design and implementation of fractional order pid controller for aerofin ...

DESIGN AND IMPLEMENTATION OF FRACTIONAL ORDER PID CONTROLLER FOR AEROFIN CONTROL SYSTEM VENU KISHORE KADIYALA1, RAVI KUMAR JATOTH2, SAKE POTHALAIAH3 1 DEPARTMENT OF MECHANICAL ENGG, 2,3DEPARTMENT OF ECE 1,2 NATIONAL INSTITUTE OF TECHNOLOGY WARANGAL 3 SRI INDU COLLEGE OF ENGG & TECHNOLOGY-HYDERABAD INDIA [email protected], [email protected], [email protected]

Abstract— This paper proposes the tuning of Fractional

comparing the results of PSO based tuning of FOPID controllers with the other conventional tuning methods. Dynamic systems based on fractional order calculus have been a subject of extensive research in recent years since the proposition of the concept of the fractionalorder PI λ D μ controllers and the demonstration of their effectiveness in actuating desired fractional order system responses by Podlubny[3].Classical optimization techniques cannot be used here because of the roughness of the objective function surface [4]. We, therefore, use a derivative-free optimization technique –– particle swarm optimization (PSO) originally devised by Kennedy and Eberhart [5].

Order PID (FOPID) controller of electromagnetic actuator (EMA) system for aero fin control (AFC) using Particle swarm optimization (PSO). The EMA is realized with permanent magnet brush DC motor which is driven by a constant current driver. Using the non-linear model of EMA-AFC system which includes the non-linearities of DC motor, an FOPID position controller is designed using different soft computing techniques like PSO in SIMULINK so that the system satisfies all the design requirements. In this paper, we proposed PSO based FOPID controller which is tuned by using PSO algorithm. The design parameters which are to be optimized are rise time, peak time and percentage overshoot. Presented results show that the transient response and closed loop response of EMA-AFC system using PSO based tuning of FOPID is better when compared to that of conventional methods and untuned system.

Electromagnetic actuation is becoming more useful in aerospace industry as more importance is placed on its maintainability. Electromagnetic actuators (EMAs) are being used in the actuation of flight critical control surfaces and in thrust vector control [6]. A good understanding of the dynamic properties of these actuators is critical in their successful application [7]. The EMAAFC system in this application is driven by a permanent magnet brush DC motor. But in this application we developed a constant current motor driver. The control signal was pulse width modulated (PWM), clipping it to maximum allowed current. Physical implementation of such actuator is usually simpler and cheaper than the conventional voltage driver. In this application, we introduced the SIMULINK model of the EMA-AFC system, which is a real time implementation. The simulink model also takes into account non-linearities due to mechanical limitations of the limited motor torque and angular velocity, fin deflection, friction in gears and bearing, backlash in gears and lever mechanism, etc. The effects produced by these non linearities can be easily studied by non linear SIMULINK model. Using the non linear SIMULINK model, an FOPID controller for the EMA-AFC system has been designed.

keywords- FOPID controller, Electromagnetic actuator, Current motor driver, PWM, PID control, Simulink, Aero fin control, Particle swarm optimization

I.

INTRODUCTION

PID controllers have been used for several decades in industries for process control applications .The reason for their wide popularity lies in the simplicity of design and good performance including low percentage overshoot and small settling time for slow process plants [1]. The performance of the PID controllers can be improved by making use of fractional order derivatives and integrals. This greatest flexibility makes us possible to design more robust control system. In fractional order PID (FOPID) controller, the integral and derivative orders are usually fractional. In FOPID besides Kp, Ki, Kd we have two more parameters λ and μ , the integral and derivative orders respectively. If λ =1 and μ =1, then it becomes integer PID. The five parameters Kp, Ki, Kd, λ , μ are to be optimized in five-dimensional hyper-space to obtain an optimal solution that satisfies all the user specifications. It is necessary to understand the theory of fractional calculus in order to understand the significance of FOPID controller [2]. This paper mainly focuses on the better way of tuning by

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numbers). If λ =1 and μ =1, it becomes PID controller with only three parameters. Therefore FOPID controller generalizes the PID controller and expands it from point to plane, as shown in fig.2 providing more flexibility to controller design and we can control the real processes more accurately.

