A Hybrid Adaptive Scheme of Fuzzy-Neural Iterative Learning ...

A Hybrid Adaptive Scheme of Fuzzy-Neural Iterative Learning Controller for Nonlinear Dynamic Systems ∗ Y.-C. Wang1, C.-J. Chien2, and D.-T. Lee1 1

Institute of Information Science, Academia Sinica, 115, Nankang, Taipei, Taiwan. 2

Corresponding Author Department of Electronic Engineering, Huafan University, 223, Shihtin, Taipei County, Taiwan. E-mail : [email protected], Fax : +886-2-86606675

Key words : Iterative learning control; adaptive control; initial state errors; fuzzy neural network; nonlinear plants

Abstract To deal with an iterative learning control (ILC) problem of unknown nonlinear systems with initial state errors and state dependent input gain, a hybrid adaptive scheme of fuzzy neural iterative learning controller is presented in this work. A new time-varying boundary layer with variable initial layer width is first introduced to design an error function in order to overcome the uncertainty from initial state errors. The hybrid adaptive learning controller is then constructed based on the above error function. This hybrid learning controller combines the features of an indirect and a direct adaptive learning control such that the tradeoff between plant knowledge and control knowledge can be realized by adjusting a weighting factor. As in general the nonlinearities of the dynamic system are unknown, a fuzzy neural network (FNN) is applied for the compensation of nonlinearities and certainty equivalent controller. The linguistic descriptions about the plant and linguistic control rules are appended into the indirect and direct component, respectively. Since the optimal parameters of the FNN for a minimum approximation error are generally unavailable, some stable adaptive algorithms are derived along the iteration axis to search for suitable parameter values. It is shown that the proposed learning system will guarantee the boundedness of all the internal signals in time domain and ensure the convergence of learning error along iteration domain, respectively. Moreover, the norm of tracking error vector at each time instant will asymptotically converge to a tunable residual set as iteration goes to infinity even the initial state errors are varying and large. ∗

This work is supported by the National Science Council, R.O.C., under Grant NSC90-2213-E-211-002

1

1

Introduction

Constructive nonlinear control methods for nonlinear dynamic systems have been a vast developing area and achieved a lot of substantial results. However, most of the control strategies were developed for a typical infinity time tracking or regulation control purposes. It may not be suitable to such control tasks as to achieve perfect tracking performance over a finite time interval under repeatable control environment. For a repeatable control problem, iterative learning control (ILC) [1] has gained a large amount of interest in the recent years. It is a method of control that feeds the system inputs for a specific task repetitively and uses the measured system response to evaluate the quality or goodness of input. Example applications include robotics and manufacturing where a specified tracking control task is to be performed repeatedly. In fact, several contributions [2]-[7] have been made since the work of [1], toward improving ILC performance and relaxing ILC design constraints. In general, a restrict global Lipschitz condition is required for traditional D-type or P-type iterative learning control of nonlinear dynamic systems. To solve this difficulty, substantial efforts by using adaptive iterative learning control (AILC) strategies have been reported in in [8]-[15] for broader applications to uncertain robot manupulators [8]-[11], non-Lipschitz nonlinear systems [12]-[14], or high relative degree nonlinear systems [15]. Most of the adaptive learning controllers are designed based on the fact that the nonlinearities are linearly parameterizable. However, it is a difficulty to have a linearly parameterized nonlinearity for a general nonlinear function. To this end, fuzzy system or neural network based controller has become an effective approach for adaptive control of nonlinear systems if the nonlinearity can not be linearly parameterizable as in [8]–[15]. In [16]–[17], multilayer feedforward neural networks or recurrent neural networks were used for adaptive control design. On the other hand, fuzzy system, Gaussian neural network and fuzzy neural network are another good choices since they can be expressed as a series expansion of basis functions [18]–[24]. Recently, the concept based on fuzzy system, neural network or recurrent fuzzy neural network design was applied to the iterative learning control of nonlinear dynamic systems [25]-[27] such that the Lipschitz condition in a typical ILC problem can be relaxed. By applying the fuzzy system approximation technique, an adaptive nonlinear compensation ILC and a fuzzy system based AILC were presented in [25] and [26], respectively. Since the fuzzy system and neural network are employed to model the nonlinear plant, both schemes can be considered as an indirect AILC (IAILC) similar to that defined in [19]. On the contrary, a direct AILC (DAILC) is proposed in [27] where a recurrent fuzzy neural network is introduced for compensation of the unknown certainty equivalent controller. 2

As shown in [28], the hybrid adaptive fuzzy control scheme possesses both advantages of direct and indirect structures. The so called hybrid scheme uses a weighting factor to sum together the control inputs from a direct input and an indirect input. It expands the basic ability of direct controller and indirect controller to copy with dynamic systems via approximate models of the plant and controller. With the tradeoff between plant knowledge and control knowledge, the weighting factor can be adjusted. Hence, a very interesting question is to ask if it is possible to combine the fuzzy system based IAILC such as in [25, 26] and DAILC such as in [27] to form a hybrid adaptive iterative learning control (HAILC)? One of the main goals of this paper is to prove that it is feasible for a HAILC scheme. But it needs to note that the controller design and convergence analysis of AILC problem is quite different from those in typical adaptive fuzzy control. The design concept or approach for time domain hybrid adaptive fuzzy control (such as that in [28]) can not be directly applied to our problem. Actually, for a typical stable adaptive fuzzy control system, the control objective is to show that tracking error can asymptotically converge to zero as time approaches infinity. It doesn’t have to worry about any possible initial state errors. But for an iterative learning control design, a so called two-dimensional control problem, i.e., a finite time interval t ∈ [0, T ] in time domain and an infinity repetitions j ∈ {1, 2, · · · , ∞} along iteration domain is given. In general, the controller must guarantee boundedness along time domain at each iteration and asymptotic convergence through iteration domain. It requires that the tracking error should be exactly zero for all t ∈ [0, T ] at the final iteration if there is no initial state errors. Hence, a HAILC should be carefully investigated. In addition to the contribution of a new HAILC for repetitive control tasks, we show that by applying the current error to the design of parameter adaptive laws, the HAILC can be easily reduced to an IAILC or a DAILC scheme whose control structure is much simpler than those presented in [25]-[27]. With the projection mechanism in the adaptive law, it is shown that all the adjustable parameters and internal signals are bounded for all iterations. Moreover, the norm of tracking error vector at each time instant will asymptotically converge to a tunable residual set as iteration goes to infinity even the initial state errors are varying and large. This paper is organized as follows. The plant description, control objective and design concept of the proposed FNN based HAILC are presented in section 2. Analysis of stability, convergence and learning performance will be studied extensively in section 3. To demonstrate the effects of the learning controller, a repetitive tracking control of a Duffing forced-oscillation system is shown in section 4. Finally a conclusion is made in section 5.

