J Control Theory Appl 2010 8 (3) 309–316 DOI 10.1007/s11768-010-0019-6
On iterative learning control design for tracking iteration-varying trajectories with high-order internal model Chenkun YIN 1 , Jianxin XU 2 , Zhongsheng HOU 1,3 (1.Advanced Control Systems Laboratory, School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing 100044, China; 2.Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576; 3.State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China)
Abstract: In this paper, iterative learning control (ILC) design is studied for an iteration-varying tracking problem in which reference trajectories are generated by high-order internal models (HOIM). An HOIM formulated as a polynomial operator between consecutive iterations describes the changes of desired trajectories in the iteration domain and makes the iterative learning problem become iteration varying. The classical ILC for tracking iteration-invariant reference trajectories, on the other hand, is a special case of HOIM where the polynomial renders to a unity coefficient or a special first-order internal model. By inserting the HOIM into P-type ILC, the tracking performance along the iteration axis is investigated for a class of continuous-time nonlinear systems. Time-weighted norm method is utilized to guarantee validity of proposed algorithm in a sense of data-driven control. Keywords: ILC; High-order internal model; Iteration-varying; Nonlinear systems; Continuous-time
1
Introduction
Iterative learning control (ILC) is an effective control method for improving tracking performance and transient response of uncertain dynamics systems. Control problems generally treated under ILC framework must have some repeatable properties. Systems with repetitively operated dynamics are common research objects, such as hard disk drive [1], robotic manipulator in manufacturing applications [2,3], and traffic flow in consecutive sections of a freeway [4]. Utilizing the repetitiveness in control problem, ILC extends the classical tracking problem from time domain to the second dimension: discrete iteration domain. This enables ILC tracking error to converge in a pointwise sense as enough iterations execute. Another characteristic for ILC is its simple controller versus complicated system dynamics. ILC is an intelligent control algorithm only based on input and output data but not on model. As a data-driven control, it can be applied to systems with unmodeled dynamics, uncertain parameters, strong nonlinearity, or unknown dynamics structure. In this regard, ILC attracted continual attention over the past two decades. Previous ILC works focus on two important issues. One is to revise the initial resetting condition [5]. The other concerns eliminating the need of time-weighted norm. In [6], Xu and Tan use concept of composite energy function to design ILC for continuous nonlinear systems. This method can not only avoid using time-weighted norm but also be applied to local Lipschitz continuous nonlinearity. Similarly, with the help of Lyapunov function, reference [7] gives a unified adaptive ILC framework for un-
certain nonlinear time-varying system. By adjusting a parameter constructed in ILC, the updating law is allowed to switch among a pure time-domain adaptation, a pure iteration-domain adaptive ILC, and a combination of both. Besides the two issues, there are some other attractive topics on ILC research, such as different updating rules, robustness against uncertainty and monotonic convergence of tracking error, etc. Reference [8] summarizes various ILC updating rules including D, P, I, PD, and PID-type and introduces nonlinear type ILC, such as Newton-type and secanttype. References [9, 10] discuss high-order iterative learning control (HOILC), which uses information from more than one previous trial in learning algorithm. Disturbance rejection is considered in [11] with feedback law embedded in ILC. Reference [12] adopts a projection of continuoustime I/O signals onto a finite dimensional parameter space and achieves noise reduction by H2 optimization. Reference [13] lists ILC literature published in recent years elaborately and introduces their results and methodologies briefly. Actually, the most fundamental restriction on iterative learning problem is that the system’s dynamics and other components must meet the assumption of iterationinvariance, which is called strict repetitiveness along the iteration axis. As such, most of the existing ILC works including [1∼12, 14] are based on the iteration-invariance property. However, strict repeatability never happens in the real world. For example, a robotic manipulator can be modeled with time-varying dynamics during one operation interval [0, T ]. In the next trial, it may not be possible for the manipulator to work under the same environment and the
