Processing, vol. 44, no. 6, pp. 1492-1503, June 1996. Berndt Pilgram and Daniel T. Kaplan, “A comparison of estimators for 1/f noise”, Physica D: Nonlinear Phenomena, vol. 114, no. 1-2, pp. 108122, March 1998. Richard J. Barton and H. Vincent Poor, “Signal detection in fractional Gaussian noise”, IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 943-959, Sept. 1988. Panagiotis Tsakalides and Chrysostomos L. Nikias, “Maximum likelihood localization of sources in noise modeled as a stable process”, IEEE Transactions on Signal Processing, vol. 43, no. 11, pp. 27002713, Nov. 1995. Xinyu Ma and Chrysostomos L. Nikias, “Joint estimation of time delay and frequency delay in impulsive noise using fractional lower order statistics”, IEEE Transactions on Signal Processing, vol. 44, no. 11, pp. 2669-2687, Nov. 1996. Saptarshi Das, Sumit Mukherjee, Indranil Pan, Amitava Gupta, and Shantanu Das, “Identification of the core temperature in a fractional order noisy environment for thermal feedback in nuclear reactors”, TechSym 2011 - Proceedings of the 2011 IEEE Students' Technology Symposium, art. no. 5783821, pp. 180-186, Jan. 2011, Kharagpur. Saptarshi Das, Basudev Majumder, Indranil Pan, Amitava Gupta, and Shantanu Das, “A new fractional Fourier transform based design of a band-pass FIR filter for power feedback in nuclear reactors under noisy environment”, 2011 International Conference on Emerging Trends in Electrical and Computer Technology, ICETECT 2011, art. no. 5760199, pp. 655-660, March 2011, Nagercoil. Basudev Majumder, Saptarshi Das, Indranil Pan, Shantanu Das, and Amitava Gupta, “Denoising SPND signal by discrete wavelet analysis for efficient power feedback in regulating system of PHWRs under noisy environment”, Proceedings - 2011 2nd National Conference on Emerging Trends and Applications in Computer Science, NCETACS2011, art. no. 5751403, pp. 159-163, March 2011, Shillong. Petre Stoica, Torsten Soderstrom, and Benjamin Friedlander, “Optimal instrumental variable estimates of the AR parameters of an ARMA process”, IEEE Transactions on Automatic Control, vol. 30, no. 11, pp. 1066-1074, Nov. 1985. YangQuan Chen, Rongtao Sun, and Anhong Zhou, “An overview of fractional order signal processing (FOSP) techniques”, Proceedings of 2007 ASME Design Engineering Technical Conference, DETC ’07, Sept. 2007, Las Vegas, USA. VersaMotion Servo Motors and Amplfiers: User’s Manual, GFK-2480, GE Fanuc Intelligent Platforms, Dec. 2007. Haiyang Chao, Ying Luo, Lon Di, and Yang Quan Chen, “Roll-channel fractional order controller design for a small fixed-wing unmanned aerial vehicle”, Control Engineering Practice, vol. 18, no. 7, pp. 761772, July 2010. Suman Saha, Saptarshi Das, Ratna Ghosh, Bhaswati Goswami, R. Balasubramanian, A.K. Chandra, Shantanu Das, and Amitava Gupta, “Design of a fractional order phase shaper for iso-damped control of a PHWR under step-back condition”, IEEE Transactions on Nuclear Science, vol. 57, no. 3, pp. 1602-1612, June 2010. Saptarshi Das, Suman Saha, Shantanu Das, and Amitava Gupta, “On the selction of tuning methodology of FOPID controllers for the control of higher order processes”, ISA Transactions, vol. 50, no. 3, pp. 376-388, July 2011. Ying Luo, YangQuan Chen, and Youguo Pi, “Fractional order ultra lowspeed position servo: improved performance via describing function analysis”, ISA Transactions, vol. 50, no. 1, pp. 53-60, Jan. 2011.
