A Generalized Iterative Learning Controller against Initial ... - CiteSeerX

A Generalized Iterative Learning Controller against Initial State Error Kwang-Hyun Park and Zeungnam Bien1

Department of Electrical Engineering Korea Advanced Institute of Science and Technology 373-1 Kusong-dong, Yusong-gu, Taejon, 305-701, Korea e-mail : [email protected]

Abstract In this paper, the previous results that the performance of iterative learning control(ILC) algorithm can be improved by adding a proportional term and/or an integral term of error in D-type ILC algorithm are generalized using an operator. Then, a sucient condition for convergence and robustness of the generalized ILC algorithm are investigated against initial state error. As a special case of the operator, a nonlinear ILC algorithm is also proposed and it is shown that the eect of initial state error can be reached to zero in a given nite time. It is shown that the bound of error reduction can be eectively controlled by tuning gains of the proposed nonlinear ILC algorithm. In order to con rm validity of the proposed algorithms, two examples are presented.

1 Introduction The iterative learning control(ILC) method has been found to be a good alternative to overcome the limitations of conventional control method against Professor to whom all the correspondences should be addressed, ( e-mail : [email protected] ) 1

1

inaccuracy in modeling and/or uncertainty of some system parameters when the required task is repetitive(Arimoto et al. 1984). In most ILC algorithms, it is assumed, however, that the initial state value of the plant is equal to that of the desired trajectory for perfect tracking, even though it is hard to set the initial state value of the plant to that of the desired trajectory exactly. In recent years, there has been a great deal of study on the robustness against initial state error(Heinzinger et al. 1989, Lee and Bien 1991, Heinzinger et al. 1992, Saab 1994). Lee and Bien (1996) showed that a proportional term of error can be positively solicited to get a better performance against initial state error. To be more speci c, consider the linear system described by (1).

x_ (t) = Ax(t) + Bu(t) y(t) = Cx(t)

(1)

Here, x 2 Rn ; u 2 Rr and y 2 Rm denote the state, the input and the output, respectively. A; B and C are matrices with appropriate dimensions and it is assumed that CB is a full rank matrix. Let xd (
) be the desired state trajectory and yd(
) be the corresponding output trajectory. Assume that xd (
) is continuously dierentiable on [0; T ]. It is shown by Lee and Bien(1996) that when the below ILC algorithm (2) is applied to the system (1), the output trajectory converges to the form in (3).

uk+1(t) = uk (t) + ; (e_k (t) ; Rek (t)) lim y (t) = yd(t) + eRt C (x0 ; xd (0)) k!1 k

(2) (3)

Here, yk (
); uk (
) and ek (
) are the output trajectory, the control input trajectory, and the output error trajectory at k-th iteration, respectively. Equation (3) shows that the eect of initial state error can be controlled by gain R of the 2

ILC algorithm, and the error asymptotically converges to zero if R is chosen such that all the eigenvalues have negative real parts. Later, Park and Bien(1999) generalized the previous result to the PID-type ILC algorithm and showed that the performance can be improved by adding an integral term. It is shown that when the ILC algorithm (4) is applied to the system (1), the output trajectory converges to the form in (5). 

uk+1(t) = uk (t) + ; e_k (t) + Q0 ek (t) + Q1 lim y (t) = yd(t) + CR eARt 0 k!1 k

t

Z

0

ek ( )d



(4) (5)

Here, 2

AR = CR =

6 4 

2

0 =

6 4

0 I ;Q1 ;Q0

I 0

3 7 5



3

I C (x0 ; xd (0)) : ;Q0 7 5

Equation (5) implies that the output trajectory can be controlled in a variety of ways by introducing the integral-term. Most of the results up to now on the ILC are in the form of linear control algorithm which is generally expressed in the following form:

uk+1(t) = uk (t) + (Lek (
)) (t); 0  t  T;

(6)

where L is a linear operator of error function ek (t), 0  t  T , between the actual output and the desired output at the k-th iteration. In general, linear control algorithms are more simple to implement and easier to analyze than nonlinear algorithms. However, a nonlinear controller may possess capability to overcome limitations of a linear controller and improve the system performance. 3

