Investigating Stability Comparison of a Conventional Controller and Fuzzy Controller on a Non-Linear System

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1 International Journal of Research in Engineering and Management Technology (IJREMT), Volume 01 Issue 03, October, 2015 Available at 1 Investigating Stability Comparison of a Conventional Controller and Fuzzy Controller on a Non-Linear System Neethu Mary, Clint Augustine, Sonsy Heartson M.Tech, EIE Dept, Vimal Jyothi Engineering College, Kannur Assistant Professor, EIE Dept, Vimal Jyothi Engineering College, Kannur PhD Scholar, Chemical Dept, NIT, Calicut Abstract-This paper considers the trajectory tracking control of linear inverted pendulum (IP) system. The controller is designed based on the mathematical model of the inverted pendulum system. Non-linear Inverted Pendulum is a multivariable system having angle of pendulum and position of cart are two variables to be controlled. For this purpose we used fuzzy logic controller and compare its response with conventional controller. The controller design procedure is comparatively simple and natural. The fuzzy logic controllers (FLC) have the ability to control a system by using some limited expert knowledge. Through simulation in Matlab by selecting appropriate fuzzy rules are designed to the controller, the performance of the inverted pendulum system has improved significantly compare to conventional PID controller. Index Terms Inverted pendulum, Fuzzy Logic System, PID controller, Fuzzy controller. I. INTRODUCTION Inverted pendulum is one of the most difficult systems to control in the field of control engineering, because it is a nonlinear as well as unstable system [1]. It provide a platform to test various control techniues and used to simulate experiments such as walking robots, missile guidance and flying objects in space etc. Here the aim of the study is to design a control system that keeps the pendulum balanced and tracks the cart to a commanded position [2]-[6]. In general, the control problem consists of obtaining dynamic models of systems, and using these models to determine control laws to achieve the desired system response and performance. The conventional PID controller is still used in industries, because of its simple in control structure, not too expensive and effective for a linear system [7]-[11]. The conventional PID controller can be used for virtually any process condition. If an accurate mathematical model of a system is available to us then we can design a conventional PID controller. But the real world is not linear as well as uncertain and contains always incomplete data. So we cannot use the conventional PID controller if the system has a high level of complexity such as high order modeling non linearity, time delay and structural uncertainties [12]. *Corresponding author. addresses: n.m.neeths@gmail.com (Neethu Mary) Fuzzy is always best to handle the real world or imprecise data, so we are used the fuzzy logic controllers. The fuzzy logic controllers (FLC) have the ability to control a system by using some limited expert knowledge. Fuzzy logic theory is known as intelligent computational techniue which has given novel solutions to the various control system problems [13]-[15]. Intelligent control is a viable recent approach which has been emerged from the integration of control methodologies with intelligent computational techniues. The structures of the FLC consist of fuzzifier, rule base, fuzzy interference engine and a defuzzifier. The controller is designed based on the mathematical model of the system. The performance of the systems such as rise time, overshoot, settling time and error steady state can be improved by Fuzzy controller. In fuzzy sets the membership functions are certain or crisp, here the input variables in fuzzy system are mapped by a set of membership functions. A FLS is characterized by IF-THEN rules, its antecedent and conseuent sets are fuzzy set [16]-[17]. A digital computer of the stored program concept was created to perform a variety of tasks in seuence. The operation can be easily changed by changing the program [18]. Through simulation in Matlab by selecting appropriate fuzzy rules are designed to the controller. In this study, we propose two controllers conventional PID and Fuzzy controller to balance the pendulum at upright position and analysis the result. We find the improvement in system performance over the conventional PID controller by the influence of the external disturbance. This paper is organized as follows: section II presents the analytic study of conventional PID and Fuzzy controller. In section III, the description of the mathematical model of the nonlinear inverted pendulum system and design of fuzzy controller is contained. Finally, section IV presents the simulation results followed by the conclusion and the references. II. ANALYTIC STUDY OF CONTROLLERS The main objectives of controller design are as follows: (1) To stabilize the pendulum at its upright position, (2) To uphold the cart position at the origin, (3) Tracking of desired position by pendulum cart, (4) To use minimum control effort reuired to control the pendulum angle and cart position. A. Conventional Controller PID (Proportional, Integral and Differential) controller is the most common form of feedback. In PID controller the basic idea is the examination of signals from sensors placed in the 40 International Journal of Research in Engineering and Management Technology, (ISSN: ), Vol. 01, Issue 03, October, 2015