II) PROBLEM DESCRIPTION

Figure 1: Block Diagram of PSO tuned FOPID controller Here, we consider the plant model as EMA-AFC system. When the missile is moving, the disturbances which can affect its motion are aerodynamic forces, friction torque and viscous forces and measurement noises also affect the system. In presence of these noises, the missile’s trajectory has to be maintained. The role of the FOPID controller is to minimize the error that is caused due to the noises. The function which is to optimised in PSO tuned FOPID controller is given by f = α ..R + β .abs ( po) + γ ..S + δ .e ss (1) where R is the rise time po is the percent overshoot S is the settling time ess is the steady-state error

Fig.2: Generalization of PID controller from point to plane The design requirements like rise time, maximum peakovershoot, settling time , steady state error depends on five parameters Proportional constant (Kp), integral proportional constant (Ki),derivative constant (Kd), λ and μ . IV. AERO FIN CONTROL SYTEM

α , β , γ , δ are proportionality constants This function mainly depends on the design specifications required by the user. The value of these proportionality constants depends upon the ranges of the design specifications and also depends on, which design specifications are mainly considered. Suppose if the rise time is mainly considered, then α = 100 and other proportionality constants are taken as 1(if all the design requirements are of same ranges).The function is then optimised using the standard PSO algorithm and the best fit value is obtained as output.

Aerofin control system considered here consists of control of missile using four grid fins. The grid fins configuration is shown in figure1.By deflecting grid fins, moments are generated about the center of mass, which in turn rotates the missile. The resulting incidence angles generate aerodynamic forces, which accelerate the vehicle in the desired direction [8].

III. FOPID CONTROLLER To construct a more robust control system, the integral and differential parts should be of fractional order. The differential equation of the FOPID controller is

u (t ) = K p e(t ) + D − λ e(t ) + D μ e(t )

(2)

The transfer function of the FOPID controller is represented by

K p + K i s −λ + K d s μ Where λ and μ are the orders of integration and differentiation respectively (both are any positive real

Figure 3: Aerofin Control System

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In the figure shown above, the missile auto pilot sends roll, pitch and yaw commands ( ∂ x , ∂ y and∂ z ) to the AFC system. Before they are to be implemented, they have to be separated into individual fin commands, i.e. angles α i where i=0, 1, 2, 3. Each actuator module should contain tight, independent position control of surface deflection, usually less than 10 degrees [9]. V. PARTICLE SWARM OPTIMIZATION PSO is population based on stochastic optimization technique inspired by social behavior of bird flocking or fish schooling etc. The swarm of particles indicates estimation of multiple parameters involved in the problem. We can begin with initializing a random swarm of particles. During each iteration fitness of the particle is evaluated with the help of fitness of each function. The algorithm progressively replaces most fit parameters of each particle i.e. pbest. Pbest is the best position of particle itself. There exists another best position gbest which is the global gbest i.e. the best position in the swarm. Each particle has the influence of these two bests in their trajectories [10].The parameters of each particle are updated with the following equations.

The trajectory of the particle is dependent on three factors: its previous position, pbest and gbest. Greater the strain of particle in searching food, smaller is the acceleration coefficients. The inertial weight factor w signifies the importance of the particle’s previous position in further search.[11] Thus each particle tends to move towards gbest to reach food early. If gbest has less number of values then the particles will reach food early. The algorithm comes to an end when all the particles converge at the gbest i.e. food position [12]. In our problem i.e.attaining minimum possible value for steady state error signal. The implementation of PSO for the given system can be shown like this:

Velocity updation

vi (t + 1) = w.vi (t ) + c1rand ( pbest (t ) − x(t )) + c 2 .rand .( gbest (t ) − xi (t )) Position updation P=P+V Where P - Instantaneous position of the particle V - Instantaneous velocity of the particle Pbest - positional best of the particle gbest – global best position of the swarm of the particles W – Inertial weight factor C1, C2 – acceleration coefficients

(3) (4)

Figure 5: Implementation of PSO based PID tuning of EMA-AFC system VI. MODELING Fig 6. Presents nonlinear model of EMA-AFC system. The model incorporates several non-linear effects: gear backlash, static friction, motor shaft rate limiter and control current saturation. Figure 4: Trajectory of particle after velocity updation

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Figure 6.Simulink Block Diagram of EMA-AFC system The motor torque

Tm is proportional to the magnetic flux,

which is proportional to the armature current I A .

