MODEL BASED ITERATIVE LEARNING CONTROL (MILC) FOR ...
14th World Congress ofTFAC
C-2a-13-6
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
MODEL BASED ITERATIVE LEARNING CONTROL (MILC) FOR UNCERTAIN DYNAMIC NON-LINEAR SYSTEMS MuhaIIllnad Arif, Tadashi Ishihara and Hikaru Inooka
Graduate School of Information Sciences, Tohoku University, Japan an!, ishihara,
[email protected]
Abstract: Model based Ite:rative Lea:rning Control (MILe) is proposed for a class of time varying nonlinear uncertain systems. Convergence of MILC is analyzed and the uniform boundedness of tracking error is obtained in the presence of unce:rtainty and disturbances. It is shown that the learning algorithm not only guarantees the robustness, but also improves the learning rate despite of the presence of disturbances and slowly varying desired trajectories in succeeding iterations. The effectiveness of the proposed MILe is presented by simulations. Copyright © 1999IFAC Keywords~
Iterative learning control, Nonlinear control, Uncertain systems, Model based control
1. INTRODUCTION
Various type of learning control methods have been proposed to deal with the uncertainties of the dynamic systems in which the past experience history is utilized to improve the control quality. However if the control task is of repetitive nature as in many robotics applications in the industry, then iterative learning cont:rol is found to be effective. Such kind of learning cont:rol method is proposed by Uchiyama[l] and is elaborated as a formal theory by Arimoto[2] in 1984. Since then, there has been a great deal of efforts by the researchers to synthesize a better iterative Jearning control scheme[3,4]. Although sufficient conditions are derived for the convergence of the learning process, the convergence rate is often slow in most of the proposed control laws. Chen et. al. [5] has described a current iteration tracking error assisted iterative learning control method. They have used the static extrapolation method to predict the current error. However, the effect of the error in the prediction of the current
iteration error data has not been incorporated in the convergence analysis of the control algorithm. In this paper, we present a new model based iterative learning control (MILC) scheme for unce:rtain time varying nonlinear systems. In our view, it is better to use a nonlinear model of the system for the prediction of error. Future error can be predicted by using the model of the system. This model of the system can be obtained by using the nonlinear system identification technique. However, if the nonlinear system can be linea:rized around some operating point, we can identify the linearized system by the identification technique for linear systems. It gives us the advantage of using more simpler and easily implement able identification technique. In this way, we can restore the simplicity of the learning algorithm by using an approximate linear model of the system to predict the future error data. In the simulations section, it is shown that the linearized model can be used to predict the error and we can still get a better convergence. Necessary and sufficient conditions are stated for the convergence of the proposed method in the presence of disturbances. It is also
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Copyright 1999 IF AC
ISBN: 008 0432484
MODEL BASED ITERA TTVE LEARNING CONTROL (MILC) FOR .. .
shown that the convergence rate of the learning process can be increased using MILe technique. To illustrate the applicability of MILe, we have demonstrated an example of nonlinear system.
14th World Congress ofTFAC
model based iterative learning control law MILC is presented in this paper. MILe is defined as
Ui(t) = Ui - l (t)
+ r}
Ci- l
Ct) + r 2
e; et)
(6)
where 2. PROBLEM FORMULATION Consider a nonlinear time varying dynamic system subjected to the disturbances described as, Xi (t) = f(Xi(t), t) Vi(t)
= C(t)x.(t)
+ B(t)u.(t) + diet) + Vi(t)
(1)
where i corresponds to the ith iteration of the system; Xi(t) E Rnxl, Ui(t) E Rm x l and Vi(t) E R r x 1 are the states, control input and output of the system respectively for t E [0, T]; c.4(t) and Vi(t) are the unknown but bounded disturbances to the system and to the output respectively; f(Xi(t),t) : Rn X [O , T] t-+ Rn is a nonlinear, piecewise continuous function of Xi(t) and t.
f (.) satisfies the Lipschitz condition as follows IIf(X2(t), t) - f(xt(t),
t)11
~
Ll
Ilx2(f) -
xdt) 11 (2)
where Ll > 0 is the Lipschitz constant. Let Yd(t) be the desired trajectory, which is achievable, and continuously differentiable on the finite time interval t E [0, T], such that for bounded desired trajectory Yd(t), there exists a unique bounded input Ud(t) for t E [0, T], for which system has unique bounded states Xd(t) and Yd(t) = C(t)Xd(t) at t E [O,T]. The problem can be stated as follows: For a desired trajectory Yd(t) starting with some initial control input uo(t), modify the control input Ui(t) sequentially for ith iteration such that for i -t 00, Vi(t) -t Yd(t) Vt E [0, T], in the presence of bounded disturbances.
Definition 1. Norms of function get), introduced in this paper, are as follows.
Ilg(t) lb. ::::
sup e- At
Ilg(t)1I
(3)
IIg(t) 1100 = sup IIg(t)1I
::::Yd (t)- Yi-l
.*
(7)
The matrices fk E Rmxr (k = 1,2) are the learning matrices. In the above equation, uHt) is defined as the output of the nonlinear system's model which is identified by using the inputoutput data of i - Ith iteration. Identification of the system can be done on-line in each iteration to ensure the tracking of the parameter drift of the system during the operation.
