Direct Adaptive Output Feedback CMAC Control with Unknown Control Gain for a Class of Nonlinear Systems

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Procedia

Engineering

www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE)

Direct Adaptive Output Feedback CMAC Control with

Unknown Control Gain for a Class of Nonlinear Systems

Chun-Sheng Chen

∗

Department of Electronic Engineering, China University of Science and Technology, Taipei 115, Taiwan

Abstract

A direct adaptive output feedback CMAC (cerebellar model articulation controller) control based on only output measurements is proposed with unknown control gain function for a class of affine nonlinear systems. Therefore, it is needed to design a state observer to estimate unmeasured states of the systems. By using strictly positive-real Lyapunov theory, the stability of the closed-loop system can be guaranteed. Finally, simulation results for an inverted pendulum system show that the effect of the approximation error on the tracking error can be attenuated efficiently. © 2012 Published by Elsevier Ltd.

Keywords: CMAC neural network; Direct adaptive controller; Observer; Nonlinear systems

1. Introduction

The control of general nonlinear system has been a widely investigated problem because of its wide applications in practical systems. Many practical systems may be so complex or, even, model-free that to construct a mathematical model or identify its parameters is difficult or even impossible. There has been considerable attention over the years on researches using neural networks (NNs) based on human heuristic and learning algorithms [1,2]. Especially, the cerebellar model articulation controller (CMAC) was first developed by Albus in the 1970s [3], and is regarded as a nonfully connected perceptron-like associative memory network with overlapping receptive-fields. Compared with the general multiplayer neural network with back-propagation algorithm, the CMAC has been applied widely to the closed-loop control of complex dynamical systems because of its simple computation, fast learning property and good

∗

Corresponding author.

E-mail address: [email protected].

Open access under CC BY-NC-ND license.

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generalization capability [4]. Therefore, the application of CMAC is not only for the control problem but also for the model-free function approximator.

However, these control methods are based on the assumption that the system states are known or available for feedback. If system states are not available, an observer needs to estimate the unmeasured states and the SPR-Lyapunov design approach has been used to derive the adaptive and control laws [5,6], and the stability of the closed-loop system can be guaranteed.

2. System formulation

Consider the nth-order nonlinear dynamic system described in the following form: d

u g f

y(n) = (x)+ (x) + (1)

where f(x) is an unknown continuous function, g(x)is also unknown but strictly positive function, d is the unknown external bounded disturbance, u and y are the control input and output, respectively. Let

T n T n x x x x x x , , , ] [ , , , ] [ ( 1) 2 1 = − = L &L

x . Rewriting (1) in the state space representation

x x x x x T C y d u g f B A = + + + = ( ( ) ( ) ) & (2) where ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − × − × − ) 1 ( 1 1 1 ) 1 ( 0 _n n n A 0 I 0 , B=[0,0,L,0,1]T, C=[1,0,L,0]T.

Give the desired trajectory y_d and define the output tracking error e=yd −y, desired trajectory

vector [ , , , (n 1)]T, d d d

d = y y& L y −

y output tracking error vector e= y_d −x. The control objective is to force the output y for tracking a given bounded desired trajectory y_d. If f(x) and g(x) are known and the system is free of the external disturbance d , the certainty equivalent controller can be chosen as

] ) ( [ ) ( 1 ( ) * _x _x T_e c n d f K y g u = − − + ₍₃₎ where c T n c c c k k k

K =[ ₁, ₂,L, ] . Let _u₌_u*_{, substituting (3) into (1), the error dynamics is derived as}

0 1 2 ) 2 ( 1 ) 1 ( ) ( ₊ ₊ − ₊ ₊ ₌ − − _k _e _k _e _k _e e k en _nc n _nc n L c& c (4)

If K_c to be chosen such that the characteristic polynomial is Hurwitz, then it implies that the tracking error trajectory will converge to zero when time tends to infinity, i.e. lim_t_→_∞e(t)=0.

However, f(x) and g(x) are unknown functions and only the system output is available for measurement, the optimal control (3) cannot be implemented. An observer will be designed to estimate the state vector and determine a direct adaptive CMAC controller such that the closed loop is stable. 3. Design of CMAC system

The architecture of the CMAC system includes input space, association memory space, receptive-field space, weight memory space and output space. The Gaussian function is chosen as the receptive-field basis function which can be expressed as

] / ) ( exp[ ) ( i i ik 2 ik2 ik I I m σ φ = − − for k=1 L,2, ,Nb. (5)

where φik(Ii) represents the kth block receptive-field basis function of the input Ii with the mean mik and the variance σ ;_ik Nb is the number of block for each input dimension. The multidimensional receptive-field function bp(I) is associated with pth hypercube and can be expressed as

) ( )

( ni 1 ik i

p I

b I =∏= φ for p=1 L,2, ,Nh where Nh is the number of memory.

