Published in IET Control Theory and Applications Received on 16th August 2007
Revised on 29th March 2008 doi: 10.1049/iet-cta:20070315
ISSN 1751-8644
Robust fuzzy-neural sliding-mode controller
design via network structure adaptation
P.-Z. Lin
1
C.-F. Hsu
2
T.-T. Lee
3
C.-H. Wang
1
1Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan, Republic of China 2Department of Electrical Engineering, Chung Hua University, Hsinchu 300, Taiwan, Republic of China
3Department of Electrical Engineering, National Taipei University of Technology, Taipei 106, Taiwan, Republic of China
E-mail: [email protected]
Abstract: A robust fuzzy-neural sliding-mode control (RFSC) scheme for unknown nonlinear systems is proposed. The RFSC system is composed of a computation controller and a robust controller. The computation controller containing a self-structuring fuzzy-neural network (SFNN) identifier is the principle controller, and the robust controller is designed to achieve L2 tracking performance. The SFNN identifier uses the structure- and parameter-learning phases to perform the estimation of the unknown system dynamics. The structure-learning phase consists of the growing of membership functions, the splitting of fuzzy rules and the pruning of fuzzy rules, and thus the SFNN identifier can avoid the time-consuming trial-and-error tuning procedure for determining the network structure of fuzzy neural network. Finally, the proposed RFSC system is applied to three nonlinear dynamic systems. The simulation results show that the proposed RFSC system can achieve favourable tracking performance by incorporating SFNN identifier, sliding-mode control and robust control techniques.
1
Introduction
The fuzzy-neural network (FNN) that incorporates the advantages of fuzzy inference and neuro-learning has been an interesting topic. The FNN possesses the merits of the low-level learning and computational power of neural network and the high-level human knowledge representation from fuzzy theory[1, 2]. Recently, the FNNs are increasingly receiving attention in solving the control problems [3 – 7]. For the FNN-based adaptive control approaches in [3 – 7], the structure of FNN should be determined in advance by trial-and-error. However, it is difficult to consider the balance between the rule number and the desired performance. If the number of fuzzy rules is chosen too large, the computational load is heavy so that they are not suitable for practical applications. If the number of fuzzy rules is chosen too small, the learning performance may not be good enough to achieve desired performance.
To solve the problem of structure determination in FNN approaches, much interest has been focussed on the self-structuring fuzzy-neural network (SFNN) approach
[8–10]. The self-structuring approach demonstrates the property of automatically generating rules of FNN without the preliminary knowledge. In general, the mathematical descriptions of the existing rules can be expressed as a fuzzy cluster where each cluster corresponds to a fuzzy IF-THEN rule[10]. As usually seen in other self-structuring approaches, a new membership function is generated when a new input signal is far away from the current clusters, and an existing rule is cancelled when the fuzzy rule is insignificant.
Recently, the SFNNs also have been widely adopted for the control of complex dynamic systems owing to its good generalisation capability, structure adaptation and simple computation [11–14]. Some of them use the gradient descent method to derive the parameter learning algorithms
[11, 14]; however, stability analysis has not been performed yet. Some of them use the Lyapunov function to derive the parameter-learning algorithms; however, the complex design procedure is not suitable for practical applications [12, 13]. Lin et al. use a similarity measure method to avoid a newly generated membership function being similar to the existing ones; however, the network structure would grow large, as
the input data has large variations[11, 14]. In [12, 13], an error reduction ratio with orthogonal-triangular (QR) decomposition is used to prune rules; however, the computational load is too heavy. In summary, a comparison between the proposed robust fuzzy-neutral sliding-mode control (RFSC) and [10, 14, 15] is made. In [10], an independent component analysis (ICA)-mixture-model-based self-constructing FNN is proposed. It is not suitable for controller design. In[14], a similarity measure method is used to avoid a newly generated membership function being similar to the existing ones. However, the computational load of pruning algorithm is too heavy. In [15], a self-structuring neural network is proposed; however, the proposed approach cannot avoid the structure of neural network growing endlessly. The proposed RFSC can grow the membership functions, split the fuzzy rules and prune the fuzzy rules of SFNN automatically. The structure of neural network can avoid growing endlessly and the computational load can be reduced.
