Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006

ThB11.5

Fractional Order PID Control of A DC-Motor with Elastic Shaft: A Case Study Dingy¨u Xue and Chunna Zhao

YangQuan Chen

Faculty of Information Science and Engineering CSOIS, Dept. of Electrical and Computer Engineering Northeastern University Utah State University Shenyang 110004, China Logan, UT 84322-4160, USA xuedingyu@ise.neu.edu.cn; chunnazhao@163.com yqchen@ece.usu.edu

Abstract— In this paper, a fractional order PID controller is investigated for a position servomechanism control system considering actuator saturation and the shaft torsional flexibility. For actually implementation, we introduced a modified approximation method to realize the designed fractional order PID controller. Numerous simulation comparisons presented in this paper indicate that, the fractional order PID controller, if properly designed and implemented, will outperform the conventional integer order PID controller. Index Terms— Fractional order calculus, fractional order control, PID control, servomechanism, elastic shaft, robustness.

I. I NTRODUCTION There is an increasing interest in dynamic systems of noninteger orders. Extending derivatives and integrals from integer orders to noninteger orders has a firm and long standing theoretical foundation. For example, Leibniz mentioned this concept in a letter to L Hospital over three hundred years ago and the earliest more or less systematic studies have been made in the beginning and middle of the 19th century by Liouville, Riemann and Holmgren [1]. In the literature, people often use the term “fractional order calculus”, or “fractional order dynamic system” where “fractional” actually means “non-integer”. Clearly, for closed-loop control systems, there are four situations. They are 1) IO (integer order) plant with IO controller; 2) IO plant with FO (fractional order) controller; 3) FO plant with IO controller and 4) FO plant with FO controller. In this paper, we focus on using FO-PID controller for an IO plant - “DC-Motor with elastic shaft”, a benchmark system from [2]. Intuitively, with noninteger order controllers for integer order plants, there is a better flexibility in adjusting the gain and phase characteristics than using IO controllers. This flexibility makes FO control a powerful tool in designing robust control system with less controller parameters to tune. The key point is that using few tuning knobs, FO controller achieves similar robustness achievable by using very high-order IO controllers. Since the tradeoff between the stability and other control specifications always exists, introducing fractional order control makes it more straightforward to achieve a better tradeoff. The possible advantages The work done in Northeastern University was supported by the Key Laboratory Project and the work by YangQuan Chen was supported in part by the VPR’s TCO Technology Bridging Grant of Utah State University. Corresponding author: YangQuan Chen. Tel. (435)7970148; Fax: (435)7507390.

1-4244-0210-7/06/$20.00 ©2006 IEEE

of fractional order control in modeling and control design have motivated renewed interest in various applications of fractional order control [3], [4]. Some MATLAB tools of the fractional order dynamic system modeling, control and filtering can be found in [5]. Reference [6] gives a fractional order PID controller by minimizing the integral of the squared errors. Some numerical examples of the fractional order PID control were presented in [7], [8]. In [9], a PIα controller was designed to ensure that the closed-loop system is robust to gain variations and the step responses exhibit an iso-damping property. For speed control of two-inertia systems, some experimental results were presented in [10] by using a fractional order PID controller. A comparative introduction of four fractional order controllers can be found in [11]. It is also believed that FO calculus will be the right tool for biomimetic control [12]. The major contributions of this paper include 1) A newly modified approximate realization method for fractional order derivative; 2) An extensive simulation study using a benchmark system with code available online to make our results reproducible1; 3) Show for the first time that, under the same optimization condition, the best FO PID controller outperforms the best IO PID controller. Moreover, for the first time, we show that the achieved robustness using FO PID is robust against the approximate error in FO controller realization. The remaining part of this paper is organized as follows. In Sec. II, the benchmark position servomechanism system is introduced with detail model and model parameters. In Sec. III, the fractional order PID controller and its mathematical foundation are presented. Section IV presents a new modified finite dimensional realization method for FO derivatives. In Sec. V, extensive simulation investigations of the position servomechanism controlled by optimal IO PID/PI controllers and optimal FO PID/PI controllers are presented to illustrate the superior robustness achieved by using fractional order controller. Finally, conclusions are drawn in Sec. VI. II. T HE B ENCHMARK P OSITION S ERVO S YSTEM The Benchmark position servomechanism system is a “DC-motor with elastic shaft” from [2]. To make this paper self-containing, in this section, we re-state from [2] the whole model used in our simulation, as depicted in Fig. 1 where

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1 http://mechatronics.ece.usu.edu/foc/code/acc06.zip

TABLE I S ERVOMECHANISM S YSTEM â€™ S PARAMETERS

we can see the benchmark system consists of a DC motor, a gearbox, an elastic shaft and a load. R Î¸M V

)

motor model Fig. 1.

