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obtained from (1) by using the Leibniz–Newton formula
(t) 0 (t 0 (t)) =
t
0
t (t)
_() d:
The initial condition for (35) is given by the vector-valued function on the set
E
01
=
ft 2 R : t = 0 () 0; 0g [ ft 2 R : t = 0 () 0 ( 0 ()) 0; 0g:
It should be observed that each solution of (1) is also a solution of (35); see, e.g., [18]. We also introduce the following differential equation:
y_ (t) = 0( 0 k )y(t) + q(t)y(t 0 (t))
(36)
where
q(t) = 0 k 0
(t)
exp
0
t
0
t (t)
d ()
(37)
and the proof runs similarly to the proof of Theorem 1 for the “new” functional differential equation (35).
[16] M. T. Nihtila, “Finite pole assignements for systems with time-varying input delay,” in Proc. 30th IEEE Conf. Decision and Control, Brighton, U.K., 1991, pp. 927–928. [17] S. Phoojaruenchanachai and K. Furuta, “Memoryless stabilization of uncertain linear systems including time-varying delay,” IEEE Trans. Automat. Contr., vol. 37, pp. 1022–1026, 1992. [18] V. I. Rozkhov and A. M. Popov, “Inequalities for solutions of certain systems of differential equations with large time-lag,” Diff. Eq., vol. 7, pp. 271–278, 1971. [19] T. J. Su and C. G. Huang, “Robust stability of delay dependence for linear uncertain systems,” IEEE Trans. Automat. Control, vol. 37, pp. 1656–1659, 1992. [20] R. Stefan, ¸ “The development of toolbox for time-delay systems modelization. Application to the control of a tank system (in French),” Int. Rep. L.A.G., 1994. [21] E. I. Verriest, “Robust stability of time varying systems with unknown bounded delays,” in Proc. 33rd IEEE Conf. Decision and Contr., Lake Buena Vista, FL, 1994, pp. 417–422. [22] J. A. Walker, Dynamical Systems and Evolution Equations. New York: Plenum, 1980. [23] S. S. Wang, B. S. Chen, and T. P. Lin, “Robust stability of uncertain time-delay systems,” Int. J. Contr., vol. 46, pp. 963–976, 1987. [24] L. Xie and C. E. de Souza, “Robust stabilization and disturbance attenuation for uncertain delay system,” in Proc. 1993 European Contr. Conf., Groningen, The Netherlands, pp. 667–672. [25] L. A. Zadeh, “Operational analysis of variable-delay systems,” in Proc. IRE, 1952, vol. 40, pp. 564–568.
REFERENCES [1] E. Cheres, S. Gutman, and Z. J. Palmor, “Robust stabilization of uncertain dynamic systems including state delay,” IEEE Trans. Automat. Contr., vol. 34, pp. 1199–1203, 1989. [2] C. A. Desoer and M. Vidyasagar, Feedback System: Input–Output Properties. New York: Academic, 1975. [3] L. E. Els’gol’ts and S. B. Norkin, “Introduction to the theory and applications of differential equations with deviating arguments,” Math. Sci. Eng., vol. 105, 1973. [4] J. Hale and S. M. Verduyn Lunel, “Introduction to functional differential equations,” Appl. Math. Sci., vol. 99, 1991. [5] M. Ikeda and T. Ashida, “Stabilization of linear systems with timevarying delay,” IEEE Trans. Automat. Contr., vol. AC-24, pp. 369–370, 1979. [6] R. A. Johnson, “Functional equations, approximations and dynamic response of systems with variable time-delay,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 398–401, 1972. [7] V. B. Kolmanovskii and V. R. Nosov, “Stability of functional differential equations,” Math. Science Eng., vol. 180, 1986. [8] B. Lehman