Procedia Technology 10 ( 2013 ) 581 – 586

2212-0173 © 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of the University of Kalyani, Department of Computer Science & Engineeringdoi: 10.1016/j.protcy.2013.12.398

ScienceDirect

International Conference on Computational Intelligence: Modeling Techniques and Applications (CIMTA) 2013

Gravitational Search Algorithm and Lyapunov Theory based Stable Adaptive Fuzzy Logic Controller

Ananya Roya and Kaushik Das Sharmab**

aDepartment of Electrical Engineering, Bengal Engineering and Science University, Shibpur, India bDepartment of Applied Physics, University of Calcutta, 92, APC Road, Kolkata-700009, India

Abstract

In this paper a new hybrid design methodology for stable adaptive fuzzy controllers for a non-linear system is proposed. The proposed design strategy utilizes the gravitational search algorithm (GSA) based heuristic global search technique and Lyapunov theory based local adaptation. The objective is to design a self-adaptive fuzzy controller, optimizing the free parameters of the fuzzy controller, so that the designed controller can guarantee desired stability. The GSA based design and hybrid GSA-Lyapunov concurrent design methodologies are implemented for a nonlinear plant with different reference signals and the results demonstrate the superiority of proposed hybrid design methodology. © 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the University of Kalyani, Department of Computer Science & Engineering.

Keywords: Gravitational Search Algorithm; Lyapunov Theory; Adaptive Fuzzy Logic Controller.

1. Introduction

Fuzzy controllers appear in consumer products – such as washing machines, video cameras, cars – and in industry, for controlling cement kilns, underground trains, and robots [2]. A fuzzy controller is an automatic controller, a self-acting or self-regulating mechanism that controls an object in accordance with a desired behaviour. The object can be, for instance, a robot set to follow a certain path. A fuzzy controller acts or regulates by means of rules in a more

A Roy. Tel.:+91 9062160052; E-mail address:ananya.roy2009@gmail.com; * K Das Sharma Tel.:+919831498689; E-mail address:kdassharma@gmail.com.

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© 2013 The Authors. Published by Elsevier Ltd.Selection and peer-review under responsibility of the University of Kalyani, Department of Computer Science & Engineering

582 Ananya Roy and Kaushik Das Sharma / Procedia Technology 10 ( 2013 ) 581 – 586

or less natural language, based on the distinguishing feature: fuzzy logic. The rules are invented by plant operators or design engineers, and fuzzy control is thus a branch of intelligent control [3]. The heart of designing of FLC lies in the selection of efficient membership functions that represents the human expert's version of the linguistic variables, because different membership functions determine different extent to which the rules affect the action and the performance of the controller. But with fixed central positions of input membership functions, fixed positions of output singletons and fixed values of the free parameters obtained from trial and error method not only violates the autonomy of the controller but also gives unsatisfactory results. To deal with this problem so many approaches such as Genetic Algorithm based approach, Particle Swarm Optimization based approach, Ant Colony Optimization based approach, have been proposed in last few years with the development in the optimization techniques. Lyapunov Theory based approach (LTBA) is used in control systems for long time to ensure the closed stability of the system. But the main problem of this approach in case of tracking problem is that it takes long time to track. Gravitational Search Algorithm (GSA) based on the law of gravity and mass interactions provides excellent results without having much assurance of the closed loop stability of the system. In this paper a new design methodology is proposed such that problems of both approaches is eliminated giving excellent results with confirming closed loop stability of the system. In present work, a hybridization of locally operative LTBA and global version of GSA based approach have been proposed in such a way that both the approaches work simultaneously and simulated for a benchmark non-linear plant [1][3][4].