2 2 system, called feedback signals. Let s consider Fig. 1 the below given unity feedback system, Fig. 1: system with PID controller The PID controller is usually implemented as follows: Fig. 2. FLS consist fuzzifier, rules, inference engine and output processor (defuzzifier) and that are interconnected. The fuzzifier converts the crisp value into Fuzzy Sets. It is needed to activate rules that are in terms of linguistic variables. The rules are the heart of an FLS. The rules are expressed as a collection of IF-THEN statements. The IF-part of a rule represents antecedent and the THEN part represents conseuent. The fuzzified inputs activate the inference engine and the rule base to produce a Fuzzy Set output. The commonly used inferential procedure is minimum and product implication method. Defuzzification is necessary to obtain the crisp number as the output. u t = k p e t + k i e t + k d de t dt (1) e t = y r t y m t (2) Where k p,k i, and k d are the proportional, the integral, and the derivative gains respectively. The controller output, the process output, and the set point are denoted as u(t), y r t and y m t respectively. A proportional controller (k p ) will have effect of reducing the rise time, but never eliminates the steady-state error. An integral controller (k i ) will reduce the steady-state error but may make the transient response worse. A derivative controller (k d ) will have an effect on stability of the system, it reduces the overshoot, and improving the transient response. Effects of these three controllers can be summarized as shown in the Table 1 Table 1: Effects of three controller parameters Gain Effect of increasing gain parameters Rise time Overshoot Settling time Steady Sate error k p Decrease Increase Small change Decrease k i Decrease Increase Increase Eliminate k d Small Change Decrease Decrease Small change To build a self-balancing pendulum on a cart we first derived the system euation then check its real time response (both time and freuency). Then we designed a PID controller to control the close loop function. The easiest way to tune a PID controller is to auto-tune the P, I and D parameters one at a time. The obtained final values are k p = 796.6,k i = , and k d = Fig. 2: A fuzzy control system All the rules that have any truth in their premises will fire and contribute to the output fuzzy set - the one that will represent the force action control variable in the inverted pendulum example. Suppose that at given time, t, the system sensors determine the pendulum angle (Error) to be E0 and the change of angle (Change of error) to be de0. As the Fig. 3 shows, de0 fall into a single region of the change of error variable-namely Zero (Z). The Error E0 has degree of membership in Zero (Z) region. This combination cause rule IF error is Zero and change of error is Zero THEN output force is Zero. The rule has somehow to be combined to form a system output. The following three-step procedure shows how: (1) For each premise expression connected by an AND, take the minimum of the truth of the expressions as the truth level of the premise. (2) Truncate the output fuzzy set being built at the truth level of the premise. (3) Copy the newly modified fuzzy set into the output variable fuzzy set. If that region is not empty, combine it with the current contents by taking the maximum of the new fuzzy region and the currently existing fuzzy region at each point in the domain. B. Fuzzy Controller In implementation of fuzzy control the mathematical model euations of systems are not needed but the expert knowledge of the system behaviour is reuired. The performance specifications of the systems such as rise time, overshoot, settling time and error steady state can be improved by using Fuzzy controllers. Lotfi Zadeh, the father of fuzzy logic is extend two valued logic, defined by the binary pair {0, 1}, to the whole continuous interval [0, 1]. Fuzzy controllers use heuristic information in developing design the control of nonlinear dynamic system. A fuzzy control system is shown in Fig. 3: Firing Interval calculation Fig. 3 shows that the change of error has a 0.48 degree of membership in Zero and error has a 0.57 degree of membership in Zero. In accordance with the first step of the procedure, the 41 International Journal of Research in Engineering and Management Technology, (ISSN: ), Vol. 01, Issue 03, October, 2015

