Performance Analysis of PID and Fuzzy PD+I Controller on Nonlinear Systems

Bharat Bhushan1, Nupur Jha2, Sangeeta Devra 3, Sarath S. Pillai 4

Department of Electrical Engineering Delhi Technological University

Delhi, India Email1: bharatdce@yahoo.co.in

Email2: nj.nupurjha@gmail.com Email3: sangeeta.devra@gmail.com Email4: sspillai305@gmail.com

Abstract—This paper analysis the performance of Fuzzy PD+I and conventional PID controller on nonlinear systems. The nonlinear systems used are level control of a surge tank and cart pole system. Level control of a surge tank level system and angular position of cart pole system have been tracked using Fuzzy PD+I controller and conventional controller. Four type of membership functions Bell, Pi, Gaussian and Psigmoid are used in the fuzzy PD+I control of the nonlinear systems. Effects of different membership functions on the systems control have been compared with conventional PID controller. For four different membership functions besides control performance stability criterion also have been implemented using phase-plane method.

Keywords— Fuzzy PD+I controller; Surge tank level system; cart pole system; PID controller; Phase plot.

I. INTRODUCTION The Fuzzy logic gives a simple way to reach definite

conclusion even when the input is based on vague, noisy, imprecise, ambiguous or missing information. It provides a multi-valued truth space in [0,1]. It is capable of generating inferences even when a partial matching occurs. Here we make use of Conventional PID and Fuzzy PD+I controller to control liquid level in a nonlinear Surge tank system [13] and to control the angular position in cart pole system.

Conventional proportional integral derivative (PID) controller is widely made use of in processes due to its simplicity in structure, ease to design and robust performance in wide range of operating conditions [9]. Conventional PID controller is the most popular control tool in many industrial applications as they can improve both the transient response and steady state error of the system at the same time [18].

Traditionally the parameters of Conventional PID controller, i.e. KP, KI& KD, are usually fixed during operation. Such a controller is thus inefficient for controlling a system while a system is disturbed by the changes in the surrounding environment of the system or due to unknown facts [6, 17]. Integral action is necessary whenever a sustained error in steady state occurs. It will increase or decrease the control signal if there is a positive or negative error, even for small magnitudes of error. Thus, a controller with integral action will always return to the reference in steady state [12].

A fuzzy controller acts on three inputs: error, integral error and change in error. The integral action in the crisp PID controller combined with a fuzzy PD rule base as fuzzy PD+I (FPD+I) controller is an effective way to control the nonlinear systems.

In recent years, fuzzy logic control has been applied successfully in the area of nonlinear process control [1-4]. In direct fuzzy control applications fuzzy logic controller (FLC) estimates the necessary control action through rule inference by taking the linguistic description of the error behavior [5]. The literature tells that it has been successfully applied to various problems viz. temperature control of glass furnace [10-11], controlling the Yaw system for wind turbine generator [14] robot manipulator [16], Electro-hydraulic servo system [17, 18] inverted pendulum [19-20].

In this paper four types of membership functions have been considered. FPD+I controller have been implemented on nonlinear systems using four type of membership functions.

This paper is divided into several sections wherein Section II introduces the Conventional PID controller, Section III gives an account on Fuzzy PD+I controller section IV describes the nonlinear systems and section V shows the simulation results and discussion followed by conclusion in section VI and then references.

II. CONVENTIONAL PID CONTROLLER The combinative action of proportional, integral and

derivative [8] action is an effective tool for controlling processes. It is a generic control loop feedback mechanism (controller) widely used in industrial control system. An error value is calculated by the PID Controller as the difference between a desired set point and measured process variable. By adjusting the process control inputs the error is minimized by the controller.

The output of the controller is denoted by

(1)

The control signal u is a linear combination of the error e, its derivative and its integral. The parameter Ti is the integral

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time, KP is the proportional gain, Td is the derivative time and Ts is the sampling time.

III. FUZZU PD+I CONTROLLER In this section, Fuzzy PD+I controller [12] have been

described. Three inputs had been used, error(e), change in error(ce) that will be fed to the fuzzy controller while integral error will be used as conventional integral action . Figure 1 shows the structure of Fuzzy PD+I controller.

