Adaptive Fuzzy Sliding Mode Control of Uncertain Nonlinear SISO Systems

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Procedia Engineering

Procedia Procedia Engineering 00 (2011) Engineering 24000–000 (2011) 33 – 37 www.elsevier.com/locate/procedia

2011 International Conference on Advances in Engineering

Adaptive fuzzy sliding mode control of uncertain nonlinear SISO systems Shaojiang WANGa, Li HOUa, Lu DONGa, Huajun XIAOa, a* a

School of Manufacturing Science and Engineering, Sichuan University, Chengdu and 610065, China.

Abstract In order to improve the performance of single-input single-output (SISO) nonlinear systems with uncertainties, an adaptive fuzzy sliding mode controller (AFSMC) that combines linearization feedback is presented in this paper. The fuzzy logic system is used to approximate the unknown system function and the AFSMC algorithm is designed by used of sliding mode control techniques. Based on the Lyapunov theory, the continuous function is designed to eliminate the chatting action of the control signal. The simplicity of the proposed scheme facilitates its implementation and the overall control scheme guarantees the global asymptotic stability if all the signals involved are uniformly bounded. Simulation results have shown that the proposed controller shows superior tracking performance. AFSMC can effectively achieve desired performance and have much more advantages over conventional SMC.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of ICAE2011. Keywords:Adaptive control, fuzzy system, sliding mode control, nonlinear system, linearization feedback;

1. Introduction Nomenclature θ g

angle of the pendulum m mass of the pole mc mass of the cart acceleration due to gravity L half-length of pole u applied force Over the past two decades, fuzzy logic control has found extensive applications for plants that are complex and ill-dened. In most of these applications, the rule base of the fuzzy controller is constructed from expert knowledge. It has been proven that fuzzy logic can approximate any nonlinear function to any desired accuracy because of the universal approximation theorem [1]. Fuzzy control schemes have * Corresponding author. Tel.: +86-28-85403687; Fax: +86-28-85460940. E-mail address: [email protected].

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.11.2597

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been found to be particularly useful to model unknown functions in nonlinear systems rather that only unknown parameters. There have been signicant research efforts on adaptive fuzzy control for nonlinear systems [2–4]. The major advantage in all these adaptive fuzzy control schemes are that the developed controllers can be handled the complex nonlinear functions instead of linearly parameters. Sliding mode control (SMC) ,which aims to provide as a popular robust strategy to treat system parameter uncertainties and external disturbances, is widely accepted as a powerful control method to solve the tracking control of uncertain nonlinear systems [5]. The main feature of SMC offer good robustness against model uncertainties and external disturbances, provide that the uncertainties and disturbances lie within a bound. However, it inherits a discontinuous control action and hence chattering phenomena will take place when the system operates near the sliding surface. Sometimes this discontinuous control action can even cause the system performance to be unstable. The fuzzy logic controller integrates the fuzzy approximation theory and the SMC have been developed. Moreover, adaptive fuzzy sliding mode control (AFSMC) schemes have been proposed [6–11]. The adaptive fuzzy controller incorporated with a SMC is developed. In other words, the adaptive fuzzy logic systems are utilized to approximate the unknown system functions in designing the SMC of nonlinear system. In order to improve the performance of AFSMC, an adaptive fuzzy logic control combining linearization feedback and SMC is considered in this paper. It is proved that the closed-loop system is globally stable in the Lyapunov sense, if all the signals are bounded and the system output can track the desired reference output asymptotically with uncertainties. 2. Problem formulation A simple approach to robust control is the sliding mode methodology in which the nth-order problem is replaced by an equivalent rst-order system. Consider the nth-order nonlinear dynamic system with a single-input signal-output:  i 1,...., n  1,  xi x i 1 (1)    xn f ( x, t )  g ( x, t )u

