Model reference fuzzy adaptive learning system

IFAC Workshop on Adaptation and Learning in Control and Signal Processing, and IFAC Workshop on Periodic Control Systems, Yokohama, Japan, August 30 – September 1, 2004

MODEL REFERENCE FUZZY ADAPTIVE LEARNING SYSTEM Martin Kratmüller, Peter Fodrek, Ján Murgaš and Eva Miklovičová Department of Automatic Control Systems, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19 Bratislava, Slovak Republic E-mail: mailto:[email protected] E-mail: mailto:[email protected] E-mail: mailto:[email protected] E-mailmailto:[email protected]

Abstract: Fuzzy control has revealed as a practical alternative to several conventional control schemes since it has shown good results in some application areas. However, there are several drawbacks of this approach: (i) the design of fuzzy controllers is usually performed in an ad hoc manner where it is often difficult to choose some of the controller parameters (e.g., the membership functions), and (ii) the fuzzy controller constructed for the nominal plant may later perform inadequately if significant and unpredictable plant parameter variations occur. A “learning system” possesses the capability to improve its performance over time by interacting with its environment. A learning control system is designed so that its “learning controller” has the ability to improve the performance of the closed-loop system by generating command inputs to the plant and utilizing feedback information from the plant. Learning controllers are often designed to mimic the manner in which a human in the control loop would learn how to control a system while it operates. Some characteristics of this human learning process my include: (i) after learning how to control the plant for some operating condition, if the operating conditions change, then the best way to control the system may have to be relearned; (ii) a human with significant amount of experience at controlling the system in one operating region should not forget this experience if the operating condition changes. To mimic these types of human learning behavior, we introduce strategy that can be used to learning controller onto the current operating region of the system. Copyright © 2004 IFAC Keywords: Adaptive fuzzy control, Riccati equation, uncertain system, nonlinear systems, Lyapunov.

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1. INTRODUCTION ym

In recent years, there has been a significant growth in the commercial and industrial use of fuzzy logic systems. Subway systems, automobile transmissions, air-conditioners, washing machines, autofocus cameras and camcorders have employed fuzzy control (Tong, 1977), and these products have been advertised to have “intelligence” as compared to their “non-fuzzy” counterparts. While there is significant marketing hype about fuzzy systems being intelligent, in reality the fuzzy controller is no more than a nonlinear controller. Hence, there exists the same type of problems in design and implementation of fuzzy controllers as exist in conventional control. For instance, there is the need to specify a set of performance objectives (e.g. stability, rise-time, overshoot etc.) and show that these objectives are achieved. In addition, as with conventional fixed (i.e., non-adaptive) control, (i) it can be difficult to tune the parameters of the controller and (ii) there always exists the possibility that upon implementation plant parameter variations will result in performance degradation. A multitude of approaches to adaptive control have been introduced to address these two problems. For instance, there exist many approaches to model reference adaptive control (MRAC) that seek to tune the parameters of a linear controller in response to plant variations so that the behavior specified in a “reference model” is achieved (Sastry, Bodson, 1989). In analogous fashion to MRAC, the work in (Layne, et. al., 1993; Layne, and Passino, 1993; Kwong, et. al., 1995; Moudgal, et. al., 1995) shows how fuzzy adaptive controllers can achieve and maintain the performance specified in a reference model by synthesizing and tuning a fuzzy controller. In this paper, we investigate the possibility of enhancing the MRFALC learning capabilities that seek to optimally allocate the fuzzy controller rules to the operating region of the system.

REFERENCE MODEL e

Rule base modificator

r

u

DFAC

PLANT y

Fig. 1. Block scheme of MRFALC. Mechanism of rule base modification : The rule base modificator modifies the rule base of fuzzy controller so that a desired criteria is satisfied. The modification can be done in many ways. Our approach is to change of sharp of highest significant rules fuzzy membership functions for the controller output value. If the consequence part of fuzzy rule has symmetrical convex membership function its modification is reduced to a simple change of its size. 3. PRINCIPLE OF THE METHOD, ITS STABILITY AND ROBUSTNESS ANALYSIS Consider the following n-th order SISO system

(

  , x ( n −1) x ( n ) = f x, x,

(

 , x +g x, x,

)

( n −1)

)u +d

(1)

y=x where f and g are unknown (uncertain) but bounded continuous functions and u ∈ R and y ∈ R are the input and output of the system, respectively. d denotes the external disturbance which is unknown but bounded. The external disturbances are due to system load, external noise, ect. Let

(

)

x = x, x , ,x (n −1) ∈ R n be the state vector of the system which is assumed to be available. For (4) to be controllable, we require that g(x ) ≠ 0 for x in certain