Symbol

gearbox JM Î˛M

Ď

T

load

)

JL

Î¸L

i

The benchmark system in Fig. 1 can be represented by the following differential equations: kÎ¸ Î¸ M Î˛L Ď‰Ë™ L = âˆ’ Î¸L âˆ’ âˆ’ Ď‰L (1) JL Ď JL kT V âˆ’ kT Ď‰M Î˛M Ď‰M kÎ¸ Î¸M Î¸L âˆ’ âˆ’ (2) + Ď‰Ë™ M = JM R JM Ď JM Ď where V is the applied voltage, T is the torque acting on the load, Ď‰L = Î¸Ë™L is the loadâ€™s angular velocity, Ď‰M = Î¸Ë™M is the motor shaftâ€™s angular velocity, kÎ¸ and kT the torsional rigidity and motor constant, JM and JL the motor and nominal load inertia, Î˛M and Î˛L the motor viscous friction coefficient and load viscous friction coefficient, Ď the gear ratio and R the armature resistance. Defining the state variables as xp = [Î¸L Ď‰L Î¸M Ď‰M ]T , the above model can be converted to an LTI state-space form: âŽ¤ âŽĄ 0 1 0 0 kÎ¸ Î˛L kÎ¸ âŽĽ âŽ˘ âˆ’ 0 âŽĽ âŽ˘ âˆ’ âŽĽ âŽ˘ JL JL Ď JL xË™ p = âŽ˘ 0 âŽĽ xp 0 0 1 âŽĽ âŽ˘ 2 âŽŁ kÎ¸ kÎ¸ Î˛M + kT /R âŽŚ 0 âˆ’ 2 âˆ’ Ď JM Ď JM JM âŽ¤ âŽĄ 0 âŽ˘ 0 âŽĽ âŽĽ âŽ˘ +âŽ˘ 0 âŽĽV (3) âŽŁ kT âŽŚ RJM Î¸L = [1 0 0 0] xp

kÎ¸ 0 xp . T = kÎ¸ 0 âˆ’ Ď

The only measurement available for feedback is Î¸L . The loadâ€™s angular position must be set at a desired value by adjusting the applied voltage V . The elastic shaft has a finite shear strength, so the torque T must stay within specified limits. From an input/output viewpoint, the plant has a single input V , which is manipulated by the controller. It has two outputs, one measured and fed back to the controller Î¸L and one unmeasured T . Parameters of the experimental position servomechanism system are shown in Table. I. The designed controller must set the loadâ€™s angular position Î¸L at a given value. The elastic shaft has a finite shear strength, so the torque, T , must stay within specified limits |T | â‰¤ 78.5Nm. Also, the applied voltage must stay within the range |V | â‰¤ 220V [2].

Value (SI units)

1280.2

Ď

20

kT

10

Î˛M

0.1

JM

0.5

Î˛L

25

JL

50JM

R

20

III. F RACTIONAL O RDER PID C ONTROLLER : A N I NTRODUCTION To study the fractional order controllers, the starting point is of course the fractional order differential equations using fractional calculus. A commonly used definition of the fractional differointegral is the Riemann-Liouville definition

m t d f (Ď„ ) 1 Îą D f (t) = dĎ„ (6) a t Î“(m âˆ’ Îą) dt (t âˆ’ Ď„ )1âˆ’(mâˆ’Îą) a for m âˆ’ 1 < Îą < m where Î“(Âˇ) is the well-known Eulerâ€™s gamma function. An alternative definition, based on the concept of fractional differentiation, is the GrÂ¨unwaldLetnikov definition given by Îą a Dt f (t)