In this study, a novel SFNN that includes the growing of membership functions, the splitting of fuzzy rules and the pruning of fuzzy rules is developed. The idea of these algorithms is given as follows. The growing method can be considered when the input values will be far away from the edge of existing membership functions as shown in[10]. The split algorithm[15] is performed when the weight updating rate of certain neuron is larger, since larger weight updating rate will cause a larger updating of certain weight values, and the precise approximation is difficult to capture. The pruning algorithm [16] will eliminate those rules that make a very small contribution to the neural network output. Because the error between the real function and the estimation function gradually converges to a small bound after several iterations, and some fuzzy rules that are generated in the transient period may be less or never used. Therefore the structure learning phase of the proposed SFNN possesses several techniques, including the growing of membership functions, the splitting of fuzzy rules and the pruning of fuzzy rules.
For the conventional FNN-based adaptive control approaches, the structure of FNN should be determined in advance by trial-and-error. By the proposed SFNN approach, this paper develops a novel FNN-based adaptive control approaches, which the structure of FNN can automatically generate fuzzy rules to avoid the time-consuming trial-and-error tuning procedure. This paper proposes a RFSC system, which is composed of a computational controller and a robust controller. The computation controller containing the SFNN identifier is the principle controller, and the robust controller with desired attenuation level is designed to achieve L2
tracking performance. The structure-learning phase of the SFNN identifier possesses the ability of on-line generation and elimination of fuzzy rules to achieve optimal network structure, and the parameter-learning phase adjusts the parameters of membership functions and fuzzy rules to achieve favourable approximation performance. By the proposed structure adaptation, the annoyance will be solved
where the balance between the rule number and the desired control performance is difficult to consider. To investigate the effectiveness of the proposed control scheme, the RFSC system is applied to three nonlinear dynamic systems. Simulation results demonstrate that the proposed RFSC scheme can achieve favourable tracking performance without unknown system dynamics. The SFNN identifier can estimate the system dynamics and achieve approximation performance in the presence of external disturbance in a real-time environment.
2
Sliding-mode control design
Consider the nth-order nonlinear dynamic system of the form
x(n) ¼f (x) þ u þ d y ¼ x
(1) wherex ¼ [x _x x(n1)]T is the state vector of the system, y is the output of system, f (x) is the system dynamics, d is the external disturbance and u is the control effort. Assume that the system dynamics is well known, (1) can represent the nominal model of the nonlinear dynamic system
x(n) ¼fn(x) þ u þ dn
y ¼ x (2)
where fn(x) is a mapping that represents the nominal
behaviour of f (x) and dn is the nominal value of d. If
uncertainties occur, that is, parameters of the system deviate from the nominal value, the controlled system can be modified as
x(n)¼fn(x) þ u þ dnþz
y ¼ x (3)
where z is the lumped uncertainty which is defined as z ¼ Df (x) þ Dd with the assumption jzj Z in which Z is a given positive constant, Df (x) denotes the system uncertainties and Dd denotes the disturbance uncertainties. The control objective is to find a control law so that the system output y can track a command trajectory yc. A
tracking error and a sliding surface are defined as
e ¼ ycy (4) and s ¼ e(n1)þkne(n2)þ þk2e þ k1 ðt 0 e d t (5) where ki, i ¼ 1, 2, . . . , n are the positive constants. The sliding-mode control law is given as[17]
where the equivalent controller ueqis represented as
ueq¼ fn(x) dnþx(n)c þk
Te (7)
in which e ¼ [e _e e(n1)]T and k ¼ [k1k2 kn]T, and
the hitting controller uht is designed to guarantee the
system stability as
uht¼Zsgn(s) (8)
where sgn(.) is the sign function. Substituting (6) –(8) into (3) yields
e(n)þkne(n1)þ þk2_e þ k1e ¼ z Zsgn(s) ¼ _s (9)
An important concept of sliding-mode control is to make the system satisfy the reaching condition and guarantee sliding condition. Consider the candidate Lyapunov function in the following form as
V1¼
1 2s
2 (10)
Differentiating (10) with respect to time and using (9), we obtain
_
V1¼s_s ¼ zs Zjsj jzjjsj Zjsj
¼ (Z jzj)jsj 0 (11)
In summary, the sliding-mode controller in (6) can guarantee the stability in the sense of the Lyapunov theorem [17]. Because the system dynamics and the external disturbance may be unknown or perturbed, the sliding-mode control law usccannot be implemented. Moreover, a large control gain Z
is often required in order to minimise the reaching time from the initial state to the switching surface. The selection of the control gain Z, which is related to the magnitude of uncertainties, keeps the trajectory on the sliding surface[17]. Nevertheless, the parameter variations of the system are difficult to measure for practical applications. A conservative control law with a large control gain Z is usually considered, but the unnecessary jumping movement on the switching surface may yield and cause an outcome of a large amount of chattering. The chattering phenomena in control efforts will wear the bearing mechanism and excite unmodelled dynamics.
