DIRECT FUZZY CONTROL FOR NONLINEAR SERVOMECHANISM

International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm) Volume 1, Number 1, Pages 79–103, March 2003 Publisher Item Identifier S 1542-5908(03)10103-0/$20.00 Article electronically published on October 17, 2002 at http://www.YangSky.com/ijcc11.htm. Please cite this paper as: hRong-Jong Wai, Chih-Min Lin and Chun-Fei Hsu, “Direct Fuzzy Control for Nonlinear Servomechanism Using Adaptive Tuning Algorithm (Invited Paper)”, International Journal of Computational Cognition (http://www.YangSky.com/yangijcc.htm), Volume 1, Number 1, Pages 79–103, March 2003i.

DIRECT FUZZY CONTROL FOR NONLINEAR SERVOMECHANISM USING ADAPTIVE TUNING ALGORITHM (INVITED PAPER) RONG-JONG WAI, CHIH-MIN LIN AND CHUN-FEI HSU

Abstract. In this study an adaptive fuzzy control (AFC) system is proposed to control a nonlinear motor-mechanism coupling system, that is a toggle mechanism driven by a permanent magnet (PM) synchronous servo motor. In the proposed AFC system, a fuzzy controller is the main tracking controller that is used to mimic an ideal feedback linearization control law and a compensated controller is proposed to compensate for shortcomings in the fuzzy control process. Moreover, an on-line tuning methodology, which is derived to tune the premise and consequence parts of fuzzy rules in a fuzzy basis function (FBF) network, is proposed to increase the learning capability of the fuzzy controller. In addition, to relax the requirement for the uncertain bound in the compensated controller, an AFC system with bound estimation is investigated to control the motor-mechanism coupling system. Furthermore, simulated and experimental results due to periodic step and sinusoidal commands verify that the proposed control systems can achieve favorable tracking ability under different reference trajectories, and robust with regard to parameter variations and c external disturbance. Copyright °2002 Yang’s Scientific Research Institute, LLC. All rights reserved.

1. Introduction Since fuzzy control (FC) is a model free design method and is more insensitive to plant parameter variations and external disturbances, it has been successfully applied to many engineering systems. As for the FC system, Received by the editors October 16, 2002 / final version received October 17, 2002. Key words and phrases. adaptive control, fuzzy control, toggle mechanism, PM synchronous servo motor. This paper is supported by the National Science Council of Republic of China under Grant NSC-90-2213-E-155-014. All corresponds address to Rong-Jong Wai, Tel: 886-34638800 ext:429, Fax: 886-3-4639355. c °2002 Yang’s Scientific Research Institute, LLC. All rights reserved.

79

80

RONG-JONG WAI, CHIH-MIN LIN AND CHUN-FEI HSU

though it is one of the most effective methods using expert knowledge without knowing the parameters and structure of the controlled systems, the major drawback of the FC system is lack of adequate analysis and design techniques [1]. To tackle this problem, some researchers have been focused on the use of the Lyapunov synthesis approach to construct a stable adaptive fuzzy controller (AFC) [2-6]. Based on the universal approximation theorem [2], the AFC design method can provide stabilizing controllers in Lyapunov sense. However, most of these approaches can only tune the parameters of the consequence part of the fuzzy rules. Some approaches can tune the parameters of the premise and consequence parts of the fuzzy rules; however, they do not guarantee global stability and tracking performance [7, 8]. The toggle mechanism has many applications, for example, clutches, rock crushers, truck tailgates, vacuum circuit breakers, pneumatic riveters, punching machines, forging machines and injection modeling machines [9, 10]. Lin et al. [9] presented a fuzzy sliding-mode controller to control the motor-toggle servomechanism. However, when the rules base is not built well, the response may be not well enough. Lin et al. [10] derived a fuzzy neural network controller with adaptive learning rates was implemented to control the motor-toggle servomechanism. However, many rules and a pretraining process were required for this control strategy to achieve the best control performance. The main topic of this study is to present a tuning algorithm based on the Lyapunov stability theorem to tune all parameters of the fuzzy controller with unknown control system dynamics to overcome the drawbacks of the previous works. Moreover, the control system with the tuning algorithm can guarantee stability and favorable tracking performance for the motor-toggle servomechanism. If the plant model is well known, there exists an ideal feedback linearization control law to achieve favorable control performance [11, 12]. Since the system parameters and the external load disturbance may be unknown or perturbed, the ideal feedback linearization control law can not be implemented in practical applications. The motivation of this study is to design an adaptive fuzzy control (AFC) system to overcome the mentioned drawbacks. The AFC system combines a fuzzy controller that is used to mimic an ideal feedback linearization control law and a compensated controller is proposed to compensate for shortcomings in the fuzzy control process. All parameters in the proposed AFC system are tuned in the Lyapunov sense, thus the stability of the closed-loop system can be guaranteed. The on-line tuning methodology, which is derived to tune the premise and consequence parts of fuzzy rules, is proposed to increase the learning capability of the

