Direct Adaptive Iterative Learning Control of
Nonlinear Systems Using an Output-Recurrent
Fuzzy Neural Network
Ying-Chung Wang, Chiang-Ju Chien, and Ching-Cheng Teng
Abstract—In this paper, a direct adaptive iterative learning
con-trol (DAILC) based on a new output-recurrent fuzzy neural net-work (ORFNN) is presented for a class of repeatable nonlinear sys-tems with unknown nonlinearities and variable initial resetting er-rors. In order to overcome the design difficulty due to initial state errors at the beginning of each iteration, a concept of time-varying boundary layer is employed to construct an error equation. The learning controller is then designed by using the given ORFNN to approximate an optimal equivalent controller. Some auxiliary control components are applied to eliminate approximation error and ensure learning convergence. Since the optimal ORFNN pa-rameters for a best approximation are generally unavailable, an adaptive algorithm with projection mechanism is derived to up-date all the consequent, premise, and recurrent parameters during iteration processes. Only one network is required to design the ORFNN-based DAILC and the plant nonlinearities, especially the nonlinear input gain, are allowed to be totally unknown. Based on a Lyapunov-like analysis, we show that all adjustable parameters and internal signals remain bounded for all iterations. Further-more, the norm of state tracking error vector will asymptotically converge to a tunable residual set as iteration goes to infinity. Fi-nally, iterative learning control of two nonlinear systems, inverted pendulum system and Chua’s chaotic circuit, are performed to verify the tracking performance of the proposed learning scheme.
Index Terms—Direct adaptive control, iterative learning control,
nonlinear systems, output-recurrent fuzzy neural network.
I. INTRODUCTION
D
URING the past two decades, iterative learning control (ILC) has been known to be one of the most effective control strategies in dealing with repeated tracking control or periodic disturbance rejection for nonlinear dynamic systems. For instance, the backing up control of a vehicle is a very dif-ficult exercise for most of the beginners because of its high de-gree of nonlinear uncertainty. In order to become a skilled ve-hicle driver, the beginner learns how to successfully control the vehicle through a learning process over and over again. This simple example explains the feature and objective of iterative learning control. In general, the ILC system improves its con-trol performance by a self-tuning process without using anac-Manuscript received December 24, 2002; revised July 22, 2003. This work was supported by the National Science Council of Taiwan, R.O.C., under Grant NSC91-2213-E-211-001. This paper was recommended by Associate Editor H. Qiao.
Y.-C. Wang and C.-C. Teng are with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu, Taiwan, R.O.C.
C.-J. Chien is with the Department of Electronic Engineering, Huafan Univer-sity, Shihtin, Taipei County, Taiwan, R.O.C. (e-mail: cjc@huafan.hfu.edu.tw).
Digital Object Identifier 10.1109/TSMCB.2004.824525
curate system model and can be applied to practical applica-tions such as the control of robotics, servo motors, etc.. To begin with, the D-type, P-type, or PID-type iterative learning con-trollers [1]–[7] were developed for nonlinear plants with non-linearities satisfying global Lipschitz continuous condition. Ba-sically, the control input is directly updated by a learning mech-anism using the information of error and input in the previous iteration. Due to the difficulty to apply these ILCs for non-Lips-chitz nonlinear plants and design of the learning gain depends on the input–output coupling matrix, other new types of ILC algo-rithms have been widely studied in recent years. One of the most interesting and important developments is the so-called adaptive iterative learning control (AILC) [8]–[11]. The control parame-ters, instead of the control input itself, are updated between suc-cessive iterations in the design of AILC. This concept is similar to typical adaptive control problem. The main difference is that control parameters are tuned along iteration axis, but not time axis. Substantial efforts of AILC have been reported for broader applications to non-Lipschitz nonlinear plants, high relative de-gree plants, etc. However, as most of the traditional time-domain adaptive controls of nonlinear systems [12], the plant unknown parameters must be linear with respective to some known non-linear functions in those AILC schemes. The dependence on the structure of plant model unfortunately losses the most important feature of iterative learning system.
