Cuckoo PID-P Controller for Non-Linear Process

Swapnil Nema1 and Prabin Kumar Padhy2

1,2Department of Electronics and communication engineering Indian institute of information technology design and manufacturing

Jabalpur (Madhya Pradesh), 482005

1swapnil.nema@iiitdmj.ac.in 2prabin16@iiitdmj.ac.in

Abstract— A PID-P controller is designed for a class of non-Linear processes using Cuckoo Optimization (CO) techniqe, where a proportional controller is used in the inner feedback loop which increases the stability of the non-linear process and the PID controller is connected in feed forward path for desired response. The objective function for CO optimization includes both the Integral Time Squared Error (ITSE) and the peak overshoot. Different non-linear processes are analyzed through simulation to show the effectiveness of the CO based PID-P controller. Real time experiment on three tank conical system is also performed to illustrate the robustness of the proposed method.

Keywords— Cuckoo Optimization (CO), Proportional-integral-derivative (PID), non-linear process, stability etc.

I. INTRODUCTION

Proportional-Integral-Derivative (PID) controller is the commonly used controller in different applications like process control, motor drives, magnetic & optic memories, automotive, flight control, instrumentation etc. [1-4] because of its potentiality and simple control strategy even though many new control techniques have been proposed. It is very difficult to design a controller accurately because of non-linearity, complexity and time variance in the industrial processes. So, proper tuning of controllers is needed to obtain desired output of the process [5]. Non-linearity is a common behavior exists in all the processes and it disturbs in the performance of the system. Special attention is required to design a controller for non-linear system. The non-linearity can be understood through the following forms like Static Non-linearity, Dynamic Non-linearity, Intentional Non-linearity and Incidental Non-linearity. Several methods are proposed in literature to design PID controller for non-linear systems [6-8]. In 1995, Kennedy and Eberhart introduced Particle Swarm Optimization (PSO) for unconstrained continuous optimization problems [9]. This technique has gained much attention and wide applications in various fields [10-12]. Several methods based on Fuzzy, Neural and Genetic Algorithm [13-15] have been proposed in literature for

parameter optimization. It is observed that a non-linear system with multiple steady states can be effectively controlled by a gain-scheduled control scheme [16-17]. Jhunjhunwala and Chidambaram [18] analyzed an optimized PID controller tuning for non-linear systems such as the biochemical reactor and the continuous stirred tank reactor (CSTR) of chemical process industries. Chang and Shih [19] proposed an improved PSO-based PID controller design for a class of non-linear system.

The non-linear systems are very difficult to analyze and control explicitly. Recently different evolutionary methods for controller design are the topic of interest of many researchers. In [20], a new evolutionary optimization algorithm which is inspired by lifestyle of a bird family called cuckoo is proposed. Specific egg laying and breeding of cuckoos is the basis of this optimization algorithm. In this paper, a cuckoo optimization approach is used to design the controller processes. The instability in the process disturbs the performance. So, stabilization is required to improve the performance of the process. The design of PID controllers for unstable processes has attracted attention recently [21]. The performances specifications that are normally obtained for stable process cannot be obtained for unstable processes. The proportional-derivative controller in the feedback path helps to improve the stability of the process [22]. In this paper, PID-P structure is used where PID controller is connected in the feed forward path and P controller in the feedback path. The PID-P controller is designed based on the new evolutionary random search cuckoo optimization technique. This method is based on the social behavior of Cuckoo’s egg laying process. The survival of the mature cuckoo from the egg is the basic idea of this method.

The paper is organized as follows. Section 2 describes the design of PID-P controller using Cuckoo optimization (CO) method in detail. Section 3, presents simulation of three non-linear examples and the comparison with existing method. Experiment on real time system (three tank conical system) is presented in Section 4. Finally conclusion is given in Section 5.

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II. DESIGN OF PID-P CONTROLLER USING CUCKOO OPTIMIZATION

A. Cukoo Optimization (CO) The cuckoo search algorithm [23] is a recently developed meta-heuristic optimization algorithm, which is suitable for solving optimization problems with high accuracy and convergence rate.

The CO algorithm is based on the egg laying behavior of common cuckoos, which is constrained by common radial distance related to the cuckoos flight capability. This radial distance is called as egg laying radius, which is different for each cuckoo. The cuckoo is then constrained to lay egg in that particular radius. The cuckoo lay one egg at a time in the other bird’s nests. According to the similarity behavior of the egg’s with the host bird’s egg will be replaced. These replaced eggs are then developed and mature if host bird could not detect the replacement otherwise it will be destroyed by the host. The subsistence of the egg is the fundamental characteristics of the algorithm. The rate of subsistence will be the term governed by the CO algorithm. To optimize the rate of subsistence, cuckoos search for the optimum area for laying the egg. The eggs surviving will then develop to mature cuckoos. These cuckoos are then forming their societies, having their own environment. This environment attracts all those cuckoos looking for the best environment, forcing them to immigrate near to them. Then, cuckoo starts laying eggs in random nests within the egg laying radius starting the process which continues until the optimum position with maximum subsistence rate is obtained and then most of the cuckoo’s inhabitant is gathered around the same position. The cuckoo evolutionary optimization algorithm, consider 20 cukoos, with a maximum egg laying value of 100 to tune the PID-P controller parameters of non-linear processes.

B. Design of PID-P Controller

Fig. 1. Proposed control scheme

The proposed structure to design the PID-P controller is shown in Fig. 1. The transfer function of nonlinear plant is represented by Gnl and the structure of PID controller, which connected in feed forward path, is given as follows.

s (1)

where, Kp is the proportional gain, Ki is the integral gain and Kd is the derivative gain. In this proposed structure, the inner feedback gain “q” stabilizes the non-linear system which helps to design optimal controller. Now, the controlled input to the non-linear system can be expressed as by using the following equation.