3
Robust fuzzy-neural
sliding-mode control design
3.1 Description of SFNN
The structure of the SFNN has four layers of neural network: the input, the membership function, the rule and the output layers. Nodes at layer 1 are input nodes (linguistic nodes) that represent input linguistic vector. Nodes at layer 2 are term nodes which act as membership functions to represent the
terms of the respective linguistic vector. Each node at layer 3 is a fuzzy rule. Layer 4 is the output layer, where the node in this layer is the output of SFNN. The interactions for those layers are given as follows.
Layer 1 – Input layer: For every node i, in this layer, the net input and the net output are represented as
1 neti ¼ 1 xi (12) 1 yi¼ 1 fi 1 neti ¼1neti, i ¼ 1, 2, . . . , L (13)
where1xirepresents the ith input to the node of layer 1 and L
is the total number of input variables. They mean that output equals input in this layer. This layer of SFNN just executes the transmission work.
Layer 2 – Membership layer: In this layer, each node performs a membership function and acts as a unit of memory. The bell-shaped function is adopted as the membership function. For the ith input, the corresponding net input and output of the jth node can be expressed as
2 netij¼ 2 xi 2 mij 2 2s ij 2 (14) 2 yij ¼ 2 fij 2 netij ¼exp 2netij , j ¼ 1, 2, . . . , M (15) where2mijis the mean,
2s
ij is the standard deviation and M
is the total number of membership functions with respect to the respective input node. In this study, the input linguistic variable is the tracking error vector.
Layer 3 – Rule layer: Each node k in this layer is denoted by P which multiplies the incoming signals and outputs the result of the product. For the kth rule node, the operation of the net input and output of this layer is presented as
3 netk ¼ Y3 wij 3 xij (16) 3 yk ¼ 3 fk 3 netk ¼3netk, k ¼ 1, 2, . . . , N (17)
where3xij represents the i, jth input to the kth node of layer
3, 3wij between the membership and the rule layers are
assumed as unity and N is the total number of fuzzy rules. Layer 4 – Output layer: The single node o in this layer is labelled as S, which computes the overall output as the summation of all incoming signals. It executes the sum-of-weighting defuzzification. The description of the net input and output is expressed as
4 neto¼ X k 4 wk 4 xk (18) 4 yo¼ 4 fo 4 neto ¼4neto (19)
where4wkis the output action strength of the output associated
with the kth rule,4xk represents the kth input to the node of
layer 4 and 4yo is the output of SFNN. For the FNN
approaches in[3–7], the structure of the FNN is determined in advance by trial-and-error, but it is difficult to consider the balance between the rule number and the desired control performance. In this paper, the structure-learning algorithm, which is composed of the growing of membership functions, the splitting of fuzzy rules and the pruning of fuzzy rules, is proposed to solve this problem. The descriptions for those algorithms are given as follows.
For the method of the growing of membership functions, the concept, which decides whether to add a new node (membership function) in layer 2 and the associated fuzzy rule in layer 3, will be introduced. In order to construct fuzzy rules of the SFNN, a new rule is generated when a new input signal is far away from the current clusters. A cluster corresponds to a fuzzy IF-THEN rule. Each cluster in the product space of the input – output data represents a rule in the rule base. The firing strength of a rule can be represented as the degree that the incoming data 1xi
belongs to the cluster. If the value of firing strength is too small, it represents that the input value is on the edge of range of the existing membership functions. Under this situation, the output will cause a bad output performance. Therefore a new node (membership function) should be added at this moment. The firing strength obtained from (17) is used as the degree measure[10]
bk¼3yk, k ¼ 1, 2, . . . , N (20)
where N is the number of the existing rules. Define the maximum degree bmaxamong bkas
bmax ¼ max
1kNbk (21)
It can be observed that if bmaxGthwhere Gth[(0, 1) is a
pre-given threshold, the incoming data is far away from the edge of range of the existing membership functions. Hence, a new membership function should be generated. The mean and the standard deviation of the new membership function and the corresponding weight are selected as follows mnewi ¼ 1 xi (22) snewi ¼si (23) wnew ¼0 (24)
where1xi is the new incoming data and siis a pre-specified
constant. If the unknown control system dynamics is complex, more membership functions can be created by choosing a large Gth.