To solve the ILC problem for nonlinear systems whose nonlinearities are not Lipschitz continuous or not linearly parameterizable, a powerful strategy is to apply fuzzy system [13] or neural network [14] as a nonlinear approximator when designing the iterative learning controller. Actually, the fuzzy system [15]–[17] or neural-network-based [18]–[20] control strategy has been well known and studied extensively in the time-domain adaptive control of nonlinear systems for many years. Very few results were found in the literature of ILC. Recently, the learning controller based on adaptive fuzzy or neural compensation force were proposed by [21], [22]. As the other existing AILC schemes [8]–[11], the parameters of fuzzy system or neural network are tuned along iteration domain. The adaptive learning controller must guarantee both time domain boundedness and iteration domain convergence due to the special repeated property of ILC. So the controller structure, adaptive law, and stability analysis of fuzzy system or neural-network-based AILC are different from those in tradi-tional adaptive control. Basically, the AILC systems proposed in [21] and [22] can be classified into an indirect scheme [16] since the fuzzy system or neural network is used to model the
plant nonlinearities. In this paper, we aim to present a direct AILC (DAILC) based on a new output-recurrent fuzzy neural network (ORFNN). The ORFNN-based DAILC scheme is dif-ferent from [21] and [22] as the ORFNN is used to approximate the certainty equivalent controller but not to model the plant nonlinearities. The main features of this learning controller and its contributions relative to the related works (especially compared with the works in [21] and [22]) are summarized as follows.
1) The robustness against initial state error at the beginning of each iteration is a very special and important issue in ILC. It has been extensively studied and solved for tradi-tional PID-type iterative learning controllers [2]–[7]. For example, in [4] the learned tracking error can be estimated in terms of initial state error and parameters of PD-type ILC algorithm, and in [7] the learning controller can be robust to variable initial shifting by using a finite ini-tial rectifying action. For all the existing AILC systems [8]–[11], [21], [22], initial state error is still a challenging work since it must be exactly zero for stability analysis. It is hard to directly apply those techniques in [2]–[7] to AILC design. In this paper, the proposed DAILC scheme will relax this critical requirement. By introducing a con-cept of time-varying boundary layer, the initial state er-rors can be allowed to be nonzero. Moreover, they could be varying at each iteration and not necessarily small. It is shown that the norm of state tracking errors will asymp-totically converge to a residual set whose size depends on the width of boundary layer.
2) Consider a simple first-order nonlinear system in the form
of where is the state
and is the input (control) gain function. For the fuzzy system or neural-network-based AILC in [21] and [22] (or even the fuzzy system or neural network-based adaptive control in [15]–[18]), the input gain is always a critical issue for controller design. In addition
to the well-known necessary condition for
a controllable plant, some special constrains are usually required on for stability and convergence anal-ysis in the literature. For example, a constant lower bound of should be known in [21] and [22], the lower and upper bounding functions of should exist and be known in [16], or the time derivative of is bounded by a known positive function in [17] and [21], and so on. Furthermore, two fuzzy systems or neural net-works are often used to estimate both and when designing the adaptive iterative learning controller or adaptive controller. This leads to a more complicated control structure since two sets of parameters are needed to be tuned. In our ORFNN-based DAILC scheme, only one network is required for nonlinear compensation and is allowed to be totally unknown. The only mild requirement on is that it is bounded away from zero.
3) As we know, both the fuzzy system and neural network are to mimic human-like knowledge processing capa-bility. Therefore, the fuzzy neural network (FNN) [23], [24], a combination of the above two approaches, has
be-come a popular research topic in a variety of applications. However, the application domain of FNN is limited to static problems, and it usually processes dynamic prob-lems inefficiently. Hence, the concept of recurrent FNN (RFNN) [27]–[30] was then proposed in the literature. The RFNN, which in general has the internal feedback connections, captures the dynamic response of system without using external feedback through delays so that it has the superior dynamic mapping capability than FNN. In this paper, besides the main contributions to the field of ILC, we also present a new ORFNN to design our learning controller. This ORFNN is a dynamic mapping function with global feedback structure and has a smaller number of network parameters when compared with some existing RFNN [27], [29], [30] if the network input, output, and rule numbers are the same. The ORFNN will reduce to be a static FNN if the recurrent weight is set to be zero. By using the technique of Taylor series ex-pansion, the approximation error between the estimated ORFNN and an optimal ORFNN can be expressed in a linearly parameterized form modulo a residual term linear with some known functions. This fact enables us to derive adaptive laws such that all the consequent, premise, and recurrent parameters can be tuned during the iteration processes.