Received 21 January 2010. This work was supported by the General Program (No.60774022), the State Key Program of National Natural Science Foundation of China (No.60834001), and the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (No.RCS2009ZT011). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010
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manipulator dynamics may vary as well. In the field of ILC, repetitiveness holds only conditionally. From this point, some groups recently begin to consider iteration-varying phenomenon, or nonrepetitiveness, a more essential topic in iterative learning problem. Moore et al. [16] point out that iteration-varying phenomenon could appear not only in system dynamics as the above example indicates but also in reference trajectory, parameter, disturbance, and noise. Every component involved in iteration can be iteration varying. With the help of delay operator in time domain (standard Z-transformation), reference [16] converts the dynamics of linear discrete-time systems into a dynamic process in iteration domain; this extension can be applied to timeinvariant parameters, input variable, output variable, and error. This skill is called supervector transformation [17] or lifting technique, which originates from 2-D property possessed by iterative learning systems [18]. By the lifting technique, a linear discrete-time system at finite time interval can be remodeled by a Markov parameter matrix Hp linking input supervector with output supervector, that is, Yk = Hp Uk . Then, original 2-D process becomes a static mapping in the iteration domain. In this regard, convergence can be analyzed using transfer function approach in iteration domain. Following this idea, reference [19] considers that parameters of linear discrete-time system for different iterations are uncertain at some close intervals with known upper and lower bounds (vertices). It is proven that the convergence condition for this iteration-varying problem only depends on the property of vertex parameters. Robustness is also explored in iteration domain but not in classical time domain. The direct benefit from the static mapping is the convenience in ILC design, which can not only focus on learning performance in iteration domain but also eliminate the use of time-weighted norm and initial resetting condition. In the past few years, some ILC schemes have been developed to track various iteration-varying reference trajectory [20∼22]. Regarding the analogy between the discretetime axis and the iterative learning axis, reference [20] extends recursive least square method often used in adaptive control into parameter estimation in iteration domain, and the learning scheme can track iteration-varying trajectory with random initial condition. It should be noted that all of the works [20∼22] use random number or noise to characterize and produce iteration-varying phenomenon in reference trajectory. The results are conservative because the worst case is considered. On the other hand, algorithms proposed in [20] and [22] need more knowledge about nonlinearity in system dynamics, but the significance for ILC as a model-free control approach is abandoned. This paper deals with iteration-varying tracking problem whose reference trajectory is imposed with high-order internal model (HOIM). Harnessing HOIM in reference trajectory can formulate iteration-varying phenomenon quantitatively in the iteration domain. However, the lifting or supervector method is not applicable to continuous-time nonlinear systems. The aim of this work is to explore the implementation of HOIM in the ILC design, while the property for ILC as model-free control are retained, and to provide limit analysis of HOIM-based ILC using the time-weighted
norm method. The paper is organized as follows. Section 2 presents problem formulation and new HOIM-based P-type ILC. In Section 3, main theorem and its proof are given. A numerical example is provided in Section 4. Section 5 concludes the paper.