DIRECT FUZZY CONTROL FOR NONLINEAR SERVOMECHANISM

81

AFC. Moreover, the validity of the proposed AFC design method can be illustrated by the simulated and experimental results. This study is organized as follows. The nonlinear motor-mechanism coupling system is described in Section 2, and a fuzzy basis function (FBF) network is introduced in Section 3. Section 4 presents an algorithm for tuning the FBF-based controller in the sense of the Lyapunov stability theorem. Section 5 exhibits a simple estimated algorithm to estimate the uncertain bound in the compensated controller. In Section 6, the effectiveness of the developed AFC systems for the motor-toggle servomechanism is verified by simulated and experimental results. Finally, a conclusion is drawn in Section 7. 2. Motor-Mechanism Coupling System Gear Mechanism

PC

Ball Screw

Slider C

m5 Link 5 r5

PM Synchronous Servo Motor

q5

q3 r1

r2

f

h

Link 3

m2

r4

Link 2

r3

Slider B

m3

PB

mB

q2

PE

f

XB

Figure 1. Toggle mechanism driven by PM synchronous servo motor. A toggle mechanism driven by a PM synchronous motor is depicted in Fig. 1 [9, 10], in which the most important parameters that affect the control performance of the toggle mechanism are the external force (PE ) and the parameter variation of the mass of slider B (mB ). Moreover, m2 , m3 and m5 are the mass of links 2, 3 and 5; r1 , r2 and r4 are the length of link 2; r3 and r5 are the length of links 3 and 5; θ2 , θ3 and θ5 are the angles of links 2, 3 and 5; PB and PC are the forces acting on the slider B and C; h is the height between the two horizontal guides where sliders B and C move along; f is an offset between links 2 and 3. The Hamilton’s principle and the Lagrange multiplier have been used to derive the differential-algebraic equation for the toggle mechanism [9, 10]. The implicit method must be employed to solve

82


the differential-algebraic equation of mechanical motion. The result is a set of differential equations with only one independent generalized coordinate. The motor-mechanism coupling system shown in Fig. 1 can be represented by the following equation [9, 10]: (1)

v¨(t) = f (Θ ;t) + G(Θ ;t) u(t) + d(Θ ;t)

where f (Θ ;t) denotes a nonlinear dynamic function; G(Θ ;t) is a control gain; d(Θ ;t) represents disturbance and friction functions; u(t) is the con> > trol input and Θ = [ v v˙ ] = [ θ2 θ˙2 ] . Moreover, f (Θ ;t), G(Θ ;t) and d(Θ ;t) include the uncertainties introduced by system parameters and external disturbance. In addition, the control gain G(Θ ;t) is a negativesign function and invertible [10]. Assume that the uncertainties are absent and rewrite (1) as (2)

v¨(t) = fn (Θ ;t) + Gn (Θ ;t) u(t) + dn (Θ ;t),

where fn (Θ ;t), Gn (Θ ;t) and dn (Θ ;t) are the nominal values of f (Θ ;t), G(Θ ;t) and d(Θ ;t). If the uncertainties occur, i.e., the parameters of the system are deviated from the nominal value or an external disturbance is added into the system, the dynamic equation of the motor-mechanism coupling system can be modified as v¨(t) (3)