3

2

Plant Description and Design of the HAILC Algorithm

2.1 The Nonlinear Systems and Control Objective In this paper, a class of nonlinear dynamic systems is considered as follows, x˙ j1 (t) = xj2 (t) x˙ j2 (t) = xj3 (t) .. . x˙ jn (t) = −f (X j (t)) + b(X j (t))uj (t)

(1)

where X j (t) = [xj1 (t), xj2 (t), · · · , xjn (t)]⊤ ∈ Rn×1 × [0, T ] is the state vector of the system, uj (t) is the control input, f (X j (t)) and b(X j (t)) are unknown real continuous nonlinear functions of states. Here, j ∈ {1, 2, · · · , ∞} denotes the index of iteration and t ∈ [0, T ] is the time index. The nonlinear systems to be controlled are required to perform some given control tasks repeatedly over a finite time interval [0, T ]. In other words, the control process starts at an initial states from time 0 and stops when time reaches a given time instant T . The control objective is to force the system state vector X j (t) to follow, as close as possible for all t ∈ [0, T ], a desired trajectory vector Xd (t) = (n−1)

[xd (t), x˙ d (t), · · · , xd

it is desired that lim

(t)]⊤ with possible initial resetting state error Xd (0) 6= X j (0). More precisely,

Z

T

j→∞ 0

kX j (t) − Xd (t)kdt ≤ ǫ for some small positive error tolerance bound ǫ. In

order to achieve the above control objective, the nonlinear dynamic system (1) and desired trajectory are assumed to satisfy : A1) Let the state errors ej1 (t), · · · , ejn (t) be defined as ej1 (t) = xj1 (t) − xd (t), ej2 (t) = x˙ j1 (t) − x˙ d (t), (n−1),j

· · · , ejn (t) = x1

(n−1)

(t) − xd

(t). The initial state errors at each iteration are not necessarily

zero, small and fixed, but assumed to be bounded. A2) The nonlinear functions f (X j (t)) and b(X j (t)) are bounded if X j (t) is bounded. The control gain function satisfies b(X j (t)) > 0 for X j (t) in a certain controllable compact set Ac × [0, T ] ⊂ Rn × [0, T ] and for all j ≥ 1. (n−1)

A3) The desired state trajectory vector Xd (t) = [xd (t), x˙ d (t), · · · , xd

(t)]⊤ is bounded and contained

in the compact set Ac × [0, T ]. Assumption A1) implies that the condition on initial resetting state errors can be varying and large. It is well known in the field of ILC that the issue of initial state error is a special practical robustness problem since the iterative learning controller is possibly divergent even for a small value of initial 4

error. We remark that ej1 (0), · · · , ejn (0) at the beginning of each iteration are available since the states are measurable. Assumption A2) describes that the system (1) is controllable for X j (t) ∈ Ac × [0, T ], j ≥ 1. This condition can be found in most of the researches dealing with the similar control problems in time domain [18, 22, 23, 24] or in iteration domain [25, 26, 27]. The boundedness of the desired trajectory is given in assumption A3). 2.2 A Boundary Layer with Varying Layer Width As discussed in the section of introduction, the problem of initial state error is an important robustness issue for our learning control design. To this end, we first design a control function sj (t) as a linear combination of all the state tracking errors, i.e., sj (t) = c1 ej1 (t) + c2 ej2 (t) + · · · + cn−1 ejn−1 (t) + ejn (t)

(2)

where c1 , · · · , cn−1 are the coefficients of a Hurwitz polynomial ∆(D) = Dn−1 + cn−1 Dn−2 + · · · + c1 . It is clear that if the learning controller can drive sj (t) to zero for all t ∈ [0, T ], then the state tracking errors will also asymptotically converge to zero for all t ∈ [0, T ]. However, it is impossible since sj (0) 6= 0 due to initial state errors. To solve this difficulty, let εj be the known constant satisfying |sj (0)| = |c1 ej1 (0) + c2 ej2 (0) + · · · + ejn (0)| ≡ εj , and introduce the following error function sjφ (t) similar to that in [26] as sjφ (t)

= sj (t) − φj (t)sat

!

sj (t) , φj (t) = εj e−kt , k > 0 φj (t)

(3)

where sat is the saturation function defined as sat

sj (t) φj (t)

!

=

    

if sj (t) > φj (t) if |sj (t)| ≤ φj (t) if sj (t) < −φj (t)

1 sj (t) φj (t)

−1

According to (3), it can be easily shown that sjφ (0) = 0, ∀j ≥ 1. This implies if lim sjφ (t) = 0, ∀t ∈ [0, T ] j→∞

and φj (t) is small, then the learning performance will be satisfied since lim |sj (t)| ≤ φj (t). To find the j→∞

approach for the controller design later, we first derive the time derivative of (sjφ (t))2 as follows :     d  j 2 sφ (t) = 2sjφ (t) s˙ j (t) − sgn sjφ (t) φ˙ j (t) dt

=

2sjφ (t)

(n−1 X

ci eji+1 (t)

−

(n) xd (t)

j

j

j

− f (X (t)) + b(X (t))u (t) − sgn

i=1





sjφ (t)

˙j

)

φ (t) (4)

where sgn is the notation for sign function. If the nonlinear functions f (X j (t)) and b(X j (t)) are completely known, the certainty equivalent controller can be derived as uj⋆ (t)

"

#

n−1 X 1 (n) j = f (X (t)) + x (t) − ci eji+1 (t) − ksj (t) d j b(X (t)) i=1

5

(5)

with the positive constant k the same as that in (3). Substituting (5) into (4), it yields     d  j 2 sφ (t) = 2sjφ (t) −ksj (t) − sgn sjφ (t) φ˙ j (t) dt ( ! )   sj (t) j j j j j ˙ = 2sφ (t) −ksφ (t) − kφ (t)sat − sgn sφ (t) φ (t) φj (t) 