3.1 Non Linear System Identification The main issue in the system identification is to search for a suitable model structure, within which a good model can be found. Parameter estimation, once the structure is known is a lesser problem. To estimate the model, experimental data z(t) E R 1x (p+m) can be arranged as,
z(t)
= [UI (t)......
Vv(t) UI (t)..........
Urn
(t)]
(8) It is very important that the data should reflect all important features of the underlying system. The excitation signal u(t) should carefully be chosen, so that the system dynamics show up in the output signal yCt).
Selection of a model structure M is also an important aspect of system identification. The selection comprises of type and size of model. Once the size and type is defined, one has to search for the model in the model set M*. The next step is to parameterize the model set. Let the members of M- is parameterized by a finite dimensional vector () E D M C Rd x 1. The model structure is defined by the mapping M: DM 3
tE[O ,T]
Ilgll:::: l::;.::;n max 19i1
Ct)
(t), . .* e i (t) = Yd (t)- Yi (t)
ei-l
(J ~
M(9) EM"
(9)
(4) (5)
t E[O, T ]
where M (0) is a particular model corresponding to O. We denote a family of functions parameterized by (J as follows,
3. MODEL BASED ITERATIVE LEARNING CONTROL (MILC) For a nonlinear time varying system with the disturbances diet) and Vi(t) as given in (1), a new
y' (t, 0)
=
h('P(t), 9)
(10)
where
(11) 1470
Copyright 1999 IF AC
ISBN: 008 0432484
MODEL BASED ITERATIVE LEARNING CONTROL (MILC) FOR ...
We call r.p (t) as regression vector. Regression vector can itself be parameterized: 'P(t, Tj) = r.p(yt-l, u t -
1,
'TI)
(12)
In practice, measured and model output differs and do not match exactly.
= yet) - y' (t, 0)
t(t,O)
(13)
where c(t,O) is an error due to the unmodelled dynamics and noise. An error criterion can be defined as,
14th World Congress ofTFAC
3.2 Analysis of Convergence MILe is a modified version of standard D type ILC technique to increase the convergence rate. In this technique, we have used predicted error to speed up the convergence. Predicted error is not supposed to be accurate but bounded. The effect of the uncertainty in the prediction will be compensated as the actual error will be available in the next iteration.
Assumptions: Al:System (1) is stable for 1ft E [0, T] and Vi. A2: The disturbances d(t) and vet) are bounded such that for 'tit E [0, T] and Vi
The problem of identifying a nonlinear dynamic system is a vast field of research and is not in the scope of this paper. As long as the error criterion is minimum, any technique can be used for the identification of the nonlinear system. Instead of using identification technique for nonlinear systems, the system identification techniques for linear systems can alElo be used for the prediction provided that the prediction error is bounded. It is because of the fact that predicted error at ith iteration will be corrected in the i + Ith iteration of the operation. A nonlinear system can be written as,
- di - 1 (t)IIA S bd IIvi(t) - vi-l(t)II A S bt}
(20)
A3: Prediction error is also bounded to a certain constant bm such that,
(21) where
ei =
Predicted error.
= Actual error at ith iteration. bm == Modelling error bound. ei
d
dt;£(t) =
lC;£,y, t) + Byet)
(15)
It can be linearized around some operating states. For the nonlinear system described in the above equation, expressing the dependent state variables and independent forcing functions as deviations from some operating state, ~ and Yo, gives ;r(t)
(16)
A4:
+ by(t)
(17)
Vi.
Substituting these expression in equation (15) and writing the nonHnear terms in a first order taylor series expansion about the operating point, we get the expression that represents a linearization of the original nonlinear system.
d
dt ox(t) = J", (2:0, Yo)ax(t)
+ Bau(t)
(18)
where £h..
£h..
!lli.
!lli
aXl
17""
~ aXl
Remark 1. Prediction error bound bm is a measure to represent the deviation of et from ei, which means that the higher the bm value, the poorer will be the identification of the system, causing the wrong estimation of the error.
= ~(t) + 5~(t)
yet) = Yo(t)
Jx =
II~ (t)
aX2 aX2
I!..b>.. ()X2
~l
[Ill,. + CBr 2 I1 oo ]-1 exists for Vt
E [0, T] and
A5: Yd(t) is the desired trajectory which is reachable such that for bounded Yd(t), there exists a bounded Ud(t), for which system has unique bounded states Xd(t) and Yd(t) = CXd(t) at t E
[O,T]. A6: l(x.Ct), t) : Rn X [0, T] I-t Rn is a piecewise continuous function and satisfies the Lipschitz condition. FUrthermore G(t) and B(t) are uniformly bounded for all t E [0, T] and for all iterations i E [0,00], such that
aXn
!!..li. a"'n
!!..b... &Xn
be =
(19)
bb
~,!fo
Once the nonlinear system is linearized, we can apply some simple system identification technique for linear system to predict the error.