Tm = K m I A

(5)

Due to motor shaft rotation, there is induced back electromotor force E m in the rotor coils. Back electromotor force is proportional to rotor angular velocity ω m .

K e is motor electric constant. The torque constant

dI A + Em dt

Where R A is armature resistance, while inductance. Combining equation (5)-(7), we get

moments of inertia of rotating motor components and the actuator assembly moment of inertia, respectively. The inertia for each of the of the rotating components in the actuator assembly to the motor shaft results in

N 2 .η g

2

θm

+ J screw + J pg

J = J m + J L is the total moment of inertia.

(11)

be angle of motor shaft rotation, then:

dθ m = ωm dt

(9)

(12)

Two kinds of load torques are taken into account: a friction torque and an aerodynamic torque. The friction torque can be modeled by where,

(8)

L A / R A defines motor electrical time constant τ e . The two moments of inertia J m and J L are equivalent

JL =

(10)

T fric = Csign(θm ) + Bθm

Ratio

J fin + J lever

where

(7)

K L A dTm + Tm = m U m − K eω m R A dt RA

the efficiency of

dω m = Tm − TL dt

J

Let

L A is the armature

η g is

screw drive system. Now, the following differential equations describe dynamics of the motor.

(6)

and the motor electric constant are same for ideal motor. Voltage equation for rotor circuit is

U m = RA I A + LA

J lever is the moment of inertia of the lever in screw driver mechanism, J pg is the planetary

gear head moment of inertia, and

E m = K eω m

J screw is the

screw moment of inertia,

where Km is motor constant.

Where

where J fin is the fin moment of inertia ,

(13)

°1............ forθm ≥ 0 (14) sign(θm ) = ® °¯− 1.......... forθm < 0 Tf C= . (15) N The load torque Taf induced by the aerodynamic

forces is proportional to the fin deflection angle

Taf =

Ta α N


α , thus

(16)

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R A =611 Ω , L A = 1.2 ⋅ 10 −4 H, I max =6.5 A

Nm/A,

ω max = 9500 min −1

Figure 7. Simulink block diagram of the motor driver The factor of probability Ta depends on maximum aerodynamics torque and maximum fin deflection angle.

Ta =

Ta max

(17)

α max

, mechanical time constant of the

motor is 3 ms, and gear backlash is 0.2 deg. Design Specifications for EMA-AFC: 1) Rise time < 1sec 2) Peak-overshoot < 5% 3) Settling time < 2sec 4) Steady-state error < 5% The limits on the position vectors of the particles In PSO (i.e. the controller parameters) are set by us as follows. Inertia factor w=0.45, C1=2.05,C2=2.05 The range of Kp, Ki, Kd are taken as 0 to 10 and that of λ and μ are from 0 to 1. Step response of untuned EMA-AFC system 25

Finally, for the screw lever mechanism, the total gear ratio is:

2Π.l N= cos(α ).h

20

Where, l is the normal distance from the roller nut and the output shaft axes. The gear ratio depends upon fin deflection angle, but for small angles cos(α ) =1. For small angles this approximation is not allowed.

finangle(in degrees)

(18)

VII. IMPLEMENTATION IN SIMULINK

15

10

5

0

-5

0

1

2

3

4

5 time(sec)

6

7

8

9

10

Figure 9: Step response of untuned EMA-AFC system In the above figure, the closed loop step response is very unstable without FOPID controller. Step response of conventional tuned EMA-AFC system 12

Fig.8: Plant with PSO tuned FOPID controller in Simulink

VIII. SIMULATIONS AND RESULTS According to mechanical design parameters and motor specifications provided by vendor, we have following model coefficients:

J fin = 1.5 ⋅ 10 −2 kg.m 2 , J lever = 5 ⋅ 10 −3 kg.m 2 , J pg = 0.15.10 −6 kg .m 2 , J screw = 5.10 −6 kg .m 2 , J m =

700

=10Nm, B=0,

8

6

4

2

0

0

1

2

3

4

5 time(sec)

6

7

8

9

10

Figure 10: Step response of conventional tuned EMA-AFC

J m = 3.33 ⋅ 10 −6 kg .m 2 , l = 50 mm , h=3mm, N=395,

δ max = 10 deg, Ta max =19 Nm, T f

10


In this plant, the parameters in of FOPID controller are optimized using PSO algorithm. This type of tuning is done for EMA-AFC plant and the results obtained are also compared with that of conventional tuning methods and unturned system and are shown in table below.