2 Problem formulation and HOIM-based ILC 2.1 System description Consider the following continuous-time nonaffine nonlinear systems: x˙ i (t) = f (xi , ui , t), (1) y i (t) = C(t)xi (t) + D(t)ui (t),
where x ∈ X ⊆ Rm , u ∈ U ⊆ Rq , and y ∈ Y ⊆ Rq are state, input, and output vectors, respectively; C(t) ∈ C([0, T ], Rq×m ), D(t) ∈ C([0, T ], Rq×q ); and f is unknown vector function. The dynamics process is repeatable over finite time interval [0, T ], as the subscript i ∈ Z+ denotes the ith learning iteration. Assumption 1 The nonlinear function f is global Lipschitz continuous (GLC), namely, for all xa , xb ∈ X and ua , ub ∈ U , there exists constant Lf independent of X and U such that f (xa , ua , t) − f (xb , ub , t) Lf [xa − xb + ua − ub ]. (2) Assumption 2 The nonlinear function f (·) satisfies f (0, 0, t) = 0. (3) Assumption 3 An initial resetting condition is satisfied for state variable xi (0) = 0, ∀i = 1, 2, · · · . (4) It is stricter than the condition xi (0) = xj (0), which is widely assumed in conventional iterative learning problem. However, assumption condition (4) is still easy to implement. 2.2 Iteration-varying reference trajectory Denote y ri (t) ∈ Y the time-varying reference trajectory. The subscript i has the same meaning as described in system dynamics (1). Therefore, the task considered here is to track iteration-varying reference trajectory. Actually, a cluster of time-varying functions {y ri (t)}i=1 are assigned as different desired outputs for different iterations. Although classical ILC can perform perfect tracking when it tracks an iterationinvariant desired trajectory y r (t) as iteration i tends to infinity, it means that the system merely execute the same task in different trials, which can be thought as a simple case of our iteration-varying tracking problem. The essence of ILC algorithm is to seek and approximate the reference model in the time domain without any knowledge of the system’s time-varying model. However, it does not mean that there is not any model for reference trajectory or system dynamics building in iteration domain. Classical ILC can track a trajectory y r (t) because an internal model exists in iteration domain: y r (t) is invariant for different iterations. Next, for constructing the dynamic model in the iteration domain, we adopt HOIM to generate the iterationvarying reference signal y ri (t).
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Because the iteration domain in which the tracking problem will be described is a discrete space, a new shift operator w in the iteration domain is introduced first: (5) w−1 ui (t) = ui−1 (t). Remark 1 The shift operator used here is different from lifting technique that is essentially a Z-transformation in time domain [15∼17, 19]. Shift operation is conducted in iteration domain in which it can be viewed as another Z-transformation. This skill makes quantitative analysis of iteration-varying components possible. In what follows, compact descriptions of iteration-varying reference trajectory is addressed by a difference equation, and manipulation of the difference equation can be reduced to a purely algebraic problem. Assumption 4 The iteration-varying reference trajectory y ri (t) satisfies the following HOIM: y ri+1 = H(w−1 )y ri ,
−1
(6)
where H(w ) : Y → Y is a polynomial operator H(w−1 ) = h1 + h2 w−1 + · · · + hn w−n+1 . (7) hi , i = 1, · · · , n are known coefficients of an nth-order stable characteristic polynomial C(z) z n − h1 z n−1 − · · · − hn . In the compact set W X ×U ×Y ×[0, T ], the solution of (1) exists with respect to given desired outputs y ri (t), for all i ∈ Z+ . Remark 2 HOIM presented here is an iterative mechanism satisfied by the cluster of reference trajectories {y ri (t)}i=1 . It reflects a simple internal law that imposes on reference trajectory in iteration domain. Moreover, the law is essentially iteration varying. We call (6) nth-order internal model in reference signal. The iteration-invariant reference trajectory y ri+1 = h1 y ri is a special case of (6), where h1 = 1 and hj = 0, ∀j 2. Hence, it is categorized as a first-order internal model, and its characteristic polynomial is z − 1. Remark 3 It is important to discuss