=

[fn (Θ ;t) + ∆f (Θ ;t)] + [Gn (Θ ;t) + ∆G(Θ ;t)] u(t) +[dn (Θ ;t) + ∆d(Θ ;t)] ≡ fn (Θ ;t) + Gn (Θ ;t) u(t) + dn (Θ ;t) + w(Θ ;t)

where ∆f (Θ ;t), ∆G(Θ ;t) and ∆d(Θ ;t) denote the uncertainties; w(Θ ;t) is the lumped uncertainty and defined as w(Θ ;t) = ∆f (Θ ;t) + ∆G(Θ ;t) u(t) + ∆d(Θ ;t). 3. Fuzzy Basis Function Network Assume that there are r rules in a fuzzy rule base and each of which has the following form: ˜j , (4) Rule j: IF x is A˜j THEN y is B where x contains the input variable to the fuzzy system; y is the output ˜j are linguistic terms characterized variable of the fuzzy system; A˜j and B by their corresponding fuzzy membership function µA˜j (x) and µB˜j (y), respectively. For an FBF network, the membership function is chosen as a Gaussian function. In this study, the FBF approximation is implemented with singleton fuzzification, product inference, and defining the defuzzifier as a weight sum of each rule’s output. The scheme of the FBF network,


83

c1 , w1 c2 , w2

φ1

φ2 c3 , w3

x

φ3

α1 α2

α3

… cr , wr

y

αr

φr

Figure 2. Network representation of FBF expansion system. which is comprised of one input, r rules (hidden units) and one output, is depicted in Fig. 2. Such an FBF network implementing the procedures of fuzzification, fuzzy inference and defuzzification performs the mappings according to (5)

y=

r X

αj ϕj (kx − cj k , σj ),

j=1

where ϕj represents a Gaussian membership function in the j-th rule; cj and σj are the center and width of the Gaussian membership function in thej−th rule; the connecting weights are denoted by αj ; k · k are denoted the Euclidean norm. The FBF can be represented by (6)

ϕj = exp(−wj2 (x − cj )2 ),

where wj = 1 /σj is the inverse radius of an FBF. For ease of notation, define vector c and w collecting all centers and inverse radii of FBF as (7)

c = [c1 , c2 , ..., cr ]>

(8)

w = [w1 , w2 , ..., wr ]>

The output of an FBF network can be represented as (9)

y(x, c, w, α) = α> φ(x, c, w)

84


where α = [α1 , α2 , ..., αr ]> and φ = [ϕ1 , ϕ2 , ..., ϕr ]> . It has been proven in [2] that there exists a fuzzy system in the FBF expansion of (9) such that it can uniformly approximate a nonlinear even time-varying function Ξ. By universal approximation theorem, there exists ideal vectors α∗ , c∗ and w∗ such that Ξ = y ∗ (x, c∗ , w∗ , α∗ ) + ε = α∗T φ∗ + ε,

(10)

where ε is the approximation error. Employing an FBF network yˆ to approximate Ξ ˆ yˆ(x, ˆ c, w, ˆ α ˆ) = α ˆ> φ,

(11)

where α ˆ, ˆ c and w ˆ are the estimated values of α∗ , c∗ and w∗ . In this study, the parameters of α ˆ, ˆ c and w ˆ are all adjusted and learning rules will be stated in later. 4. Adaptive Fuzzy Control Adaptive Fuzzy Control

t 0

x*B

Transformation

vd

Reference vm+ Model − v

Compensated Control

edτ e

d/dt

Motor-Mechanism Coupling System

uvs Error Performance Function

Fuzzy ufz + Control

+

u

Permanent Magnet Synchronous Motor

Toggle Mechanism

xB

e Tuning Algorithm

Transformation

Figure 3. Block diagram of AFC system. An AFC system for motor-toggle servomechanism is depicted in Fig. 3, where x∗B , xB , vd , and v are the command slider position, slider position, command angle of link 2, and angle of link 2, respectively. Since xB is the desired control objective and v is the state of the motor-mechanism coupling system, the x∗B and xB should be transformed to vd and v using the one-to-one relationship as follow [9, 10]:


(12) v = sin

· −1

¡ 2 ¢ xB + r22 + f 2 − r32

85

¸ · Áq ¸ Á q −1 2 2 2 2 xB 2r2 xB + f − sin xB + f

The position signal of the slider Bis detected by linear scale. The control problem is to find a control law so that the system state can track the desired command. To achieve the control objective, define the tracking error e(t) = vm (t) − v(t), in which vm (t) represents the reference trajectory. If the system parameters are known well, an ideal feedback linearization control law can be obtained from (3) u∗ (t) (13)