(2)

where, e(t) is the error signal between input r(t) and output y(t). With increase in the value of “q”, the poles of the process get shifted further towards the left of the s-plane that ensures more stability of the system as illustrated in [24]. In this method, all the four parameters are obtained by using CO. The objective function F for optimization includes the Integral Time Square Error (ITSE) and Peak Overshoot (Mp) which is given by F te t α M . dt (3) where, α is a weighting factor and chosen as 5 in the simulation.

III. SIMULATION RESULTS Different non-linear processes are considered to validate the

proposed scheme. Further ±10% variation in process parameters is introduced to check the robustness of the proposed method.

A. Non-linear Ship Roll Consider the non-linear ship roll [22]. Generally it

experience three types of linear motions (heave, sway or drift, and surge) and three angular motions (yaw, pitch, and roll). The dynamics is represented by

.01 0.5 8. 10 0.362. 10 10 The PID-P controller is designed using CO method. The

unit step output responses with static disturbance of unity in magnitude at 3 second for the proposed CO based PID-P controller, PSO tuned PID controller, PSO based PID-P controller and PID-P controller [24] are shown in Fig. 2 and the controller parameters are given in Table I. For robust controller design, the process parameter varies about ±10% which is shown in Fig 3.

TABLE I

DIFFERENT VALUES OF TUNED PARAMETER

METHODS Kp Ki Kd q PID-P 25.316 9.592 16.5 2.176

PSO-PID 18.05 0.567 6.514 - PSO-PID-P 89.53 4.858 14.94 1 CO-PID-P 82.001 4.99 17.99 0.05

Gnl (s) GPID(s) Y(s)R(s)

q

CO L

+ +

e u

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It is observed from the Fig. 3 the proposed CO based PID-P controller gives better response in presence of small variation in the dynamics of ship roll dynamics.

Fig. 2. Comparison of unit step output responses

Fig. 3. Unit step output response for ±10% variation in non-linear ship roll

dynamics

B. Non-Linear Deterministic Systems considering VDPO S.R.Andersona, V. Kadirkamanathan in [6] used a test

problem of a Van-der-Pol (VDPO) oscillator for representing non-linear deterministic systems in delta domain. VDPO oscillator is an oscillator with non-linear damping and it evolves for the control of the position. The dynamics of non-linear system is given by

0.2 1 The unit step output responses with disturbance of 0.5 in

magnitude at 5 seconds for the proposed CO-PID-P controller, PSO tuned PID controller and PSO based PID-P controller are shown in Fig. 4 and the controller parameters are given in Table II.

For robust controller design, the process parameter varies about ±10% and the output response is shown in Fig. 5.

Fig. 4. Comparison of unit step output responses

Fig.5 Comparison Unit step output response for ±10% variation

TABLE II DIFFERENT VALUES OF TUNED PARAMETERS

C. Immune Feedback System The Immune Feedback system is considered in this example

where PID-type immune feedback controller is discussed in [25]. The immune system enables human survival of infection and disease. The most important cells of the immune system are lymphocytes. There are mainly two classes of lymphocytes, namely T cells and B cells. Antibody (Ab) molecules are synthesized and secreted by B cells, and the process is regulated by T cells. T cells can either help (TH) or suppress (TS) B cell’s response to a stimulus. The antigen (Ag) is recognized by T cell’s receptors and activated to secrete interleukin. The interleukin is the second signal to B cells. B

0 2 4 6 8 100

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METHODS Kp Ki Kd q PSO-PID 3.439 1.191 2.654 -

PSO-PID-P 7.049 3.363 3.814 1.367 CO-PID-P 37.502 3.99 17.98 0.701

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cells are stimulated by the interleukin. It is required to control the synthesis of Antigen (Ag) as per the amount of Antibody (Ab) in the system. The dynamics of the Immune response is given by 0.5

Fig. 6. Comparison of unit step output responses

TABLE III DIFFERENT VALUES OF TUNED PARAMETER

The unit step responses with the disturbance of 0.5 at 10

seconds using the proposed CO PID-P, the general tuned PSO-PID and PSO-PID-P are shown in Fig. 6. The controller parameters obtained from all the methods are shown in Table III. The response is faster and the disturbance is readily recovered in the proposed CO-PID-P controller in comparison to other methods.

IV. EXPERIMENTAL SETUP A real time system is used which includes Interacting / Non

Interacting Conical Tank System. Many research works has been carried out for such conical tank systems [26-28].

In this experiment, single input single output (SISO) non-linear system is analyzed. The experiment is performed for a single tank system in a non-interacting mode. Since the tank is conical in shape, the variation of level of water is non-linear. The photograph of conical Tank system is shown in Fig. 7. Matlab 7.10.0 / Simulink is used for experimental purpose.

The controller is designed using proposed CO PID-P method. The PID-P controller parameters are designed using CO for the control of level of water and the values are given as Kp=12; Ki=3.2; Kd=0.01and q=5.04. The experimental result for a set value of 26 cm is shown in Fig. 8.It is clear from the experimental result that the proposed CO based PID-P controller gives fast response with zero overshoot.

Fig. 7. Three tank conical system

Fig.8 Experimental result

V. CONCLUSION A design method for non-linear system has been

successfully presented. The controller is designed using cuckoos optimization technique. The robustness of the proposed scheme is studied through simulation of stable and unstable non-linear systems along with ±10% variation in parameters. The proposed controller design method is validated by experimental study. It is observed from both the simulation and experimental results that proposed method gives improved result in comparison to existing controller techniques.

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