This paper is organized as follows. In Section II, the plant de-scription, control objective, and design steps of the proposed ORFNN-based DAILC are presented. Analysis of closed-loop stability and learning performance will be studied extensively in Section III. To demonstrate the learning effects of this DAILC, two examples, including a repetitive tracking control of an in-verted pendulum system and Chua’s chaotic circuit, are used for computer simulation in Section IV. Finally, a conclusion is made in Section V. The detailed description of the proposed ORFNN is given in the Appendix.
II. ORFNN-BASEDDIRECTADAPTIVEITERATIVE
LEARNINGCONTROLLER
In this paper, we consider a nonlinear system which can per-form a given task repeatedly over a finite time interval as follows:
.. .
(1)
where is
the state vector of the system, is the control input,
and and are unknown real continuous
nonlinear functions of state, respectively. Here, denotes the index of iteration and . The iterative learning control objective for a repeatable system is different from a typical control problem since the control domain is only a finite time interval. Given a specified desired trajectory and a
the control objective is to force the state vector to follow
such that for some
small positive error tolerance bound . In order to achieve this control objective, some assumptions on the nonlinear system and desired trajectory are given as follows.
A1) The input gain function satisfies for all
and .
A2) Define state errors as
. The initial state errors at each iteration are not neces-sarily zero, small, and fixed, but are assumed to
sat-isfy for some known positive constants
.
A3) The desired state trajectory vector is bounded.
In order for the system (1) to be controllable, we require
for , and without
loss of generality we assume that in assumption
(A1). This is almost a necessary condition for most of the researches dealing with the similar control problem in iteration domain [21], [22] or in time domain [15], [17], [18], [26], [29], [31], [32]. One of the main contributions in this paper will be to show that the estimation of or the bounding infor-mation on is not required in our learning scheme. The condition on initial state errors is given in assumption (A2). In general, they can be varying and large. The only requirement is their upper bounds which are used for controller design. Now, in order to illustrate the idea of the learning controller, we use the following three steps to explain the design approach.
Step 1: Using the well-known design approach in most of the
adaptive control for this class of nonlinear affine system (1), we first design a control function as a linear combination of all the state tracking errors, i.e.,
(2)
where are the coefficients of a Hurwitz polynomial
. It is clear that if the learning controller can drive to zero for all , then the state tracking errors will also asymptotically converge to
zero for all . However, it is impossible since .
To overcome the uncertainty from initial state errors, let be the upper bound on , i.e.,
, and define a new error function as follows:
(3) In (3), is the saturation function defined as
and is the width of the boundary layer which is
time-varying depending on time , but not related to the
iteration number . Now will play the main role
in our controller design since it can be easily shown that
. If and
is small, then the learning performance will be satisfied since . To find the approach for the controller design later, we first derive the time derivative of as follows:
(4)
where is the notation for the sign function. If the nonlinear
functions and are completely known, we
can define the certainty equivalent controller as
(5) with the positive constant the same as that in (3). Then substi-tuting (5) into (4), it yields
(6)
since .
This implies for all and since
. However, and are in general
unknown or only partially known. Hence, the result of (6) can not be achieved. According to the certainty equivalent controller (5), (4) can only be rewritten as
Remark 1: Equation (7) will be considered as the error
equa-tion for our controller design later. The main control objective will be achieved if we can drive to zero for all
as learning iterations are large enough. It is emphasized that the careful choice of boundary layer with is quite dif-ferent from traditional approaches for variable structure control and will be very important to ILC analysis. Also different from typical adaptive control design, the control purpose of ILC is to
ensure , but not
. In the next step, an ORFNN-based DAILC will be proposed to compensate for both of the unknown certainty equivalent con-troller and the ORFNN approximation error in order to achieve the control purpose.
Step 2: Consider the ORFNN given in Appendix A and
Ap-pendix B, the proposed ORFNN-based DAILC is designed as
(8)
with
(9)
(10)
where are the learning gains,
are the network parameter vectors and is the robust control parameter vector. In this controller, is designed to compensate for the un-known certainty equivalent controller and approximation error given in (35). is a feedback stabilization com-ponent which will be clear in later convergence and stability
analysis. Now if we do not consider the effect of in this moment and substitute (9) into (7), we have
(11) by using the results of (35) and (36) given in Appendix B.