3 3 lesser 0.48 is taken as the truth level of the premise. Then, in accordance with the second step, the level of the output of rule Zero is truncated at that level, and copied to the output variable fuzzy set. One of the several techniues of Defuzzification is applied to produce an expected value for the output action. In this case, using the weighted average of the combined region, a value of +150 mm/s is produced. This value is used to adjust the force and keep the pendulum at upright position. After that, the error and change of error sensing sensors will make new measurements, starting the cycle over again. The most successful fuzzy logic development projects follow an iterative development cycle like shown in the Fig. 4. moments on the inverted pendulum and keeps the pendulum upright. Fig. 6: The Free Body Diagram of the Inverted Pendulum System The Free Body Diagram of the system is used to obtain the euations of motion. Below given Fig. 6 shows the free body diagrams. The meaning and values of parameters are listed in Table 2 Table 2: Values of parameters of Inverted Pendulum system Cart mass (m c ) Pendulum mass (m p ) Friction co-efficient of cart (b) Distance from pendulum rotation axis centre to pendulum mass centre (l) Gravity acceleration (g) Pendulum inertia (I) kg kg 0.1 N/m/s 0.25 m 9.8 m/ s kg.m2 The variables are, F- Force acting on the cart, x- Cart position, φ- Angle between the pendulum and vertically upward direction and θ- Angle between the pendulum and vertically downward direction. To obtain the transfer function of the linearized system euation analytically, we must first take the Laplace transform of the system en. (3) to en. (4) [2-6]. Fig. 4: Fuzzy logic development project iterative development cycle. III. MODEL OF THE SYSTEM A. Mathematical Model of Inverted Pendulum (m p + m c )x + bx m p lφ = u (3) (I+m p l 2 )φ m p glφ = m p lx (4) The obtained Transfer function is as: φ S U S = S 3 + b l+m p l 2 m p l S S 2 m p gl m c +m p S bm p gl (5) Where = m c + m p l + m p l 2 (m p l) 2 (6) Fig. 5: Inverted Pendulum System The system consists of mathematical model of inverted pendulum and model of fuzzy controller. Model of the pendulum and controller was created in Matlab- Simulink program. The inverted pendulum is mounted on a moving cart is shown in the Fig. 5. A servomotor is controlling the translation motion of the cart, through a belt/ pulley mechanism. A rotary potentiometer is used to feedback the angular motion of the pendulum to servo electronics to generate actuating signal. The controller circuits provide the controlling signal which then drives the cart through the servomotor and driving pulley/ belt mechanism. To and fro motion of the cart applies φ S U S = m p l S S 3 +b l+m p l 2 S 2 m p gl m c +m p S bm p gl The obtained Transfer Function in term of values of parameters is as: s s s s B. Design of Fuzzy Controller In fuzzy structure, there are two inputs to fuzzy inference: error e t and change of error de t and one output and contain 25 rules. Fuzzy inference block of the controller design is shown in Fig. 7 below. The steps for designing aimed controller for the inverted pendulum are as follows: (7) (8) 42 International Journal of Research in Engineering and Management Technology, (ISSN: ), Vol. 01, Issue 03, October, 2015

4 4 a) Select the input and output parameters for the fuzzy controller. Here we choose the error signal and the change of error signal as the input parameters and output parameters for the fuzzy controller as force that reuire for the movement of cart position. b) Then divide the universe of discourse into FSs. Fig. 8 and 9 show the input membership functions for the error signal and change of error signal respectively. Here the universe of discourse is divided as Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). Fig. 10 show, the output membership function for the force, whereas the universe of discourses is divided as Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS) and Positive Big (PB). Fig. 9: Membership functions for the change of error signal Fig. 10: Membership functions for the force. Table 3: Rule base for the force Fig. 7: Fuzzy inference block of the controller c) Write the rule base for the Fuzzy controller, based on experience and it is described in the below given Table 3 correspondingly. d) Use the algorithm of the amied controller :Weighted average defuzzification is the best techniue to obtain the crisp output. It can be further described by following en. (9). K x = f i M i=1 M i=1 fi K i c i K c is the extremum value of ith output membership function f i is the weight associated with ith rule (9) Change of error signal Error signal NB NS Z PS PB NB NB NB NS NS Z NS NB NS NS Z PS Z NS NS Z PS