−1

= Gn (Θ ;t) [−fn (Θ ;t) − dn (Θ ;t) - w(Θ ;t) + v¨m (t) + k1 e(t) ˙ + k2 e(t)]

where k1 and k2 are positive gains. Substituting (13) into (3) gives (14)

e¨(t) + k1 e(t) ˙ + k2 e(t) = 0

If k1 and k2 are chosen to correspond to the coefficients of a Hurwitz polynomial, that is a polynomial whose roots lie strictly in the open left half of the complex plane, then lim e(t) = 0. t→∞

Since the lumped uncertainty w(Θ ;t) is difficult to measure in practical application, the ideal feedback linearization control law u∗ (t) can not be implemented. Therefore, an AFC system is proposed to mimic the ideal feedback linearization control law in this study. Define an error performance function as Z (15)

t

s(t) = e(t) ˙ + k1 e(t) + k2

e(τ )dτ 0

to be an input variable of the fuzzy controller. Thus, there exists an optimal fuzzy controller u∗f z (s(t), c∗ , w∗ , α∗ ) to approximate the ideal feedback linearization control such that (16)

u∗ (t) = u∗f z (s(t), c∗ , w∗ , α∗ ) + ε

The control law for the AFC system is assumed to take the following form: (17)

u(t) = u ˆf z (s(t), ˆ c, w, ˆ α ˆ) + uvs (s(t))

86


where u ˆf z is the main tracking controller to approximate the ideal feedback linearization control law u∗ (t), and the compensated controller uvs is designed to compensate the difference between the ideal feedback linearization control law and the fuzzy controller. By substituting (17) into (3), it is revealed that (18)

v¨(t) = fn (Θ ;t) + Gn (Θ ;t)( u ˆf z + uvs ) + dn (Θ ;t) + w(Θ ;t)

After some straightforward manipulation, the error equation governing the closed-loop system can be obtained through (13), (15) and (18) s(t) ˙ = Gn (Θ ;t) [u∗ (t) − u ˆf z − uvs ]

(19)

Moreover, u ˜f z is defined as (20)

ˆ+ ε u ˜f z = u∗ − u ˆf z = u∗f z − u ˆf z + ε = α∗T φ∗ − α ˆ> φ

˜ = φ∗ − φ ˆ to obtain For simplicity of discussion, define α ˜ = α∗ − α ˆ and φ a rewritten form of (20) ˜+ α ˆ+ ε u ˜f z = α∗T φ ˜> φ

(21)

In this study, a control methodology is proposed to guarantee closedloop stability and tracking performance, and to tune center and radii of fuzzy basis functions on line. To achieve this goal, a linearization technique is employed to transform the nonlinear fuzzy basis functions into partially ˜ in a Taylor series, linear form so that the expansion of φ ¯    ∂ϕ1 > ¯  ∂ϕ1 > ¯¯ ϕ˜1 ¯ ¯ ∂c ∂w  ϕ˜2   ∂ϕ2  ¯¯  ∂ϕ2  ¯¯   ∂c  ¯  ∂w  ¯ ˜ =  φ ˜ c+  . w ˜ +H  ..  =  .  ¯  ¯  .   ..  ¯  ¯  .. ¯ ¯ ∂ϕr ∂ϕr ϕ˜r ¯ ¯ ∂c ∂w c=ˆ c

(22)

≡

w=w ˆ

A˜ c + Bw ˜ +H

£ where H is a vector of higher-order terms; A = £ ∂ϕ2 1 ˜ c = c∗ − ˆ c; w ˜ = w∗ − w; ˆ B = ∂ϕ ··· ∂w ∂w are defined as (23)

∂ϕj =[ ∂c

∂ϕj ∂c1

∂ϕj ∂c2

···

∂ϕ1 ∂c ∂ϕr ∂w

∂ϕj ∂cr

]>

¤ r |c=ˆc; · · · ∂ϕ ∂c ∂ϕj ∂ϕj |w=w ˆ ; ∂c and ∂w

∂ϕ2 ¤∂c


∂ϕj ∂ϕj ∂ϕj = [ ∂w ··· ∂w2 1 ∂w Substituting (22) into (21), it is revealed that (24)

u ˜f z (25)