Remark 2: Since the optimal parameters , and for an optimal approximation are generally unknown,
the weights of ORFNN, and
pa-rameters at time of the th iteration will be tuned via some suitable adaptive laws between successive iteration. A set of stable adaptation algorithms is necessary to update the parameters such that closed loop stability is guaranteed and learning performance is improved.
Step 3: The adaptation algorithms for control parameters
, and at the (next)
th iteration are given as
(12) (13) (14) (15) (16) (17) (18) (19)
(20) (21) where proj denotes the projection mechanism
with being the upper bound of ( belongs to an element
of ).
Remark 3: According to the projection algorithm, it is noted
that the parameter errors will be bounded for all iterations and for all . The adaptation process is realized between successive iteration so that the convergence could be guaranteed along the iteration domain. This is the main reason to develop an ORFNN with iteration domain delay for our control task. Due to this ORFNN, the network parameters at the th iteration will not only depend on the parameters and system information at the th iteration, but also depend on those at the th,
th iterations.
Before analyzing stability and convergence of the ORFNN-based DAILC system, we summarize the design procedures as follows.
D1) Construct the control function as in (2) and error function as in (3).
D2) Design the controller as in (8) with the iterative learning components as in (9) and (10). The fuzzy rule
base of the ORFNN in is given by
D3) Update the control parameters
and for the next iteration by using the adaptation algorithms as in (12)–(21).
III. ANALYSIS OFSTABILITY ANDCONVERGENCE
Define the projected parameter error as
and unprojected parameter error as where
. Then we have , and
. Furthermore, it is easy to show by subtracting the optimal control gains on both sides of (12)–(16), such that
The main results about the closed-loop stability and learning convergence for our proposed ORFNN-based DAILC are now shown in the following theorem.
Main Theorem: Consider the nonlinear system (1) satisfying
the assumptions A1)–A3). If we design the ORFNN-based DAILC following the design steps D1)–D3), and define , then the following facts will hold.
t1) .
t2)
.
t3) All adjustable control parameters
and internal signals are bounded
and .
t4) Let be the positive constant such that is
still a Hurwitz polynomial. Then there exists a constant such that for all
(22) (23)
Proof: In this proof, the time argument inside the
integra-tion will be omitted for the notaintegra-tion brevity unless otherwise specified.
t1) Define the cost functions of performance as
(24) In order to further analyze (24), we integrate (11) over time
in-terval , as follows:
(25) After some direct manipulations on (25) and using , we can find
(26) The feedback stabilization component in step 2 is now clear if we substitute (10) and (26) into (24), and show that
(27) Thus, we have
for any iteration . Note that is bounded
due to projection algorithms (17)–(21) and con-verges to some positive function since is positive defi-nite and monotonically decreasing by the fact of (27). Hence,
converges to zero and
Therefore, we have ,
according to assumption A1) that for all
and . This proves t1) of the main theorem.
t2) The boundedness of at each iteration over can be concluded from (3) because is always bounded and the
bound of will satisfy
This proves t2) of the main theorem.
t3) Boundedness of implies boundedness of
. Together with the fact that all the ad-justable parameters are bounded, t3) is guaranteed.
t4) To find the learning performance of each state tracking error
at the final iteration, we consider the following state–space equation:
where
..
. ... ... . .. ...
.. .
by using assumption (A2) and the definition of control function in (2). Solution of (28) in the time domain is given by
(29)
where the state transition matrix satisfies
for some suitable positive constant . Taking norms on (29), it yields
which shows (22). Finally, tracking performance of given in (23) can be easily found by using (2). This concludes t4) of the main theorem.
Remark 4: In the main theorem, we show that
converges to zero as , and hence,
. Since the initial state errors (or equivalently, ) may be large and the requirement of is necessary, it is not practical to set the boundary
layer as a constant, i.e., , when
is large. Actually, should be as small as possible because
. If is small enough, the learning
performance of each , which is directly
related to , can be guaranteed. This is the main reason for choosing a time varying boundary layer
which will decrease along the time axis. In this design, the parameter plays an important role of reducing tracking error
at the final iteration. In general, and hence,
can be small if is large.
Remark 5: In addition to the parameter , the design
param-eters and , defined as the learning gains, play
another important roles in this ORFNN-based DAILC system. In general, it is only required to set these gains as positive con-stants. Due to the result shown in (27), the convergent speed of
and will increase if , and are large.
