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Procedia Computer Science 86 (2016) 83 - 86

2016 International Electrical Engineering Congress, iEECON2016, 2-4 March 2016, Chiang Mai,

Thailand

Optimal Design of I-PD Controller for DC Motor Speed Control

System by Cuckoo Search

D. Puangdownreong^*, A. Nawikavatana, C. Thammarata

aDepartment of Electrical Engineering, Faculty of Engineering, South-East Asia University, 19/1 Petchkasem Rd., Nongkhangphlu, Nongkhaem, Bangkok 10160, Thailand

Abstract

The I-PD controller, one of the modified versions of the PID controller, was proposed for eliminating the set-point kick caused by proportional and derivative terms appeared during set point change. In this paper, the optimal I-PD controller design for DC motor speed control system by the cuckoo search (CS), one of the most efficient metaheuristic optimization techniques, is proposed. The simulation results show that the I-PD parameters can be optimized by the CS. The controlled system with the I-PD provides better responses with smaller overshoot and faster regulating time once compared to that with the parallel PID controller. © 2016 The Authors. Published by ElsevierB.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of iEECON2016 Keywords: I-PD controller; DC motor speed control; Cuckoo search; Metaheuristic optimization

1. Introduction

The proportional-integral-derivative (PID) controller was firstly proposed in 1922 by Minorsky1 and firstly applied for industrial applications in 19392. The PID controller has been played the most important role as the heart of control engineering practice in the feedback control system due to ease of use and simple realization. However, the major drawbacks of the parallel PID controllers are the effects of proportional and derivative kick (or shortly set-point kick). In order to reduce these effects, one of the modified versions of the PID controller called the I-PD controller was

* Corresponding author. Tel.: +66-2807-4500; fax: +66-2807-4530. E-mail address: deachap@sau.ac.th

1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Organizing Committee of iEEC0N2016 doi:10.1016/j.procs.2016.05.021

proposed3. By literatures, many analytical design methods for the I-PD were consecutively launched, for examples, the coefficient ratio assignment (CRA) method4 and the coefficient diagram method (CDM)5.

Recently, the control design framework has been changed to new paradigm known as the parameter optimization problem. The metaheuristic optimization search techniques have been widely applied to design the I-PD controller, such as particle swarm optimization (PSO)6 and bacterial foraging algorithm (BFA)7. Among the nature-inspired population-based metaheuristic techniques, the cuckoo search (CS) is one of the most powerful techniques proposed by Yang and Deb in 20098. The CS is very promising and could outperform existing popular population-based metaheuristic algorithms. The global convergent property of the CS algorithms has been proved9. Moreover, the CS has been applied to solve several real-world engineering problems. In this work, the CS is applied to design an optimal I-PD controller for a DC motor speed control system.

2. I-PD Control Problem Formulation

A conventional feedback control loop with the parallel PID controller is represented by the block diagram in Fig. 1, where Gp(s) and Gc(s) are the plant and the PID controller transfer functions, respectively. The PID controller receives error signal E(s) and generates control signal U(s) to controlled output C(s) and regulate disturbance D(s) referring to referent input R(s). The time-domain function of the PID controller are stated in (1), where Kp, Ki and Kd are the proportional, integral and derivative gains, respectively.

u(t)\PDD = Kpe(t) + Ki J e(t)dt + Kd ^ (1)

u(t)|,_PD = KJ e(t)dt - ^Kpc(t) + Kd d-f j (2)

Gp (s) =_3 25ai20__(3)

p 0.0001486s3 + 1.487s2 + 27.48s + 86.15

Fig. 1. Control loop with parallel PID controller. Fig. 2. Control loop with I-PD controller.

By using the parallel PID controller, a step change in the reference input R(s) will cause an immediate spiky change in the control signal U(s). This abrupt change in the controller output is known as the set-point kick (proportional and derivative kick). This kick effects rapidly change the command signal to the actuator which controls the entire operation of the plant Gp(s)3,7. The I-PD controller is developed to avoid the set-point kick and reduce undesirable overshoot. The feedback control loop with the I-PD controller is represented by the block diagram in Fig. 2. By this scheme, only the integral term Ki responds on the error signal E(s). An abrupt change in the reference input R(s) will not affect the proportional Kp and derivative Kd terms. The time-domain function of the I-PD control signal is stated in (2). In this work, the model of a DC motor was developed as stated in (3)10 conducted as the plant Gp(s) in Fig. 2.