1 = lim hâ†’0 Î“(Îą)hÎą

(tâˆ’a)/h

k=0

Î“(Îą + k) f (t âˆ’ kh). (7) Î“(k + 1)

One can observe that by introducing notion of the fractional order operator a DtÎą f (t), the differentiator and integrator can be unified. Another useful tool is the Laplace transform. It is shown in [13] that the Laplace transform of an nth (n âˆˆ R+ ) derivative of a signal x(t) relaxed at t = 0 is given by: L Dn x(t) = sn X(s). So, a fractional order differential equation, provided both the signals u(t) and y(t) are relaxed at t = 0, can be expressed in a transfer function form

(4) (5)

Symbol

kÎ¸

Î˛L The benchmark position servomechanism model

Value (SI units)

G(s) =

a1 sÎą1 + a2 sÎą2 + Âˇ Âˇ Âˇ + amA sÎąmA b 1 s Î˛ 1 + b 2 s Î˛ 2 + Âˇ Âˇ Âˇ + b m B s Î˛ mB

(8)

2 where (am , bm ) âˆˆ R2 , (Îąm , Î˛m ) âˆˆ R+ , âˆ€(m âˆˆ N ). The most common form of a fractional order PID controller is the PIÎť DÎź controller [14], involving an integrator of order Îť and a differentiator of order Îź where Îť and Îź can be any real numbers. The transfer function of such a controller has the form KI Gc (s) = KP + Îť + KD sÎź , (Îť, Îź > 0). (9) s The integrator term is sÎť , that is to say, on a semi-logarithmic plane, there is a line having slope âˆ’20ÎťdB . /dec. The control signal u(t) can then be expressed in the time domain as

u(t) = KP e(t) + KI Dâˆ’Îť e(t) + KD DÎź e(t).

(10)

Clearly, selecting Îť = 1 and Îź = 1, a classical PID controller can be recovered. The selections of Îť = 1, Îź = 0,

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and λ = 0, μ = 1 respectively corresponds conventional PI & PD controllers. All these classical types of PID controllers are the special cases of the fractional PIλ Dμ controller given by (9). It can be expected that the PIλ Dμ controller may enhance the systems control performance. One of the most important advantages of the PIλ Dμ controller is the better control of dynamical systems, which are described by fractional order mathematical models. Another advantage lies in the fact that the PIλ Dμ controllers are less sensitive to changes of parameters of a controlled system [14]. This is due to the two extra degrees of freedom to better adjust the dynamical properties of a fractional order control system. However, all these claimed benefits were not systematically demonstrated in the literature. In this paper, from practical application point of view, we attempt to illustrate the benefits in a reproducible manner. It was pointed out in [15] that a band-limit implementation of fractional order controller is important in practice, and the finite dimensional approximation of the fractional order controller should be done in a proper range of frequencies of practical interest. This is true since the fractional order controller in theory has an infinite memory and some sort of approximation using finite memory must be done. In the next section, we will present a modified approximation scheme whose performance is better than Oustaloup’s method [16]. IV. A M ODIFIED A PPROXIMATE R EALIZATION M ETHOD Here we introduce a new approximate realization method for fractional derivative in the frequency range of interest [ωb , ωh ]. Our proposed method here gives a better approximation than Oustaloup’s method in both low frequency and high frequency parts. Let ⎛ s ⎞α 1+ d ⎜ ⎟ b ωb ⎟ sα ≈ ⎜ (11) s ⎝ ⎠ 1+ b d ωh where 0 < α < 1, s = jω, b > 0, d > 0. Thus

α

α bs −ds2 + d sα ≈ . 1+ 2 dωb ds + bsωh

(12)

Then, within [ωb , ωh ], using Taylor series expansion ⎛ s ⎞α 1 + d ⎟ ⎜ (dωb )α b−α b ωb ⎟ ⎜ sα ≈ s ⎠ ⎝ 1 + αp(s) + α(α−1) p2 (s) + · · · 1+ b 2 d ωh (13) where −ds2 + d p(s) = 2 . (14) ds + bsωh Truncating the Taylor series to the first order term, ⎛ 1+