Figure 1.Fuzzy PID controller, FPD+I

The control signal U(n) after the gain GU, at the time instant n, is a nonlinear function of error, change in error, and integral error,

(2)

The function f is again the control surface of a PD rule base. The mapping is usually non-linear, but with a favorable choice of design, a linear approximation is

(3)

(4)

Comparing equation (1) and (4) the gains are related as follows:

(5)

(6)

(7)

IV. NONLINEAR SYSTEMS

A. Surge Tank Level System This section describes the Nonlinear Surge tank system [7,

13] in which the liquid level is to be controlled. The system can be modeled by the equation:

(8)

where u(k) is the control input (input flow) and it can both pull liquid out of the tank and put it in (can be positive or negative), h(k) is the output of the plant (liquid level) and T=0.1.

A(h(k)) = | h(k)+ |describes the cross-sectional area of

the tank , >0 and >0. Here = 0.01 and = 0.2; g=9.8; is the clogging factor [0.9, 1] for a filter in the pump actuator and symbolically if =1, there is no clogging as the filter is clean. is related to the diameter of the output pipe and its nominal value is taken as 1. We assume that the input to the plant saturates at +/- 50 for that controller generates an input (k).

B. Inverted Pendulum System The aim of the controller is to stabilize the pendulum [7] in

the upright position by applying the right amount of force. The system is described by the following equations:

(9)

(10)

By approximating the system and converting into state space following state equations are obtained:

(11)

M (Mass of cart)=3Kg; m(mass of bob)=0.2Kg; l(length)=0.31m; b(frictional force)=0.1N/ms-1.

V. SIMULATION RESULTS AND DISCUSSION Here, liquid level in a surge tank system is controlled using

both Conventional PID and Fuzzy PD+I controller. FPD+I controller has been implemented with four different membership functions viz. Bell shaped, Pi, Gaussian andPsigmoid functions. The result of each controller has been projected along with a comparative analysis. Here a plane is spanned by variables E and CE, which is a phase plane between error and its derivative. The sequence of errors and its derivative can be plotted; CE(t) against E(t), to form a phase trajectory. If the trajectory is ended in the centre (origin), then the plant stays on reference and not moving and that system is stable. For each membership function, phase plots have also been shown for stability analysis.

Four membership functions have been used: Gaussian, Bell, Pi and Psigmoid. The FPD+I controller uses these for different iterations. The result for each function applied on to the surge tank system and inverted pendulum were obtained. Three linguistic variables for each input have been used: Neg (negative), Pos (positive), and Zer (zero). There are two inputs: e and ce.The rules are as shown in the Table I.

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TABLE I. FUZZY RULE BASE

Neg

Zer

Pos

Neg

Neg

Neg

Zer

Zer

Neg

Zer

Pos

Pos

Zer

Pos

Pos

Simulation results are obtained for surge tank level system and inverted pendulum system using conventional and FPD+I controller. Four membership functions were used for the FPD+I controller to study the effect on the nonlinear systems.

Reference input for surge tank is shown in Fig. 2 below and the reference for cart position and inverted pendulum is zero.

Figure 2.Reference Input

A. Conventional PID The system was controlled with the Conventional PID

controller with a fixed value for Kp, Ki & Kd.

The controlled output for the level of the liquid inside the tank is shown in Fig. 3. While angle and position of inverted pendulum by using the conventional PID controller are shown in Fig. 4 and 5 respectively.

Figure3. The Controlled level of the liquid using PID

Figure4. The Controlled angle of pendulum using PID

Figure5. The Controlled position of cart using PID

B. FPD+I controller with Gaussian Membership Function The Gaussian membership function is shown in Fig. 6.

Figure6. Gaussian Membership Function

The output of the surge tank level and inverted pendulum systems obtained by using the FPD+I controller with gaussian membership function with respect to the reference input are shown in Fig. 7, 8 and 9.

Figure7. The Controlled level output using Gaussian Membership function

ce e

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Figure8. The angle of pendulum using Gaussian Membership function

Figure 9 The position of cart using Gaussian Membership function

It can be seen clearly that the controller is able to control the level and angle of the systems efficiently and is also successfully able to track the reference input given to the systems.