2

= −2k sjφ (t) 

2



= −2k sjφ (t) where we use the fact that sjφ (t)sat



sj (t) φj (t)





− 2|sjφ (t)| φ˙ j (t) + kφj (t) 

(6)



= sjφ (t)sgn sjφ (t) = |sjφ (t)|. This implies sjφ (t) = 0 for all

t ∈ [0, T ] and j ≥ 1 since sjφ (0) = 0. However, f (X j (t)) and b(X j (t)) are in general unknown or only partially known. 2.3 The FNN Based Hybrid Adaptive Iterative Learning Controller Based on the error function sjφ (t), the basic structure of the proposed FNN based HAILC will be defined as follows : uj (t) = αujI (t) + (1 − α) ujD (t) , α ∈ [0, 1]

(7)

where ujI (t) and ujD (t) are the IAILC component and DAILC component of the HAILC, respectively. Obviously, if the plant knowledge is more important and reliable than the control knowledge, a larger α can be chosen, otherwise, a smaller α should be used. In order to illustrate the main idea of design for ujI (t) and ujD (t), we first substitute (7) into (4) and use the certainty equivalent controller (5) to yield that  2 n o d  j 2 sφ (t) = −2k sjφ (t) + 2αsjφ (t)b(X j (t)) ujI (t) − h(X j (t)) − g(X j (t))aj (t) dt n o

+ 2(1 − α)sjφ (t)b(X j (t)) ujD (t) − uj⋆ (t)

where aj (t) = −

n−1 X

(8)

(n)

ci eji+1 (t) + xd (t) − ksj (t), h(X j (t)) = f (X j (t))/b(X j (t)), g(X j (t)) = 1/b(X j (t)).

i=1

In order to overcome the unknown nonlinear functions f (X j (t)), b(X j (t)) (or equivalently, the unknown nonlinear functions h(X j (t)), g(X j (t))), and the certainty equivalent controller uj⋆ (t), two FNNs (4)

(4)

(4)

Oh (X j (t), Whj (t)) and Og (X j (t), Wgj (t)) are used in ujI (t) and another FNN OD (X j (t), WDj (t)) is applied in ujD (t). In this paper, we adopt a four-layer FNN (as shown in Figure 1), which consists of input layer, premise layer, rule layer and output layer, respectively. The FNN performs a fuzzy reasoning from an input linguistic vector X = [x1 , · · · , xn ]⊤ ∈ IRn×1 in input layer to an output variable O(4) ∈ IR in output layer. The FNN O(4) (X, W ) which is an universal approximator can be further written in a matrix form as follows : O(4) (X, W ) = W ⊤ O(3) (X) 6

(9)

where W = [w1 , w2 , · · · , wM ]⊤ ∈ IRM ×1 is the consequent parameter vector in output layer and (3)

(3)

(3)

O(3) (X) = [O1 (X), O2 (X), · · · , OM (X)]⊤ ∈ IRM ×1 is the fuzzy basis function vector in rule layer. (3)

The elements Oℓ (X) is determined by the selected input membership functions in premise layer. Note (3)

that 0 < Oℓ (X) ≤ 1, ℓ = 1, · · · , M with M being the numbers of rule nodes in rule layer. All the three FNNs will take the form of (4)

(3)

Oh (X j (t), Whj (t)) = Whj (t)⊤ Oh (X j (t))

(10)

Og(4) (X j (t), Wgj (t)) = Wgj (t)⊤ Og(3) (X j (t))

(11)

(4)

(3)

OD (X j (t), WDj (t)) = WDj (t)⊤ OD (X j (t)) (3)

(3)

(12)

(3)

where Oh (X j (t)) ∈ IRMh ×1 , Og (X j (t)) ∈ IRMg ×1 , OD (X j (t)) ∈ IRMD ×1 are the fuzzy basis function vectors, Whj (t) ∈ IRMh ×1 , Wgj (t) ∈ IRMg ×1 , WDj (t) ∈ IRMD ×1 are the corresponding consequent parameter vectors with Mh , Mg , MD being the numbers of rule nodes . It is well known that the FNNs (10), (11) and (12) can uniformly approximate real continuous nonlinear functions h(X j (t)), g(X j (t)), uj⋆ (t) on a compact set Ac ⊂ Rn×1 . An important aspect of the above approximation property is that there exist optimal weights Wh∗ , Wg∗ , WD∗ such that the function approximation er(4)

(4)

(4)

rors between the optimal Oh (X j (t), Wh∗ ), Og (X j (t), Wg∗ ), OD (X j (t), WD∗ ) and functions h(X j (t)), g(X j (t)), uj⋆ (t) can be bounded by prescribed constants ǫ∗h , ǫ∗g , ǫ∗D on the compact set Ac . More (4)

(4)

precisely, if we let h(X j (t)) = Oh (X j (t), Wh∗ ) + ǫh (X j (t)), g(X j (t)) = Og (X j (t), Wg∗ ) + ǫg (X j (t)), (4)

and uj⋆ (t) = OD (X j (t), WD∗ ) + ǫD (X j (t)), then the approximation errors will satisfy |ǫh (X j (t))| ≤ ǫ∗h , |ǫg (X j (t))| ≤ ǫ∗g , |ǫD (X j (t))| ≤ ǫ∗D , ∀X j (t) ∈ Ac . O ( 4 ) (t )

∑

∏

G

G

•••••

G

G

∏

Layer 4 (Output Layer)

•••••

•••••

G

Layer 3 (Rule Layer)

∏

G

G

Layer 1 (Input Layer)

•••••••••••••

x1 (t )

•••••••••••••

x n (t )

Figure 1 : Structure of the FNN 7

Layer 2 (Premise Layer)

Based on the FNNs given in (10) and (11), we propose the IAILC component ujI (t) as follows : ujI (t)

=

(3) Whj (t)⊤ Oh (X j (t))

+

Wgj (t)⊤ Og(3) (X j (t))aj (t)

!

sj (t) − sat θ j (t)(1 + |aj (t)|) φj (t) I

(13)

Similarly, the DAILC component via the FNN in (12) is given by ujD (t)

=

(3) WDj (t)⊤ OD (X j (t))