Next, the split of the kth fuzzy rule is executed if the following condition is satisfied[15]
j4_^wkj PN k¼1 j4_^w kj Sth, k ¼ 1, 2, . . . , N (25)
where Sthdenotes a split threshold value and the tuning law
4_^w
k will be derived in the next subsection. On the left-hand
side of (25), the denominator implies the total sum of the positive weight variations and the numerator represents the positive weight variations connecting to the kth node. The proposed split algorithm is derived from the observation that if the updating of the weight values is relatively large, the precise approximation is difficult to capture. If (25) is satisfied, a new neuron is duplicated to spread the relatively large variation of the weights. If the designer wants more neurons in the SFNN, because the nonlinear plant is complex, more neurons can be created by choosing a small Sth, and vice versa. The weight value connecting the k0th
node is considered as
4
wk0 ¼(1 a) 4wk (26)
where a is a positive constant. The original weight value connecting the kth node is changed as
4
wk¼a 4
wk (27)
This method is induced from the fact that the weights connected to the newly created neuron will share the large variations of the weights. When the estimated parameters are all the optimal parameters in the training procedure, that is, the learning algorithm searches the optimal solutions, the tuning law 4_^wk will be equal to zero for all k.
Note that if the condition satisfies 4_^wk¼0 for all k, the
operation of split does not be executed.
To avoid the endless growing of network structure and the overload computation, the algorithm of the pruning of fuzzy rules is developed to eliminate inappropriate fuzzy rules. When the rth firing strength br is smaller than a threshold
value Pth, it means that the relationship becomes weak
between the input and the rth rule. This fuzzy rule may be less or never used. Then, we will gradually reduce the value of the rth referring index when the rth firing strength br
satisfies our setting condition continuously. The referring index is determined as[16] Ir¼ Irpexp (t), if br, Pth Irp, if brPth , r ¼ 1, 2, . . . , N (28)
where Iris the referring index of the rth rule and its initial value
is 1, Pth is the pruning threshold value, t is the decayed
constant, and Irp denotes the most recent Ir. If IrIth is
fuzzy rule is pruned. Moreover, the computational load should also be decreased. If the computational load is the important issue for practical implement, we can choose a large Pth so
that more fuzzy rules can be pruned. In summary, the flow chart of the structure-learning algorithm is shown in Fig. 1. The major contributions of using these algorithms for training SFNN structure are: (1) SFNN can be operated directly without spending much time on pre-determining membership functions and fuzzy rules; (2) the computational load can be reduced simultaneously.
3.2 Parameter-learning algorithm
Since the system dynamics f (x) and the external disturbance d are unknown in practical applications, the equation of the
nonlinear dynamic system in (1) can be rewritten as x(n)¼F (x) þ u
y ¼ x
(29) where f (x) is called the lumped system dynamics and defined as f (x) ¼ f (x) þ d. As shown inFig. 2, the proposed RFSC system is composed of a computational controller and a robust controller, that is
ufsc¼uccþurc (30)
where the computational controller is designed as ucc¼ F (e, ^Q) þ x(n)c þk
Te
¼ ^F þ x(n)c þkTe (31)
in which the SFNN identifier F (e, ^Q) ¼ ^F is used to on-line estimate the lumped system dynamics F(x) in (29). e and
^
Q ¼[4w^2k m^ 2 ijs^ij]
T