Furthermore, most of ILC algorithms have a nite time interval at each iteration. It is, however, impossible to impose such a restriction that the initial state error reaches to zero in nite time on existing ILC algorithms since the error asymptotically converges to zero(see (3) and (5)). In this paper, a generalized ILC algorithm is proposed and a sucient condition of the operator is investigated for convergence. As a special case, a nonlinear ILC algorithm, which is in the form of terminal attractor(Man et al. 1994), is proposed. It is shown that the error can be reached to zero in a nite time and can be bounded by an exponential function. In the sequel, for n-dimensional Euclidean space Rn, kxk denotes the Euclidean norm of a vector x. For the vector x = (x1 ; x2;


; xn)T , the i-th component of x is denoted as either xi , or as fxgi as appeared in section 4. For a matrix A, kAk denotes its induced matrix norm. kxk1 denotes the sup-norm de ned by

kxk1 = 1sup jxi j: in

For an n  r matrix A with components aij , kAk1 is de ned by

kAk1 = sup

r

X

1in j =1

jaij j:

For a function h : [0; T ] ! Rn and a real number  > 0, kh(
)k denotes the -norm de ned by

kh(
)k = sup e;t kh(t)k1: t2[0;T ]

Let F n be the normed function space of all h : [0; T ] ! Rn with the norm being kh(
)k. Then, for an operator P of F n into F n, kP k denotes its induced operator norm de ned by )(
)k : kP k = nsup k(kPh h2F ; h(
)6=0 h(
)k 4

As in the notation uk (t), the subscript k denotes the iteration number.

2 Generalized ILC for LTI dynamic systems In this section, the eect of initial state error for LTI systems is shown. Consider the ILC algorithm described by (7).

uk+1(t) = uk (t) + ; [e_k (t) + (Pek (
)) (t)] :

(7)

Here, P is an operator of error function ek (t), 0  t  T , which is continuous in ek (
) and can be linear or nonlinear. We rst show that, if the initial state of the system (1) can be made to be same at each iteration, as in forced resetting case, then the output trajectory is converged and can be estimated from the information about the desired output trajectory, the initial output value and the operator P .

Theorem 1 Suppose that the update law (7) is applied to the system (1) and

that the initial state at each iteration can be dierent from the desired initial state, i.e., xk (0) = x0 6= xd (0), for k = 0; 1; 2;


. If

kI ; ;CB k1   < 1

(8)

then,

lim y (t) = yd(t) ; e~(t) k!1 k where e~(t) is the solution of

e~_ (t) + (P e~(
)) (t) = 0 e~(0) = C (xd (0) ; x0 ) : 5

(9)

Proof:

Let ua(t) and xa (t) be the control input and the state that satisfy (10). Z

t

xa (t) = x0 + [Axa ( ) + Bua ( )] d 0 yd(t) ; e~(t) = Cxa (t)

(10)

Let
uk (t) = ua(t) ; uk (t)
xk (t) = xa (t) ; xk (t): It follow from (7) and (10) that
uk+1(t) =
uk (t) ; ; [e_k (t) + (Pek (
)) (t)] =
uk (t) ; ; [C
x_ k (t) ; (P e~(
)) (t) + (P (C
xk (
) + e~(
))) (t)] = (I ; ;CB )
uk (t) ; ;CA
xk (t) ;; [(P (C
xk (
) + e~(
))) (t) ; (P e~(
)) (t)] :

(11)

Taking the norm k
k on both sides of (11), we have

k
uk+1(
)k  k
uk (
)k + k;CAk1k
xk (
)k +k;k1k (P (C
xk + e~)) (
) ; (P e~) (
)k

(12)

Since P is a continuous operator, there exists a constant k0 such that(Ahn et al. 1993)

k (P (C
xk + e~)) (
) ; (P e~) (
)k  k0kC
xk (
)k: Then, from (12), we can have

k
uk+1(
)k  k
uk (
)k + k;CAk1k
xk (
)k +k0k;k1kC k1k
xk (
)k: 6

(13)

From (10), we can obtain
xk (t) =

t

Z

[A
xk ( ) + B
uk ( )] d:

0

(14)

Taking the norm k
k1 on both sides of (14) and applying the GrownwallBellman inequality(Rugh 1993), we nd that

k
xk (t)k1 

Z

0

t a(t; ) e kB k

1 k
uk ( )k1 d

(15)

where

a = kAk1: Multiplying both sides of (15) by e;t to obtain the -norm expressions of
xk (
) and
uk (
), and substituting the result into (13), we further nd that ;(;a)T k
uk+1(
)k   + k1 1 ;e ; a k
uk (
)k !