2. MRFALC STRUCTURE

T

controllability region U c ⊂ R n . The control objectives is to force y to follow a given bounded reference signal y m . Let us denote the output tracking error e and the parameter tracking ~ error θ e = ym − y (2) * θ = θ − θ

MRFALC contains adaptive mechanisms, which observes output data of closed loop with fuzzy controller, counts actual value of tracking error and automatically tunes parameters of fuzzy controller to achieve a desired control criterion. This criterion is defined by reference model, comparable to conventional MRAC. The aim of controller is a modification of knowledge rules base of fuzzy controller so that output y of closed loop is as close as possible to the reference model output y m which is a response to the same reference signal Fig. 1. MRFALC structure consists of two levels. First, lower one, is a simple control loop with fuzzy adaptive controller, Second, observing level, compares the reference model output of and closed loop output and based on a difference between them adapts the knowledge base of the fuzzy controller on the first level .

for some parameter estimate θ of fuzzy logic system and optimal parameter estimate θ of fuzzy logic system. Let w denote the sum of matching errors due to fuzzy approximations of f (x ) , g (x ) and the external disturbance d. Then our design objective is to impose an adaptive fuzzy control algorithm so that the following asymptotically stable tracking (3) e (n ) + k 1 e (n −1) +  + k n e = 0 *

512

is achieved while w=0 (i.e., in the case of perfect fuzzy approximation and free of external disturbance). In this study, we assume that w is unknown but bounded. However, the effect of w will deteriorate the control performance of fuzzy control system. Therefore, how to eliminate the effect of w to guarantee the control performance is an important issue in the control systems. H ∞ control is the most important control design to efficiently eliminate the effect of w on the control system. Let us consider the following H ∞ control performance

∫ ∫

tf

0 tf

M

∑ ( Δe )

2

i

1 π

i=0

M

∑e

f κ= e = fv

2 i

i =0

M

∑(Δ e ) 2

2

i

1 π

i =0 M

∑ ( Δei )

(8)

2

i=0

M

∑ ( Δe )

2

i

e Qedt w T wdt

0

< ρ2

i =0

=

T

(4)

M

M

∑ e ∑(Δ e ) i=0

2 i

2

i=0

2

i

or

∫

tf

0

e T Qedt < ρ 2

∫

tf

0

w T wdt

The inequality in (5) can be seen as boundeddisturbance and bounded-state but with a prescribed gain ρ . If the initial condition is also considered, the inequality (5) can be modified as (Chen, et. al., 1994)

(5)

where t f denotes the terminal time of the control, ρ is a prescribed value which denotes the worst case effect of w on e, and Q is a positive-definite weighting matrix. The physical meaning of (5) is that the effect of w on e must be attenuated below a desired level ρ from the viewpoint of energy, no

∫

be equal to or less than a perscribed value ρ 2 . In general, ρ is choosen as a positive small value less than 1 for attenuation of w. The basic idea of width change of membership function is very simple. Observing the zero crossing frequencies of the tracking error f e and of its first

for

matrices

Q = QT ≥ 0

,

(

)

e = e, e ,  , e (n −1) . Direct adaptive fuzzy control algorithm is employed in this paper to control the nonlinear system in (1) to guarantee the tracking performance (9) with a prescribed disturbance attenuation level ρ . Remark 1. The roots of polynomial (n −1) n in the h (s ) = s + k 1s +  + k n −1s + k n characteristic equation of (3) are all in the open lefthalf plane via an adequate choice of coefficients k 1 , k 2 , , k n . If ρ = ∞ , this is the case of minimum error tracking control without disturbance attenuation. If g(x ) is known in (4), the direct adaptive fuzzy control in (Wang, 1994; Wang, 1993) has attempted to directly approximate the following control law 1 T u= − f (x ) + y (mn ) + k e (10) g (x )

fe . However, it is always valid that fv 0 ≤ κ ≤ 1 . For the zero-crossing frequency of the tracking error we obtain (Maršík, 1983) (6)

In analogous manner, we get for v = Δe (the difference of the tracking error, “velocity”) 1 σ2a π σ 2v

weighting

P = P ≥ 0 , an adaptation gain η > 0 , and a prescribed attenuation level where ρ ,

ratio κ =

fv =

given

(9)

T

difference f v we can find the dependence of their

1 σ 2v π σ 2e

eT Qedt < eT ( 0 ) Pe ( 0 )

tf 1 T + θ ( 0 ) θ ( 0 ) + ρ 2 ∫ w T wdt 0 η ∀t f ∈ [0, ∞ ) w ∈ L 2 [0, t f ]

matter what w is, i.e., the L 2 gain from w to e must

fe =

tf

0

(7)

where σ2e , σ 2v and σa2 are dispersions of e and of its derivatives. Thus the frequency of zero crossings corresponds to the number of half-waves. For sampled signals analogous formulas may be applied. Instead of derivatives the differences and instead of integrals the sums are used. Using the steady-state values κ we can adapt formulas (6) and (7) as follows

T

[

]

In this paper, an additional H ∞ control u h is employed to attenuate both the effects of fuzzy approximation error and external disturbance. More precisely, when f (x ) and g(x ) in the nonlinear system (1) are well known, the following control law would be used u u* = u + h (11) g (x )

513

where the H ∞ control u h will be discussed later.