are the input vector and estimated parameter vector of the SFNN identifier, respectively. In the sliding-mode control approach, if the sliding condition is satisfied, the error terms will converge to zero as time approaches to infinite. Furthermore, bounds on s can be directly translated to bounds on the tracking error, because the sliding surface is composed of the error terms. In particular, once the system is on the surface, the system trajectories will remain on the surface and the tracking error tends exponentially to zero. Consequently, the stability of closed-loop system can be guaranteed[17]and Section 2 has the proof of stability. This motion is the same as minimising the error function if the error function is selected as the cost function. Hence, the sliding condition, which substitutes for the error function, becomes the cost function. The on-line learning algorithm is a gradient descent algorithm in the space of network parameters and aims to minimise s_s for
Figure 1 Flow chart of the proposed structure adaptation
achieving fast convergence of s. Substituting (30) into (29) and using (5) yield
e(n)þkne(n1)þ þk2_e þ k1e ¼ ^F F urc¼_s (32)
Multiply both sides of (32) by s to yield
s_s ¼ ( ^F F urc)s (33)
According to the gradient descent method[3], the adaptive law of weights in the output layer can be derived as
4_^w k ¼ hw @s_s @(4w^ k) ¼ hw@s_s @ ^F @ ^F @(4w^ k) ¼ hws 4xk (34)
where the positive constant hwis a learning rate. In order to
easily derive the update laws of mean and variance, the parameter j is derived in advance as
j ¼@s_s @ ^F @ ^F @(4net o) @(4neto) @(3y k) @(3yk) @(3net k) @(3netk) @(2y ij) @(2yij) @(2net ij) ¼s 4wk 3 yk (35)
The update laws of means and variances,2mij and 2
sij, can be obtained by the gradient search algorithm[3]as
2_^m ij¼ hm @s_s @(2m^ ij) ¼ hm@s_s @ ^F @ ^F @(4net o) @(4neto) @(3y k) @(3yk) @(3net k) @(3netk) @(2y ij) @( 2 yij) @(2net ij) @(2netij) @(2m^ ij) ¼ hmj 2 2xi 2 ^ mij 2s^ ij 2 (36) and 2_^s ij¼ hs @s_s @(2s^ ij) ¼ hs@s_s @ ^F @ ^F @(4net o) @(4neto) @(3y k) @(3yk) @(3net k) @(3netk) @(2y ij) @( 2 yij) @(2net ij) @(2netij) @(2s^ ij) ¼ hsj 2 2xi 2 ^ mij 2 2s^ ij 3 (37)
where hm and hs are the learning rates with positive
constants.
3.3 Robust controller design
Approximating linear or nonlinear mapping through learning is one of most useful property of a neural network. The relationship between F(x) and F (e, ^Q) is described as
F (x) F (e, ^Q) ¼ 1 (38)
where 1 denotes the modelling error. Substituting (38) into (32), (32) can be rewritten as follows
_s ¼ 1 urc (39)
If 1 exists, a robust controller will be considered to satisfy a specified L2tracking performance[18 – 20]
ðT 0 s2(t) dt s2(0) þ d2 ðT 0 12(t) dt (40)
where d is a prescribed attenuation constant. If the system starts with an initial condition s(0) ¼ 0, the L2 tracking
performance in (40) can be rewritten as sup
1[L2[0,T ]
ksk
k1kd (41)
where ksk2¼Ð0Ts2(t) dt and k1k2¼Ð0T12(t) dt. If d ¼ 1, this is the case of minimum error tracking control without disturbance attenuation. Define a Lyapunov function as
V2 ¼
1 2s
2
(42) Differentiating (42) with respect to time and using (39), we obtain
_
V2¼s_s ¼ s( 1 urc) (43)
The robust controller is designed as
urc¼
d2þ1
2d2 s (44)
and thus (43) can be rewritten as
_ V2¼s 1 d2þ1 2d2 s ! ¼ s1 s 2 2 s2 2d2 ¼ s2 2 1 2 2s1 þ s2 d2 ! ¼ s 2 2 1 2 s2 d2þ2s1 þ 1 2 d2 ! þ1 21 2 d2 ¼ s 2 2 1 2 s dþ1d 2 þ1 21 2d2 1 2s 2þ1 21 2d2 (45)
Assume 1 [ L2[0, T ], 8T [ [0, 1). Integrating the above
equation from t ¼ 0 to t ¼ T yields
V2(T ) V2(0) 1 2 ðT 0 s2dt þ1 2d 2 ðT 0 12dt (46)
Since V2(t) 0, we can rearrange (46) as 1 2 ðT 0 s2dt V2(0) þ 1 2d 2ðT 0 12dt (47)
Using (42), this inequality is equivalent to inequality (40). Since V2(0) is finite, if the approximation error 1 [ L2,
that is Ð0T12(t) d t , 1, it implies that limt!1jsj ¼ 0. Then, the RFSC can achieve L2tracking performance with
attenuation of disturbances including approximation errors and external disturbances.