(16)

where

k1 = (k;CAk1 + k0k;k1kC k1) kB k1: Since 0   < 1 by assumption, it is possible to choose  suciently large so that

0 =  + k1 1 ;e ; a

;(;a)T

< 1:

From (16), we can show that lim k
uk (
)k = 0:

k!1

From (10) and (15), we can nally conclude the following: lim y (t) = yd(t) ; e~(t): k!1 k 7

This completes the proof. Theorem 1 implies that if the initial state error is the same at each iteration, the output trajectory can be exactly estimated from the information about the desired output trajectory, the initial output error value and the operator P as stated in (Lee and Bien 1996, Park and Bien 1999). If the operator P is chosen as (Pe(
)) (t) = ;Re(t); the algorithm becomes PD-type ILC algorithm as in (Lee and Bien 1996) and the output trajectory converges to (3) since

k ; Rh1 (
) ; (;Rh2 (
))k  kRk1kh1(
) ; h2 (
)k; 8 (h1 (
); h2(
)) 2 F m  F m: If P is chosen as (Pe(
)) (t) = Q0e(t) + Q1

Z

0

t

e( )d;

the algorithm becomes PID-type ILC algorithm as in (Park and Bien 1999) and the output trajectory converges to (5) since sup e;t kQ0h1 (t) + Q1

t2[0;T ]

t

Z

0

h1( )d ; Q0h2 (t) ; Q1

Z

0

t

h2 ( )d k1

;T  kQ0k1 + kQ1k1 1 ; e kh1(
) ; h2(
)k  (kQ0k1 + kQ1 k1) kh1(
) ; h2 (
)k; 8 (h1 (
); h2(
)) 2 F m  F m: !

From this observation, the proposed generalized ILC algorithm can be considered as an extension of the PD-type and/or PID-type ILC algorithm. 8

In the proposed ILC algorithm, the eect of the initial state error can be controlled in a variety of ways. For example, consider the following operators: (i) (P1 e(
)) (t) = ;tRe(t) (ii) (P2 e(
)) (t) = ;(t)e(t) (iii) (P3 e(
)) (t) = ;(t)e(t): Here, (t) is a real-valued function ( : [0; T ] ! R) and (t) is a matrix-valued function ( : [0; T ] ! Rm  Rm ) which are assumed to be bounded as

k(t)k1  b < 1 k(t)k1  b < 1: It can be easily shown that each operator is continuous and the output trajectories converge, respectively, to the following functions at each case: (i) limk!1 yk (t) = yd(t) + e 21 Rt2 C (x0 ; xd (0)) Rt (ii) limk!1 yk (t) = yd(t) + e 0 ( )d C (x0 ; xd (0)) Rt (iii) limk!1 yk (t) = yd(t) + e 0 ( )d C (x0 ; xd (0)) : To study a more realistic situation, we are going to examine the case of random initial state error in the following. Assume that the initial state value at each iteration is in a neighborhood of x0 such that

kxk (0) ; x0 k1 
0 :

(17)

We rst consider the case that the operator P is linear. As in (Lee and Bien 1996, Park and Bien 1999), we can obtain the following result that shows robustness of the ILC algorithm (7) against random initial state error.

Theorem 2 Suppose that the continuous operator P is linear. Also, suppose that Equation (8) holds and the update law (7) is applied to the system (1). Let

9

ua(t) and xa (t) be the control input and the state that satisfy (10). If the initial state at each iteration satis es (17), then, as k ! 1, the error between ua (t) and uk (t) is bounded and this bound continuously depends on kCA + PC k and
0 .