and K. Shujaee, “Dealy independent stability conditions and decay estimates for time-varying functional differential equations,” IEEE Trans. Automat. Contr., vol. 39, pp. 1673–1676, 1994. [9] J. Louisell, “A stability analysis for a class of differential-delay equations having time-varying delay,” in Delay Differential Equations and Dynamical Systems, Lecture Notes in Math., vol. 1475, S. Busenberg and M. Martelli, Eds. Berlin, Germany: Springer Verlag, 1991, pp. 225–242. [10] M. Malek-Zavarei and M. Jamshidi, Time-Delays Systems: Analysis, Optimization and Applications. North-Holland, 1987. [11] T. Mori, “Criteria for asymptotic stability of linear time-delay systems,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 158–160, 1985. [12] S. I. Niculescu, C. E. de Souza, J. M. Dion, and L. Dugard, “Robust stability and stabilization of uncertain linear systems with state delay: Single delay case (I),” in Proc. Workshop Robust Control Design, Rio de Janeiro, Brazil, 1994, pp. 469–474. , “Robust stability and stabilization for uncertain linear systems [13] with state delay: Multiple delays case (II),” in Proc. Workshop Robust Control Design, Rio de Janeiro, Brazil, 1994, pp. 475–480. , “Robust 1 memoryless control for uncertain linear systems [14] with time-varying delays,” in Proc. 3rd European Control Conf., Rome, Italy, 1995, pp. 1802–1808. [15] S. I. Niculescu, “On the stability and stabilization of linear systems with delayed state (in French),” Ph.D. dissertation, INPG, Laboratoire d’Automatique de Grenoble, Feb. 1996.
H
A Discrete Iterative Learning Control for a Class of Nonlinear Time-Varying Systems Chiang-Ju Chien
Abstract—A discrete iterative learning control is presented for a class of discrete-time nonlinear time-varying systems with initial state error, input disturbance, and output measurement noise. A feedforward learning algorithm is designed under a stabilizing controller and is updated by more than one past control data in the previous trials. A systematic approach is developed to analyze the convergence and robustness of the proposed learning scheme. It is shown that the learning algorithm not only solves the convergence and robustness problems but also improves the learning rate for discrete-time nonlinear time-varying systems. Index Terms—Discrete-time, iterative learning control, nonlinear timevarying system.
I. INTRODUCTION The iterative learning control (ILC) method has been proposed by Arimoto et al. [1] for the control systems which can perform the same task repetitively. To date, most of the proposed learning algorithms have been used in applications on robot control where the robot system is required to execute the same motion repetitively, with a certain periodicity. The basic learning controller for generating the present control input is based on the previous control history and a learning mechanism. A recent textbook [2] about ILC for Manuscript received March 13, 1996. This work was supported by the National Science Council, R.O.C., under Grant NSC86-2213-E-211-004. The author is with the Department of Electronic Engineering, Hua Fan University, Shihtin, Taipei Hsien, Taiwan, R.O.C. (e-mail:
[email protected]). Publisher Item Identifier S 0018-9286(98)02087-X.