PA + AP T + Q −

This control law u * forces the error dynamics to become e = Ae + Bu h − Bd (12) where A and B are the same as in (Wang, 1994). In our case, however, f (x ) is unknown. Therefore,

Define

Ωδ

where θ is a parameter vector, and ξ(x ) is a regressive vector with the regressor ξ l (x ) defined as ξ (x ) =

n i =1

∑ (∏ M

l=1

μ F l (x i ) i

n i =1

(14)

)

μ F l (x i ) i

Then, we get the control law as u T (15) u = ξ (x )θ − h g(x) After some straightforward manipulations, the tracking error dynamics of the nonlinear system (4) with the fuzzy control law u in (15) to replace u * in (11) is of the following form e = Ae + Bu h − Bg ( x ) ( u ( x | θ ) − u ) − Bd (16)

⎛ (x − x) ⎞ (27) μ ( x ) = exp ⎜ − ⎟ 2σ 2 ⎠ ⎝ where x is the center of the membership function where the membership grade is equal to 1, and σ > 0 is the spread of membership function. We are changing center of the input membership functions depending on zero crossing frequencies rate.

on θ and x , respectively. Let the minimum approximation error be denoted by

) )

(18)

The error dynamic equation (16) can be rewritten as

(

(

* e = Ae + Bu h − Bg ( x ) u ( x | θ ) − u x | θ

− Bd + Bw c From (17), (19) becomes T * e = Ae + Bu h − Bg ( x ) ξ ( x ) θ − θ +B( wc − d )

(

) ) (19)

)

(26)

In our basic setup, a fuzzy set is characterized by a Gaussian membership function μ f ( x ) expressed as

]

*

}

⎛ θˆ 2 − M ⎞ e T PBg x ξT x ( ) ( ) ⎜ ⎟ ⎠ −⎝ θˆ 2 ˆ δ θ

where Ω θ and Ω x denote the sets of desired bounds

((

}

Pr ⎡⎣ηξ ( x ) g ( x ) BT Pe ⎤⎦ = ηξ ( x ) g ( x ) BT Pe

*

w c = −g ( x ) u x | θ − u

(24)

≤ M+δ

{

Define the optimal parameter vector θ for which the fuzzy logic system can approximate control law u optimally * θ = arg min θ∈Ω θ sup x∈Ω x u (x | θ) − u (17)

[

2

where M is positive design parameter. Take the parameter adaptation law as ⎧ ηξ ( x ) g ( x ) BT Pe ⎪ ⎪ ⎛ ⎞ ⎜ ⎟ ⎪ ⎜ ⎟ ⎪ ⎜ ⎟ ⎪ θˆ ∈ Ω ⎜ ⎟ ⎪ ⎟ ⎪if ⎜ or  ⎪ ⎜ ⎟ θˆ = ⎨ (25) ⎜⎛ θ = M ⎟ ⎞ ⎪ θ ⎜⎜ ⎟⎟ ⎪ ⎜ ⎜ and ⎟⎟ ⎪ ⎜⎜ ⎟ T ⎪ ⎜ ⎜ ηξ ( x ) g ( x ) B Pe ≤ 0 ⎟⎟ ⎟ ⎝ ⎠ ⎝ ⎠ ⎪ T ⎪ Pr ηξ ( x ) g ( x ) B Pe ⎪ ⎪ otherwise ⎩ where Pr [∗] is the projection operator defined as

u in (11) can not be obtained. Adopt u = u (x | θ ) − u a / g (x ) , and let the fuzzy control law be in terms of the following fuzzy logic system as follows T T u (x | θ) = θ ξ(x ) = ξ (x )θ (13)

∏

}

2

Ω = θ| θ ≤ M

*

l

{ = {θ | θ

2 1 PBB T P + 2 PBB T P = 0 (23) λ ρ

4. SIMULATION EXAMPLE The above described adaptive fuzzy control algorithm will now be evaluated using the unstable system. The dynamic equation of this system is given by (Wang, 1994)

(20)

and we will specify u h and an adaptive law for θ to

x ( t ) =

1− e

−x( t)