PS PS NS Z PS PS PB PB Z PS PS PB PB IV. SIMULATION RESULTS Here we compare the results of Fuzzy controller and PID controllers. The parameters of an inverted pendulum on a cart system are given in Table 2. Two different tasks are described below. The degree of each membership function which was computed in the previous step of fuzzifications encountered by the subprogram called defuzzify and this after certain process it returns defuzzified output. Fig. 11: system with Fuzzy controller Fig. 8: Membership functions for the error signal A. Task 1: Normal Case Fig. 12 shows the response of input at normal case for initial condition y m = 1 rad and y r = 0 rad/sec. The controllers output are moved to the set point value without steady state error. Fuzzy controller takes less settling time, better set-point tracking, less overshoot and thereby producing minimum integral absolute error. The performance measures are tabulated in Table International Journal of Research in Engineering and Management Technology, (ISSN: ), Vol. 01, Issue 03, October, 2015

5 5 Controller Normal Case External disturbance ISE PID Fuzzy ITAE PID Fuzzy IAE PID Fuzzy Fig. 12 The response of input at normal case B. Task 2: External Disturbance The Fig. 13 shows the response of the inverted pendulum system using PID and Fuzzy controllers. Here we add the angle disturbance value as d= 1 rad after 10 sec from starting the simulation. In the presence of disturbances, the Fuzzy controller still outperforms PID controller and in Fuzzy case less control effort is reuired. The Fuzzy controller is made significantly better than the PID controller. V. CONCLUSION In this paper, we design two kinds of controllers (i.e. PID and Fuzzy controller) and analyse the responses from it by using MATLAB simulation to track the inverted pendulum system at commanded position. The result from the figures show that the performance of using PID and Fuzzy controllers, when normal case and external disturbance is presented in the system. In normal case, the controllers output are moved to the set point value without steady state error. Fuzzy controller takes less settling time, better set-point tracking, less overshoot and thereby producing minimum integral absolute error. In the presence of disturbances, the Fuzzy controller still outperforms PID controller and in Fuzzy case less control effort is reuired. Hence, the fuzzy controller capable of handle the plant with system external disturbances. On the other hand, the Fuzzy controller has a comparatively uniformed performance than the classical PID controller, no matter how large the external disturbance is presented in the system. Thus the Fuzzy controller can be considered more robust. REFERENCES Fig. 13: The response of the inverted pendulum system when the value of disturbance value is d= 1 rad To show the visual indications of the control performance, an objective measure of an error performance was made using the Integral of Suare of Errors (ISE), the Integral Timeweighted Absolute Error (ITAE) and Integral Absolute Error (IAE) criteria. They are set in en. (10) to en. (12) given below ISE = 0 e t 2 dt ITAE = t e t dt 0 IAE = e t dt 0 (10) (11) (12) The Table 4 list out the the Integral of Suare of Errors (ISE), the Integral Time-weighted Absolute Error (ITAE) and Integral Absolute Error (IAE) values respectively for the PID and in Fuzzy controllers for all the above experimental tasks. From the table we can conclude that the values of the ISE, ITAE and IAE for the proposed Fuzzy controller are lower than that of the value obtained for the PID controller. That is, the response of the Fuzzy controller is faster and superior to respond the uncertainties than the PID controller. Table 4: ISE, ITAE and IAE values for the PID and Fuzzy controller [1] S. Mori, H. Nishihara and K. Furuta, Control of unstable mechanical system control of pendulum, Int. J. Contr., vol. 23, no. 5, pp , [2] K. Ogata, Modern Control Engineering, 4th ed, Pearson Education (Singapore) Pvt. Ltd., New Delhi, 2005, ch. 12. [3] K. Ogata, System Dynamics, 4th ed, Pearson Education (Singapore) Pvt. Ltd., New Delhi, [4] J. R. White, System Dynamics: Introduction to the Design and Simulation of Controlled Systems, Online literature. [5] Ajit K. Mandal, Introduction to Control Engineering, New Age International Pub., New Delhi, 2000, ch. 13. [6] CTM Example: Inverted Pendulum Modeling pen.html [7] Bennett S. Development of the PID controller. IEEE Control Syst Mag 1993; 13: [3] Chen G. Conventional and fuzzy PID controllers: an overview. Int J Intell Control Syst 1996;1: [8] Li W. Design of a hybrid fuzzy logic proportional plus conventional integral- derivative controller. IEEE Trans Fuzzy Syst 1998; 6: International Journal of Research in Engineering and Management Technology, (ISSN: ), Vol. 01, Issue 03, October, 2015

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