0018–9286/98$10.00 1998 IEEE
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deterministic systems surveys the literature until 1992. For the ILC of continuous-time systems, a popular design [1], [3], [4] is the socalled D-type ILC. However, since the derivative action destroys the noise suppression of the control system, a P-type ILC has been studied recently for the control of robotic motion [5], [6] and for a class of nonlinear systems [7]–[9]. For real implementation of an iterative learning controller, it is necessary to discretize the systems and store the sampled-data of desired output, system output, and control input in memory. Therefore, it is more practical to design and analyze the ILC systems in discrete-time domain. The optimality of the discrete learning control scheme is studied in [10]. Later, [11] proves that the tracking error in [10] will converge to zero if and only if the input–output coupling matrices are full row rank. In [12], another learning algorithm for the discrete-time system is proposed with the assumption that the state variables are measurable. Recently, the robustness problem has been discussed in [13] for discrete-time linear systems, and the extension to discrete-time nonlinear systems can be found in [14]. Although sufficient conditions are given to ensure the convergence of the learning process, the rate of convergence is often slow since most of the continuous or discrete learning algorithms are designed based on a feedforward structure, and the ith current plant input is generated only by the previous data at the i 0 1th trial. The learning algorithm using more than one past control data for the improvement of learning speed is proposed in [15]. However, the learning structure is only a feedforward learning. The learning process utilizing the advantage of the current feedback error or the feedback configuration can be found in [9] and [16]. In [9], a P-type ILC using current error for the construction of the learning mechanism is proposed. The plant input is updated by the previous plant input and the current error. In [16], a D-type ILC is done in a feedback configuration. The learning process is performed in the feedforward input which is updated by the previous plant input and the derivative of previous error. The rapid convergence is shown in both papers either by technical proof [9] or by simulation [16] when compared with the traditional feedforward learning. In the discrete-time systems little work has been done on the performance improvement for ILC, which is certainly an important issue in practical applications. In this paper, we aim to solve the discrete ILC problem for a class of discrete-time nonlinear time-varying systems with initial state error, input disturbance, and output measurement noise. In order to achieve the control objective, some assumptions, e.g., the global Lipschitz conditions on the nonlinearities, are needed for the technical analysis. These assumptions are not crucial in the field of ILC and are similar to those for ILC design of continuous-time systems [3], [4]. In addition to convergence and robustness problems, the strategies of improving the learning performance are also presented. A feedforward discrete learning algorithm is designed under a stabilizing controller and is updated by more than one past control data in the previous trials. The stabilizing controller is designed to accommodate state error and input disturbance such that the closed-loop output tracking error in the absence of feedforward learning is within a reasonable bound, and the feedforward iterative learning controller is then updated to meet the performance requirement. It is shown that under some sufficient conditions on the feedforward learning operators, the convergence of the learning system can be guaranteed, and these conditions are independent of the stabilizing controller. This means that the feedforward learning can still work in these uncertain environments without the stabilizing controller, but the learning rate can be improved if a suitable one is used. Furthermore, the tracking error on the final iterate will be a class K function of the bounds on the uncertainties. When all the uncertainties tend to zero, the system output will converge uniformly to the desired one. A
numerical example is given to demonstrate the learning performance of the proposed discrete ILC. II. PROBLEM FORMULATION We consider the class of discrete-time nonlinear time-varying systems described by the following difference equation: xi (t + 1) = f (xi (t); t) + B(xi (t); t)ui (t) + wi (t) yi (t) = C(xi (t); t) + i (t)
(1)
where “i”, “t” denote the iteration index and discrete time, respectively, and xi (t) 2 Rn ; ui (t) 2 Rn ; yi (t) 2 Rn for all t 2 [0; N ] and for some positive integer N . Here, wi (t) 2 Rn and i (t) 2 Rn denote some random input disturbance and output measurement noise, respectively. In the following discussion, the notation k1k will denote the Euclidean norm or any consistent norm. Furthermore, we shall assume the following properties for our class of systems (1). A1) For any realizable output trajectory yd (t) and an appropriate initial condition xd (0), there is a unique control input ud (t) generating the trajectory for the nominal plant. In other words, the following difference equation is satisfied with uncertainties wi (t) = 0; i (t) = 0: xd (t + 1) = f (xd (t); t) + B(xd (t); t)ud (t) yd (t) = C(xd (t); t)
(2)
where ud (t) is uniformly bounded for all t 2 [0; N ] with the bound d = supt 2 [0;N ] kud (t)k. (x (t);t) A2) f (xi (t); t); B(xi (t); t); C(xi (t); t) and @C@x are (t) uniformly globally Lipschitz in xi on [0; N ]. That is,
k
0
k
k
0
k 8 2 1 2
h(x1 (t); t) h(x2 (t); t) `h x1 (t) x2 (t) ; t ; (h [0; N ] and for some positive constant `h
0. The functions q : Rn n n 2n s : R R are designed to be uniformly bounded qo ; s(zi (t)) so for some for all zi (t) with q(zi (t)) qo ; so > 0.
k
!
!