the stability of a reference trajectory subject to iterative variations. A polynomial is called stable if all roots of its characteristic equation lie inside the unit circle or the roots with unit modulus are single (simple, not repeated). If HOIM (7) is unstable, i.e., there exists zero of C(z) lying outside the unit circle or multiple root on the unit circle, the desired trajectory y ri (t) will become unbounded as i tends to infinity. This unrealistic extreme case is excluded. Actually, we do not pay much attention to the case when all roots of C(z) lie inside the unit circle, called asymptotically stable, since y ri (t) will tend to zero as i approaches to infinity. The paper focuses on such HOIM that its stable characteristic polynomial C(z) has at least one single zero lying on unit circle. In such circumstance, repetitiveness will emerge again in iteration domain, namely, that reference trajectory repeats after several iterations. Take a special second-order internal model Ha (w−1 ) = w−1 for example. Its characteristic equation is z 2 − 1 = 0 with two distinct roots z1 = 1 and z2 = −1. Iteration-varying reference trajectory with r r = yi−1 for all i 2. It this internal model satisfies yi+1 implies that the reference trajectories at the iterations with odd ordinal are described with one time-varying function and another function at the iterations with even ordinal. Hb (w−1 ) = −w1 is another example, whose characteristic
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equation is z 2 +1 = 0 with two pure imaginary roots z1 = j and z2 = −j, where j is unit imaginary number. Iterationvarying reference trajectory with the internal model Hb satr r = −yi−1 for all i 2. This means that reference isfies yi+1 trajectory is repetitive for every four iterations. According to (6) and (7), the reference trajectory with nth-order internal model is y ri+1 (t) = h1 y ri (t)+h2 y ri−1 (t)+· · ·+hn y ri−n+1 (t). (8) Note that n initial trajectories y r1 (t), · · · , y rn (t) are required to propagate the regressor (8). 2.3 HOIM-based ILC design The control objective is to design an iterative learning law, ui (t), such that system output y i (t) tracks the desired output y ri (t) as close as possible ∀t ∈ [0, T ] when i → ∞. The tracking error is defined as ei (t) = y ri (t) − y i (t), ∀ i ∈ Z+ . To achieve this objective, a simple P-type ILC law based on nth-order internal model is proposed as follows: ui+1 (t) = h1 ui (t) + h2 ui−1 (t) + · · · + hn ui−n+1 (t) +γ1 ei (t) + γ2 ei−1 (t) + · · · + γn ei−n+1 (t), (9) or equivalently (10) ui+1 = H(w−1 )ui + Γ (w−1 )ei , −1 where Γ (w ) = γ1 + γ2 w−1 + · · · + γn w−n+1 , and γj are learning gains. Remark 4 The HOIM-based ILC (9) is similar to highorder iterative learning control (HOILC) proposed in references [9, 10], where ILC laws are “higher orderly” developed to improve learning performance with more information from past trials. In HOILC scheme, the coefficients of H(w−1 ) is chosen such that n hj = 1. (11) j=1
This relation can guarantee the stability of polynomial z n − h1 z n−1 − · · · − hn and tracking iteration-invariant reference (first-order internal model) by corresponding ILC. For the ideal case, if past tracking errors ej are zero and past control inputs uj converge to a desired one ur for j up to i, then (9) becomes n hj ur (t) = ur (t), ui+1 (t) = j=1
and the ideal input retains at new learning cycle. As for the HOIM-based ILC, the relation (11) does not hold in general and desired control uri varies at different iterations as y ri is iteration varying. Remark 5 Conventional first-order P-type ILC can track a reference trajectory with first-order internal model and so can HOILC. However, when HOIM is employed in reference trajectory, HOILC cannot tackle new tracking problem because the task becomes more complicated.
3 Convergence analysis A property associated with the stability of polynomial will be used in the following discussion. Proposition 1 A polynomial P (z) = z n − p1 z n−1 − · · · − pn with nonnegative coefficients pj , j = 1, · · · , n, is
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ρj = Lf max γj C(t), ∀j = 1, 2, · · · , n,
asymptotically stable, if and only if n pj < 1.