Proof

As in the proof of Theorem 1, one nds from (7) and (10) that
uk+1(t) = (I ; ;CB )
uk (t) ; ;CA
xk (t) ;; [(P (C
xk (
) + e~(
))) (t) ; (P e~(
)) (t)] :

(18)

Taking the norm k
k on both sides of (18), we have

k
uk+1(
)k  k
uk (
)k + k;k1kCA + PC kk
xk (
)k:

(19)

From (10), we can obtain
xk (t) = (x0 ; xk (0)) +

t

Z

0

[A
xk ( ) + B
uk ( )] d:

(20)

Taking the norm k
k1 on both sides of (20) and applying the GrownwallBellman inequality(Rugh 1993), we nd that Z

t

k
xk (t)k1  eat
0 + 0 ea(t; ) kB k1k
uk ( )k1d

(21)

where

a = kAk1: Multiplying both sides of (21) by e;t and substituting the result into (19), we further nd that (a;)T ! 1 ; e k
uk+1(
)k   + k2  ; a k
uk (
)k +k;k1kCA + PC k
0 (22) 10

where

k2 = k;k1kCA + PC kkB k1: Since 0   < 1 by assumption, it is possible to choose  suciently large so that

1 =  + k2 1 ;e; a

(a;)T

< 1:

From (22), we can show that lim k
uk (
)k  k!1

1 k;k kCA + PC k
:  0 1 ; 1 1

This completes the proof. As shown in the proof of Theorem 2, the bound of control error, k
uk (
)k, depends on kCA + PC k as well as on
0 . This implies that the bound of control error can be adjusted by P . As commented in (Lee and Bien 1996) and (Park and Bien 1999), we can choose P as follows:  ;1 (Pe(
)) (t) = ;C~ A~C~ T C~ C~ T e(t)

or ;1 (Pe(
)) (t) = ;C~ A~C~ T C~ C~ T e(t)   Z t  ;1 T T T ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ; C A ; C AC C C C AC e( )d 



0

where A~ and C~ are some models of A and C of system (1), respectively. In case that the operator P is not linear, it is easily shown that Equation (19) becomes

k
uk+1(
)k  k
uk (
)k + k;k1 (kCAk1 + k0kC k1) k
xk (
)k 11

and Equation (22) becomes (a;)T k
uk+1(
)k   + k3 1 ;e; a k
uk (
)k + k3
0 !

where

k3 = k;k1 (kCAk1 + k0 kC k1) kB k1: So, one can easily nd that 1 k;k (kCAk + k kC k ) kB k
lim k
u (
) k  k  1 1 0 1 1 0 k!1 1; 2

where

2 =  + k3 1 ;e; a : (a;)T

This implies that the bound of control error depends on the operator P and
0 .

3 Generalized ILC for a Class of Nonlinear Dynamic Systems In this section, the eect of initial state error for a class of nonlinear systems is shown. Consider the nonlinear system described by (23).

x_ (t) = f (x(t); t) + B (x(t); t) u(t) y(t) = g (x(t); t)

(23)

Here, x 2 Rn ; u 2 Rr and y 2 Rm denote the state, the input and the output, respectively. The followings assumptions are made for the system (23). 12

A1 The functions f : Rn  [0; T ] ! Rn and B : Rn  [0; T ] ! Rnr are piecewise continuous in t, and g : Rn  [0; T ] ! Rm is dieren-

tiable in x and t with partial derivatives denoted as gx (x(t); t) and gt (x(t); t).

A2 The functions f ((x(t); t) ; B (x(t); t) ; gx (x(t); t) and gt (x(t); t) sat-

isfy the uniformly global Lipschitz condition in x on the interval [0; T ]. That is, for all pair (x1 (t); x2(t)) 2 Rn  Rn , there exists k'(0 < k' < 1) such that

k' (x1 (t); t) ; ' (x2(t); t) k1  k'kx1 (t) ; x2 (t)k1; 8t 2 [0; T ]; ' 2 ff; B; gx; gtg:

A3 The functions B (x(t); t) and gx (x(t); t) are bounded on Rn  [0; T ]. A4 gx (x(t); t) B (x(t); t) is a full rank matrix for any (x; t) 2 Rn  [0; T ] that can be achieved by the system (23).

Consider the ILC algorithm described by (24) which is similar to (7).

uk+1(t) = uk (t) + ;(t) [e_k (t) + (Pek (
)) (t)]

(24)

Then, we can obtain the following result for the nonlinear system (23).