α

⎜ dωb ds2 + bsωh α ⎜ s ≈ b d(1 − α)s2 + bsωh + dα ⎝ 1 +

then s ⎞α d ⎟ b ωb ⎟ s ⎠ . b d ωh (15) Note that (15) is stable iff all the poles are on the left half s-plane. It is easy to observe that the expression (15) has 3

poles. One is −bωh /d, which is negative because ωh , b, d > 0. The other two are roots of d(1 − α)s2 + asωh + dα = 0

(16)

whose real parts are negative as 0 < α < 1. Based on the well known zig-zag line approximation idea in Bode plot, let ⎛ s ⎞α 1+ d N ⎜ ⎟ 1 + s/ωk b ωb ⎟ ⎜ (17) s ⎠ = Nlim ⎝ →∞ 1 + s/ωk 1+ b k=−N d ωh

where −ωk and −ωk are zero and pole of rang k

α−2k α+2k dωb 2N +1 bωh 2N +1 ωk = , ωk = . b d Hence sα ≈ K

ds2 + bsωh d(1 − α)s2 + bsωh + dα

where

K=

dωb b

N s + ωk s + ωk

(18)

(19)

k=−N

α N ωk ω k=−N k

(20)

In this paper, we used b = 10 and d = 9. The procedures for the modified approximation can be briefly summarized in the following: • Given the frequency range [ωb , ωh ] and N • Based on the fractional order α, calculate ωk and ωk according to (18) • Compute K from (20) • Obtain the approximate rational transfer function from (19) to replace sα V. C OMPARATIVE S IMULATIONS A. Best IO PID vs. Best FO PID Simulations of position servomechanism controlled by normal PID controller and fractional order PID controller are carried out based on the parameters setting in Table. I with maximum output torque limitation ±78.5Nm. We used constrained optimization routine to search for the best controller parameters. Two optimization criteria are used. One is ITAE (integral of time-weighted absolute error) and another one is ISE (integral of squared error), where the constraint is |T | < 78.5Nm. The reference signal is the unit step function. For the optimally searched IO PID using ITAE, 21.13 − 8.26s; s For optimally searched IO PID using ISE, Gc1 (s) = 41.94 +

(21)

10.65 + 30.97s (22) s Fig. 2 shows the responses to unit step of the angular position controlled by two integer order PID controllers Gc1 (s) and Gc2 (s), respectively with the Bode plots of the open-loop controlled system shown in the same figure.

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Gc2 (s) = 110.09 +

Bode Diagram Magnitude (dB)

Step Response 1.5

Phase (deg)

0.5

50

50

40

0

30

−50

20

−100 360

10 0 1.5

180 0

6 4 2

1.3

1.1

0.9

−180

0 0

20

40

−360 −2 10

60

Time (sec)

(a) Step responses Fig. 2.

10

−1

0

10 Frequency (rad/sec)

10

1

0.01 − 31.6s0.6 (23) s0.7 while the best ISE is 0.87 and the corresponding fractional order controller is 91.95 Gc (s) = 61.57 + 0.5 + 2.33s0.6 . (24) s The step responses are compared in Fig. 3 with corresponding Bode plots. Magnitude (dB) 0

10

20

30 Time (sec)

40

(a) Step responses Fig. 3.

50

60

0.5 0.5

0.7

0.9

1.1

1.3

1.5

8 6

40

50

4

20

2

0 1.5

0 1.5

0

1.3

−50

1.1

0.9

0.7

0.5 0.5

0.7

0.9

1.1

1.3

1.5

1.3

1.1

0.9

(a) ITAE(λ,μ)

180

0.7

0.5 0.5

0.7

0.9

1.1

1.3

1.5

(b) ISE(λ,μ)

0

Fig. 5.