!

sj (t) − sat θ j (t) φj (t) D

(14)

We would like to remark that in addition to the network parameters Whj (t), Wgj (t), WDj (t), there are j another two control parameters θIj (t) and θD (t) in the controller (13) and (14). Both control pa-

rameters will be used to compensate for the network approximation error. However, the optimal ∗ for minimum weights Wh∗ , Wg∗ , WD∗ of the FNNs and the optimal parameters θI∗ = max{ǫ∗h , ǫ∗g }, θD

approximation errors are generally unknown. In order to see how the adaptive learning system can guarantee both time domain stability and iteration domain convergence, we define the parameter erf j (t) = W j (t) − W ∗ , W f j (t) = W j (t) − W ∗ , W f j (t) = W j (t) − W ∗ , θej (t) = θ j (t) − θ ∗ and rors as W g g g D I h D D I I h h

j j ∗ where θ ∗ = max{ǫ∗ , ǫ∗ } and θ ∗ = ǫ∗ . Then, substituting (13) and (14) into (8), θeD (t) = θD (t) − θD I h g D D

we have

1 d  j 2 s (t) b(X j (t)) dt φ  2 2k j = − s (t) b(X j (t)) φ f j (t)⊤ O (3) (X j (t)) + W f j (t)⊤ O (3) (X j (t))aj (t) − ǫh (X j (t)) − ǫg (X j (t))aj (t) + 2αsjφ (t) W g g h h

sj (t) − sat φj (t)

+ 2(1 − ≤ −

!

θIj (t)(1

α)sjφ (t)

j

!

+ |a (t)|)

f j (t)⊤ O (3) (X j (t)) − ǫD (X j (t)) − sat W

 2 2k j s (t) b(X j (t)) φ

D

D

!

!

sj (t) θ j (t) φj (t) D

(3)

f j (t)⊤ O (X j (t)) + 2αsj (t)W f j (t)⊤ O (3) (X j (t))aj (t) + 2αsjφ (t)W g g h h φ 



− 2α|sjφ (t)| θIj (t)(1 + |aj (t)|) − θI∗ (1 + |aj (t)|) (3)

f j (t)⊤ O (X j (t)) − 2(1 − α)|sj (t)|(θ j (t) − θ ∗ ) + 2(1 − α)sjφ (t)W D D D φ D

≤ −

 2 2k j s (t) b(X j (t)) φ 

(3)







f j (t)⊤ sj (t)O (X j (t)) + 2αW f j (t)⊤ sj (t)O (3) (X j (t))aj (t) + 2αW g g h φ h φ

− 2αθeIj (t)|sjφ (t)|(1 + |aj (t)|) 

(3)



f j (t)⊤ sj (t)O (X j (t)) − 2(1 − α)θej (t)|sj (t)| + 2(1 − α)W D D D φ φ

8

(15)

Since the optimal parameters Wh∗ , Wg∗ , WD∗ for a minimum function approximation and the optimal ∗ for error compensations are in general unknown or only partially known, control parameters θI∗ , θD j a set of stable adaptation algorithms for the estimated value Whj (t), Wgj (t), WDj (t) and θIj (t), θD (t) is

necessary to update the parameters such that closed loop stability can be guaranteed and learning performance can be improved as the iteration is large enough. Let γ be a positive constant, the adaptive algorithms for the design parameters are given as follows : (3)

j−1 Whj (t) = Wh,p (t) − γsjφ (t)Oh (X j (t))

(16)

j−1 Wgj (t) = Wg,p (t) − γsjφ (t)Og(3) (X j (t))aj (t)

(17)

j−1 θIj (t) = θI,p (t) + γ|sjφ (t)|(1 + |aj (t)|)

(18)

(3)

j−1 WDj (t) = WD,p (t) − γsjφ (t)OD (X j (t))

(19)

j j−1 θD (t) = θD,p (t) + γ|sjφ (t)|

(20)

and 



j−1 (t) = proj Whj−1 (t) Wh,p





j−1 Wg,p (t) = proj Wgj−1 (t)









j−1 θI,p (t) = proj θIj−1(t)





j−1 WD,p (t) = proj WDj−1 (t) j−1 j−1 θD (t) = proj θD (t)

= =

=

h

i⊤

j−1 j−1 proj(wh,1 (t)), · · · , proj(wh,M (t)) h

(21)

j−1 j−1 proj(wg,1 (t)), · · · , proj(wg,M (t)) g

(22)

h

i⊤

h

i⊤

j−1 j−1 proj(wD,1 (t)), · · · , proj(wD,M (t)) D

(23) (24) (25)

where proj denotes a projection mechanism : 

proj z

j−1



(t)

   z¯

if z j−1 (t) ≥ z¯ = −¯ z if z j−1 (t) ≤ −¯ z   z j−1 (t) otherwise

∗ }), respectively. with z¯ being the upper bound of |z ∗ | (z ∗ belongs to an element of {Wh∗ , Wg∗ , θI∗ , WD∗ , θD o (t) = W 0 (t) = θ 0 (t) = W 0 (t) = θ 0 (t) = 0. That is, for example For the first iteration, we set Wh,p g,p I,p D,p D,p (3)

adaptive law (16) becomes Wh1 (t) = −γs1φ (t)Oh (X 1 (t)). Remark 1 : We would like to point out the differences between the adaptive laws in this work and those in the relating papers. In this work, the parameters are updated by the error function sjφ (t) at jth iteration (the current error) instead of sj−1 φ (t) at (j − 1)th iteration (the previous error). The previous error based adaptive laws [25]-[27] lead to a more complex control structure, especially when compared with the controllers of (13) (the indirect case) and (14) (the direct case) given in this paper.