t∈[0,T ]
(12)
j=1
Proof Necessity can be found in [14]. We prove this sufficiency by contradiction. Suppose (12) does not hold, n n pj 1. Then P (1) = 1 − pj 0. On the namely, j=1
j=1
other hand, we can choose a sufficiently large real number a > 1 such that P (a) > 0 for limited nonnegative coefficients pj . The continuity of P (z) in the real domain implies that there exists at least one root of P lying on [1, a). This contradicts the asymptotical stability of polynomial P (z). In this paper, Euclidean norm x and induced matrix norm A are employed for a vector x ∈ Rm and a matrix A ∈ Rm×m , respectively. For a trajectory x(t) ∈ Rm [0, T ], the time-weighted norm is defined as xλ = sup e−λt x(t), where λ is a positive constant. An int∈[0,T ]
equality that reveals the significance of the time-weighted norm will play an important role in the proof of main theorem. Proposition 2 The time-weighted norm has the following property for an integral t 1 − e(α−λ)T f (·) eα(t−τ ) dτ f (·)λ . sup e−λt 0 λ−α t∈[0,T ] (13) 1 − e(α−λ)T on the right-hand λ−α side of (13) is bounded by positive constant β for any 1 λ>α+ . β Proof Inequality (13) is straightforward. For compar1 − e(α−λ)T ing and β, we consider function (λ − α)β − λ−α 1 1 + e(α−λ)T . If λ − α > , i.e., (λ − α)β − 1 > 0, then β (λ − α)β − 1 + e(α−λ)T > 0 always holds with the help of positiveness of exponential function. Hence, it implies that 1 1 − e(α−λ)T < β, for any λ > α + . λ−α β Our argument on error convergence is a bit similar to conventional HOILC. However, the iteration variance in reference trajectory will make the progress and result different. Theorem 1 For the nonlinear time-varying plant (1) under Assumptions 1∼4 and reference trajectory (8), when applying the P-type ILC with nth-order internal model (9), the tracking error is estimated as n (1−e−λT )2Lf (M1 +M2 ) |hj | Moreover, the coefficient
lim ei λ
i→∞
λ 1−
n j=1
j=1
(ηj + δρj )
, (14)
where M1 = sup xi λ , M2 = sup ui λ , δ=
i∈Z + (Lf −λ)T
1−e λ − Lf
i∈Z +
> 0, ηj = max hj I − γj D(t), t∈[0,T ]
provided the learning gains γj are chosen such that polynomial (15) R(z) = z n − η1 z n−1 − · · · − ηn is asymptotically stable. Proof Insert the reference trajectory with HOIM (8) into tracking error at the (j + 1)th learning cycle, ei+1 =y ri+1 − y i+1 =H(w−1 )y ri − H(w−1 )y i − y i+1 + H(w−1 )y i =H(w−1 )ei − ((Cxi+1 + Dui+1 ) − H(w−1 )(Cxi + Dui )) =H(w−1 )ei − D(ui+1 − H(w−1 )ui ) − C(xi+1 − H(w−1 )xi ). Using control law (9), we obtain error dynamics in iteration domain ei+1 =h1 ei + · · · + hn ei−n+1 − D(γ1 ei + · · · + γn ei−n+1 ) − C(xi+1 − H(w−1 )xi ) n = (hj I − γj D)ei−j+1 j=1
− C(xi+1 − H(w−1 )xi ). Taking the norm on both sides of the above equation and noting that ηi = hi I − γi D, it follows that n ei+1 ηj ei−j+1 j=1