The effects of design parameters and learning gains on the improvement of learning performances will be clearly shown in the following simulation examples.
IV. SIMULATIONEXAMPLES
Example 1: Consider the inverted pendulum system [16]
with a state equation of
where and denote the angular displacement (rad)
and velocity (rad/s) of the pole, respectively. In this simulation,
the plant parameters are set as m/s kg,
kg, and m. It is assumed that the angular dis-placement is limited to rad (about 85 ) so that the
input gain satisfied , i.e.,
assump-tion A1) in Secassump-tion II is valid. The control objective is to control
the state to track the desired trajectory
for . The design
steps are given in the following.
D1) The control and error functions are set to be and
with .
D2) Design the controller as in (8) with the two
itera-tive learning control components and as
in (9) and (10), respectively. There are four fuzzy rules for the ORFNN in with the initial parameters are chosen as
for all . Besides, the initial value of in
the robust learning component (9) is chosen as
for all .
D3) Finally, the parameters
and are updated for the next iteration by using the projection type adaptation algorithms (12)–(21). In general, the upper bounds on the optimal parameters for an arbitrary certainty equivalent control function
Fig. 1. (a)sup js (t)j versus iteration j; 3 for learning gains = 10 and for learning gains = 50. (b) s (t) (solid line) and (t); 0 (t) (dotted lines) versus timet; k = 10; learning gains = 50. (c) s (t) with k = 10 (dotted line) ands (t) with k = 20 (solid line) versus time t. (d) x (t) (solid line) andx (t) (dotted line) versus time t; k = 10 and learning gains = 50. (e) x (t) (solid line) and _x (t) (dotted line) versus time t; k = 10 and learning gains = 50. (f) u (t) versus time t; k = 10 and learning gains = 50.
are not easy to estimate. For real implementation of this DAILC, suitable values of the upper bounds are usually selected as large as possible. In this simulation, these upper bounds are all set to be 10.
To begin this simulation, the parameters of and are chosen as and . We assume there always exist fixed initial displacement and velocity errors at the beginning of each
itera-tion. That is, for all so that
and . Since
, the initial value of
is chosen as . Fig. 1(a) shows
the supremum value of versus iteration with two
dif-ferent learning gains and
. The asymptotic
con-vergence of clearly proves the technical
re-sult t1) of the main theorem. In addition, it is found that faster convergent speed is achieved by larger learning gains as com-mented in Remark 5. In order to demonstrate t2) of the main theorem, we show the trajectory of for the fifth iteration in Fig. 1(b), where the trajectory of is confined between and . This fact not only satisfies t2), but also implies that the transient response of in time domain can be improved
by increasing since . We show the
results of for and in Fig. 1(c), respectively.
The nice learning performance of directly gives the con-tribution to the tracking of each state as shown in t4) of the main theorem. The actual tracking behaviors of both states at the fifth
iteration for and are
shown in Figs. 1(d) and (e), respectively. Finally, the bounded control input is demonstrated in Fig. 1(f).
Remark 6: Theoretically, it is more interesting and attractive
if all the free parameters of ORFNN can be tuned during the learning process. The price we pay is the complexity of the con-troller and adaptation law. Actually, as with most of the works, it is easy to simplify our learning controller such that only con-sequent parameters are updated during learning processes. That is, the basis functions of the ORFNN are fixed. This will dras-tically reduce the complexity of the controller. We have simu-lated example 1 again by using this approach. It is interesting to find that the learning system still works well, except the learning speed is a little slower than our proposed scheme. This observa-tion leads to a suggesobserva-tion for practical implementaobserva-tion. A smart realization is to use an ORFNN with only consequent parameter adaptation; then choose a larger adaptation gain such that a rea-sonable convergent speed can be guaranteed.
Example 2: The typical Chua’s chaotic circuit is a nonlinear
oscillator circuit, which displays very rich bifurcation and chaotic phenomena. In this example, we adopt the transformed state equation of Chua’s circuit in [32]
As Chua’s chaotic circuit is sensitive to the initial dition, it is a challenge for an iterative learning con-trol problem. The concon-trol objective is to concon-trol the
state vector to track
the desired trajectory
for . The design
steps of the ORFNN-based DAILC are roughly described as follows.
D1) Define where
, and
with .