4
Simulation results
Chaotic dynamic system is a nonlinear deterministic system that displays complex, noisy-like and unpredictable behaviour. It can be observed in many nonlinear circuits and mechanical systems[21]. For control engineers, control of a chaotic dynamic system has become a significant research topic in physics, mathematics and engineering communities
[22 – 25]. However, some of them cannot achieve favourable control performance[22 – 24]and some of them require overly complex design procedures [25]. In this section, the proposed RFSC scheme is applied to two chaotic dynamic systems and a simple nonlinear system to verify its effectiveness. It should be emphasised that parameters and network structure of the SFNN can be tuned on-line by the proposed algorithm. Moreover, the appropriate network structure will be obtained by the methods of the growing of membership functions, the splitting of fuzzy rules and the pruning of fuzzy rules during the process of training adjustable parameters.
Example 1:
Consider a second-order chaotic system such as the Duffing’s equation describing a special nonlinear circuit or a pendulum moving in a viscous medium as follows[22]€x ¼ f (x) þ u (48)
where f (x) ¼ p_x p1x p2x 3
þq cos (vt) is the system dynamics, t is the time variable, v is the frequency, u is the control force, and p, p1, p2 and q are real constants.
Depending on the choice of these constants, the system is known that the solutions of (48) may exhibit periodic, that is it is almost periodic and chaotic behaviour. In order to observe the chaotic unpredictable behaviour, the open-loop system behaviour, that is, u ¼ 0, is simulated with p ¼ 0.4, p1¼ 21.1, p2¼ 1.0, q ¼ 1.95 and v ¼ 1.8. The phase
plane plots with an initial condition point (0.5, 0) for q ¼ 1.95 and q ¼ 7.00 are shown in Figs. 3a and 3b, respectively. The time responses of the uncontrolled chaotic system with initial conditions (0.5, 0) and (0.6, 0) are shown in Figs. 3c and 3d, respectively. It is shown that the uncontrolled chaotic system has different trajectories for different q values and different initial points.
The parameters of RFSC are selected as k1¼ 2, k2¼ 1,
hw¼40, hm¼hs ¼2, Gth¼0:3, Sth¼0:5, Ith¼0:2,
a ¼0:9, Pth¼0:15, t ¼ 0:01 and d ¼ 0:5. All the gains in
the proposed control system are chosen to achieve the best transient control performance considering the requirement of Figure 3 Behaviour of uncontrolled chaotic system
stability and possible operating conditions. The simulation results of RFSC for q ¼ 1.95 and q ¼ 7.00 are shown in
Figs. 4and5, respectively. The tracking responses of state x are shown inFigs. 4aand5a; the tracking responses of state _x are shown inFigs. 4band5b; the associated control efforts are shown inFigs. 4c and5c; the number of fuzzy rules are shown in Figs. 4d and5d; the responses of neural network
are shown in Fig. 4e and 5e, respectively. The numbers of fuzzy rules for q ¼ 1.95 and q ¼ 7.00 are 5 and 4, respectively. It shows that the proposed network structure adaptation can grow membership functions, split fuzzy rules and prune fuzzy rules automatically for different nonlinear systems. From the simulation results, not only the perfect tracking responses can be achieved without any knowledge of system dynamics, but also the concise SFNN structure Figure 5 Simulation results of RFSC system for q ¼ 7.00
can be obtained by applying the proposed self-structuring mechanism and the on-line learning algorithms.
Example 2:
The third-order Chua’s chaotic circuit, shown inFig. 6, is a simple electronic system which consists of one linear resistor (R1), two capacitors (C1, C2), one inductor(L1), and one nonlinear resistor (l). It owns very rich
nonlinear dynamics such as chaos and bifurcations [23]. l is defined as a cubic function as
l(vC1) ¼ avC1þcv
3
C1 (a , 0, c . 0) (49)
The standard canonical form of Chua’s circuit can be expressed as[24]
x(3)¼f (x) þ u þ d (50) where f (x) ¼ (14=1805)x (168=9025)_x þ (1=38)€x (2=45)((28=361)x þ (7=95)_x þ €x)3 is the system dynamic function, and a square-wave disturbance d is added at 10 sec with +2:0 amplitude and period 12 sec. The parameters of RFSC are selected as k1¼ 3, k2¼ 3, k3¼ 1,
hw¼20, hm¼hs¼1, Gth¼0:3, Sth¼0:5, Ith¼0:2,
a ¼0:9, Pth¼0:15, t ¼ 0:01 and d ¼ 0:5. The choice of
these parameters is also through some trails. To illustrate the effectiveness of RFSC design method, a comparison between the FNN-based adaptive control in[3]and SRFC systems is made. The parameters of the FNN-based adaptive control with three membership functions are selected as k1¼3, k2¼3, k3 ¼1, hw¼20, hm¼hs¼1.