K M =26

The above figure is obtained by the tuning of the EMA-AFC using simulink.The values of the tuned parameters are Kp=5.1, Ki=4.6, Kd=7.8, λ =0.5 and μ =0.9.


Step response of EMA-AFC system 12

10

10

8

8



Step response of PSO tuned EMA-AFC system 12

6

4

2

0

PSO tuned conventional tuned

6

4

2

0

1

2

3

4

5 time(sec)

6

7

8

9

0

10

0

1

2

3

4

5 time(sec)

6

7

8

9

10

Figure 11: Step response of PSO tuned EMA-AFC system

Fig 12. Step response of EMA-AFC system

The above figure shows the transient response of PSO based tuned EMA-AFC system. The values of the tuned parameters are Kp=6.2359, Ki=1.7859, Kd=1.9648, λ =0.1236 and μ =0.9246.

From the fig.12, we can say that the rise time and overshoot of PSO tuned FOPID controller are very much improved compared to that of conventional tuned EMAAFC system .

Conventional Tuned FOPID PSO Tuned FOPID

Rise time(sec)

Overshoot (%)

Steadystate error (%)

0.9

1.6539

5.5969

5.4

0.9246

0.351

2.8488

2

Kp

Ki

Kd

λ

μ

5.1

4.6

7.8

0.5

6.2359

1.7859

1.9648

0.1236

Table showing comparison of values IX. CONCLUSIONS In this paper, the PSO algorithm is implemented on FOPID controller of EMA-AFC system using simulink. From the above simulations and results, we can conclude that Particle Swarm Optimization based tuning of FOPID controller is performing better compared to that of conventional tuned and unturned, since the rise time, overshoot and steady-state error are very much improved in PSO based tuned FOPID. Further scope can be carried by the real time implementation of these algorithms on DSP processor. REFERENCES [1] Astrom, K., Hagglund, T.: PID Controllers; Theory, Design and Tuning., Instrument Society of America, Research Triangle Park, 1995. [2] Chen, Y.Q., Xue, D., and Dou, H.: Fractional calculus and biomimetic control, in: Proceedings of the First IEEE International Conference on Robotics and Biomimetics (RoBio04), Shengyang, China, August 2004, IEEE. [3] I. Podlubny, “Fractional-order systems and PI λ D μ controllers”, IEEE Trans. on Automatic Control, vol. 44, no. 1, pp. 208 - 213, 1999.

[4] Deepyaman Maiti, Sagnik Biswas, Amit Konar, ”Design of a Fractional Order PID Controller Using Particle Swarm Optimization Technique”.Proc. 2nd National Conference on Recent Trends in Information Systems (ReTIS-08) [5] J. Kennedy and R. C. Eberhart, “Particle swarm optimization”, Proc. of IEEE Int. Conf. on Neural Networks, pp.1942-1948, 1995. [6] Schinstock, D.E., Douglas S.A., Haskew, T.A.: Identification of Continuous-Time, Linear, and Nonlinear Models of an Electromechanical Actuator. Journal on Propulsion and Power, Vol. 13, No.4, p.p. 683-690. 1997. [7] Schinstock, D.E., Douglas S.A., Haskew, T.A.: Modeling and Estimation for Electromechanical Thrust Vector Control of Rocket Engines. Journal of Propulsion and Power, Vol. 14, No.4, p.p. 440446.1999. [8] Schinstock, D.E., Haskew, T.A.: Transient Force Reduction in Electromagnetic Actuators for Thrust-Vector Control. Journal of Propulsion and Power, Vol.17 No. 1,p.p. 65-72.2001. [9] Zipfel, P.H.: Modeling and Simulation of Aerospace Vehicle Dynamics,AIAA, Reston, VA, 2000. [10] K.E.Parsopoulos and M.N.Vrahatis. Recent approaches to global optimization problems through particle swarm optimization. Natural Computing 1: 235-306,2002. [11] Y. Shi, R.C. Eberhart, “Empirical study of particle swarm optimization”, Proc. 1999 congress on Evolutionary Computation, July 6-9 1999,pp. 1945-1950. [12] Kennedy, J., and Eberhart, R.C.(1995). Particle Swarm Optimization. Proc. IEEE International conference on Neural Networks (Perth, Australia), IEEE Service center, Piscataway, NJ,pp.IV: 1942-1948.


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