9

3

Analysis of Error Convergence and Learning Performance In this section, we will analyze the time domain stability and iteration domain convergence for the

f j (t) = W j (t)− W ∗ , proposed learning system. At first, we define the projected parameter errors as W h h,p h,p

f j (t) = W j (t) − W ∗ , θej (t) = θ j (t) − θ ∗ , W f j (t) = W j (t) − W ∗ and θej (t) = θ j (t) − θ ∗ , then W g,p g,p g I D D I,p I,p D,p D,p D,p D,p

f j (t)⊤ W f j (t)⊤ W f j (t)⊤ W f j (t)⊤ W f j (t) ≤ W f j (t), W f j (t) ≤ W f j (t), (θej (t))2 we have inequalities of W g,p g,p g g h,p h,p h h I,p

f j (t)⊤ W f j (t) ≤ W f j (t)⊤ W f j (t) and (θej (t))2 ≤ (θej (t))2 . Furthermore, it is easy to ≤ (θeIj (t))2 , W D,p D,p D D D,p D

show by subtracting the optimal control gains on both side of (16)–(20) that (3)

f j (t) = W f j−1 (t) − γsj (t)O (X j (t)) W h h,p φ h

(26)

f j (t) = W f j−1 (t) − γsj (t)O (3) (X j (t))aj (t) W g g,p g φ

(27)

f j (t) = W f j−1 (t) − γsj (t)O (3) (X j (t)) W D D,p D φ

(29)

j−1 θeIj (t) = θeI,p (t) + γ|sjφ (t)|(1 + |aj (t)|)

j j−1 θeD (t) = θeD,p (t) + γ|sjφ (t)|

(28)

(30)

The boundedness of internal signals at the first iteration will be established in the following lemma. Lemma : Consider the nonlinear dynamic system (1) satisfying assumptions A1)–A2). If the control task repeats over a finite time interval [0, T ] with a given desired trajectory Xd (t) which satisfies assumption A3), then the adaptive iterative learning controller (13), (14) and adaptive laws (16)–(20) 1 (t), u1 (t), at the first will ensure that all the internal signals, s1φ (t), s1 (t), Wh1 (t), Wg1 (t), θI1 (t), WD1 (t), θD

iteration are bounded. Proof : Consider (15) for j = 1. By substituting the parameter adaptive laws (16)–(20) into (15) and after some simple manipulations, we can find that for j = 1,  2 1 d  1 2 2k sφ (t) ≤ − s1φ (t) 1 1 b(X (t)) dt b(X (t)) 2α f 1 ⊤ f 1 2α f j ⊤ f 0 − W (t) Wh (t) + W (t) Wh,p (t) γ h γ h 2α f 1 ⊤ f 1 2α f 1 ⊤ f 0 − W (t) Wg (t) + W (t) Wg,p (t) γ g γ g 2α  e1 2 2α e1 e0 − θI (t) + θ (t)θI,p (t) γ γ I 2(1 − α) f 1 ⊤ f 1 2(1 − α) f 1 ⊤ f 0 − WD (t) WD (t) + WD (t) WD,p (t) γ γ 2(1 − α)  e1 2 2(1 − α) e1 e0 − θD (t) + θD (t)θD,p (t) γ γ

(31)

f 0 (t) = W 0 (t)−W ∗ = −W ∗ , W f 0 (t) = W 0 (t)−W ∗ = −W ∗ , θe0 (t) = θ 0 (t)−θ ∗ = −θ ∗ , Note that W g,p g,p g g h,p h,p h h I,p I,p I I

f 0 (t) = W 0 (t) − W ∗ = −W ∗ , and θe0 (t) = θ 0 (t) − θ ∗ = −θ ∗ are bounded for all t ∈ [0, T ] so W D,p D,p D D D,p D,p D D

10

that (31) can be rewritten as 1 d  j 2 s (t) b(X 1 (t)) dt φ  2 2k 1 ≤ − s (t) b(X 1 (t)) φ 2α f 1 ⊤ f 1 2α f 1 ⊤ ∗ − Wh (t) Wh (t) − W (t) Wh γ γ h 2α f 1 ⊤ f 1 2α f 1 ⊤ ∗ − Wg (t) Wg (t) − W (t) Wg γ γ g 2α  e1 2 2α e1 − θI (t) − θ (t)θI∗ γ γ I 2(1 − α) f 1 ⊤ f 1 2(1 − α) f 1 ⊤ ∗ − WD (t) WD (t) − WD (t) WD γ γ 2(1 − α)  e1 2 2(1 − α) e1 ∗ − θD (t) − θD (t)θD γ γ  2 2k α f1 ⊤ f1 α f1 ⊤ f1 α  e1 2 1 = − s (t) − W (t) W (t) − W (t) W (t) − θ (t) h g b(X 1 (t)) φ γ h γ g γ I (1 − α) f 1 ⊤ f 1 (1 − α)  e1 2 − WD (t) WD (t) − θD (t) γ γ  ⊤   ⊤   α f1 f 1 (t) + W ∗ − α W f 1 (t) + W ∗ f 1 (t) + W ∗ − Wh (t) + Wh∗ W W h h g g g g γ γ  2 ⊤   2 α  e1 (1 − α)  f 1 f 1 (t) + W ∗ − (1 − α) θe1 (t) + θ ∗ − θI (t) + θI∗ − WD (t) + WD∗ W D D D D γ γ γ α α α (1 − α) ∗⊤ ∗ (1 − α) ∗ 2 + Wh∗⊤ Wh∗ + Wg∗⊤ Wg∗ + (θI∗ )2 + WD WD + (θD ) (32) γ γ γ γ γ f 1 (t)⊤ W f 1 (t), − α W f 1 (t)⊤ W f 1 (t), − α (θe1 (t))2 , − (1−α) W f 1 (t)⊤ W f 1 (t), − (1−α) (θe1 (t))2 in If we omit − αγ W g g D D D h h γ γ I γ γ

the right hand side of (32), then (32) can be simplified as

1 d  j 2 s (t) b(X 1 (t)) dt φ  2 2k 1 ≤ − s (t) φ b(X 1 (t)) α α α (1 − α) (1 − α) 1 − kWh1 (t)k2 − kWg1 (t)k2 − |θI1 (t)|2 − kWD1 (t)k2 − |θD (t)|2 γ γ γ γ γ α α α (1 − α) (1 − α) ∗ 2 + kWh∗ k2 + kWg∗ k2 + |θI∗ |2 + kWD∗ k2 + |θD | γ γ γ γ γ  2 2k 1 ≤ − s (t) φ b(X 1 (t)) (3)

− αγ|s1φ (t)|2 kOh (X 1 (t))k2 − αγ|s1φ (t)|2 kOg(3) (X 1 (t))a1 (t)k2 

2

− αγ|s1φ (t)|2 1 + |a1 (t)| +

(3)