!
k k
k
kk
k
k
k k !
k
k
• Feedforward iterative learning controller: m f
ui (t) = j =1
Gj ui0j (t) +
m
j =1
Li0j (t)ei0j (t + 1)
(5)
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where G1 + G2 + 1 1 1 + Gm = I . Here the Li0j (t)’s are bounded learning operators to be designed and with the bound L = supi 2 [0;1] supj 2 [1;m] supt2[0;N ] Li0j (t). • Plant input at ith iteration: b f ui (t) = ui (t) + ui (t): (6) The class of stabilizing controllers is designed such that the closed-loop output tracking error in the absence of feedforward learning is within a bound b > 3 , and the feedforward iterative learning controller is then updated to meet the performance requirement as stated in the control objective when i ! 1. In this feedforward learning controller, the previous plant inputs ui01 (t); 1 1 1 ; ui0m (t) instead of previous feedforward control inputs uif01 (t); 1 1 1 ; uif0m (t) are used to construct the learning algorithm. Before showing the main results of the proposed discrete ILC system, we define the -norm of a vector v (t) [13] as kv(t)k = supt 2 [0;N ] a0t kv(t)k with > 0 and a > 1. Furthermore, we will use the following notations for the discussion in 4 the next section for the sake of convenience: ufi (t) = ud (t) 0 uif (t); 4 4 4 xi (t) = xd (t) 0 xi (t); hi (t) = h(xi (t); t); hd (t) = h(xd (t); t); 4 4 @C (x (t);t) ; C (t) = @C (x (t);t) . h 2 ff; B; C g, Cx (t) = x x (t) x (t) III. ANALYSIS OF CONVERGENCE
AND
ROBUSTNESS
When the discrete iterative learning controller (3)–(6) is applied to the discrete-time nonlinear time-varying system (1), the analysis of convergence and robustness of the learning scheme is to show that for some constant a > 1; > 0 and > 0; kuif (t)k = supt 2 [0;N ] a0t kufi (t)k satisfies
0 1 1 1 0 m ufi0m (t) (7) (z ) = z m 0 z m0 01 1 10 m = 0 having all with the polynomial D f
ui (t)
0
1
f
ui01 (t)
1
1
its roots inside the unit circle. It can be easily shown that (7) implies
lim
i!1
f ui (t)
1 0 0 1 1 1 0 m
1
(8)
which will be the key point to guaranteeing the robustness and convergence of the proposed learning controller. Before showing the main results of the learning controller, we first derive the following fact which states the relation between the output error ei (t) and ud (t) 0 ui (t). Since @C (xi (t + 1); t + 1) @C (xi (t + 1); t + 1) @xi (t + 1) = @ui (t) @xi (t + 1) @ui (t)
= Cx (t + 1)
@ (fi (t) + Bi (t)ui (t) + wi (t)) @ui (t)
= Cx (t + 1)Bi (t) we can derive that ei (t + 1) = C (xd (t + 1); t + 1)
0 C (xi(t + 1); t + 1) 0 i (t + 1)
u (t) @C (x (t + 1); t + 1) d dud (t) = @ud (t) 0
+ C (xd (t + 1); t + 1) u (t)=0 u (t) @C (x (t + 1); t + 1) i dui (t) 0 @ui (t) 0 0 C (xi(t + 1); t + 1) u (t)=0 0 i (t + 1)
= Cx (t + 1)Bd (t)ud (t) 0 Cx (t + 1)Bi (t)ui (t) + C (fd (t); t + 1) 0 C (fi (t) + wi (t); t + 1) 0 i (t + 1)
(9)