2. Design of Stable Adaptive Fuzzy Controller

Suppose that the plant is a n'th order nonlinear system described by the differential equation

xyuxgxfx n )()()( (1)

where f and g are unknown functions, Ru and Ry are the input and output of the plant, respectively, and

nTnTn Rxxxxxxx ),...,(),...,,( )1(

21 & is the state vector of the system. For (1) to be controllable, we require that 0)( xg . Here we assume that 0)( xg . The control objective is to design a controller based on fuzzy systems and an adaptation law for adjusting the parameter vector θ, such that the plant output y follows the ideal output ym, which and its time derivatives are known and bounded. Since the functions f (x) and g(x) in the plant are nonlinear and are assumed to be unknown, we are dealing with a quite general single input single output (SISO) nonlinear control problem. Therefore, in the control objective we did not insist that the plant output y should converge to the ideal output ym, asymptotically; we only require that y follows ym, as close as possible. Let, e = ym – y, Tneeee ),...,,( )1( & and nT

n Rkkkk ),...,,( 21 be such that all roots of the polynomial

121 ... ksksks n

nn are in the open left half of the complex plane. Now the ideal control law of the system

(1) is given as ekyxfxg

u Tnm )(* )(

)(1 . Because of the choice of k, we have 0)( te as t , i.e., the

plant output y converges to the ideal output ym, asymptotically. In practical case as f (x) and g(x) is not known, u* cannot be implemented in real. The probable solution of this is to implement adaptive fuzzy logic controller which can give an optimal solution. Here zero order Takagi-Sugeno (T-S) fuzzy system is constructed with control input u(t) as

)()()( eueutu sc (2)

)( eu c is the output of the AFLC and can be written in the form

)()(

)()(

1

1 ee

eeu T

N

ll

N

lll

c

(3)

583 Ananya Roy and Kaushik Das Sharma / Procedia Technology 10 ( 2013 ) 581 – 586

where θ = the vector of the output singletons, and )(el = the firing degree of rule ‘l’, N = the total number of rules, ξ(e)= vector containing normalized firing strength of all fuzzy IF-THEN rules, us(e), the supervisory control, is added in addition to bound the error.

Consider the Lyapunov function, ePeV T21

, where P is a symmetric positive definite matrix satisfying the

Lyapunov equation, QPP cTc ,where Q is a positive definite matrix. The time derivative of V along the

closed-loop system trajectory is

)]}()()[({21 eueuuxgbPeeQeV scc

TT & (4)

us(e) should be designed in a way that V& should be negative semidefinite i.e. 0V& . us (e) can be defined as

ekyf

bubPeIeu Tn

mU

Lcc

Ts

)(1sgn)(

(5)

where,

VVifIVVifI

01 V is constant and )(xff U , bbL 0 . To minimize V, it can be shown that V& is

negative. Hence, the closed loop stability is guaranteed. Thus the adaptation law becomes

)(epen

T & (6)

where 0 is the adaptation gain or learning rate and np is the last column of P [1][8][9].

3. Gravitational Search Algorithm and Lyapunov theory based hybrid AFLC design strategy

Lyapunov theory based approach is a traditional approach used for long time in control system. This method guarantees the closed loop stability of the nonlinear system. But the main drawback of this approach is that it takes a long time to track the reference signal. On the other hand Gravitational Search algorithm is a heuristic optimization technique. It is based on the Newton’s Law of Gravity. It is a global search technique which gives excellent results without confirming closed loop stability. In the Hybrid design both the techniques run concurrently such that the system is closed loop stable and the plant is able to give excellent tracking results overcoming the problems of the both techniques.

3.1. Gravitational Search Algorithm based approach (GSABA)

In GSA based design, a mass in solution space is a vector containing all required information to construct a fuzzy controller, e.g. i) information about the positions of the MFs, ii) number of rules, iii) values of scaling gains, iv) positions of the output singletons etc. Each mass (agent) has four specifications: position, inertial mass, active gravitational mass, and passive gravitational mass [4]. The position of the mass corresponds to a solution of the problem, and its gravitational and inertial masses are determined using a fitness function. This vector can be written as: M = [center locations of the MFs | scaling gains | positions of the output singletons] (7) The different steps of the proposed algorithm are the followings:

1. Identify the search space.

584 Ananya Roy and Kaushik Das Sharma / Procedia Technology 10 ( 2013 ) 581 – 586

2. Initialize randomly. 3. Evaluate fitness of the agents. 4. Update Gravitational Constant, best and worst values of the fitness function and the values of the masses. 5. Calculate the total force in different directions. 6. Calculate acceleration and velocity. 7. Update positions of the agents. 8. Repeat steps 3 to 7 until the stopping criterion is reached. 9. Stop.