∂ϕj ∂wr

]>

=

>˜ ˆ+ ε (ˆ α+α ˜) φ +α ˜> φ

=

˜+ α ˆ+ ε α ˆ> (A˜ c + Bw ˜ + H) + α ˜> φ ˜> φ ˆ+ α α ˜> φ ˆ> A˜ c+α ˆ > Bw ˜ +κ

=

87

˜ + ε is assumed to be bounded where the uncertain term κ = α ˆ> H + α ˜> φ by |κ| ≤ χ. Define the following Lyapunov function candidate: (26)

V1 (s(t), α ˜, ˜ c, w) ˜ =

α ˜> α ˜ ˜ c> ˜ c w ˜ >w ˜ 1 2 s (t) + + + 2 2η1 2η2 2η3

where η1 , η2 and η3 are positive constants; | · | is the absolute value. Differentiating (26) with respect to time and using (19) and (35), it can obtain that α ˜> α ˜˙ ˜ c> ˜ c˙ w ˜ >w ˜˙ V˙ 1 = s(t)s(t) ˙ + + + η1 η2 η3 α ˜> α ˜˙ ˜ c>˜ c˙ w ˜ >w ˜˙ = s(t)Gn (Θ ; t) (u∗ − u ˆf z − uvs ) + + + η1 η2 η3 α ˜> α ˆ˙ ˜ c>ˆ c˙ w ˜ >w ˆ˙ = s(t)Gn (Θ ; t) (˜ uf z − uvs ) − − − η1 η2 η3 >ˆ > > = s(t)Gn (Θ ;t) (˜ α φ+α ˆ A˜ c+α ˆ Bw ˜ + κ − uvs ) α ˜> α ˆ˙ ˜ c> ˆ c˙ w ˜ >w ˆ˙ − − η1 η2 η3 >ˆ s(t)Gn (Θ ;t) (˜ α φ+˜ c> A> α ˆ+w ˜ > B> α ˆ + κ − uvs )

− =

α ˜> α ˆ˙ ˜ c> ˆ c˙ w ˜ >w ˆ˙ − − η1 η2 η3 Ã ! Ã ! α ˆ˙ ˆ c˙ > > > ˆ s(t)Gn (Θ ;t) φ − +˜ c s(t)Gn (Θ ;t)A α ˆ− α ˜ η1 η2 Ã ! w ˆ˙ +w ˜ > s(t)Gn (Θ ;t)B> α ˆ− + s(t)Gn (Θ ; t)(κ − uvs ) η3

− = (27)

If the adaptive laws for parameters in the FBF network and the compensated controller are chosen as

88


(28)

ˆ α ˆ˙ = η1 s(t)Gn (Θ ;t) φ

(29)

ˆ c˙ = η2 s(t)Gn (Θ ; t)A> α ˆ

(30)

w ˆ˙ = η3 s(t)Gn (Θ ; t)B> α ˆ

(31)

uvs = χsgn(s(t))sgn(Gn (Θ ; t))

where sgn( · ) is a sign function. Thus (27) can be rewritten as V˙ 1 (s(t), α ˜, ˜ c, w) ˜ = κs(t)Gn (Θ ; t) − χ |s(t)| |Gn (Θ ; t)| ≤ =

(32)

|κ| |s(t)| |G(Θ ; t)| − χ |s(t)| |Gn (Θ ; t)| − |s(t)| |Gn (Θ ; t)| (χ − |κ|) ≤ 0

This implies that V˙ 1 is negative semidefinite. Define the following term (33) P (t) = |s(t)| |Gn (Θ ; t)| (χ − |κ|) = −V˙ 1 Because V1 (s(0), α ˜, ˜ c, w) ˜ is bounded and V1 (s(t), α ˜, ˜ c, w) ˜ is nonincreasing and bounded, then Z t (34) P (τ ) dτ = V1 (s(0),˜ α, ˜ c, w) ˜ − V1 (s(t),˜ α, ˜ c, w) ˜ α ˜ ˜ c> ˜ c w ˜ >w ˜ χ ˜2 s (t) + + + + 2 2η1 2η2 2η3 2η4

where η4 is a positive constant. Differentiating (37) with respect to time and using (28), (29) and (30), it can obtain that V˙ 2

= =

(38)