− (1 − α)γ|s1φ (t)|2 kOD (X 1 (t))k2 − (1 − α)γ|s1φ (t)|2

α α α (1 − α) (1 − α) ∗ 2 kWh∗ k2 + kWg∗ k2 + |θI∗ |2 + kWD∗ k2 + |θD | γ γ γ γ γ

If we define λ1 =

inf

t∈[0,T ]

n

(3)

αγkOh (X 1 (t))k2 + αγkOg(3) (X 1 (t))a1 (t)k2 + αγ(1 + |a1 (t)|)2 11

(33)

(3)

+ (1 − α)γkOD (X 1 (t))k2 + (1 − α)γ

o

α α α (1 − α) (1 − α) ∗ 2 kWh∗ k2 + kWg∗ k2 + |θI∗ |2 + kWD∗ k2 + |θD | , γ γ γ γ γ

λ2 = then (32) will satisfy

  2  d  1 2 sφ (t) ≤ −2k s1φ (t) − b(X 1 (t)) λ1 |s1φ (t)|2 − λ2 dt

(34)

Now suppose that s1φ (t) tends to infinity. Since s1φ (0) = 0 as an initial condition, there exists a first time instant t∞ ∈ [0, T ] such that lim |s1φ (t)| = ∞ and an instant tb < t∞ such that |s1φ (t)| ≥ |s1φ (tb )| = t→t∞

λ2 λ1

∀t ∈ [tb , t∞ ]. This implies that  2 d  1 2 sφ (t) ≤ −2k s1φ (t) , ∀t ∈ [tb , t∞ ] dt

(35)

and hence |s1φ (t)| ≤ |s1φ (tb )| < ∞ for all t ∈ [tb , t∞ ] which contradicts the assumption that lim |s1φ (t)| = t→t∞

∞. Therefore, s1φ (t) is bounded ∀t ∈ [0, T ]. The boundedness of s1φ (t) concludes the boundedness of 1 (t) (by (20)) and hence, the Wh1 (t) (by (16)), Wg1 (t) (by (17)), θI1 (t) (by (18)), WD1 (t) (by (19)), θD

boundedness of s1 (t) (by (3)) and u1 (t) (by (13) and (14)).

Q.E.D.

Now we are ready to state the main results in the following theorem. Theorem. If the FNN based HAILC is applied to the nonlinear dynamic system (1) which satisfies assumptions A1)-A3), then we can guarantee that all adjustable control parameters and internal signals are bounded ∀t ∈ [0, T ] and ∀j ≥ 1. Furthermore, let E j (t) = [ej1 (t), ej2 (t), · · · ejn−1 (t)]⊤ , we have lim sj (t) j→∞ φ

= s∞ φ (t) = 0, ∀t ∈ [0, T ]

(36)

lim |sj (t)| = |s∞ (t)| ≤ φ∞ (t) = e−kt ε∞

(37)

j→∞

lim kE j (t)k = kE ∞ (t)k ≤ m1 e−λt kE ∞ (0)k + m1 ε∞

j→∞

lim

j→∞

|ejn (t)|

=

|e∞ n (t)|

≤

n−1 X

e−kt − e−λt λ−k

−kt ∞ ci |e∞ ε i (t)| + e

(38) (39)

i=1

where λ is the positive constant such that ∆(D − λ) is a Hurwitz polynomial, and m1 is a positive constant. Proof : The proof consists two parts. Part I : Prove the convergence of sjφ (t) and sj (t). Define the cost functions of performance as, j

V (t) =

1 γ

Z t(  0

f j (τ ) + W f j (τ ) + (θej (τ ))2 f j (τ )⊤ W f j (τ )⊤ W α W g g I h h

12





+ (1 − α) Vpj (t)

1 γ

=

Z t( 

f j (τ )⊤ W f j (τ ) W D D

+

j (θeD (τ ))2

)

dτ

f j (τ )⊤ W f j (τ )⊤ W f j (τ ) + W f j (τ ) + (θej (τ ))2 α W g,p g,p I,p h,p h,p

0



+ (1 − α)

f j (τ )⊤ W f j (τ ) W D,p D,p

Then we can derive by using the facts of (26) - (30) that

+

j (θeD,p (τ ))2

)



dτ

V j (t) − V j−1 (t) ≤ V j (t) − Vpj−1 (t) =

1 γ

Z t ( "

α

0







f j (τ )⊤ W f j−1 (τ )⊤ W f j (τ )⊤ W f j−1 (τ )⊤ W f j (τ ) − W f j−1 (τ ) + W f j (τ ) − W f j−1 (τ ) W g g g,p g,p h h h,p h,p

+



(θeIj (τ ))2 "

+ (1 − α)



−

j−1 (θeI,p (τ ))2



#

#)   j j−1 j j−1 ⊤ fj ⊤ f j−1 2 2 e e f f WD (τ ) WD (τ ) − WD,p (τ ) WD,p (τ ) + (θD (τ )) − (θD,p (τ )) dτ

# Z t ( "

2

2 2

j (3)

j

j j (3) j j j = −γ α sφ (τ )Oh (X (τ )) + sφ (τ )Og (X (τ ))a (τ ) + (sφ (τ )(1 + a (τ )) 0

+

"

2 (3)

+ (1 − α) sjφ (τ )OD (X j (τ )) + |sjφ (τ )|2

Z t( 0



− 2(1 − ≤

0



(3)

f j (τ )⊤ α)W D 





(3) sjφ (τ )OD (X j (τ ))

− 2(1 −

(3)



+ 2(1

)

j − α)θeD (τ )|sjφ (τ )|

dτ 



f j (τ )⊤ α)W D





(3) sjφ (τ )OD (X j (τ ))

+ 2(1

Integrating (15) over time interval [0, t], t ∈ (0, T ], it yields t 0





f j (τ )⊤ sj (τ )O (X j (τ )) − 2αW f j (τ )⊤ sj (τ )O (3) (X j (τ ))aj (τ ) − 2αW g g h φ φ h

+ 2αθeIj (τ )|sjφ (τ )|(1 + |aj (τ )|)

Z

dτ

f j (τ )⊤ sj (τ )O (X j (τ )) − 2αW f j (τ )⊤ sj (τ )O (3) (X j (τ ))aj (τ ) − 2αW g g h φ φ h