The phase plots are obtained for both the nonlinear systems between CE and E which shows the system is stable as shown in Figure 10.

(a) (b) Figure 10 .Phase Plot of (a) tank level; (b) Inverted pendulum for Gaussian MF

C. FPD+I controller with Bell Membership Function The Bell membership function was used secondly in the FPD+I controller to analyze the effects on the control of the system. The Bell membership function is shown in Fig. 11.

Figure11. Bell Membership Function

The output of the nonlinear systems obtained by using the Bell membership function with respect to the reference input are shown in Fig. 12, 13 and 14.

Figure12. The Controlled level output using Bell Membership function

Figure13. The angle of pendulum using Bell Membership function

Figure14. The position of cart using Bell Membership function

The phase plots obtained for both the nonlinear systems between CE and E show that the systems are stable as shown in Fig. 15.

(a)

(b)

Figure 15.Phase Plot of (a) tank level; (b) Inverted pendulum for Bell MF

D. FPD+I controller with Pi Membership Function The Pi membership function was then used in the FPD+I controller to analyze the effects on the control of the nonlinear

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systems. The Pi membership function was used is shown in Fig. 16.

Figure16. Pi Membership Function

The output of the surge tank system obtained by using the Pi membership function with respect to the reference input is shown in Fig. 17.

Figure17. The Controlled level output using Pi Membership function

The angle of pendulum and position of the cart obtained with FPD+I controller using Pi membership functions are shown in Fig. 18 and 19 respectively.

Figure18. The angle of pendulum using Pi Membership function

Figure19. The position of cart using Pi Membership function

The phase plots obtained for surge tank level system and inverted pendulum between CE and E using Bell membership

functions are shown in Fig. 20, which show that both the nonlinear systems are stable.

(a)

(b)

Figure 20.Phase Plot of (a) tank level; (b) Inverted pendulum for Bell MF

E. FPD+I controller with Psigmoid Membership Function

The Psigmoid membership function was used in the FPD+I controller to analyze the effects on the control of the nonlinear system using FPD+I controller. The Psigmoid membership function is shown in Fig. 21.

Figure 21.PSigmoid Membership Function

The output of the nonlinear surge tank system obtained by using the Psigmoid membership function with FPD+I controller with respect to the reference input is shown in Fig. 22.

Figure22. The Controlled level output using Psigmoid Membership function

The outputs of the nonlinear inverted pendulum system obtained by using the Psigmoid membership function with FPD+I controller are shown in Fig. 23 and 24.

Figure23. The angle of pendulum using Psigmoid Membership function

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Figure24. The position of cart using Psigmoid Membership function

The phase plots obtained for surge tank level system and inverted pendulum system are shown in Fig. 25 which show that both the nonlinear systems are stable.

(a)

(b)

Figure 25.Phase Plot of (a) tank level; (b) Inverted pendulum for Psigmoid MF

The results show that the response of Fuzzy PD+I controller with all membership functions used above gives low overshoot than the Conventional PID Controller. We have used four different membership functions with Fuzzy PD+I controller and they are giving better response than conventional PID controller. The performance tells that large number of processes can be efficiently controlled by using Fuzzy Controller rather than Conventional PID Controller.

On the effect of membership functions on the system, it can be commented that performance using Gaussian and Psigmoid membership functions is almost same for surge tank level system, but Fuzzy PD+I controller using Pi and Bell membership functions gives better output than the output obtained using Gaussian and Psigmoid membership functions for the same. But in the case of inverted pendulum results using Pi and Psigmoid gives better response than Gaussian and Bell. And in comparison to all performances Fuzzy Controller using Pi membership function gives best output. The FPD+I helps in determining the stability of the system which is evident from the phase plots above. The plots show that the trajectory converges to the origin which shows the system stability.

VI. CONCLUSION The Conventional PID controller and FPD+I controller with different membership functions are implemented on nonlinear surge tank level control system and cart pole system using MATLAB. It is evident from the results that FPD+I perform better in four type of membership function than the PID controller. Effect of four different membership functions were studied and it is found that Fuzzy controller using Pi membership function gives best result compare to other three

type of membership functions. . Phase plot of the system for all the four membership functions shows that both the nonlinear systems are stable.

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