sup
sup IIG(t)!1
iE[O,ooj tE[O,T]
= sup
sup IIB(t)11
(22)
iEfO,oo] tE/O,T]
Theorem 1. For the nonlinear time varying uncertain system given in (1), with the assumptions Al to A6 hold, by using the model based iterative 1471
Copyright 1999 IF AC
ISBN: 008 0432484
MODEL BASED ITERATIVE LEARNING CONTROL (MILC) FOR ...
learning control law mentioned in equation (6) incorporating the condition, 'YP
< 1 fm' "It E [0, T]
'Y =
1I11r + CBr2 li ocr
(23)
14th World Congress ofTFAC
Taking the ,\ - nm'm of the above equation and assuming that A2 and A3 holds, we get
lilT
where
P=
00
(24)
l
Ill.,. - CBr1ll oo
+ cBrzlL)() lIedl" ::; [Ill,., - cBrlll,,., such that
PROOF.
Taking the).. - norm of equation (28), we get
Let
Ilxi(t) - X;-l (t)lb ::; ei(t) = Yd(t) - Yi(t)
f] [aSbd + (aSb h1 (31)
(27)
Now integrating equation (1) and combining with equation (6), we get t
=
a3 L
- C(t)[Xi{t) - Xi-l Ct)]
-[v;(t) - Vi_let)]
Xi(t) - Xi-let)
1
+ IIBrllloo) !!e;-llb. + (asbh2 + IIBrz\l"J Ile:II),]
=Yd(t) - Yi-l(t) - (Yi(t) - Yi-l(t))
= ei-l (t)
[In _
Putting the above equation in equation (30),
lie.!!"
J
+ 1[IICBr2 1Ico bm + bv ] + bioI lIei-dl" + bhzlleilh
+'Yagbc[b d Lf + [In _ asLfl (aSbd + (agb h1
+ IIBr11Ioo) Ilei-lll" +(a gbh2 + IIBr21Ioo) liei'll>.)]
1[1(Xi(T),T) - !(Xi-l(T),T)]dT o
t
SIP Ilei-III),
(32)
t
+ j[di(T) -
di-dr)]dT
+
B(r}r2
o
e; (r}dT
0
Assuming that ).. is sufficiently large, we can neglect the terms with a3 in equation (32),
t
+/
B(T)rl
Ei-l
(r)dr
o (28) Integrating by parts the last two terms containing multiplying both side with C(t), putting in equation (27) and rearranging we get,
e (r),
t
-CBrz[ei(t) - e.(t)] - C 1[!i(T) -!i-l(T)]dT o
t
1 t
-C l[di(T) - di - 1 (r)]dr
o
+C
hI (r)ei-l(r)dr
0
I
o Remark 2. The convergence condition for standard D type iterative learning control is elaborated in [2] as
P < 1 fm' "It E [O,T] where P is given in equation (25). According to the above convergence condition, we can not select the gain matrix r l beyond certain upper limit. By incorporating the modified condition mentioned in equation (23) and including an additional parameter l' which is the defined in equation (24), it is possible to increase the gain matrix rl further to get faster convergence.
t
+0
hz(r)e;(r)dr - [Vi(t) - Vi-l (t)]
o
(29)
If the system (1) is not subjected to the disturbances, that is bd and bv are zero, then the equation (33) reduces to the following expression.
Ileill>. :5 'YP lI e i-lll>. + 'Y IIcBr2li oo bm
(34)
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Copyright 1999 IF AC
ISBN: 008 0432484
MODEL BASED ITERATIVE LEARNING CONTROL (MILC) FOR ...
According to the above expression, absolute convergence of the error is not guaranteed due to the presence of bm, which is the bound of the prediction error. Although the error will converge uniformly to the neighborhood of the desired trajectory in the absence of disturbances, the absolute convergence of the error can not be guaranteed. If the prediction of the error is very accurate, which means that bm is negligible, it is possible to neglect the last term of equation (34),
(35) The above expression guarantees the absolute convergence of error to zero as i -+ 00. In the case, when the prediction error bound bm is not negligible, MILe can be applied only in the initial iterations to obtain improved convergence rate. For the rest of iterations, r2 can be set to zero, eliminating the prediction error bound bm from equation (34) and reducing the model base iterative learning control to the standard D type iterative learning control. In this way, the absolute convergence of the error can be obtained.
4. APPLICABILITY OF MILC FOR SLOWLY VARYING DESIRED TRAJECTORY In the previous section, desired trajectory Yd(t) is fixed for each iteration and does not change in succeeding iterations. In this section, we are going to address the case when the desired trajectory Yd(t) is allowed to vary in succeeding iterations. Saab et al. [6J has addressed this problem for D type, PD type and PID type of iterative learning control algorithms. Theorem 1 of Saab's paper can be extended to our proposed algorithm.
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will be bounded for sufficiently large A, provided that the condition "tP
for '.It E [0, T]
(38)
is fulfilled.
5. NUMERICAL ILLUSTRATIONS In this section, the effectiveness of the proposed MILe is demonstrated. For this purpose, simulation of a nonlinear model of single link direct joint driven manipulator subjected to disturbance has been done. The dynamic equation of the system is
.. 1 If) +(2m + M)gl sine
= T + T