=

α ˜> α ˜˙ ˜ c>˜ c˙ w ˜ >w ˜˙ χ ˜χ ˜˙ + + + η1 η2 η3 η4 ˆ+˜ s(t)Gn (Θ ; t) (˜ α> φ c> A> α ˆ+w ˜ > B> α ˆ + κ − uvs ) >˙ >˙ > ˙ α ˜ α ˆ ˜ c ˆ c w ˜ w ˆ χ ˜χ ˆ˙ − − − − η1 η2 η3 η4 χ ˜χ ˆ˙ s(t)Gn (Θ ; t) (κ − uvs ) − η4 s(t)s(t) ˙ +

If the compensated controller is chosen as (35) and the bound estimation algorithm is designed as (39) χ ˆ˙ = η4 |s(t)| |Gn (Θ ; t)|

90


then (38) can be rewritten as V˙ 2 (s(t), α ˜, ˜ c, w, ˜ χ) ˜ = κ s(t)Gn (Θ ; t) −χ ˆ |s(t)| |Gn (Θ ; t)| − χ ˜ |s(t)| |Gn (Θ ; t)| = κ s(t)Gn (Θ ; t) − χ |s(t)| |Gn (Θ ; t)| ≤ − |s(t)| |Gn (Θ ; t)| (χ − |κ|) ≤ 0

(40)

By Barbalat’s Lemma, it can conclude that s(t) → 0 as t → ∞. In summary, the AFC system presented in (17) with the parameters α ˆ, ˆ c and w ˆ adjusted by (28), (29) and (30) to approximate the ideal feedback linearization control law, and the compensated controller uvs is given in (35) with bound estimation algorithm design in (39), then the stability of the AFC system with bound estimation can be guaranteed. 6. Simulated and Experimental Results By using of Runge-Kutta fourth order numerical integration method, (1) is solved for the motor-mechanism coupling system. For numerical simulations, the parameters of the toggle mechanism are designed as follows: m2 = 0.98kg m3 = 0.91kg m5 = 0.3kg mB = 1.46kg mC = 1.84kg r1 = 0.07m r2 = 0.145m r3 = 0.19m r4 = 0.1m r5 = 0.06m f = 0.025m h = 0.08m Ld = 0.005m µ = 0.1 g = 9.8 (41) in which Ld is the lead of screw; mC is the mass of slider C; µ and g are the friction coefficient and gravity acceleration, respectively. The parameters of the proposed control schemes are selected as follows: (42) k1 = 16,

k2 = 64,

η1 = 30,

η2 = 30,

η3 = 30,

η4 = 0.1,

χ = 0.2

All the gains in the proposed control systems are chosen to achieve the best transient control performance in both simulation and experimentation considering the requirement of stability and possible operating conditions. Moreover, a second-order transfer function of the following form with rise time 0.63s is chosen as the reference model for the periodic step command: (43)

S2

wn2 35.6 = 2 + 2ξwn S + wn2 S + 11.93S + 35.6

where S is the Laplace operator; ξ and wn are the damping ratio (set at one for critical damping) and undamped natural frequency. On the other hand, when the command is a periodic sinusoidal reference trajectory, the


91

xB ( m)

reference model is set to be one. Three simulation cases due to periodic step and sinusoidal commands are addressed as follows: Case 1: nominal case (mB = 1.46kg and PE = 0), Case 2: parameter variation case (7.4kg weight is added to the mass of slider B and PE = 0), Case 3: external disturbance case (mB = 1.46kg and PE = 100N ). The control objective is to control the slider Bto move 0.04m periodically. The initial position of xB is 0.1016m for a periodic step command and the controlled stroke of the slider B, ∆xB , is equal to 0.04m. Moreover, the initial position of xB is 0.1216m for a periodic sinmsoidal command and the controlled stroke of the slider B, ∆xB , is set to be 0.02m. Substituting the slider position xB into (12), the angle v of link 2 can be obtained.