(28) + u ( t) −x t 1+ e ( ) Equation (28) shows that system is unstable without input because if u ( t ) = 0 , then x > 0 for x > 0 and x < 0 pre x < 0 . Choose h ( s ) = s and Q=12 then A=-1, B=1 a P=2. As well as we have choosen reference model as 1st order linear system 1 (29) M (s) = 2s + 1

achieve the H ∞ tracking performance. Let us define * θ = θ − θ and w = w c − d , then (20) can be rewritten as T (21) e = Ae + Bu h − Bg ( x ) ξ ( x ) θ + Bw Assume that the control u h in (15) is chosen as

1 (22) u h = − e T Pb λ where λ is a positive constant to be determined, and P is a positive definite matrix, which is a solution of the Riccati-like equation

514

Status membership functions μ N3 ( x ) , μ N 2 ( x ) ,

1

μ N1 ( x ) , μ P1 ( x ) , μ P2 ( x ) , μ P3 ( x ) and controller

0.9

output membership functions μ PB ( x ) and μ NB ( x ) have been chosen as in the article (Wang, 1994). Adaptation gain has been chosen as η = 0.2 . Integration method ode23 has been used to simulate the overall control system. Figures 2-4 show the comparison of our new method to the Direct Fuzzy Adaptive Control (Wang, 1994) and Figure 5 shows the crossing frequencies ratio, the parameter that is used in adaptation or learning mechanism.

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

200

400

600

800

1000

1200

1400

4.5

Fig. 5. Zero crossing frequencies ratio in the time

4 3.5

5. CONCLUSION

y,ym,r

3

Direct fuzzy adaptive model reference control based on the heuristic learning is proposed in this paper. Combined with H ∞ robust control technique, this adaptive fuzzy learning control algorithm not only guarantees the stability of the closed-loop, but also maintains the desired H ∞ tracking performance. Therefore, it enhances the existing adaptive fuzzy control algorithms (e.g. (Wang, 1993)). We have verified that our method compensates a steady state tracking error that is the main problem of the Direct Fuzzy Adaptive Control described in (Wang, 1993).

2.5 2 1.5

reference signal output of MRFALC reference model output of DFAC

1 0.5 0

0

20

40

60 80 time[sec]

100

120

140

Fig. 2. The state x 1 , its desired reference model value y m (t ) and reference signal, for DAFC (chap. 9 in (Wang, 1994)) and MRFALC.

REFERENCES 4.1

Tong, R. M. (1977). A control engineering review of fuzzy systems. Automatica, Vol. 13, pp. 559-569. Sastry, S. and M. Bodson (1989). Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice Hall. Layne, J. R., K. M. Passino and S. Yurkovich (1993). Fuzzy Learning Control for Antiskid Systems. IEEE Trans. on Control Systems Technology, Vol. 1, no. 2, 122-129. Layne, J. R. and K. M. Passino (1993). Fuzzy Model Reference Learning Control for Cargo Ship Steering. IEEE Control Systems Magazine, Vol. 13, no. 6, 23-34. Kwong, W. A., K. M. Passino, E. G. Laukonen and S. Yurkovich (1995). Expert Supervision of Fuzzy Learning Systems for Fault Tolerant Aircraft Control. Proc. of the IEEE, Special Issue on Fuzzy Logic in Engineering Applications, Vol. 83, no. 3, 466-483. Moudgal, V. G., W. A. Kwong, K. M. Passino and S. Yurkovich (1995). Fuzzy Learning Control for a flexible-link Robot. IEEE Trans. on Fuzzy Systems, Vol. 3., no. 2, 199-210. Maršík, J. (1983). A new conception of digital adaptive PSD control. Problems of Control and Information Theory, Vol. 12, no. 4, 267-279. Chen, B. S., T.-S. Lee and J.-H. Feng (1994). A nonlinear H ∞ control design in robotic systems under parameter perturbation and external disturbance. Int. J. Control, Vol. 59, 439-461.

reference signal output of MRFALC reference model output of DFAC

4 3.9

y,ym,r

3.8 3.7 3.6 3.5 3.4 3.3 95

100

105

110

115

120

time[sec]

Fig. 3. Detail of Figure 2. 100

DFAC MRFALC

u

50

0

-50

0

20

40

60 80 time[sec]

100

120

140

Fig. 4. Control signals.

515

Wang, L. X. (1994). Adaptive fuzzy systems and control, design and stability analysis. Englewood Cliffs, NJ: Prentice Hall. Wang, L. X. (1993). Stable adaptive fuzzy control of nonlinear systems. IEEE Trans. Fuzzy Syst., Vol. 26, 146-155.

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