−180 −360 −2 10

10

−1

0

10 Frequency (rad/sec)

10

1

10

Searching the best fractional orders (N = 6)

2

(b) Bode plots

Best FO PID Controllers. Solid line: ITAE; Broken line: ISE

The observation is clear. The best FO PID performs better than the best IO PID. This is not surprising but this may not be fair since FO PID has two more extra parameters in optimal search. B. How To Decide λ and μ? In the last section, we got a flavor that FO PID performs better in side by side comparison. We simply fixed λ = 0.5 and μ = 0.6. But in reality, how to decide these two magic orders? To our best knowledge, this research question is not well answered in the literature. In this paper, we only show a brutal force search result to partially justify why we fixed λ = 0.5 and μ = 0.6. Here, we build two tables of optimal ITAE and ISE, respectively, with respect to λ and μ which are enumerated from 0.5 to 1.5 with step of 0.1. In other words, we did 2 × 11 × 11 optimal searches. These two tables are visualized in Fig. 4. Note that, in this investigation, we used the approximate order N = 4. It is unfortunate to observe that there is no definite relationship can be established in Fig. 4. However,

We conclude that, the difference between Fig. 4 and Fig. 5 is very small. To illustrate, let us run two examples with an emphasis on the effect of different approximation order N . Fig. 6 suggest that the changes in N will not contribute to the differences in this benchmark problem. Step Response

Step Response

1.2

Amplitude

Phase (deg)

0 −0.2

0.7

10

60

−150 360

0.2

0.9

(b) ISE(λ,μ)

80

−100

0.4

1.1

C. Which N Is Good Enough? In implementing FO PID, we need to decide what the finite order is for the finite order approximation. In our case, we need to decide the N . Let us first repeat the Fig. 4 using N = 6.

Bode Diagram

0.6

1.3

in general, we can qualitatively tell that, the integer case λ = 1 and μ = 1 is not optimal. In other words, the optimal case most like corresponds to noninteger case. Moreover, we can tell that, in this benchmark system, we prefer low order integral and lower order derivative.

100

0.8

1.1

Searching the best fractional orders (N = 4)

Fig. 4.

Gc3 (s) = 135.12 +

1

0.9

0 1.5

(a) ITAE(λ,μ)

Best IO PID Controllers. Solid line: ITAE; Broken line: ISE

1.2

0.5 0.5

0.7

1.5

2

(b) Bode plots

Step Response

0.7

1.3

10

Now let us look at the best FO PID controllers. As the first attempt, let us first fix λ = 0.5 and μ = 0.6. Doing the numerical search, we get the best ITAE of 2.22 and the corresponding fractional order PID controller is

Amplitude

8

1.2

1

1

0.8

0.8 Amplitude

Amplitude

1

100

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2 0

10

20

30 Time (sec)

40

(a) N = 2 Fig. 6.

50

60

0

10

20

30 Time (sec)

40

50

60

(b) N = 6

Step responses comparisons with different N ’s

D. Robustness Against Load Variations In the last subsection, we have already seen the robustness with respect to N . Here, the position servomechanism control system is controlled by the best fractional order PID controller (23) and the best IO PID controller (22) when load changes ±50%. The results are summarized in Fig. 7. The robustness against load variations is clearly seen from Fig. 7.

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Step Response 1.2

1

1

0.8

0.8 Amplitude

0.6

198.26 . (28) s0.2 Fig. 10 summarizes the comparison of step responses and Bode plots. Gc (s) = −48.38 +

0.6

0.4

0.4

0.2

0.2

0

0

−0.2

−0.2 0

10

20

30 Time (sec)

40

50

60

0

(a) +50% load variation

10

20

30 Time (sec)

40

50

60 Step Response

(b) −50% load variation

Bode Diagram 50

1.4 Magnitude (dB)

Amplitude

However, if we use ISE, we should choose λ = 0.2. After search, the optimal fractional order PI controller becomes

Step Response

1.2

1.2

Fig. 7. Step responses comparison with N = 4 (Dotted: Best PID; Solid line: Best FO PID)

Amplitude

1

−150 360 Phase (deg)

0.4

3.36 . s Fig. 8 shows the step responses and the Bode plots. Gc (s) = 106.82 +

0

Fig. 10.