In this technique, each state variable is quantized and the problem space is divided into discrete states. A vector of quantized input values specifies a discrete state and is used to generate addresses for retrieving information from memory for this state. The areas formed by blocks (or hypercubes) are the

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addresses that store data. CMAC uses a set of indices as an address in accordance with the current state to extract the stored data. By considering the input vector v_k, the output y_v_k of CMAC with Gaussian basis function can be mathematically expressed as

k k h k v T T v N T k v Diag b b b y =a ( ₁, ₂,L, )w=ζ w=w ζ (6)

where the association index with Gaussian basis function T vk ζ is defined as ( ₁, ₂, , ) m k N T k T v a Diag b b L b ζ =

and the weight vector w of CMAC is expressed as T

Nh w w w, , , ] [ ₁ ₂ L = w .

4. Direct adaptive CMAC output feedback controller design

In order to achieve the control objective, the overall controller is chosen a

s

D u u

u

u= + + (7)

where u_D is a CMAC-neural controller designed to approximate the optimal control law (3), and u_s and a

u are employed to compensate the external disturbance and modeling error. Based on the certainty equivalence approach, an optimal control law can be rewritten as

] ˆ ) ( [ ) ( 1 ( ) * _x _x T_e c n r K y f g u = − − + + (8)

Applying (8) and (7) into (2) and after some manipulation, one has the error dynamic equation

e x e e e T a s D T c C e d u u u u g B BK A = − − − − + − = 1 * ₎ _] )( ( [ ˆ & (9) Design the following observer to estimate the error vector e in (9)

e e e e ˆ ˆ ) ˆ ( ˆ ˆ ˆ 1 1 1 T o T c C e e e K BK A = − + − = & (10) where o T n o o o k k k

K =[ 1, 2,L, ] is the observer gain vector, which is selected such that the characteristic

polynomial of T

oC K

A − is strictly Hurwitz because (C,A) is observable. Define the observation errors

e e e ˆ ~₌ ₋ _and 1 1 1 ˆ ~ _e _e

e = − . Subtracting (10) from (9), the observable error vector is given by

e x x x x x x e e ~ ~ ] ) ˆ ( ˆ ) ˆ ( ˆ )) ˆ ( )( ˆ ( ˆ [ )] ˆ ( ˆ ) ( )( ( [ ~ ) ( ~ 1 * T a s D b T o C e d u g u g u u g B g g x u B C K A = − − − − + − + − = & (11) where u_b(x)=u*(x)−u_D(xˆ)−u_s−u_a . By replacing uD(xˆ) and gˆ(xˆ) in (11) with specific CMAC system as (6). Define the optimal parameter vector w*_D and w*_g as

|] ) | ˆ ( ) ( | sup [ min arg * ˆ , * 2 1 D U U D D w D u u D w x x w x x − = ∈ ∈ Ω

∈ , arg minf[ sup₁,ˆ ₂| ( ) ˆ(ˆ| ) ]| * g U U w g g g g w x x w x x − = ∈ ∈ Ω ∈

where x , xˆ , w and D w belong to compact sets g U1,U2, ΩD and Ωg, respectively, which are defined asU₁={x: x ≤M₁},U₂ ={xˆ: xˆ ≤M₂}, Ω_D={w_D: w_D ≤M_D} and Ω_g ={w_g: w_g ≤M_g}. Lemma 1 [7]: Given an arbitrary ε*>0, there exist ξ(xˆ)=[ξ₁(xˆ),L,ξM(xˆ)] and an ideal parameter

vector T gM g g g (w*1,w*2, ,w* ) * ₌ _L w such that − =

∑

M₌ − ξ +ε j j gj gj j g b g g c w w u (x)( (x) ˆ(xˆ|w )) ₁ ( * ) (xˆ) , where ε ≤ε*, cj are some positive constants, w*_gj and w_gj are jth elements of w*_g and w_g.