Slider Position

Reference Model

Time (sec) (a)

u (A)

Control Effort

Time (sec) (b)

Figure 5. Simulated results of AFC system due to periodic step command: (a), (b) Case 1; (c), (d) Case 2; (e), (f) Case 3. The simulated results of the AFC system due to periodic step commands for Case 1, Case 2 and Case 3 are depicted in Fig. 5. From the simulated results, accurate tracking control performance of the motor-toggle servomechanism can be obtained through on-line learning of AFC system,

xB ( m)

92


Slider Position

Reference Model

Time (sec) (c)

u(A)

Control Effort

Time (sec) (d)

Fig. 5 (Continued).

and the robust characteristics can be obtained under the occurrence of uncertainties. To demonstrate the control performance of the proposed control systems with different reference trajectory and initial condition, a periodic sinusoidal command is examined here. The tracking responses and associated control efforts with AFC system at Case 1, Case 2 and Case 3 are depicted in Fig. 6. The robust control performance of the proposed control system is obvious under the occurrence of the parameter variations, external disturbance and different reference trajectories. Although favorable tracking responses are obtained by the AFC system, the chattering phenomena of the control efforts shown in Figs. 5(b), 5(d), 5(f), 6(b), 6(d) and 6(f) are undesirable. Now, an AFC system with bound estimation is applied to control the nonlinear motor-mechanism coupling system for comparison. The simulated results of the tracking responses, control efforts and estimated bound values due to periodic step commands for Case 1, Case 2 and Case 3 are depicted in Fig. 7. From the simulated results, accurate tracking control performance of the motor-toggle servomechanism can be obtained through on-line tuning of AFC system with bound estimation; the robust characteristics can be achieved under the occurrence of uncertainties. To demonstrate the control performance with different reference trajectory and

xB ( m)


Slider Position

93

Reference Model

Time (sec) (e)

u(A )

Control Effort

Time (sec) (f)

Fig. 5 (Continued).

initial condition, a periodic sinusoidal command is examined here. The simulated results of the tracking responses, control efforts and estimated bound values due to periodic sinusoidal commands for Case 1, Case 2 and Case 3 are depicted in Fig. 8. The robust control performance of the proposed AFC system with bound estimation is obvious under the occurrence of the parameter variations, external disturbance and different reference trajectories. From the simulated results, the chattering phenomena are much reduced in the associated control efforts of the AFC system with bound estimation according to the on-line adjustment of the bound value in the compensated controller. Some experimental results are provided to further demonstrate the effectiveness of the proposed control systems. Two test conditions are provided, which are the nominal case and the parameter variation case. In the experimentation, the parameter variation case is the addition of two iron disks with 7.4kg weight to the mass of the slider B. The tracking responses and control efforts using the AFC system of the two test conditions due to periodic step and sinusoidal commands are depicted in Figs. 9 and 10, respectively. The tracking responses due to periodic step command with the nominal and parameter variation cases are depicted in Figs. 9(a) and 9(c); the associated control efforts are depicted in Figs. 9(b) and 9(d). Moreover,

xB (m)

94


Slider Position

Reference Model Time (sec) (a) u(A )

Control Effort

Time (sec) (b)

Figure 6. Simulated results of AFC system due to periodic sinusoidal command: (a), (b) Case 1; (c), (d) Case 2; (e), (f) Case 3. the tracking responses due to periodic sinusoidal commands with the nominal and parameter variation cases are depicted in Figs. 10(a) and 10(c); the associated control efforts are depicted in Figs. 10(b) and 10(d). Due to the on-line learning control characteristics of the AFC system, favorable tracking responses can be obtained under the occurrence of the parameter variations and different reference trajectories. However, the chattering phenomena in the control efforts are due to large bound in the compensated controller. The undesired chattering control efforts will wear the bearing mechanism and might excite unstable system dynamics. Now, the AFC system with bound estimation shown in Fig. 4 is implemented to control the motor-toggle servomechanism for comparison. The tracking responses due to periodic step commands with the nominal and parameter variation cases are depicted in Figs. 11(a) and 11(c); the associated control efforts are depicted in Figs. 11(b) and 11(d). Moreover, the tracking responses due to periodic sinusoidal commands with the nominal and parameter variation cases are depicted in Figs. 12(a) and 12(c); the associated control efforts are depicted in Figs. 12(b) and 12(d). From the experimental results, the tracking errors converge quickly, and the robust control characteristics under the


95

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u (A)

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Fig. 6 (Continued). occurrence of the parameter variations and different reference trajectories can be clearly observed. In addition, the chattering phenomena of the AFC system are much reduced owing to the on-line adjustment of the bound in the compensated controller. The merits of the proposed AFC system with bound estimation can be verified by the simulated and experimental results. 7. Conclusions This study has successfully demonstrated the application of an AFC system with bound estimation to control the position of a slider of the motortoggle servomechanism. All parameters of the FBF network can be adjusted by the tuning algorithm based on the Lyapunov stability theorem. Thus, the stability of the developed AFC system can be guaranteed. Moreover, to relax the requirement for the uncertain bound in the compensated controller, an AFC system with bound estimation is investigated to control the motor-mechanism coupling system. The effectiveness of the proposed AFC design methods has been verified through simulated and experimental results. From the results, the AFC with bound estimation design method is more suitable to be applied in the control of the motor-toggle servomechanism.