D2) We choose eight fuzzy rules for the ORFNN in in this case. The initial parameters of the controller are chosen as
for all .
D3) The projection-type adaptation algorithms are adopted to update the parameters with upper bounds on the op-timal parameters being 10.
Due to the chaotic feature of Chua’s circuit, we espe-cially study the effect of varying initial state errors on the learning performance in this example. Also, the magnitude
Fig. 2. (a)sup js (t)j versus iteration j. (b) s (t) (solid line) and (t); 0 (t) (dotted lines) versus time t. (c) x (t) (solid line) and x (t) (dotted line) versus timet. (d) x (t) (solid line) and _x (t) (dotted line) versus time t. (e)x (t) (solid line) and _x (t) (dotted line) versus time t. (f) u (t) versus time t.
of initial errors are chosen to be large to demonstrate the robustness of the learning system. All the other design parameters of this DAILC are fixed. For example, we let
, and .
Five different initial state values are given at each iteration as
and .
Hence, the initial value of is selected as
. As in example 1, we first show the supremum value of with respective to iteration in Fig. 2(a) and observe the asymptotic convergence given in (t1) of the main theorem. The learning process is almost completed after the fourth iteration since the value is less than . We demonstrate in Fig. 2(b) to prove that
the trajectory of satisfies .
Fig. 2(c)–(e) are the comparisons between system states
and desired states . It
is emphasized that the proposed ORFNN-based DAILC can achieve a successfully iterative learning control objective even the variable and large initial state errors exist. The bounded control input at fifth iteration is finally shown in Fig. 2(f).
V. CONCLUSION
For adaptive fuzzy control [16] of nonlinear systems, it is usu-ally classified into two categories: indirect adaptive control and direct adaptive control. The fuzzy system is used to describe the plant knowledge for indirect scheme or control knowledge for direct scheme, respectively. This kind of control strategy was re-cently applied to an iterative learning control problem where the control domain contains both typical time domain and learning iteration domain. Such a control problem is quite different from the traditional control task along time axis. For example, the initial state errors become a critical robustness issue for the ILC
Fig. 3. Configuration of ORFNN.
design. Up to now, the fuzzy system or neural-network-based AILC are all indirect schemes [21], [22]. In this paper, a direct scheme is first adopted. In addition to a simpler control struc-ture, there are three major contributions in this paper compared with the related works [21], [22]. First, the robustness issue of variable initial state errors is solved by using a technique of time-varying boundary layer. Second, the nonlinear input gain can be allowed to be totally unknown before controller design. Third, a new ORFNN is proposed to compensate for the unknown certainty equivalent controller. The ORFNN ap-proximation error is also derived such that we can design an adaptation law to update all the ORFNN parameters. In sum-mary, this paper not only extends the study and application of fuzzy adaptive control to iterative learning control problems, but also provides some new design techniques. Rigorous anal-ysis of the learning system is given and all the technical results are demonstrated by computer simulations for an inverted pen-dulum system and Chua’s chaotic circuit. For practical applica-tions, this approach can be applied to repeatable tasks of control systems such as typical rotational servo motors and linear syn-chronous motors [29].
APPENDIX A STRUCTURE OF THEORFNN
The configuration of the proposed ORFNN is shown in Fig. 3, which consists of input linguistic variables, input term nodes, rule nodes, recurrent nodes and output nodes. Herein, we assume that is the number of fuzzy sets of the
th input variable . Therefore,
and there are nodes in this proposed ORFNN.
It is noted that dependency on the iteration index “ ” and time index “ ” will be omitted unless emphasis on the temporal or iterative relationships is required. The layered operation of the ORFNN is briefly described below.
Layer 1 (Input Layer): Each node in this layer represents an
input linguistic variable, which only transmits input value to the next layer directly. For the th input node,
Layer 2 (Premise Layer): Each node in this layer represents a
membership function. In this layer, there are two kinds of nodes whose inputs come from input layer and output layer, respec-tively. For the first kind of nodes, the Gaussian membership function is adopted as a membership function such that the th term node of the th input linguistic variable
will be represented as
where and are the mean and
variance, respectively. For the second kind of nodes, another Gaussian membership function is used for the th recurrent node as follows:
where is the recurrent weight and denotes the delay
of . We will discuss later.