According to Lemma 1, (11) can be rewritten as

e w x w x x ξ w w x x e e T a g s g T D g M j j gj j T o C e u g u g g w c B C K A = + − − + + − =

∑

₌ 1 1 ~ ] ) | ˆ ( ˆ ) | ˆ ( ˆ ) ˆ ( ~ ) | ˆ ( ˆ ) ˆ ( ~ [ ~ ) ( ~& _ξ _θ (12) where w~_D =w_D* −w_D_, gj gj gj w w

w~ = * − _, _j₌_{1 L}_, _,_M _{, and the total approximation error is defined as} d u u g _g − _D _D − + =_ε ˆ(_xˆ|_w )[ *(_x) (_xˆ|_w*)]

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] ) | ˆ ( ˆ ) | ˆ ( ˆ ) ˆ ( ~ ) | ˆ ( ˆ ) ˆ ( ~ )[ ( ~ 1 1=

∑

₌ ξ + g TDξ − g s− g a +θ M j cjwgj j g g u g u s H e x x w w x x w x w (13)

where s denotes the Laplace operator d /dt and H(s)=CT(sI−(A−KoCT))−1B is the transfer function of (12). In order to use the SPR-Lyapunov design approach, (13) can be written as

] ) | ˆ ( ˆ ) | ˆ ( ˆ ) ˆ ( ~ ) | ˆ ( ˆ ) ˆ ( ~ )[ ( ) ( ~ 1 1 1 1 1 1 1 =

∑

₌ ξ + g DTξ − g s − g a +θ M j cjwgj j g g u g u s L s H e x x w w x x w x w (14) were L s =sm+bsm−1+_L+b_m 1 )

( , m < , is chosen such that n L−1(s) is a proper stable transfer function and H(s)L(s) is a proper SPR transfer function, u_s₁=L−1(s)u_s , u_a₁=L−1(s)u_a , ξ₁(xˆ)=L−1(s)ξ(xˆ) andθ₁=L−1(s)θ . Because (14) is a SPR dynamic system, there exists _P₁=_P₁T >0_, ₀

2 2 =PT >

P are the

solutions of the following matrix equation

1 1 1 ( ) ) (A−BKcT TP +P A−BKcT =−Q , (A−KoCT)TP2+P2(A−KoCT)=−Q2, P2Bc=Cc (15) where T m c b b

B =[0,0,L, 1,L, ] ,Cc=[1,0,L,0]T,Q1=Q1T,Q2 =Q2T is the given positive definite matrix. Theorem 1: Consider the nonlinear system (1) and assume that only output variable is measurable. Construct the adaptive CMAC controller as (6), uD(xˆ|wD)=wTDξD(xˆ) and gˆ(xˆ|wg)=wTgξg(xˆ), and the auxiliary compensation control and the adaptation laws are chosen as

) | ˆ ( ˆ ˆ ) ( _oT ₁ T 1 _g a L s K P g u ₌ e − x w _, ₍ ₎_sgn(~ ₎_ˆ 1₍_ˆ_| ₎ 2 c g T s kL s PB g u ₌ e − x w ₍₁₆₎ ) ˆ ( ) | ˆ ( ˆ ~ 1 2 x w x e w γ T _c _g ξ D D =− P B g & , w&g =−γge~TP₂Bcξ₁(xˆ) (17)

whereγ_D >0 and γ_g >0 are positive adaptive gain constants to be designed, P1=P1T >0 and 0

2 2 =PT >

P are the solutions of the matrix equation (15), respectively. The proposed control scheme can ensures that all the closed-loop signals are bounded, and the tracking errors converge to zero.

Proof: Choose the Lyapunov function candidate as

D T D M j j gj gj T T D g c w w P P V eˆ eˆ ~e e~ ~ ~ ₂1 w~ w~ 1 21 2 2 1 1 2 1 γ γ + + + =

∑

₌ (18)

Taking the time derivative of V yields

D T D M j j gj gj T T T T D g c w w P P P P

V& e&ˆ eˆ eˆ eˆ& e~& ~e e~ e~& ~ ~& 1 w~ w~& 1 1 2 2 1 2 2 1 1 2 1 1 2 1 γ γ + + + + + =

∑

₌ (19)

Substituting (10) and (12) into (19) and the fact w~&_D =−w&_D, w~&_g =−w&_g, the above equation becomes )] [( | ~ | ~ ~ ˆ ˆ )] | [(| | ~ | ~ ~ ˆ ˆ ) ) ˆ ( ˆ ( ~ ~ ~ ˆ ˆ 2 2 1 1 2 1 1 2 2 1 1 2 1 1 1 2 2 1 1 2 1 k M PB Q Q k PB Q Q u g PB Q Q V c T T T c T T T s g c T T T − + − − ≤ − + − − ≤ − + − − = e e e e e e e e e e w x e e e e e θ θ & (20) According to k ≥M , then (20) becomes eˆ eˆ ₂1e~ ₂e~