3.2 Hybrid design based approach (HDBA)

In the hybrid design methodology both GSA based approach and Lyapunov theory based approach complement each other. This method confirms the closed loop stability as well as gives excellent tracking result without

(a) (b) (c)

Fig. 1. (a) Output response with triangular input, (b) error response, (c) controlled input response for GSABA

(a) (b) (c)

Fig. 2. (a) Output response with sinusoidal input, (b) error response, (c) controlled input response for GSABA

(a) (b) (c)

Fig. 3. (a) Output response with triangular input, (b) error response, (c) controlled input response for HDBA

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

time(sec)

amlit

ude

0 100 200 300 400 500 600 700 800 900 1000-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 100 200 300 400 500 600 700 800 900 1000-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

time(sec)

amlit

ude

0 100 200 300 400 500 600 700 800 900 1000-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700 800 900 1000-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

time(sec)

amlit

ude

0 100 200 300 400 500 600 700 800 900 1000-0.15

-0.1

-0.05

0

0.05

0.1

0 100 200 300 400 500 600 700 800 900 1000-100

-80

-60

-40

-20

0

20

40

60

80

585 Ananya Roy and Kaushik Das Sharma / Procedia Technology 10 ( 2013 ) 581 – 586

(a) (b) (c) Fig. 4. (a) Output response with sinusoidal input, (b) error response, (c) controlled input response for HDBA

consuming long time. Both techniques run simultaneously such that global optimization and local optimization of the variables is possible. The candidate vector, ]|[ M has two parts γ = [center locations of the MFs | scaling gains] and Θ = [positions of the output singletons] (8) In this method γ part is optimized using GSA technique and Θ is optimized using both the techniques. At each iteration the positions of the output singletons are optimized locally as well as globally in parallel manner. Table 1: Comparison between the approaches

Input signal Control Approach MFs

IAE CE FF Best

Value Avg. Value

Best Value

Avg. Value

Best Value

Avg. Value

Triangular Wave

GSABA 3X3 0.1611 0.1662 2117.4 2345.8 0.3744 0.3756 HDBA 0.1591 0.1602 1994.5 2015.7 0.3622 0.3623 GSABA 5X5 0.1319 0.1357 4268.1 4336.4 0.6121 0.6514 HDBA 0.1205 0.1285 3403.1 3516.2 0.5909 0.5979 GSABA 7X7 0.1221 0.1222 5421.4 5941.8 0.6012 0.6241 HDBA 0.1187 0.1190 5221.4 5324.2 0.5534 0.5924

Sinusoidal Wave

GSABA 3X3 0.1669 0.1685 1985.7 2076.7 0.4599 0.4912 HDBA 0.1505 0.1510 1770.1 1985.1 0.3455 0.3940 GSABA 5X5 0.1188 0.1189 2899.8 2948.8 0.4121 0.4436 HDBA 0.1160 0.1171 2668.8 2742.8 0.3988 0.4087 GSABA 7X7 0.1150 0.1158 4541.4 4781.4 0.4325 0.4682 HDBA 0.1133 0.1135 3121.1 3215.4 0.4299 0.4417

4. Simulations and Studies

To prove the effectiveness of the proposed method, inverted pendulum which is a benchmark nonlinear system, is taken as the plant. The plant is simulated using a fixed step 4th order Runge-Kutta method where sampling time T = 0.01 sec. The state space model of this plant can be written as