where x =[x1,...,xn]T ∈ Rn, is the state vector of the system in the normal form which is assumed available for measurement, u ∈ R is the scalar control input, y=x1 is the scalar system output. The nonlinear functions f(x,t) and g(x,t) are unknown but smooth. The control objective is to design an adaptive fuzzy sliding mode controller for system (1) such that the system output x(t) follows a desired trajectory xd(t) while all signals in the closed-loop system remain bounded. The tracking error vector is defined as:  e x - xd [e e e( n 1) ]T (2) The sliding surface in the error state space is dened as: s ( x, t )  ce (3) where c =[c1,c2, ... ,c(n-1),1]are the coefcients of the Hurwitiz polynominal h(s)=sn-1+cn-1sn-2+···+c1, i.e. all the roots are in the open left half-plane and s is the Laplace operator. Using the input/output feedback linearization approach and Lyapunov method, the following sliding mode control law is designed, which guarantees that the system output tracks the given desired output.  u ( R  f ( x, t )) / g ( x, t ) (4) n 1

R 1 ( x, t )  k sgn( s), k  0; 1 ( x, t )  xd( n )   ci e(i ) ; i 1

(5)

3. Fuzzy logic system In this section, the fuzzy logic system is briey described. The basic conguration of the fuzzy logic system[1] consists of a collection of fuzzy IF–THEN rules, which can be written as: (6) R(j): IF x1 is Fj1 and ··· xn is Fjn THEN y is Bj The fuzzy logic system performs a mapping from U∈Rn, where the input vector x =[x1, ... ,xn]T∈Rn, and the output variable y∈R denote the linguistic variables associated with of the fuzzy logic system. Fji

35 3

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and Bj are labels of the input and output fuzzy sets, respectively. Let i = 1,2, ... ,n denotes the number of input for fuzzy logic system and j = 1,2, ... ,m denotes the number of the fuzzy IF–THEN rules. By using the singleton fuzzication, product inference and center average defuzzication, the output value of the fuzzy system is n

m

n

m

y ( x)  (  y j (  F j ( xi ))) / (  (  F j ( xi )))

i j 1 j 1 i  i 1 

1

(7)

i

where  F j ( xi ) is the membership function of the linguistic variable xi, and yj represents a crisp value i

at which the membership function  B j for output fuzzy set reaches it maximum value. As a usual practice, we assume that  B j ( y j )  1 . By introducing the concept of fuzzy basis function[1], (7) can be

(8) rewritten as: y ( x)   T  ( x) 1 m T 1 m T where θ =[y , ... ,y ] is the parameter vector and ξ(x)=[ξ (x), ... , ξ (x)] is a regressive vector with n

m

n

the regressor dened as:  ( x)    F j ( xi ) / (  (  F j ( xi ))) i

i j 1 i 1

1

(9)

i

4. Adaptive fuzzy sliding mode control(AFSMC)

The result in (4) is realizable only while f (x, t) and g (x, t) are well known. However, f (x, t) and g (x, t) are generally unknown and the ideal controller (4) cannot be implemented. In order to derive the SMC law (4), we use fuzzy logic system to approximate the unknown functions f (x, t) and g (x, t). Hence, the resulting control law is as follows:  u ( R  fˆ ( x |  f )) / gˆ ( x |  g ) (10) T  fˆ ( x |  )   ( x); gˆ ( x |  )  T  ( x); (11) f

f

g

g

Theorem1. Consider the control problem of the nonlinear system (1). If the control action (11) is used, the functions fˆ and gˆ are estimated by (11) and the parameters vector θf and θg are adjusted by the adaptive law (12). The signals of closed-loop system will be bounded and the tracking error will converge to zero asymptotically. f   r1 s ( x); g   r2 s ( x)u; (12) Proof. The optimal parameters vector is dened as: ( x, t )];  g* arg min[sup | gˆ ( x |  g )  g ( x, t )];   *f arg min [sup | fˆ ( x |  f )  f  f  f

 g  g

xR n

where Ωf and Ωg are constraint sets for θf and θg , respectively. The minimum approximation error is dened as: w  ( f ( x, t )  fˆ ( x |  *f ))  ( g ( x, t )  gˆ ( x |  g* ))u