First, the simulation results of the FNN-based adaptive control with fix-structure neural network are shown in Fig. 7. The tracking response of state x is shown in
Fig. 7a; the tracking response of state _x is shown in
Fig. 7b; the tracking response of state €x is shown inFig. 7c; the associated control effort is shown in Fig. 7d; the response of neural network is shown inFig. 7e, respectively. The simulation results show that the favourable tracking performance can be achieved. However, the degenerate tracking responses are caused when the disturbance occurs. Then, the proposed RFSC with SFNN identifier is applied to the Chua’s chaotic circuit again. The simulation results of the RFSC are shown inFig. 8. The tracking response of state x is shown in Fig. 8a; the tracking response of state _x is shown in Fig. 8b; the tracking response of state €x is shown in Fig. 8 c; the associated control effort is shown in
Fig. 8d; the number of fuzzy rule is shown in Fig. 8e; the
response of neural network is shown inFig. 8f. Comparing with Fig. 7, Fig. 8 shows that the proposed RFSC system can achieve better tracking performance even under disturbance. The number of the constructed fuzzy rules is 5, and the maximum of the constructed fuzzy rules is 7. The system dynamics with the external disturbance can be estimated well by the SFNN identifier. A concise network structure can be obtained by the proposed self-structuring method.
Figure 6 Chua’s chaotic circuit
Figure 7 Simulation results of fix-structuring FNN-based adaptive control system for Chua’s chaotic circuit
Example 3:
Consider a plant as follows[26]_x ¼ f (x) þ u (51)
where f (x) ¼ (1 ex)=(1 þ ex) is the system dynamics, and u is the control force. The parameters of RFSC are selected as k1¼1, hw¼40, hm¼hs ¼2, Gth¼0:3,
Sth¼0:5, Ith¼0:2, a ¼ 0:9, Pth¼0:15, t ¼ 0:01 and
d ¼0:5. The choice of these parameters is also through some trails. The simulation results of RFSC are shown in
Fig. 9. The tracking response of state x are shown in
Fig. 9a; the associated control effort is shown in Fig. 9b; the number of fuzzy rules is shown inFig. 9c; the response of neural network is shown in Fig. 9d. The number of the constructed fuzzy rules is 5. The results illustrate that the proposed RFSC scheme can achieve satisfied tracking performance for unknown nonlinear systems, and a concise
Figure 8 Simulation results of RFSC system for Chua’s
chaotic circuit Figure 9 Simulation results of RFSC system for nonlinear dynamic system
network structure can be obtained by the proposed self-structuring method.
5
Conclusions
For the FNN-based adaptive control approaches, the structure of FNN should be determined in advance by trial-and-error. It is difficult to consider the balance between the rule number and the desired performance. Therefore this paper proposes a RFSC system with structure adaptation approach. The proposed RFSC system is composed of a computational controller and a robust controller. In the computational controller design, a SFNN with the structure- and parameter-learning is utilised to on-line estimate the unknown control dynamics equation. The robust controller with desired attenuation level is designed to achieve L2 tracking
performance. The structure-learning phase of SFNN possesses several techniques, including the growing of membership functions, the splitting of fuzzy rules and the pruning of fuzzy rules, to organise FNN automatically. The parameter-learning phase of SFNN adjusts the parameters of the membership functions and fuzzy rules based on the Lyapunov function to achieve favourable approximation performance.
The major contributions of this paper are: (1) the SFNN with the growing, splitting and pruning algorithm of the membership functions, and fuzzy rules has been developed to achieve favourable learning performance of network structure. (2) the RFSC, in which the gradient descent method is used to derive the on-line parameter-learning algorithms, has been derived and developed well.
6
Acknowledgments
The authors appreciate the partial financial support from the National Science Council of Republic of China under grant NSC 96-2218-E-216-001. The authors would like to express their gratitude to the reviewers for their valuable comments and suggestions.
7
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