= Cx (t + 1)Bi (t)(ud (t) 0 ui (t)) + Cx (t + 1)Bd (t) 0 Cx (t + 1)Bi (t) ud (t) + C (fd (t); t + 1) 0 C (fi (t) + wi (t); t + 1) (10) 0 i (t + 1): We now state the main results in the following theorem. Theorem 1: Consider the discrete-time nonlinear time-varying system (1) satisfying assumptions A1)–A3) and give a desired output trajectory yd (t) on the time sequence [0; N ] with ud (t) satisfying assumption A1). If the discrete iterative learning controller (3)–(6) is used and the learning operator Li0j (t)’s are designed such that j =
sup
sup
i 2 [0;1] t 2 [0;N ]
kGj 0 Li0j (t)Cx
(t + 1)Bi0j (t)k
111;m 0 k
j = 1;
(11)
with all the roots of the polynomial D(z ) = z m 1 z m01 m = 0 inside the unit circle, then we have limi!1 yd (t) for some suitably defined constant > 0 whose level yi (t) of magnitude depends on 1 ; 2 ; and 3 . If 1 = 2 = 3 = 0, then we guarantee that limi!1 yd (t) yi (t) = 0. Proof: The proof consists of three parts. Part I: Derive the main inequality: Using (1), (2), (5), and (10), we have m m f ui (t) = ud (t) Gj ui0j (t) Li0j (t)ei0j (t + 1) j =1 j =1 m Gj Li0j (t)Cx (t + 1)Bi0j (t) = j =1
111 0 k
k
0
0
0 0
k
0
0
2 (ud (t) 0 ui0j (t)) m 0 Li0j (t) Cx (t + 1)Bd (t) j =1
0 Cx
(t + 1)Bi0j (t) ud (t) + C (fd (t); t + 1) 0 C (fi0j (t) + wi0j (t); t + 1) (12) 0 i0j (t + 1) :
Taking norms and using Assumptions A2) and A3), (12) yields m f ui (t) Gj 0 Li0j (t)Cx (t + 1)Bi0j (t) j =1 2 ufi0j (t) 0 uib0j (t) m + L `C bd + `B cd + `C `f kxi0j (t)k j =1 m (L`C 2 + L3 ): (13) + j =1 By (4), we can find that uib0j (t) satisfies b ui0j (t) ro kzi0j (t)k + so `C kxi0j (t)k + so 3 :
(14)
Substituting (14) into (13), we get m f f ui (t) j ui0j (t) j =1 m + `1 (kzi0j (t)k + kxi0j (t)k) + `2 2 + `3 3 j =1 (15)
f
where = maxj 2[1;m] j ; `1 = max ro ; so `C + L(`C bd + `B cd + `C `f ) ; `2 = mL`C ; `3 = mL + mso . Now, investigate
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the properties of kzi0j (t)k and kxi0j (t)k as follows. According to the design of (3), kzi0j (t)k satisfies the following inequality:
kzi0j (t)k po kzi0j (t 0 1)k + qo `C kxi0j (t 0 1)k + qo : 3
(16)
xi0j (t) = fd (t 0 1) + Bd (t 0 1)ud (t 0 1) 0 fi0j (t 0 1) 0 Bi0j (t 0 1)ui0j (t 0 1) 0 wi0j (t 0 1) (17) we can take norms on (17) and use Assumption A2) to yield
3
2
(18) Combining (16) and (18), we have
kzi0j (t)k + kxi0j (t)k a(kzi0j (t 0 1)k + kxi0j (t 0 1)k) + b ufi0j (t 0 1) + + a (19) where a = maxfpo + bro ; qo `C + `f + `B d + bso `C g; a = 2
1 3
1
qo + bso . Without loss of generality, we will assume a > 1 for the following discussion so that recursively (19) can imply the following inequality:
kzi0j (t)k + kxi0j (t)k at (kzi0j (0)k + kxi0j (0)k) +
k=0
t01 k=0
at010k b ufi0j (k)
at010k (2 + a1 3 )
t01
aN 0 1 at010k b ufi0j (k) + aN 1 + ( + a1 3 ) a01 2 k=0 (20)
where zi (0) is chosen to be zero for all i. Now, substituting (20) into (15), we have the following main inequality, which is used to guarantee the robustness and convergence of the learning control system:
ufi (t)
m
j =1
t01 k=0
at010k `1 b ufi0j (k)
+
(21)
sup a0t ufi (t)
j sup a0t ufi0j (t)
j =1
t 2 [0;N ]
`1 b sup a0t t 2 [0;N ]
k=0
t01 k=0
at010k ufi0j (k)
at010k ufi0j (k) t01
a0
1
a0k ufi0j (k) a(01)(k0t) sup a0k ufi0j (k)
sup
t 2 [0;N ] k=0
a0
k 2 [0;N ]
ufi0j (t) 2 sup t 2 [0;N ]
+ sup a0t : t 2 [0;N ]
(22)
t01
k=0 a0(01)N
a(01)(k0t)
a(01)(k0t)
10 = ufi0j (t) 2 a 0 a we can now conclude that (22) implies
ufi (t)
m
j =1 m
+
(23)
j ufi0j (t)
j =1
`1 b
1 0 a0(01)N
a 0 a
ufi0j (t) + : (24)
If we let j = j + `1 b 10aa 0a , then f f ui (t) 1 ui01 (t) + 1 1 1 + m ufi0m (t) + : (25) Because the polynomial D(z ) = z m 0 1 z m01 0 1 1 1 0 m = 0 has all roots inside the unit circle, there always exists a large enough (z ) = z m 0 1 z m01 0 1 1 10 m = 0 has such that the polynomial D all roots inside the unit circle either. Hence, (25) readily concludes that (8) is achieved. Part III: Prove the convergence of yd (t) 0 yi (t) in the sense of -norm: Applying -norm to (19) and using the similar arguments given in Part II, we can get 0(01)N kzi (t)k + kxi (t)k b 1 0aa 0 a ufi (t) + 1 1 + 2 2 + 3 3 (26) N a 0 1 0 1 a where 1 = a ; 2 = a01 ; 3 = a1 a01 . As the iteration approaches infinity 1 0 a0(01)N lim kxi (t)k b i!1 a 0 a 1 0 1 0 1 1 1 0 m
+ 1 1 + 2 2 + 3 3
lim kyd (t) 0 yi (t)k
t 2 [0;N ] m
+
t01
t 2 [0;N ] k=0 t01
i!1
where = 1 1 + 2 2 + 3 3 and 1 = `1 maN ; 2 = `2 + `1 m aa0011 ; 3 = `3 + `1 ma1 aa0011 . Part II: Prove the convergence of ufi (t) by using -norm Multiplying both sides of (21) by a0t , and taking the supremum over [0; N ] we get
j =1 m
t 2 [0;N ]
and hence
j ufi0j (t)
j =1 m
+
sup a0t
1
kxi0j (t)k (`f + `B d)kxi0j (t 0 1)k + b ufi0j (t 0 1) + b(ro kzi0j (t 0 1)k + so `C kxi0j (t 0 1)k + so ) + :
+
Since
= a01 sup
Furthermore, since
t01
751
ilim !1 `C kxi (t)k +
3
4 = (1 ; 2 ; 3 ):
This implies that the output error is bounded 8i in [0; N ], and even the uncertainties that exist will converge to a residual set whose size will depend on the bounds of 1 ; 2 and 3 . Furthermore, if 1 = 2 = 3 = 0, we have limi!1 kyd (t) 0 yi (t)k = 0. Remark: It is found that the convergence conditions (11) of the learning algorithm are independent of the design of the stabilizing controller. However, if the plant output trajectory can stay within a neighborhood of the desired output trajectory by the stabilizing controller, the convergence of the feedforward learning can be very quick. The utilization of more than one past control data in the feedforward learning is another strategy for performance improvement. Intuitively, the tracking speed will be faster if the number m in (5) increases. But it is hard to choose the weighting matrices Gi and the learning operators Li0j to satisfy the convergence condition (11) when the number m is large. This would be the main limitation on the design of the feedforward iterative learning controller using large past control data.