(13)

xR n

Assumption 1:  f  { f  R n | |  f | M f };  g { g  R n | 0   |  g | M g };

(14) (15)

where Ωf , Ωg and ε are pre-specied parameters for estimated parameters’ bound. It is assumed that the fuzzy parameters θf and θg never reach the boundaries. Then, we have n 1

n 1

n 1

n) (i ) (n) (n) (i ) (n)  s  ci e(i )  e(  ci e  x  xd  ci e  f ( x, t )  g ( x, t )u  xd i 1 i 1 i 1  f ( x, t )  g ( x, t )u  1 ( x, t )  f ( x, t )  ( g ( x, t )  gˆ ( x |  g )  gˆ ( x |  g ))u  1 ( x, t )  f ( x, t )  fˆ ( x |  f )  ( g ( x, t )  gˆ ( x |  g ))u  R  1 ( x, t )  f ( x, t )  fˆ ( x |  f )  ( g ( x, t )  gˆ ( x |  g ))u  1 ( x, t )  k sgn( s)  1 ( x, t )  f ( x, t )  fˆ ( x |  f )  ( g ( x, t )  gˆ ( x |  g ))u  k sgn( s )  fˆ ( x |  *f )  fˆ ( x |  f )  ( gˆ ( x |  g* )  gˆ ( x |  g ))u  k sgn( s)  w  Tf  ( x)   gT  ( x)u  k sgn( s )  w

where   *f   f ,   g*   g . f g

(16)

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Now the Lyapunov function candidate is defined as: V  ( s 2   Tf  f / r1   gT  g / r2 ) / 2

(17)

where r1 and r2 are positive constant. The time derivative of V along the error trajectory (17) is V  ss   Tf  f / r1   gT  g / r2  s ( Tf  ( x )   gT  ( x )u  k sgn( s )   )   Tf  f / r1   gT  g / r2  s Tf  ( x )   Tf  f / r1  s gT  ( x )u   gT  g / r2  sk sgn( s )  sw   Tf ( rs ( x )   f ) / r1   gT ( r2 s ( x )u  g ) / r2  k | s |  sw   k | s |  sw  0

(18)

Since w it the minimum approximation error (18) is the best result that we can obtain. Therefore, all signals in the system are bounded. Obviously, if e(0) is bounded, then e(t) is also bounded for all t. Since the reference signal xd is bounded, then the system states x is bounded as well. Moreover, we employ the continuous function Sδ replacing sgn(s) to avoid chattering problem.  S  s / (| s |   ) s / (| s |   0   1 e ) (19) where δ0 and δ1 are positive constant. 5. Simulation

In this section, the proposed AFSMC is applied to the inverted pendulum to let it to track a desired trajectory. In this example, we test the AFSMC on the tracking control of the benchmark control problem of inverted pendulum. Let x1 =θ be the angle of the pendulum with respect to the vertical line and x2 =  . The dynamic equations of such system are given by [1].  x1  x2  cos x1 / ( m  mc ) g sin x1  mLx22 cos x1 sin x1 / ( m  mc )   u 2  x 2 (4 / 3 cos / ( )) (4 / 3 L  m x m  m L  m cos 2 x1 / ( m  mc )) c 1 

(20)

The control objective is to maintain the system to track the desired angle trajectory, xd = θd = 0.1sin(πt) +0.05sin(2.5πt). The system parameters are given as mc = 1 kg, m = 0.1 kg, L= 0.5 m, g = 9.81 m/s2. Choose the sliding surface as s = c1e + e , c1 = 5. The initial values of parameters θf and θg are set by θf=0.2 and θg=0.2. The membership functions for system state xi, i = 1,2 are selected (Fig.1) as:

 PM ( s)  exp[(( s   / 6) / ( /12)) 2 ];  PS ( s)  exp[(( s   /12) / ( / 12)) 2 ];  ZO ( s )  exp[( s / ( /12)) 2 ];  NS ( s)  exp[(( s   / 12) / ( / 12)) 2 ]; 2  NM ( s )  exp[(( s   / 6) / ( / 12)) ];

(21)

Then there are 25 rules to approximate the system functions f(x, t) and g(x, t), respectively. Choose the initial condition θ=[π/60,0]T , adaptive parameters r1=5, r2=1, δ0=0.03, δ1=5, k=5 and step size 0.01 s. Fig. 2-Fig.5 show the simulation results.