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Fig. 6 (Continued). References [1] C. C. Lee, Fuzzy logic in control systems: fuzzy logic controller-part I/II. IEEE Trans. Systems, Man, and Cybernetics, vol. 20, no. 2, pp. 404-435, 1990. [2] L. X. Wang, Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1994. [3] F. C. Sun, Z. Q. Sun, and G. Feng, An adaptive fuzzy controller based on sliding mode for robot manipulators. IEEE Trans. Systems, Man, and Cybernetics- Part B: Cybernetics, vol. 29, 661-667, 1999. [4] D. L. Tsay, H. Y. Chung, and C. J. Lee, The adaptive control of nonlinear systems using the sugeno-type of fuzzy logic. IEEE Trans. Fuzzy Syst., vol. 7, no. 2, pp. 225-229, 1999. [5] C. C. Wong and J. Y. Chen, Fuzzy control of nonlinear systems using rule adjustment. IEE Proc. Control Theory Appl., vol. 146, no. 6, pp. 578-584, 1999. [6] Y. C. Chang, Robust tracking control for nonlinear MIMO systems via fuzzy approaches. Automatica, vol.36, pp. 1535-1545, 2000. [7] J. Nie and D. A. Linkens, Learning control using fuzzified self-organizing radial base function network. IEEE Trans. Fuzzy Syst., vol. 1, no. 4, pp. 280-287, 1993. [8] C. T. Lin and C. S. Lee, Reinforcement structure/parameter learning for neuralnetwork-based fuzzy logic control systems. IEEE Trans. Fuzzy Syst., vol. 2, no. 1, pp. 46-63, 1994. [9] F. J. Lin, R. F. Fung, and Y. C. Wang, Sliding-mode and fuzzy control of toggle mechanism using PM synchronous servomotor drive. IEE Proc. Control Theory Appl., vol.144, no. 5, pp. 393-402,1997.

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Reference Model

Time (sec) (a)

u ( A)

Control Effort

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Figure 7. Simulated results of AFC system with bound estimation due to periodic step command: (a), (b) Case 1; (c), (d) Case 2; (e), (f) Case 3; (g) estimated bound value of χ. [10] F. J. Lin, R. F. Fung, and R. J. Wai, Comparison of sliding-mode and fuzzy neural network control for motor-toggle servomechanism. IEEE/ASME Trans. Mech., vol. 3, no. 4, pp. 302-318, 1998. [11] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989. [12] J. J. E. Slotine and W. P. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. Department of Electrical Engineering, Yuan Ze University, Chung-Li, TaoYuan 320, Taiwan, R.O.C. E-mail address: [email protected]

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Fig. 7 (Continued).

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Fig. 7 (Continued).


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Fig. 7 (Continued).

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Figure 8. Simulated results of AFC system with bound estimation due to periodic sinusoidal command: (a), (b) Case 1; (c), (d) Case 2; (e), (f) Case 3; (g) estimated bound value of χ.

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Fig. 8 (Continued).


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Fig. 8 (Continued). Control Effort 0.1416m

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Figure 9. Experimental results of AFC system: (a), (b) nominal case due to periodic step command; (c), (d) parameter variation case due to periodic step command.

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Figure 10. Experimental results of AFC system: (a), (b) nominal case due to periodic sinusoidal command; (c), (d) parameter variation case due to periodic sinusoidal command. Control Effort 0.1416m


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Figure 11. Experimental results of AFC system with bound estimation: (a), (b) nominal case due to periodic step command; (c), (d) parameter variation case due to periodic step command.


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Figure 12. Experimental results of AFC system with bound estimation: (a), (b) nominal case due to periodic sinusoidal command; (c), (d) parameter variation case due to periodic sinusoidal command.