+Cxi+1 − H(w−1 )xi . (16) To evaluate the state-dependent term xi+1 − H(w−1 )xi in (16), Assumption 3 is used: xi+1 − H(w−1 )xi t = (f (xi+1 , ui+1 ) − H(w−1 )f (xi , ui ) dτ 0t = f (xi+1 , ui+1 ) − f (H(w−1 )xi , H(w−1 )ui ) 0 + f (H(w−1 )xi , H(w−1 )ui )−H(w−1 )f (xi , ui ) dτ . Apply Lipschitz continuity and Bellman-Gronwall Lemma, xi+1 − H(w−1 )xi t Lf xi+1 − H(w−1 )xi 0 +ui+1 − H(w−1 )ui dτ + εi t Lf ui+1 − H(w−1 )ui eLf (t−τ ) dτ + εi 0
t n Lf eLf (t−τ ) γj ei−j+1 dτ + εi , (17) 0
j=1
where T εi f (H(w−1 )xi , H(w−1 )ui )−H(w−1 )f (xi , ui )dt. 0
Substitute (17) into (16), n ei+1 ηj ei−j+1 j=1
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+
n j=1
ρj
t 0
ei−j+1 eLf (t−τ ) dτ + εi . (18)
Next, taking time-weighted norm on both sides of (18) and applying Proposition 2, we have n (ηj + δρj )ei−j+1 λ + ελ , (19) ei+1 λ j=1
where ελ = sup e−λt εi . δ(λ)ρj 0 can be made arbit∈[0,T ]
trarily small with a sufficiently large λ = Lf . More precisely, according to Proposition 2, there exists ρ j , Λ1 = Lf + 1 − ηj such that for any λ > Λ1 , n (ηj + δρj ) < 1. (20) j=1
Λ1 can not become infinite because the condition that the polynomial (15) is asymptotically stable is equivalent to n ηj < 1 by Proposition 1. Under (20), the steady tracking
j=1
error is linearly dependent on ελ , lim ei λ
i→∞
1−
n j=1
ελ (ηj + δρi )
,
(21)
provided ελ is bounded. Hence, uniform boundedness of xj λ , uj λ and ej λ for all j ∈ Z + is needed in the following proof. We use induction here. For j = 1, boundedness of initial conditions can guarantee boundedness of control input u1 (t) for t ∈ [0, T ], and its bound is denoted as bu1 = sup u1 (t). Noting the t∈[0,T ]
GLC condition on f and Assumption 2, (1) implies that t
x1 Lf x1 + Lf u1 dτ 0 t Lf u1 eLf (t−τ ) dτ 0 t Lf bu1 eLf (t−τ ) dτ 0
1 − e−Lf T . Lf bu1 Lf The boundedness of e1 can be easily derived from output equation using boundedness of x1 , u1 , and y r1 . Consequently, x1 (t) and e1 (t) are also bounded from a timeweighted norm perspective. For j = i, assume that xk λ , uk λ , and ek λ are bounded by M1 , M2 , and M3 , respectively for k = 1, · · · , i. Now, we consider xi+1 λ , ui+1 λ , and ei+1 λ . The influence of ελ in (19) will become sufficiently small after applying the time-weighted norm. According to Proposition 2, ελ is less than p(λ)f (H(w−1 )xi , H(w−1 )ui ) − H(w−1 )f (xi , ui )λ , where p(λ) (1 − e−λT )/λ. Note GLC condition on f and Assumption 2, f (H(w−1 )xi , H(w−1 )ui ) − H(w−1 )f (xi , ui )λ n
2Lf |hj | xi−j+1 λ + ui−j+1 λ . j=1
Because xk λ and uk λ are bounded by M1 , M2 , re-
spectively, for k = 1, · · · , i, then ελ 2p (λ)Lf (M1 + M2 )
n j=1
|hj |.