+ 2αθeIj (τ )|sjφ (τ )|(1 + |aj (τ )|) Z t(

#)

1 d j (s (τ ))2 dτ = j b(X (τ )) dτ φ

≤

Z t( 0

−

Z

(sjφ (t))2

(sjφ (0))2

2k (sj (τ ))2 b(X j (τ )) φ 

(3)

)

j − α)θeD (τ )|sjφ (τ )|

dτ

(40)

1 d(sjφ (τ ))2 j b(X (τ ))







f j (τ )⊤ sj (τ )O (X j (τ )) + 2αW f j (τ )⊤ sj (τ )O (3) (X j (τ ))aj (τ ) + 2αW g g h φ h φ

− 2αθeIj (τ )|sjφ (τ )|(1 + |aj (τ )|)

)   (3) j j j j ⊤ j e f + 2(1 − α)WD (τ ) sφ (τ )OD (X (τ )) − 2(1 − α)θD (τ )|sφ (τ )| dτ

13

(41)

which implies that Z t( 0





(3)





f j (τ )⊤ sj (τ )O (X j (τ )) − 2αW f j (τ )⊤ sj (τ )O (3) (X j (τ ))aj (τ ) − 2αW g g h h φ φ

+ 2αθeIj (τ )|sjφ (τ )|(1 + |aj (τ )|)

− 2(1 − ≤ −

Z

t

0

f j (τ )⊤ α)W D





(3) sjφ (τ )OD (X j (τ ))

2k (sj (τ ))2 dτ − b(X j (τ )) φ

Z

(sjφ (t))2

0

+ 2(1 −

)

j α)θeD (τ )|sjφ (τ )|

dτ

1 d(sj (τ ))2 b(X j (τ )) φ

(42)

where we use the fact of sjφ (0) = 0. Substituting (42) into (40), we can show that j

V (t) − V

j−1

(t) ≤ −

Z

0

t

2k (sj (τ ))2 dτ − b(X j (τ )) φ

Z

0

(sjφ (t))2

1 d(sj (τ ))2 b(X j (τ )) φ

(43)

Thus, we have Z


2

sjφ (t)

0

1 d(sj (τ ))2 ≤ V j−1 (t) − V j (t) ≤ V 1 (t) b(X j (τ )) φ

for any iteration j ≥ 1. This further implies that

Z (sj (t))2 φ 0

1 j d(sjφ (τ ))2 and sφ (t) are bounded b(X j (τ ))

∀t ∈ [0, T ] and j ≥ 1 since V 1 (t) is bounded ∀t ∈ [0, T ] by using the result of the Lemma. On the other hand, V j (t) will converge to some positive function since V j (t) is positive definite and monotonically decreasing by the fact of (43). Hence, V j (t) − V j−1 (t) converges to zero and lim

Z

j→∞ 0


2

sjφ (t)

1 d(sj (τ ))2 = b(X j (τ )) φ

Therefore, we have lim sjφ (t) j→∞

=

s∞ φ (t)

Z


2

s∞ (t) φ

0

=

1 b(X ∞ (τ ))

2 d(s∞ φ (τ )) = 0, ∀ t ∈ [0, T ]

0, ∀ t ∈ [0, T ], according to assumption A2) that

b(X j (t)) > 0. This proves (36) of the theorem. The boundedness of sj (t) at each iteration over [0, T ] can be concluded from equation (3) because φj (t) is always bounded. Furthermore, the bound of s∞ (t) will satisfy lim |sj (t)| = |s∞ (t)| ≤ φ∞ (t) = e−kt ε∞ , ∀ t ∈ [0, T ]

j→∞

This proves (37) of the theorem. It is noted that boundedness of sj (t) implies boundedness of ej1 (t), ej2 (t), · · · , ejn (t) for all t ∈ [0, T ] and j ≥ 1. Together with the fact that all the adjustable parameters are bounded due to projection mechanism on the adaptive laws, we can conclude that the control input uj (t), and hence all the internal signals are bounded. Part II : Prove the convergence of E j (t) and ejn (t). In order to investigate the tracking performance in the final iteration when (36) and (37) of the theorem are achieved, we consider the following state space equation : E˙ ∞ (t) = Ac E ∞ (t) + Bc s∞ (t) 14

(44)

where 

0  0  =  ..  . −c1

Ac

1 0 .. . −c2

0 1 .. . −c3

··· ··· .. . ···



 

0 0 0 0     ..  , Bc =  ..  s∞ (t) . .  −cn−1 1

which is constructed by the definition of sj (t) in (2). Solution of (44) in time domain is given by E ∞ (t) = Φ∞ (t)E ∞ (0) +

Z

t

0

Φ∞ (t − τ )Bc s∞ (τ )dτ

(45)

where the state transition matrix Φj (t) satisfies kΦj (t)k ≤ m1 e−λt for some suitable positive constant m1 . Taking norms on (45), it yields ∞

−λt

kE (t)k ≤ m1 e

−λt

≤ m1 e

∞

kE (0)k + m1 ∞

kE (0)k + m1

Z

t

0

Z

t

e−λ(t−τ ) kBc k|s∞ (τ )|dτ e−λ(t−τ ) e−kτ ε∞ dτ

0

≤ m1 e−λt kE ∞ (0)k + m1 ε∞

e−kt − e−λt λ−k

This concludes (38) of the theorem. Finally, the tracking performance of e∞ n (t) shown in (39) can be easily derived by using the definition of (2).

Q.E.D.