Theorem 3 Let the system (23) satisfy the assumptions from A1 to A4. As-

sume that the output trajectory ya(t) de ned by (25) is achievable(Heinzinger 1992) for a given desired trajectory yd (t):

ya (t) = yd(t) ; e^(t)

(25)

where e^(t) is the solution of

e^_ (t) + (P e^(
)) (t) = 0 e^(0) = g (xd (0); 0) ; g (x0 ; 0) : 13

(26)

Suppose that the update law (24) is applied to the system (23) and that the initial state at each iteration can be dierent from the desired initial state, i.e., xk (0) = x0 6= xd (0) for k = 0; 1; 2;


. If

kI ; ;(t)gx(x; t)B (x; t)k1   < 1; 8(x; t) 2 Rn  [0; T ];

(27)

then

lim y (t) = yd(t) ; e^(t): k!1 k

Proof

Let ua(t) and xa (t) be the control input and the state that satisfy (28). Z

t

xa (t) = x0 + [f (xa ( );  ) + B (xa ( );  ) ua( )] d 0 ya (t) = g (xa (t); t)

(28)

Let
uk (t) = ua(t) ; uk (t)
xk (t) = xa (t) ; xk (t): For notational convenience, function parameters are denoted in subscript notation as follows.

gk (t) = g (xk (t); t) @ g(x; t)j gxk (t) = @x x=xk(t) @ g(x; t)j gtk (t) = @t x=xk(t) fk (t) = f (xk (t); t) Bk (t) = B (xk (t); t)

; ga(t) = g (xa (t); t) @ g(x; t)j ; gxa(t) = @x x=xa(t) @ g(x; t)j ; gta (t) = @t x=xa(t) ; fa (t) = f (xa (t); t) ; Ba (t) = B (xa (t); t) 14

kgx; kgt; kf ; kB and kg are the Lipschitz constants for gx (x(t); t), gt (x(t); t), f (x(t); t), B (x(t); t) and g (x(t); t), respectively. From (24) and (28), it is observed that
uk+1(t) =
uk (t) ; ;(t) [e_k (t) + (Pek (
)) (t)] =
uk (t) ; ;(t) [gxa(t) (fa (t) + Ba (t)ua(t)) + gta (t) ;gxk (t) (fk (t) + Bk (t)uk (t)) ; gtk (t)

;(P e^(
))(t) + (P (ga(
) ; gk (
) + e^(
))) (t)] = (I ; ;(t)gxk (t)Bk (t))
uk (t) ;;(t) [(gxa(t) ; gxk (t)) (fa (t) + Ba (t)ua(t)) +gxk (t) (fa(t) ; fk (t)) + gxk (t) (Ba (t) ; Bk (t)) ua(t) + gta (t) ; gtk (t) ; (P e^(
)) (t) + (P (ga(
) ; gk (
) + e^(
))) (t)] : (29) Since P is a continuous operator, there exists a constant k4 such that(Ahn et al. 1993)   k (P (ga ; gk + e^)) (
) ; P ^(e) (
)k  k4k (ga ; gk ) (
)k: Taking the norm k
k on both sides of (29), we have

k
uk+1(
)k  k
uk (
)k + k5k
xk (
)k

(30)

where

k5 = b; (kgxbfBu + bgx kf + bgxkB bua + kgt + k4kg ) b; = sup k;(t)k1; bfBu = sup kfa (t) + Ba (t)ua(t)k1 bgx =

t2[0;T ]

sup

(xk ;t)2Rn [0;T ]

From (28), we can obtain
xk (t) =

Z

0

+

t

t2[0;T ] kgxk (t)k1; bua =

sup kua(t)k1:

t2[0;T ]

[fa ( ) ; fk ( ) + (Ba (t) ; Bk (t)) ua(t)] d

Z

0

t

Bk ( ) (ua( ) ; uk ( )) d: 15

(31)