Phase (deg)

0.4 0.2 0 0

10

20

30 Time (sec)

40

50

60

1.2

1.2

0.4

−180

0

0.2 0 0

−270

0

1

10

10

2

0.8

0.2

10 Frequency (rad/sec)

10

1

0.8

−150 −90

10

20

30 Time (sec)

40

50

(a) +50% load variation

2

60

0

10

20

30 Time (sec)

40

50

60

(b) −50% load variation

Fig. 11. Step responses comparison with N = 4 (Dashed line: Best IO PI; Solid line: Best FO PI

Best IO PI Controllers. Solid line: ITAE; Broken line: ISE

1.05

11 10

1

9 8

0.95

7 6

0.9

5 4

0.85

3 0.2

1.4

0.6

−1

1

10

Step Response

1.4

0.4

10

0

10 Frequency (rad/sec)

Best FO PI Controllers. Solid line: ITAE; Broken line: ISE

0.6

−360 −2 10

−1

10

(b) Bode plots

−50

(b) Bode plots

12

0

60

−100

Similar to Fig. 4, we can draw Fig. 9 which provides the basis for λ selection.

2

50

1

0

(a) Step responses Fig. 8.

40

Step Response

0.8 0.6

30 Time (sec)

The observation is again clear. The best FO PI performs better than the best IO PI. This is again not surprising but this may not be fair since FO PI has an extra parameter in optimal search. When load increases ±50%, the corresponding results are summarized in Fig. 11. Again the robustness against load variations can be observed.

Amplitude

Magnitude (dB)

1

20

0 −180

(a) Step responses

Bode Diagram

1.2

10

180

−360 −2 10

0

50

1.4

Amplitude

0.2

Amplitude

Step Response

(26)

−100

0.8 0.6

E. FO PI Controllers It will be interesting to check if we can see the case that “the best FO PI works better than the best IO PI”. We repeat what we performed for PID controllers and summarize briefly in the following. Under the ITAE criterion, the following optimal IO PI controller is searched 0.14 Gc (s) = 107.35 + (25) s and by the ISE criterion,

0 −50

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.8

0

(a) ITAE(λ) Fig. 9.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(b) ISE(λ)

From Fig. 9, if we use ITAE, we should choose λ = 0.05. Then, the optimal fractional order PI controller is 72.3 . s0.05

Using square wave as the reference input signal (period T = 40 sec.) and adding Coulomb friction 0.1, the output responses of the controlled angular position are shown in Fig. 12. Similarly, we simulated the case with backlash, with the deadband width of 0.5. The symmetrical square wave position tracking responses are compared in Fig. 13. Finally, we checked the case of deadzone with its parameter set as ±0.5. In this case, the responses are compared in Fig. 14. So, we can observe that best FO PID controllers perform much better than their IO counterparts. G. Robustness to Elasticity Parameter Change

Searching the best fractional order (N = 4)

Gc (s) = 39.82 +

F. Robustness to Mechanical Nonlinearities

(27)

Finally, we are interested in checking the robustness with respect to the changes of the elasticity parameter kθ in Table 1. When kθ varies ±50%, the corresponding results are summarized in Figs. 15 and 16, respectively. We can again observe that the best FO controllers perform better than the best IO controllers.

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Step Response

1

1

0.5

0.5

0.8

0

0

−0.5

−0.5

−1

−1

Step Response

1 Amplitude

1.5

1

Amplitude

1.5

0.6

0.8 0.6

0.4 0.4

0.2

−1.5 0

20

40

60

80

100

−1.5 0

20

(a) best PID

40

60

80

−0.2

100

(b) best PI

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5 0

−1.5 0

20

40

60

80

100

20

(a) best PID

40

60

80

100

Fig. 13. Responses comparison with backlash (Dashed line: Best IO Controllers; Solid line: Best FO Controllers 1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5 0

−1.5 0

20

40

60

80

100

(a) best PID

20

40

60

80

100

(b) best PI

Fig. 14. Responses comparison with deadzone (Dashed line: Best IO Controllers; Solid line: Best FO Controllers Step Response

Step Response

1

1

0.8

Amplitude

Amplitude

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 −0.2 0

20

Time (sec)

40

(a) best PID

60

0 0

20

Time (sec)

20

Time (sec)

40

60

0

0

40

20

Time (sec)

40

60

(b) best PI

Fig. 16. Responses comparison when kθ decreases 50% (Dashed line: Best IO Controllers; Solid line: Best FO Controllers

the control performance in any noticeable amount. With the rapid development of computer performances, the realization of fractional order control systems also became possible and much easier than before. Despite fractional order control’s promising aspects in modeling and control design, fractional order control research is still at its primary stage. The notable future research is to develop tuning rules for FO PID and in particular on tuning the fractional orders.