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tracking errors at 50th iteration for three cases are listed in Table I. It can be found that the convergence and robustness are achieved for all three cases. But this discrete learning algorithm, designed under a stabilizing controller and updated by more than one past control data, really improves the learning speed and the learning performance. V. CONCLUSION
Fig. 1. The supremum output tracking error e3sup;i versus iteration i. TABLE I
IV. NUMERICAL SIMULATIONS In this section, a simple discrete-time nonlinear time-varying system with input disturbance and output measurement noise is given as follows: x1i (t + 1) = sin x1i (t) + 2x2i (t) sin t + 0:01randn x2i (t + 1) = 2x1i (t) + cos x2i (t)
0
+ (0:8e
t
sin x1i (t) + 1)ui (t)
yi (t) = 0:2 sin x1i (t) + 0:1x2i (t) + 0:01randn
REFERENCES (27)
where randn is a generator of random number with normal distribution, mean = 0 and variance = 1. In this control problem, the desired output trajectory for t 2 [0; 100] is given to be yd (t) = 0:05t(6 0 0:05t), and the initial states are set to be x1i (0) = 0:5; x2i (0) = 0 so that yd (0) 6= yi (0). For the purpose of comparison, three cases of simulations are discussed as follows. Case 1: We first remove the stabilizing controller and use only feedforward learning. The learning algorithm is constructed by only one past control data, i.e., ufi (t) = ui01 (t) + Li01 (t)ei01 (t + 1). Since Cx (t + 1)Bi01 (t) = 0:08e0t sin x1;i01 (t) + 0:1, we choose the learning operator Li01 (t) as a simple constant one so that 1 =
sup
t
2 [0
k1 0 0:08e0
t
;N ]
sin x1;i01 (t)
0 0:1k < 1:
Case 2: We now design a simple proportional controller ubi (t) = Kei (t) with K = 1 which stabilizes the discrete nonlinear timevarying system (27). The feedforward learning algorithm is the same as that in Case 1. Case 3: In this case, we use the same stabilizing controller as that in Case 2 but replace the feedforward learning algorithm by uif (t) =
G1 ui01 (t) + G2 ui02 (t) + Li01 (t)ei01 (t + 1) + Li02 (t)ei02 (t + 1)
with G1 = 0:9; G2 = 0:1. The learning operators are designed as Li01 (t) = Li02 (t) = 1 so that 1 = 2 =
sup
k0:9 0 0:08e0
sin x1;i01 (t)
0 0:1k 0:88
sup
k0:1 0 0:08e0
sin x1;i02 (t)
0 0:1k 0:08
t
2 [0
t
2 [0
;N ]
;N ]
t
t
In this paper, a discrete iterative learning controller is proposed for discrete-time nonlinear time-varying systems with initial state error, input disturbance, and output measurement noise. A feedforward discrete learning algorithm is designed under a stabilizing controller and is updated by more than one past control data in the previous trials. A systematic approach is developed to analyze the robustness and convergence of the learning system. It is shown that under some sufficient conditions on the learning operators, the robustness and convergence of the learning system can be guaranteed even though the system is uncertain and nonlinear time-varying and the learning rate can be improved greatly. The main result of this paper shows that the uniform boundedness between the system output and the desired output is achieved in each iterate and the tracking error on the final iterate is a class K function of the bounds on the uncertainties. When all the uncertainties tend to zero, the system output will converge uniformly to the desired one.
and hence, the polynomial D(z) = z 2 0 1 z 0 2 = 0 has two roots z1;2 = 0:9631; 00:0831 inside the unit circle. 3 ;i = sup The supremum output tracking error esup t 2 [0;N ] kyd (t) 0 yi (t)k with respect to iteration number i is now shown in Fig. 1 for three cases, respectively. Since there is always an initial output error yd (0) 0 yi (0) = 00:096 0 0:01randn, we choose the tolerance bound to be 3 = 0:15 and find that the control objective is achieved at the 50th iteration for Case 1, 25th iteration for Case 2, and 17th iteration for Case 3, respectively. Furthermore, the supremum output
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