Taking the norm k
k1 on both sides of (31) and applying the GrownwallBellman inequality(Rugh 1993), we nd that

k
xk (t)k1 

t

Z

0

bB ek6 (t; ) k
uk ( )k1d

(32)

where

bB = sup kBk (t)k1 (xk ;t)2Rn [0;T ] k6 = kf + kB bua : Multiplying both sides of (32) by e;t and substituting the result into (30), we further nd that (k6 ;)T ! 1 ; e k
uk+1(
)k   + k5bB  ; k k
uk (
)k: (33) 6 Since 0   < 1 by assumption, it is possible to choose  suciently large so that (k6 ;)T 3 =  + k5bB 1 ;e; k < 1: 6 From (33), we can show that lim k
uk (
)k = 0:

k!1

From (28) and (32), we can nally conclude the following: lim y (t) = yd(t) ; e^: k!1 k This completes the proof. If Theorem 3 is applied to the LTI system (1), the sucient condition (27) becomes the sucient condition (8) since gx (x(t); t) = C , B (x(t); t) = B and ;(t) = ;. Also, Equation (26) of Theorem 3 becomes Equation (9) of Theorem 1 since g (xd (0); 0) ; g (x0 ; 0) = C (xd (0) ; x0). From this, one may say that Theorem 3 is an extension of Theorem 1. 16

4 Nonlinear ILC algorithm for a Finite-Time Convergence In this section, the eect of initial state error for a nonlinear ILC algorithm is shown as a special case of the operator P . Let there be given a nite time ts  T for a LTI system described by (1). The problem is to nd a control input u(t); 0  t  T , such that the error ek (t) reaches to zero in a given nite time ts and is exponentially bounded. The following ILC algorithm is proposed as a solution.

uk+1(t) = uk (t) + ; (e_k (t) + Rek (t)q )

(34)

Here, R = diagfrig, ri  0, 0  i  m, and q 2 f= : ; ( > ) are positive odd integers g. For an m-dimensional vector x = (x1;


; xm )T , the operation xq is de ned by xq = ((x1 )q ;


; (xm)q )T . Now, the convergence of the ILC algorithm (34) will be shown.

Theorem 4 Suppose that the update law (34) is applied to the system (1) and

the initial state at each iteration can be dierent from the desired initial state, i.e., xk (0) = x0 6= xd (0), for k = 0; 1; 2;


. If

kI ; ;CB k1   < 1 then,

lim y (t) = yd(t) ; e(t; q) k!1 k where

8 > > > > > > > >
> > > > > > > :

h

i 1

;q

1

; 0  t < tis ; tis  t

0 17

and

Proof

tis = r (11; q) (fC (xd (0) ; x0 )gi )1;q : i

Dierentiating e(t; q) in t, we can obtain

fe_ (t; q)gi + ri (fe(t; q)gi)q = ;ri(1 ; q) (fC (xd (0) ; x0 )gi )q  1 ;1 q 1 ; ri(1 ; q) (fC (xd (0) ; x0 )gi)q;1 t +ri (fC (xd (0) ; x0 )gi)q q  1 ; ri(1 ; q) (fC (xd (0) ; x0 )gi)q;1 t ;q h

h

i

i

= 0:

q ;q

1

1

(35)

Here, recall that the notation fxgi denotes the component of the vector x and ri's are the components of R. Substituting 0 in t of e(t; q), we can also obtain

fe(0; q)gi = fC (xd (0) ; x0 )gi :

(36)

From (35) and (36), e(t; q) is the solution of

e_ (t; q) + Re(t; q)q = 0 e(0; q) = C (xd (0) ; x0 ) : Let P be an operator de ned by (Pe(
)) (t) = Re(t)q : Then, since P is continuous in e(
), we can nally conclude the following from Theorem 1: lim y (t) = yd(t) ; e(t; q): k!1 k 18

This completes the proof. Theorem 4 implies that if the initial state error is the same at each iteration, the output trajectory can be exactly estimated from the information about the desired output trajectory, the initial output error value, and the learning controller parameters q and R in (34). Note that if the parameter q is chosen as q ! 1, the algorithm becomes PDtype as in (2) and the output trajectory converges to (3). From this observation, the proposed nonlinear ILC algorithm is considered as an extension of the PDtype ILC algorithm. Assume that the initial state error at each iteration is bounded such that

kx0 ; xd (0)k1 
1 : Then, for a given nite time ts, if we know the bound
1 , we can choose the parameters q and ri of R such that tis  ts; 1  i  m. Since jfe(t; q)gij is an increasing function of q for a xed t, it is easily seen that the converged output trajectory is satisfying the following:



jfC (xd (0) ; x0 gij ; ri t  klim fy (t) ; yd(t)gi !1 k  jfC (xd (0) ; x0 gij e;rit : From this result, one can nd that the converged output error trajectory is bounded by an exponential function and the exponential bound is determined by the learning parameter R. Similar to Theorem 4, it is easily shown that, if the ILC algorithm (34) is applied to the system (23), the output trajectory converges as follows: lim y (t) = yd(t) ; e(t; q) k!1 k 19

where 8 > > > > > > > >
> > > > > > > :

h

i 1

;q

1

; 0  t < tis ; tis  t

0

and

tis = r (11; q) (fg (xd (0); 0) ; (x0 ; 0)gi )1;q : i

Theorem 4 shows that the error can be reached to zero in a nite time. This indicates that, if one gives a pre-trajectory which has a nite interval concatenated prior to the given desired trajectory, one can make the error reach to zero in the appended interval and get the perfect tracking for the given desired trajectory(see gure 1).

5 Numerical Examples The following examples are given to illustrate the eciency of the proposed algorithms.

Example 1. Linear time-invariant system:

Consider the following dynamic system(Lee and Bien 1996). 2

x_ (t) = y(t) =

6 4 

3

2

3

0 1 7 07 6 5 x(t) + 4 5 u(t) ;2 ;3 1 

0 1 x(t)

The desired output trajectory and the initial state are given as follows.

yd(t) = 0:03t(20 ; t); 0  t  20 20

2

3

17 5 1 According to Theorem 1, one can know that the best choice of ; is 1. Under the assumption of 30% uncertainty of the system parameters, ; can be chosen as 1=1:3. Figure 2 shows the output trajectories and the error trajectories at 45th, 20th, 20th and 38th iteration, respectively. Figures 2 (a), (b), (c) and (d) are the results for the case of the following operators, respectively:

xk (0) = x0 =

6 4

(i) (Pe(
)) (t) = 2e(t) R (ii) (Pe(
)) (t) = 2e(t) + 0t e( )d (iii) (Pe(
)) (t) = 2t1=2 e(t) (iv) (Pe(
)) (t) = 2e(t)5=9 : Figure 3 also shows the results of (i) and (iv) magni ed for comparison. As mentioned in Theorem 1, the converged output trajectory depends on the operator P . In the PD-type ILC algorithm ( gure 2 (a), gure 3 (a)), the error decreases exponentially as the time increases. In the PID-type ILC algorithm ( gure 2 (b)), the behavior possesses the characteristics of a second order dynamic system, i.e., s2 + 2s + 1. In the ILC algorithm of which the learning gain is a function of time ( gure 2 (c)), the error decreases more quickly as the time increases. In the nonlinear ILC algorithm ( gure 2 (d), gure 3 (b)), the error reaches to zero in a nite time(1.125). Overall, gure 2 and gure 3 show that the eect of the initial state error can be arbitrarily controlled by the choice of an operator.

Example 2. Nonlinear Time-varying System: Consider the following dynamic system. 2 6 4

x_ 1 (t) x_ 2 (t)

3 7 5

2

=

6 4

3

2

3

7 5

6 4

7 5

x2 (t) 0 + u(t) ;2x1 (t) ; 3x2 (t) + tsinx1 (t) 1 21

2

3

6 4

7 5

x1 (t) x2 (t) Desired output trajectory and the initial state are given as follows. 3 t4 ; 0  t  2 yd(t) = 38 t2 ; 83 t3 + 32 0:005 xk (0) = x0 = 0:005 Under the assumption of 10% uncertainty of system parameters, let the learning gain ; be chosen as 1=1:3 from Theorem 3. Similar to Example 1, the following operators are chosen: y(t) =



0 1



2

3

6 4

7 5

(i) (Pe(
)) (t) = 6e(t) R (ii) (Pe(
)) (t) = 6e(t) + 9 0t e( )d (iii) (Pe(
)) (t) = 6t1=2 e(t) (iv) (Pe(
)) (t) = 6e(t)7=9 : Figure 4 shows the output trajectories and error trajectories at the 13th iteration. As mentioned in Theorem 3, the converged output trajectory is also decided by the operator P . Similar to Example 1, the error decreases exponentially in the PD-type algorithm ( gure 4 (a)). In the PID-type algorithm ( gure 4 (b)), the error changes with the characteristics of the second-order dynamic system, i.e., s2 + 6s + 9 = 0. In the third ILC algorithm ( gure 4 (c)), the error decreases more quickly as the time increases. Finally, in the nonlinear ILC algorithm ( gure 4 (d)), the error reaches to zero in a nite time(0.2311).