(b) best PI

1.5

0

(a) best PID

Fig. 12. Responses comparison with Coulomb friction (Dashed line: Best IO Controllers; Solid line: Best FO Controllers 1.5

0.2

0

60

(b) best PI

Fig. 15. Responses comparison when kθ increases 50% (Dashed line: Best IO Controllers; Solid line: Best FO Controllers

VI. C ONCLUSIONS In this paper, a fractional order PID controller is examined on a benchmark position servomechanism control system considering actuator saturation and the shaft torsional flexibility. Using numerical optimization, numerous simulation comparisons presented in this paper indicate that, the fractional order PID controller, if properly designed and implemented, will outperform the conventional integer order PID controller. It was shown that the best FO PID works better than IO PID. For actually implementation, we introduced a modified approximation method to realize the designed fractional order PID controller. We used simulation to illustrate that the order the approximation does not affect

R EFERENCES [1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York and London; 1974. [2] Mathworks Inc., Model Predictive Control Toolbox User’s Guide. [Online]: http://www.mathworks.com/access/helpdesk/ help/toolbox/mpc/ [3] J. A. Tenreiro Machado, Theory analysis and design of fractionalorder digital control systems, Journal Systems Analysis Modeling Simulation, Vol. 27, 1997, pp. 107-122. [4] A. Oustaloup, J. Sabatier and X. Moreau, From fractal robustness to the CRONE approach, ESAIM, Vol. 5, 1998, pp. 177-192. [5] Dingy¨u Xue and YangQuan Chen, Advanced Mathematic Problem Solution Using MATLAB, Beijing: Tsinghua University Press; 2004. [6] Z.F. Lv, Time-domain simulation and design of SISO feedback control systems, Doctoral Dissertation, National Cheng Kung University, Taiwan, China, 2004. [7] R. Caponetto, L. Fortuna, and D. Porto, “A new tuning strategy for a non integer order PID controller,” IFAC2004, Bordeaux, France, 2004. [8] Chunna Zhao, Dingy¨u Xue and Yangquan Chen, “A fractional order PID tuning algorithm for a class of fractional order planst,” Proc. of the IEEE International Conference on Mechatronics and Automation, Niagara Falls, Canada, 2005,pp. 216-221. [9] C. A. Monje, B. M. Vinagre, Y. Q. Chen, V. Feliu, P. Lanusse and J. Sabatier, “Proposals for fractional PIλ Dμ tuning”, The First IFAC Symposium on Fractional Differentiation and its Applications 2004, Bordeaux, France, July 19-20, 2004. [10] C. B. Ma and Y. Hori, “Design of fractional order PID controller for robust two-inertia speed control to torque saturation and load inertia variation,” IPEMC 2003, Xi’an, China,2003. [11] Dingy¨u Xue and YangQuan Chen, “A comparative introduction of four fractional order controllers,” Proceedings of the 4th World Congress on Intelligent Control and Automation, Shanghai, China, 2002,pp. 32283235. [12] YangQuan Chen, Dingy¨u Xue and Huifang Dou. “Fractional calculus and biomimetic control”. In Proc. of the First IEEE Int. Conf. on Robotics and Biomimetics (RoBio04), pages robio2004 347, Shengyang, China, August 2004. IEEE. [13] I. Podlubny, Fractional-order Systems and Fractional-order Controllers, The Academy of Sciences Institute of Experimental Physics, UEF-03-94, Kosice, Slovak Republic, 1994. [14] I. Podlubny, Fractional-order systems and PIλ Dμ -controllers, IEEE Trans. Automatic Control, vol.44,1999,pp.208-214. [15] A. Oustaloup, F. Levron, B. Mathieu, and F.M. Nanot, “Frequencyband complex noninteger differentiator: characterization and synthesis”, IEEE Trans. Circuits Syst. I, vol. 47, 2000,pp.25-39. [16] Xue Dingy¨u, Chunna Zhao, and YangQuan Chen, “A modified approximation method of fractional order systems”, submitted to IEEE International Conference on Mechatronics and Automation, 2006, Luoyang, China.

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