1

2

21

)()()(xy

tuxgxfxxx

&

& (9)

where

)(cossin)(cossin)(

34

12

1112

mMlxmlxgmMxxmlxxf

and )(cos

cos)(3

41

21

mMlxmlxxg

, g = 9.8 m/s2,

m = 0.1 kg, M = 1 kg, l = 0.5 m. In both techniques, sum of integral of absolute error (IAE) )( edt and control

energy (CE) ( dtu 2 ) is minimized. The fitness function can be written as

0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

time(sec)

amlit

ude

0 100 200 300 400 500 600 700 800 900 1000-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 100 200 300 400 500 600 700 800 900 1000-60

-40

-20

0

20

40

60

80

100

586 Ananya Roy and Kaushik Das Sharma / Procedia Technology 10 ( 2013 ) 581 – 586

Fitness function = IAE + λ * CE (10) Where λ is the multiplying factor and its value is taken as 0.0001 to convert CEs range in the range of IAE. GSA based approach In this approach M of (7) is optimized in such a way that sum of integral of absolute error (IAE) and control energy (CE) is minimized. Population size of 10 agents is taken and for each simulation the maximum no. of iteration of 100 is taken. The obtained results are given in the table 1. Hybrid design approach In this approach γ is optimized using only the GSA, and θ is optimized using both LTBA and GSA concurrently. Here also, M is optimized in such a way that sum of Integral of absolute error (IAE) and Control energy (CE) is minimized. Population size of 10 agents is taken and for each simulation the maximum iteration of 100 is taken.

001.001.001.01

Pand the value of ν is taken as 0.1. The obtained results are given in the table I.

The response of the output, error and controlled input of the both approaches are given in fig. 1, 2, 3 & 4. In these approaches sinusoidal wave and triangular wave is taken as the input signals.

5. Conclusion

The above proposed technique is proved to be effective as the results are very satisfactory. From the table 1 it is to be observed that the IAE is decreasing in nature as membership functions increases and for sinusoidal wave input, they are minimal. But if we observe the values of CE, it is increasing in nature as the no of membership functions of the FLC increases. But this is a new approach. It confirms the closed loop stability and gives excellent tracking result. It is able to overcome the disadvantages of the classical Lyapunov theory based approach as well as the new Gravitational search algorithm. References [1] Li-Xin Wang, A Course in Fuzzy Systems and Control, Prentice-Hall International, Inc. [2] S. Yasunobu, S. Miyamoto, Fuzzy control for automatic train operation in proc. 2nd IFSA Congress, Baden-Baden, April, 1983. [3] Jan Jantzen, Foundation of Fuzzy Control, John Wiley & Sons, Ltd. [4] K. Das Sharma, A. Chatterjee, and F. Matsuno, A Lyapunov theory and stochastic optimization based stable adaptive fuzzy control

methodology, In: Proc. SICE Annual Conference (2008), Japan, 1839-1844. [5] E. Rashedi, Hossein Nezamabadi-pour, Saeid Saryazdi ,Information Sciences 179 (2009) 2232–2248. [6] Kim Chwee Ng and Yun Li, Design of Sophisticated Fuzzy Logic Controllers Using Genetic Algorithms, 1994, IEEE. [7] K. Das Sharma, A. Chatterjee, and A. Rakshit, A hybrid approach for design of stable adaptive fuzzy controllers employing Lyapunov

theory and particle swarm optimization, IEEE Trans. Fuzzy System, 17 (2009), 329-342. [8] K Das Sharma, A Chatterjee, A Rakshit, Instrumentation and Measurement, A Random Spatial lbest PSO-Based Hybrid Strategy for

Designing Adaptive Fuzzy Controllers for a Class of Nonlinear Systems , IEEE Transactions on 61 (6), 1605-1612. [9] A Roy, K Das Sharma, GA and Lyapunov Theory Based Hybrid Adaptive Fuzzy Controller for Nonlinear Systems, International Journal of

Electronics, In Press.