Layer 3 (Rule Layer): Each node in this layer represents the
firing strength of a fuzzy rule. The dynamical fuzzy reasoning is performed by product of the input signals of the th rule node, which is represented as
where is the th input to the th rule node of
layer 3.
Layer 4 (Output Layer): Each node in this layer represents
an output node, which is the consequence with respect to the th output variable from each rule. For the th output node,
where and the link weight
is the action strength of the th output associated with the th rule.
Hence, this ORFNN performs a dynamic fuzzy reasoning in the traditional fuzzy rule form of
For the special feature of iterative learning control, we design
the delay mechanism as where and
denote time index and iteration index, respectively. Thus, the overall representation from input in layer 1 to the th output
in layer 4 is then given by
(30)
By using the similar techniques in [33], it is easily shown that each output of the ORFNN will depend not only on past net-work inputs, but also on past netnet-work outputs. Therefore, the ORFNN is a dynamic mapping function, which has the global feedback structure. Besides, this ORFNN has a smaller number of network parameters than the RFNN in [27] or TRFN in [30].
For example, if , and , then the network
parameters for FNN in [25], RFNN in [27] and TRFN in [30] are 192, 256, and 448, respectively. In addition, if we assume for our proposed ORFNN such that same condition can be satisfied, then we have only 84 pa-rameters in this ORFNN.
APPENDIX B
THEORFNNAS AUNIVERSALAPPROXIMATOR
From the representation (30), a multi-input single-output ORFNN can be further described in a matrix form as follows:
(31) with
In fact, the ORFNN (30) or (31) can be easily shown that it is an universal approximator. That is, for any real
continuous nonlinear function on a compact set
, and arbitrary , there exists optimal
constants and such that the functional
ap-proximation error between the optimal ORFNN
and will satisfy
. The detailed proof of universal approximation theorem for this ORFNN can be easily derived by using the
similar techniques in [27]. However, the weights of ORFNN are often tuned via suitable adaptation laws in the adaptive schemes since it is not easy to get the optimal ones for the controller design. In the following we will show how the approximation
error can be
expressed in a linearly parameterized form modulo a residual term. This enables us to tune all the network parameters via a uitable adaptive law. First, define the estimation errors of consequent, premise and recurrent parameter vectors as . For simplicity, define
and . Then
the functional approximation error will satisfy
(32) In order to deal with , we use the Taylor series expansion
of at as follows:
(33)
where denotes a sum of the high-order terms
of the argument in a Taylor series expansion and
and
are derivatives of with
respec-tive to and at . In other
words
This implies that
(34)
Substituting (34) into (32), we have
(35)
Since and are bounded
due to the bounded properties of Gaussian membership function and its derivative, it is easy to show that in (34) will satisfy
where are some bounded constants. On the other
hand, by the fact of
and , we can find that
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Ying-Chung Wang was born in Taipei, Taiwan,
R.O.C., in 1972. He received the B.S. degree in electronic engineering and the M.S. degree in mechatronic engineering from Huafan University, Taipei, in 1996 and 1998, respectively, and the Ph.D. degree in electrical and control engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2003.
He is currently a Postdoctoral Fellow at the Institute of Information Science, Academia Sinica, Taiwan. His main research interests are fuzzy systems, neural networks, fuzzy neural systems, iterative learning control, nonlinear control, adaptive control, and robot control.
Chiang-Ju Chien was born in Taiwan, R.O.C.,
in 1963. He received the B.S. degree in nuclear engineering from National Tsing Hua University, Hsinchu, Taiwan, in 1985 and Ph.D. degree in elec-trical engineering from National Taiwan University, Taipei, in 1992.
Since 1993, he has been with the Department of Electronic Engineering, Huafan University, Taipei, where he is currently an Associate Professor and the Chairman of the Department Electronic Engineering. His present research interests include iterative learning control, adaptive control, fuzzy-neuro systems, and control circuit design.
Dr. Chien is a member of the Chinese Automatic Control Society.
Ching-Cheng Teng was born in Taiwan, R.O.C.,
in 1938. He received the B.S. degree in electrical engineering from National Cheng Kung University, Taiwan, in 1961.
From 1991 to 1998, he was the Chairman of the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu, Taiwan. He is currently a Professor of the Department of Elec-trical and Control Engineering, National Chiao Tung University. His research interests includeH op-timal control, signal processing, and fuzzy neural sys-tems.