1 2 1 _Q _Q V&≤− T − T _.Denoting _[ _, _] 2 1 Q Q diag = Q and ] ~ , ˆ [ T T T

E = e e , then _V&_≤₋1₂_ET_Q_E_<0_{. Therefore, it imply that}

∞

∈ L

e e x

x,ˆ, ,ˆ . Integrating V& from 0 = t to t =T yields

∫

≤ (0)− ( )<∞ 0 2 1_E _Edt _V _V _T T _T

Q . Now because all variables in the right side of (12)

are bounded, ~e&₁(t) is bounded. According to Barbalat’s lemma, we have lim_t_→_∞E(t)=0. 5. Simulation results

Consider the dynamic equation of the inverted pendulum system [8]

(

)

(

)

u d x m m m l x x m m m l x x mlx x g m m y c c v c ₊ − + + − + − + = 1 2 3 4 1 1 2 3 4 1 1 2 2 1 cos ) ( cos cos ) ( sin cos sin ) ( & & (21)

where y=x₁=θ is the angle of the pendulum with respect to the vertical line and x₂ =θ&, .m_c is the mass of cart, m is the mass of pole, g_v=9.8 m/s2_{is the acceleration due to gravity,}_l_{is the half length} of the pole, u is the control input. In this example, m=0.1kg, m_c=1kg, l=0.5m. The control

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objective is to maintain the output y to track the desired trajectory with only output is available for measurement. The disturbance d is assumed to be a square wave with the amplitude ±1 and period the π (s). The Gaussian basis functions in each input space are chosen as φ_ik(I_i)=exp[−(I_i−m_ik)2/σ_ik2] for

2 , 1 =

i and k=1,2,L,9 , and parameters σ_ik =0.5 , [m_i₁,m_i₂,m_i₃,m_i₄,m_i₅,m_i₆,m_i₇,m_i₈,m_i₉]= ] 8 . 0 , 6 . 0 , 4 . 0 , 2 . 0 , 0 , 2 . 0 , 4 . 0 , 6 . 0 , 8 . 0

[− − − − . Given the positive matrices Q₁ = Q₂ =diag[10,10], feedback and observer gain vectors are given as T =

[

144,24

]

c

K and T =

[

60,900

]

o

K . The initial states are chosen as

T ] 3 . 0 -, 2 . 0 [ ) 0 ( = −

x , xˆ(0)=[0.2,0.1]T and let adaptation gains as γ_D=γ_g =100.

Let desired trajectory y_d =0.1sin(t). Fig.1 shows the system output can track the desired output well. In Fig.2, the trajectory of the estimation state xˆ₁ catches up to the trajectory of the system state x₁

quickly and well. Fig. 3 shows that the tracking error has been attenuated efficiently.

0 5 10 15 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 time(sec) T racki ng r esp on se ,r ad desired actual 0 5 10 15 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 T racki ng r espo nse, ra d estimated actual 0 5 10 15 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 e1 time(sec)

Fig. 1. Tracking response (y:dash, yd:solid) Fig. 2. Tracking response (x1:dash,ˆx₁:solid) Fig. 3. The tracking error

6. Conclusions

The stable direct adaptive output feedback CMAC controller has been proposed for the trajectory tracking of uncertain nonlinear systems. This design obtains robustness in the sense that the self-tuning mechanism can automatically adapt the CMAC controller by using a learning algorithm and the global asymptotic stability of the algorithm is established via Lyapunov stability criterion. The simulation example is used to achieve a good trajectory following performance.

References

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[3] J. S. Albus. A new approach to manipulator control: The cerebellar model articulation controller (CMAC). Trans. ASME

J. Dyn. Syst., Meas., Contr 1975; 97:220–227.

[4] S. H. Lane, D. A. Handelman, and J. J. Gelfand. Theory and development of higher-order CMAC neural networks. IEEE

Contr. Syst. Mag. 1992; 12:23–30.

[5] Y. G. Leu, T. T. Lee and W. Y. Wang. Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems. IEEE Trans. Systems Man Cybern. B 1999; 29:583–591.

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Chun-Sheng Chen / Procedia Engineering 29 (2012) 3966 – 3971

objective is to maintain the output y to track the desired trajectory with only output is available for measurement. The disturbance d is assumed to be a square wave with the amplitude ±1 and period the

π (s). The Gaussian basis functions in each input space are chosen as φ_ik(I_i)=exp[−(I_i −m_ik)2/σ_ik2] for 2

, 1 =