2014 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
PID Controller for Nonlinear System using Cuckoo Optimization Praveen Kumar1 , Swapnil Nema 2 , P. K. Padhy3 1 Mechatronics 2, 3 Department of Electronics and Communication Engineering Indian institute of information technology design and manufacturing Jabalpur (Madhya Pradesh), 482005 1
[email protected] [email protected]
2
3
[email protected]
Abstract— In this paper, nature inspired cuckoo based proportional-integral-derivative controller is designed for nonlinear system. The cuckoo optimization is used to tune the parameters of PID controller. Performance criteria such as IS E, ITS E and IAE are optimized by using this optimization technique. For this optimum value of performance criteria, controller parameter is selected. These parameters of controller are used to stabilize the plant or give the desired response. To generalize the proposed technique inverted pendulum, ship roll dynamics and Van der Pol oscillator (VDPO) are taken as examples of nonlinear systems. The simulation results showed the desire performance of the systems. Result has been compared with some other standard technique proposed in the past research. Keywords-Cuckoo Optimization; Nonlinear System; PID; Inverted Pendulum; VDPO.
I.
INTRODUCTION
Proportional-Integral-Derivative (PID) controllers have been used widely in process and automation industry [1]. No w mo re than 90% control system is still based on PID controller due to its simp le structure [2]. Parameters of this controller are proportional, integral, derivative gains such as Kp , Ki and Kd . PID controller can be used either in forward or feedback loop of the p lant. Responses of the systems are affected by the controller parameters Kp , Ki and Kd . In the last decade the tuning of the controller parameter was done by manually [3]. Controller parameters obtained by manual tuning did not give desired response. The manual tuning is used for the linear and simple system. The most perilous step in the application of PID controller is parameters tuning. Today self-tuning PID d igital controller is used to tune the parameters of the controller [4, 5]. Most of the practical systems are nonlinear in nature [6]. If the plant dynamics are very comp licated then the method of manual tuning is not feasible. Auto tuning [7] of the controller parameters is used to solve this problem. In the last decade many intelligence techniques were used to tune
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the controller parameters. These intelligent controls are optimal control [8], fu zzy logic [9], neural network [10], robust control [11], predict ive control [12] and adaptive control [13]. Nature inspired optimization techniques are used to optimize the parameters of the controllers such as genetic algorith m [14], ant colony optimization [15] and particle swarm optimization [16]. Particle Swarm Optimization (PSO) is one of the latest optimizat ion algorith m introduced by Kennedy and Eberhart in 1995 [17, 18] used for tuning the parameters of the PID controller. These optimizat ion techniques take more t ime in the optimization process as compared to proposed technique. The convergence rate of proposed controller has better than others. There are two kinds of performance criteria, wh ich are used in nature inspired evolutionally optimization such as frequency domain analysis and overshoot method. ISE (integral squared error), IA E (integral absolute error) and ITSE (integral of t ime square weighted method) co mes under frequency domain. Overshoot method includes rise time t r , peak time t p , maximu m overshoot M p , steady state error ess and settling time ts . This paper presents development of an optimal PID controller for control of nonlinear systems. Tracking problem o f inverted pendulum, stability analysis of Van der Pol oscillator and stability analysis of capsizing problem taken as an example for validation of proposed technique. In this work, optimal PID parameters are obtained with the help of nature inspired cuckoo optimization. The proposed cuckoo optimization algorith m is used to find the optimal parameters according to the requirement of the system such that time response specifications namely steady-state error, overshoot and settling time. Simu lation results show that the proposed method gives satisfactory response. This paper is organized as follows section II describes the controller design, section III represents the block diagram of proposed controller, section IV describes the cuckoo PID controller where as in section V some examp le of nonlinear
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2014 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
systems are taken and at last conclusion are presents in section VI. II.