The I-PD controller design optimization for DC motor speed control by the CS can be represented by the block diagram in Fig. 3. The sum-squared error between R(s) and C(s) is set as the objective function J expressed in (4). Designing the I-PD is to search for optimal parameters Kp, Ki and Kd in (2). J will be fed back to the CS tuning block to be minimized in order to find the optimal I-PD parameters satisfying to the inequality constraints stated in (4), where tr, Mp and Ess are rise time, maximum overshoot and steady-state error, respectively.

Initialized:

- Objective function fx), x = (x1,...,Xd)T.

- Generate initial population of n host nests Xi (i = 1, 2,...,n). while (t < MaxGen) or (termination criteria: TC)

- Get the j cuckoo randomly by Levy flight.

- Evaluate its quality by the given objective function Fi.

- Choose a nest j among n nests randomly. if (Fi < Fj)

- Replace j by the new solution.

- A fraction (pa) of worse nests are abandoned,

and build new ones at new locations via Levy flight.

- Keep best solution (or nests with quality solutions).

- Rank the solutions and find the current best. end

- Report the best solution found.

Fig. 3. CS-based I-PD controller optimization.

Fig. 4. Pseudo code of cuckoo search (CS).

J = £[r (i) - c(i)]2 i=1

t r < 0.1 sec, Mp < 10%, Ess < 0.1%,

10 < K < 100, 300 < Ki < 600, 0.1 < Kd < 2.0

Minimize

subject to tr < 0.1 sec, Mp < 10%, Ess < 0.1%, (4)

3. CS Algorithm

The CS algorithm mimics the obligate brood parasitic behaviour of some cuckoo species in combination with the Levy flight behaviour of some birds and fruit flies8,9. The CS algorithms can be summarized by the pseudo code as shown in Fig 4. New solutions x(t+1) for cuckoo i can be generated by using a Levy flight as stated in (5), where a> 0 stands for the step size. In most cases, a= 1. A symbol © means entry-wise multiplications, while a symbol Levy(A) expressed in (5) represents a Levy flight providing random walk with random step drawn from a Levy distribution having an infinite variance with an infinite mean. In the other hands, the step length s can be calculated by (6), where u and v are drawn from normal distribution. Standard deviations of u and v are also expressed in (6).

(t+1) _ x (t)

+ a® Lévy(A), Lévy « u = rÀ,(l <À< 3)

u « N(0. of), v « N(0. CTy),

T(1 + P) sin(^/?/ 2)

r[(1 + ^)/2]/02(^1)/2

4. Results and Discussion

The CS algorithm is coded by MATLAB. The CS search parameters are a priory set: the numbers of nests n = 25, the fraction pa = 0.2 and p= 1.5. The maximum generation MaxGen = 1000 is set as the TC. Designing proceeds 40 trials in order to obtain the best solution. After the search stopped, the optimal PID and I-PD parameters are successfully obtained as summarized in Table 1 with their responses, where ts is settling time and treg is regulating

time. The normalized step command and step disturbance responses are depicted in Fig. 5, while their control signals are plotted in Fig. 6. It was found that the controlled system with the I-PD controller gives the better response than those with the parallel PID with smaller overshoot and faster regulating time. The PID produces much greater setpoint kick than the I-PD does. It was noticed that the I-PD controller is more practical for real-world implementation.

Table 1. Optimal PID and I-PD controllers by CS and their corresponding responses

Controllers Parameters System responses

Kp K Kd tr(sec.) ts(sec.) Mp(%) Es.(%%) treg(sec.)

PID 59.6745 464.4714 1.2445 0.0202 0.2029 9.4367 0.0000 0.2245

I-PD 24.7789 542.7411 0.3704 0.0841 0.2141 4.2264 0.0000 0.1654

_ without controller — with PID controller ■with I-PD controller

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

- by PID controller ■ by I-PD controller -

0 0.05 0.1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 time(sec.)

Fig. 5. Speed responses by PID and I-PD controllers.

Fig. 6. Control signals of PID and I-PD controllers.

5. Conclusion

The optimal I-PD controller design for DC motor speed control system by the cuckoo search (CS) has been proposed in this paper. The I-PD has been proposed to avoid the proportional and derivative kick occurred during setpoint change and to reduce undesirable overshoot. The CS which is one of the most efficient nature-inspired population-based metaheuristic techniques has been conducted in this work to optimize the PID and the I-PD parameters. As results, the CS could provide the optimal PID and I-PD controllers within very short search time consumed. By comparison with the parallel PID controller, the controlled system with the I-PD controller provided better responses with smaller overshoot and faster regulating time as well as practical control signal.

References

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