Remark 2 : Although sjφ (t) converges to zero as j → ∞, it is more important to ask if the value of sj (t) can be as small as possible for all t ∈ [0, T ]. This is because the tracking errors eji (t), i = 1, · · · , n are directly related to sj (t) as shown in (2). Since the requirement of φ∞ (0) = ε∞ is necessary and |s∞ (t)| < φ∞ (t), we have to choose a time varying boundary layer φ∞ (t) = e−kt ε∞ which will decrease along the time axis if ε∞ is large. In general, s∞ (t) and hence, e∞ i (t), i = 1, · · · , n can be small if k is large. In addition to the parameter k, there is another design parameter γ, which is defined as the learning gain. The only requirement on γ is to set it as a positive constant. But according to the technical analysis, a large γ is recommended since the convergent speed of sjφ (t) and sj (t) should be increased by a large positive γ. Remark 3 : It is interesting to note that the error convergence of sjφ (t) is still guaranteed even the projection mechanism is removed from the adaptive laws. That is, if we use the following adaptive laws (3)

Whj (t) = Whj−1 (t) − γsjφ (t)Oh (X j (t)) Wgj (t) = Wgj−1 (t) − γsjφ (t)Og(3) (X j (t))aj (t) θIj (t) = θIj−1 (t) + γ|sjφ (t)|(1 + |aj (t)|) (3)

WDj (t) = WDj−1 (t) − γsjφ (t)OD (X j (t)) j j−1 θD (t) = θD (t) + γ|sjφ (t)|

15

0 (t) = 0, then the boundedness of all the internal sigwith Wh0 (t) = Wg0 (t) = θI0 (t) = WD0 (t) = θD

nals is still ensured for the first iteration. Moreover, it can be easily shown that V j (t) − V j−1 (t) in the above theorem satisfies ineqality (43). This implies the boundedness and zero convergence of sjφ (t). The only problem is that the boundedness of control parameters can not be guaranteed in this case. However, since V j (t) is bounded for all j ≥ 1 according to (43), all the parameters j Whj (t), Wgj (t), θIj (t), WDj (t), θD (t), j ≥ 2 will at least be L2 bounded.

4

Simulation Example

In this example, we apply the proposed FNN based HAILC to the Duffing forced-oscillation system. The state equation of the Duffing forced-oscillation system [19] is given by x˙ j1 (t) = xj2 (t) x˙ j2 (t) = −0.1xj2 (t) − (xj1 (t))3 + 12 cos(t) + uj (t) It is widely known that the system is chaotic when uj (t)=0 which is a very suitable challenge for our iterative learning problem with resetting behavior. The control objective is to make the state vector X j (t) = [xj1 (t), xj2 (t)]⊤ to track as close as possible the desired trajectory Xd (t) = [xd (t), x˙ d (t)]⊤ = [sin(t), cos(t)]⊤ for all t ∈ [0, 15]. In order to verify the robustness of the proposed HAILC against varying initial state errors, we assume that the initial states of the plant take the following arbitrary values for the first five iterations : [xj1 (0), xj2 (0)]⊤ = [0.35, 0]⊤ , [0.25, 0.1]⊤ , [0.13, −0.1]⊤ , [0.15, 0.12]⊤ , [0.16, 0.5]⊤ . The controller design follows the steps in section 2. To begin with this simulation, we choose α as 0.5 since the plant knowledge and control knowledge are both unknown. The other design parameters are set to be k = 5 and c1 = 5. Since sj (0) = 5ej1 (0) + ej2 (0) = 5(xj1 (0) − sin(0)) + (xj2 (0) − cos(0)), the initial value of φj (t) is chosen as φj (0) = εj = |sj (0)| = |5ej1 (0) + ej2 (0)| = |5(xj1 (0)− sin(0))+ (xj2 (0)− cos(0))|. Figure 1(a) shows the supremum value of |sjφ (t)|, i.e., sup |sjφ (t)|, t∈[0,15]

versus iteration j with two different learning gains γ = 5 and γ = 10. The asymptotic convergence of sup |sjφ (t)| clearly proves the technical result (36) of the theorem. As commented in remark 2, we

t∈[0,15]

found in Figure 1 (a) that a faster convergent speed is achieved by a larger learning gain. In order to demonstrate (37) of the theorem, we show the trajectory of s5 (t) for the fifth iteration in Figure 1(b), where the trajectory of s5 (t) is confined between φ5 (t) and −φ5 (t). This fact not only satisfies (37) of the theorem, but also implies that the transient response of s5 (t) in time domain can be improved by increasing k since |s5 (t)| ≤ φ5 (t) = |sj (0)|e−kt . We also show the results of s5 (t) for k = 5 and k = 10 in Figure 1(c), respectively. In addition, the tracking behaviors between system states xj1 (t), xj2 (t) and 16

desired states xd (t), x˙ d (t) are shown in Figure 1 (d) and Figure 1 (e) for the 5th iteration with k = 10 and γ = 10, respectively. Finally, the bounded control input u5 (t) under γ = 10, k = 10 is demonstrated in Figure 1(f). (a)

0

(b)

10

4 2

−2

10

0 −2

−4

10

1

2

3 (c)

4

−4

5

2

1

1

0.5

0

0

−1 0

5

5

10

−2

15

0

5

(e) 40

1

20

0

0

−1

−20 0

5

15

10

15

10

15

(f)

2

−2

10 (d)

1.5

−0.5

0

10

−40

15

0

5

Figure 1 : (a) supt∈[0,15] |sjφ (t)| versus iteration j; ∗ for γ = 5 and ◦ for γ = 10. (b) s5 (t) (solid line) and φ(t), −φ(t) (dotted lines) versus time t; k = 5; γ = 10. (c) s5 (t) with k = 5 (dotted line) and s5 (t) with k = 10 (solid line) versus time t. (d) x51 (t) (dotted line) and xd (t) (solid line) versus time t; k = 10 and γ = 10. (e) x52 (t) (dotted line) and x˙ d (t) (solid line) versus time t; k = 10 and γ = 10. (f) u5 (t) versus time t; k = 10 and γ = 10.

5

Conclusion

This paper presents a fuzzy neural network based hybrid adaptive iterative learning control for a class of repeatable nonlinear dynamic systems. The initial state errors at the beginning of each iteration are allowed to be varying and large. Based on the tradeoff between plant knowledge and control knowledge, an adjustable weighting factor is used to sum together the control inputs generated by a direct adaptive iterative learning control and an indirect adaptive iterative learning control. Although the hybrid design concept is motivated from the work of time domain adaptive fuzzy control, it is quite different for an iterative learning control problem. Two iterative learning components based on fuzzy 17

neural network design are utilized to compensate for the plant nonlinearity and unknown certainty equivalent controller, respectively. A boundary layer technique varying along both time domain and iteration domain is employed to solve the problem of variable initial state errors. Based on a Lyapunov like analysis, the adaptive laws are derived for tuning the weights of the fuzzy neural network during iteration process. Stability and convergence are ensured for the resulting closed-loop system. We also show that the tracking error asymptotically converges to a tunable residual set as iteration goes to infinity and all adjustable parameters as well as the internal signals remain bounded.

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