CONTROLLER DESIGN
A. Cuckoo based PID Control structure
Cuckoo Optimization
Input signal
e(t) r(t)
PID Controller
u 1 (t) Plant
egg in host bird nest. Some eggs are detected and destroyed by the host bird. When the eggs are not detected then chick grows. There is some identification problem of mature cuckoo. To overcome this problem Cuckoo form clusters. In the proposed method controller parameters are analogous to the Cuckoo’s habitat. So me of the eggs are randomly generated to calculate the ELR of each Cuckoo. Then initialize Cuckoo’s habitat by three dimensional matrix because PID controller has three constant parameters within the ELR. Flow chart o f the nature inspired cuckoo optimization is shown in the Fig.2 [19].
c(t)
Respo -nse
Start Initialize Cuckoos with Egg
Determine Eggs Lying Radius for each Cuckoo
Lay Eggs in Different Nest
Feedback Fig.1. Block Diagram of Cuckoo Based PID controller
Some of Eggs are Detected and Killed
The Fig. 1 shows the block diagram of Cuckoo based PID controller. Generalized equation for the PID controller is given below
Determine Cuckoo Society
(1)
Population is less than Maximum Value
No
In the above equation u 1 (t) is the plant input or output of the PID controller. e(t) is the error of the plant & defined by the difference of the desired output and actual input. In the equation (1) Kp is the proportional gain, Ki is the integral gain, Kd is the differential gain of the controller. To make system stable or get the desired value of response we have to assign appropriate value of controller parameters. B. Cuckoo Optimization Cuckoo search (CS) optimization algorith m has given by Xin-she and Suash Deb in 2009 [19]. This optimizat ion method has high accuracy and convergence rate and it is inspired by obligate brood parasitism of Cuckoo species. Cuckoo lay their eggs in the nests of similar host species. Cuckoo gives one egg at a time. Mature cuckoo and their egg laying is the basic concept behind this algorithm. The Cuckoo’s egg is too heavy and bigger in size as compare to others. Therefore Cuckoo cannot flies long distance with egg. In the egg laying duration, the maximu m d istance covered by the Cuckoo is known as Egg lay ing radius (ELR). The maximu m area of the cuckoo where they can put their eggs [20] is given by (2) where, Maxvari and Min vari are the maximu m and min imu m range of the controller parameters. Each cuckoo lay their
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Move all Cuckoo towards Best Environment
Kill Cuckoos in Worst Area
Yes Check Survival of Eggs in Nests (get Profit Value)
Find Nest with Best Survival
Let Egg Grow Stop Condition satisfied? No Yes
End
Fig.2. Flow Chart for the Cuckoo Optimization
III.
DESIGN OF CUCKOO PID CONTROLLER
To solve the optimization problem we have to define cost function. Cost function depends on the performance criteria of the controller. The performance criteria such as error function, overshoot, undershoot, rise t ime, settling time and steady state error are used in controller design. Generally three types of error function are used. I. ISE (integral squared error)
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2014 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
II. III.
IAE (integral absolute error) ITSE (integral of time square weighted method)
Mathematical expression for ISE (integral squared error) is ISE =
(3)
In the present work cost function is the sum of ISE and maximu m overshoot and the expression of cost function is given by Cost function = α.ISE+β.Max overshoot (4) where, α and β are constant and α= β=10 is considered. IV.
SIMULATION RESULTS
A. Inverted Pendulum The inverted pendulum is a highly unstable nonlinear system. Control of angular position of inverted bob depends on the movement of the cart. Fig.3 shows the block d iagram of inverted pendulum. Mathematical model of the inverted pendulum with cart was proposed by Wei-Der Chang & Shun-Peng Shih [21].
Fig.4. Convergence rate of inverted pendulum.
Fig.5. T racking control of inverted pendulum Fig.3. schematic diagram of inverted pendulum
(5)
(6) (7) For simu lation result some nu meric value of constant is taken such as g is acceleration constant due to gravity and its value is 9.8 m/s 2 , l = 0.5 m is the length of the Inverted Pendulum, mc = 1 kg is the mass of the cart, m= 0.1 kg is the mass of the pole of the inverted pendulum and u(t) is the control input .
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Fig.6. Error plot for inverted pendulum
The PID controller is designed using Cuckoo optimizat ion technique and the simulation results are shown in Fig.4, Fig.5 and Fig.6. The proposed method is compared with well-known PSO method [21] to determine the robustness. The controller parameters of proposed method are =56, =453 and = 79 and by PSO method are =119.70, =438.42 and = 33.56.It is clear fro m the graphs that the response obtained from cuckoo PID controller is better as compare to PSO-PID controller.