6 Concluding Remark In this paper, a generalized ILC algorithm was proposed and the robustness against initial state error was investigated for linear dynamic systems and for 22

a class of nonlinear dynamic systems. As a special case, the nonlinear ILC algorithm was also proposed, which is in the form of terminal attractor. The proposed ILC algorithm is shown to be an extended form of PD-type and/or PID-type ILC algorithm. The eect of the initial state error can be more eectively controlled in various ways and the performance can be improved according to the choice of an operator. Furthermore, the error can be reached to zero in a nite time when the proposed nonlinear ILC algorithm is applied. The robustness of the ILC algorithm against the random initial state error, noise and state disturbance is yet to be further investigated. On the other hand, when the existing linear ILC algorithms are applied to the systems with input delay, erratic estimation of delay time may cause the control input to diverge. A rigorous analysis based on the operator approach should be challenging problem.

References Ahn, H.-S., Choi, C.-H., and Kim, K.-B., 1993, Iterative learning control for a

class of nonlinear systems. Automatica, 29 (6), 1575{1578. Arimoto, S., Kawamura, S., and Miyazaki, F., 1984, Bettering operation of robots by learning. Journal of Robotic System, 1, 123{140. Heinzinger, G., Fenwick, D., Paden, B., and Miyazaki, F., 1989, Robust learning control. Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, Florida, USA, 426{440. Heinzinger, G., Fenwick, D., Paden, B., and Miyazaki, F., 1992, Stability of Learning Control with Disturbances and Uncertain Initial Conditions. IEEE Transactions on Automatic Control, 37 (1), 110{114. Lee, H.S. and Bien, Z., 1996, Study on robustness of iterative learning control with non-zero initial error. International Journal of Control, 64 (3), 345{359. 23

Lee, K.H. and Bien, Z., 1991, Initial condition problem of learning control. IEE

Proceedings-Part D., 138 (6), 525{528.

Man, Z., Paplinski, A.P., and Wu, H.R., 1994, A robust MIMO terminal sliding

mode control for rigid robotic manipulators. IEEE Transactions on Automatic Control, 39 (12), 2464{2468. Park, K.-H. and Bien, Z., 1999, A study on the robustness of a PID-type iterative learning controller against initial state error. International Journal of Systems Science, 30 (1), 49{59. Rugh, W.J., 1993, Linear System Theory (Prentice Hall). Saab, S.S., 1994, A discrete-time learning control algorithm. Proceedings of the 1994 Americal Control Conference, Baltimore, Maryland, USA, 749{753.

24

List of Figures Figure 1. the concatenated trajectories Figure 2. the output and the error for Example 1 Figure 3. the output and the error for Example 1 (Magni cation) Figure 4. the output and the error for Example 2

25

Figure 1: the concatenated trajectories

26

(a)

(b) Figure 2: the output and the error for Example 1 (a) the case of (Pe)(t) = 2e(t) R (b) the case of (Pe)(t) = 2e(t) + 0t e( )d

27

(c)

(d) Figure 2: the output and the error for Example 1 (continued) (c) the case of (Pe)(t) = 2t1=2 e(t) (d) the case of (Pe)(t) = 2e(t)5=9

28

(a)

(b) Figure 3: the output and the error for Example 1 (Magni cation) (a) the case of (Pe)(t) = 2e(t) (b) the case of (Pe)(t) = 2e(t)5=9

29

(a)

(b) Figure 4: the output and the error for Example 2 (a) the case of (Pe)(t) = 6e(t) R (b) the case of (Pe)(t) = 6e(t) + 9 0t e( )d

30

(c)

(d) Figure 4: the output and the error for Example 2 (continued) (c) the case of (Pe)(t) = 6t1=2 e(t) (d) the case of (Pe)(t) = 6e(t)7=9

31