Fig. 1.The membership functions of xi.

Fig.2. Response of the angle θ.

Fig.3. Control input signal.

Shaojiang WANG et al.Procedia / Procedia Engineering (2011) 33 – 37 Shaojiang WANG/ Engineering 0024 (2011) 000–000

Fig.4. Estimation of f(x, t) and f(x, t).

Fig.5.Estimation of g(x, t) and g(x, t).

Fig.6.Sliding mod surface s.

The simulation result for the angle of the pendulum θ is shown in Fig.2, and the control input signal is shown in Fig. 3. Fig. 4 and Fig.5 show the estimation of f(x,t) and g(x,t). The sliding mod surface s is shown in Fig.6. It can be seen that actual trajectories converge rapidly to the desired ones, the control signal and the estimated parameters are bounded. These simulation results demonstrate the tracking capability of the proposed controller and its effectiveness for control tracking of uncertain nonlinear systems. 6. Conclusions

In this paper, an adaptive fuzzy sliding control algorithm has been proposed for a class of unknown nonlinear systems. We introduced the fuzzy sliding mode control and proposed the robust control using the adaptive control strategy. Moreover, based on the Lyapunov synthesis approach, the linearization feedback parameters can be tuned on-line by the adaptive law. The drawback of chattering in sliding mode control is reduced. Finally, the proposed method has been applied to control the inverted pendulum to track a reference trajectory. The simulation results show that the adaptive controller can achieve desired performance. Simulation results show the precise angle control, which is obtained in spit of the disturbance and uncertainties in the system, and advantages over the conventional SMC. References [1] L.X. Wang, A Course in Fuzzy System and Control, Prentice-Hall, Upper Saddle River, NJ, 1997. [2] C.Y. Su, Y. Stepanenko, Adaptive control of a class of nonlinear systems with fuzzy logic, IEEE Trans. Fuzzy Syst. 2 (1994) 285–294. [3] B.S. Chen, C.H. Lee, Y.C. Chang, H1 tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach, IEEE Trans. Fuzzy Syst. (1996) 432–443. [4] Y.C. Chang, Adaptive fuzzy-based tracking control for nonlinear SISO systems via VSS and H1 approaches, IEEE Trans. Fuzzy Syst. 9 (2001) 278–292. [5]Jovan, D.B., Li, S.M., Raman K M. Robust adaptive variable structure control of spacecraft under control input saturation. Journal of Guidance, Control and Dynamics 24 (2001), 14–22. [6] J. Wang, A.B. Rad, P.T. Chan, Indirect adaptive fuzzy sliding mode control: Part I-fuzzy switching, Fuzzy Sets Syst. 122 (1) (2001) 21–30. [7] R.J. Wai, K.H. Su, Adaptive enhanced fuzzy sliding-mode control for electrical servo drive, IEEE Trans. Ind. Electron. 53 (2006) 569–580. [8] C.-C. Cheng, S.-H. Chien, Adaptive sliding mode controller design based on T–S fuzzy system models, Automatica. 42 (6) (2006) 1005–1010. [9] R.-J. Wai, Fuzzy sliding-mode control using adaptive tuning technique, IEEE Trans. Ind. Electron. 54 (2007) 586–594. [10] T.-Z. Wu, Y.-T. Juang, Adaptive fuzzy sliding-mode controller of uncertain nonlinear systems, ISA Trans. 47 (2008) 279– 285.

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