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2014 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
B. Ship Roll Dynamics Dynamics of the Non-linear ship- roll model [22] is taken as an example for nonlinear system and the mathematical model is described by
1 0.9
1
x 10
Current Cost = 990.6569, at iteration = 101
Amplitude
0.7
(8) 5
Cuckoo PID Padhy & Majhi [23]
0.5 0.3 0.1 -0.1 -0.3 -0.5
Cost Value
0.5
0
0
5
10
15 20 Cuckoo Iteration
25
30
Fig.7. Convergence rate of ship –roll- dynamics 1.4
Cuckoo PID
1.2
Padhy & Majhi [23]
Amplitude
1 0.8 0.6 0.4 0.2 0
4
8 12 Time (sec)
16
20
Fig.9. Error plot for the ship – roll model
-0.5
-1
0
0
5
10 Time (sec)
15
20
Fig.8. Unit step response for ship – roll model T ABLE I. PERFORMANCE T ABLE OF CONT ROLLER FOR SHIPROLL MODEL
Performance Parameters Settling time (t s )
Padhy & Majhi PID 11 sec
Propose cuckoo PID 7 sec
Steady state error (e ss )
0
0
Peak overshoot (M p )
13.5%
11%
Rise time (t r)
1.3 sec
0.6 sec
Parameters of the PID controller are optimized using cuckoo optimization. To optimize the performance of the system parameters of the cuckoo optimization are selected. These parameters are maximu m nu mber of cuckoo = 10, Min imu m number of eggs = 2, maximu m nu mber of eggs = 4, maximu m iteration = 100, cluster number = 1, mot ion coefficient = 9 and radius coefficient = 5. Optimu m value of the Parameters of the PID controller are Kp =60.00, Ki=24.5535 and Kd=8.7108 Padhy and Majhi [23] also suggested the PID controller parameters for the same are Kp=25.316, Ki=9.592, Kd =16.5. The unit step response for this system is given in fig.7, at 3rd iterat ion shown in the fig.6, error p lot shown in the fig.9 and the performance parameters of the system are given in Table I. After observing the output response plot, error plot and performance parameter, it is clear that cuckoo-PID controller gives better response. C. Non-Linear Deterministic System - Van der Pol Oscillator (VDPO) V. Kadirkamanathan and S.R. Andersona [24] used a test problem of Van der pol oscillator in delta domain. The dynamics of the nonlinear system is given in the Equation (9). (9)
Fig.10. Convergence rate of VDPO
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2014 International Conference on Control, Instrumentation, Communication and Computational Technologies (ICCICCT)
2
step response of the system shown in fig.11 and the error plot shown in the fig.12.The performances of the controller are given in Table II. A disturbance unit step signal of amp litude 4 unit is applied at t= 15 sec to check the robustness of the controller. Table III shows the enactment of the controller after applying the disturbance. After observing the response plot, error p lot and performance table, it is clear that cuckoo – PID controller gives better response compared to response obtained by the previous research.
Cuckoo PID Gupta & Padhy [25]
Y
1.5
1
0.5
0
0
5
10
15 Time (sec)
20
25
30
Fig.11. Unit step response for the VDPO Cuckoo PID Gupta & Padhy[25]
0.8 0.6
Error
V.
0.4
REFERENCES
0.2
[1]
0 -0.2 -0.4
CONCLUSION
In this paper, Cuckoo based PID controller has designed for nonlinear system. Tracking control of inverted pendulum, Van der Pol oscillator and ship roll dynamics are taken as examples of nonlinear system to show the effectiveness of the proposed method. The simu lation results give improved response in terms of rise time and settling time.
0
5
10
15 Time (sec)
20
25
30
Fig.12. Error plot for the VDPO T ABLE II. PERFORMANCE T ABLE OF T HE CONT ROLLER FOR VDPO (WIT HOUT DIST URBANCE)
Performance criteria
Settling time (ts ) Steady state error (ess ) Rise time (tr)
PID tuned by
Proposed
Gupta & Padhy
cuckoo PID
7 sec
1 sec
0
0
1.75 sec
0.625 sec
T ABLE III. PERFORMANCE T ABLE OF T HE CONT ROLLER FOR VDPO (AFT ER DIST URBANCE)
Performance criteria
Settling time (t s ) Steady state error (ess ) Peak overshoot (M p )
PID tuned by
Proposed
Gupta & Padhy
cuckoo PID
8 sec
1 sec
0
0
35 %
1%
Optimu m value of the Parameters of PID controller are =60.00, =24.5535 and =8.7108 whereas Gupta & Padhy [25] also reported the parameters for the controller are =25.316, =9.596 and =16.5. The convergence plot is shown in the fig.10 shows that the optimu m value of the controller parameters achieved at 8th iterat ion. The unit
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