(22)
Regarding (20), we choose a sufficient small η > 0, satisn (ηj + δρj ) 1. For positive constant ηM3 , fying η + j=1 2Lf (M1 + M2 ) |hj | such that for any there exists Λ2 = ηM3 λ > max{Λ1 , Λ2 }, n 2p(λ)Lf (M1 + M2 ) |hj | ηM3 . (23) j=1
Then, from (19), (22), and (23), we obtain that, for any λ > max{Λ1 , Λ2 }, n ei+1 λ (ηj + δρj )M3 + ηM3 M3 . (24) j=1
It also means that y i+1 λ is bounded by (M3 + br ) where br = sup y ri λ . i∈Z +
For proving the boundedness of ui+1 λ , rewrite HOIMbased ILC (10) as Γ (w−1 ) (25) ui+1 = −1 ei , H (w ) where H (w−1 ) = 1 − h1 w−1 − · · · − hn w−n is a stable polynomial by Assumption 4. If all poles of (25) lie inside unit circle, i.e., H (w−1 ) is asymptotically stable, then ui+1 will converge to the equilibrium state or oscillate around it with decreasing amplitude, provided that ei is bounded. If there exists a root α of H (w−1 ) lying on the unit circle, we can properly choose ILC gains such that α becomes a root of Γ (w−1 ). By cancelation of neutrally stable pole and zero, all poles of Γ (w−1 )/H (w−1 ) always lie inside the unit circle. Consequently, ui+1 is bounded by the finite constant M2 in both cases. We prove boundedness of xi+1 λ from (1). By GLC condition on f and Assumption 2, we have t f (xi+1 , ui+1 )dτ xi+1 0 t Lf [xi+1 + ui+1 ]dτ. 0
By virtue of Bellman-Gronwall Lemma, we have t ui+1 eLf (t−τ ) dτ, xi+1 Lf 0
(26)
Taking time-weighted norm on both sides of (26) and applying Proposition 2, we have 1 − e(Lf −λ)T Lf ui+1 λ O1 (λ−1 )M2 . xi+1 λ λ − Lf Lf M2 Using Proposition 2 again, there exists a Λ3 = Lf + M1 such that for any λ > Λ max{Λ1 , Λ2 , Λ3 }, O1 (λ−1 )M2 M1 (27) holds. Moreover, xi+1 λ M1 . Hence, the boundedness of xi+1 λ , ui+1 λ , and ei+1 λ is proven. With the help of uniformly boundedness of xj λ , (14) is straightforward from (21) and (22). Remark 6 The steady tracking error may not be eliminated as long as f and HOIM H(w−1 ) are not exchange-
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able. When f is linear with respect to x and u, then εi = 0, and tracking error ei could tend to zero as i approaches infinity. Remark 7 The convergence of tracking error depends on the selection of learning gains γj . Applying classical zplane analysis to (19), the transient response of the error in discrete iteration domain is determined by the distribution of R(z)’s roots.
4
Numerical example
Consider a nonlinear system on time interval [0, 1], ⎧ 1 2 ⎪ ⎨ x(t) ˙ = x(t) + 0.1ecos (5πx(t)) u(t), 1 + |x(t)| (28) ⎪ ⎩ y(t) = (t − cos(10t))x(t) + (1.5 − 100t )u(t), 1 + 100t 100t where D(t) = 1.5 − ∈ (0.5, 1.5] for t ∈ [0, 1]. 1 + 100t We consider a second-order internal model in the iterationvarying reference trajectory, H(w−1 ) = 2 cos(0.1) − w−1 . (29) Two initial reference trajectories are provided: y1r (t) = esin(20t) − 1, y2r (t) = sin3 (50t) + te2t . (30) Fig.1 depicts the cluster of iteration-varying reference trajectories in t − i coordinate.
Fig. 2 The maximum absolute tracking error versus iteration i.
Fig. 3 Output tracking in the 200th trial.
Fig. 1 Iteration-varying reference trajectory satisfying HOIM (29) with initial reference trajectories (30).
HOIM-based ILC is designed as ui+1 (t) = 2 cos(0.1)ui (t) − ui−1 (t) +1.2ei (t) − 1.15ei−1 (t). We assume that the initial inputs are u1 (t) = 0 and u2 (t) = 1, t ∈ [0, T ]. Maximum absolute tracking error |ei |max = max |ei (t)| is used to evaluate the control per-
Fig. 4 Input and desired input in the 200th trial.
t∈[0,T ]
formance, and the error profile in iteration domain is shown in Fig.2. Considering the steady-state error, it becomes acceptable for the tracking performance after about 60 trials. Output tracking and input profile in the 200th trial are illustrated in Figs.3 and 4. A different desired trajectory for tracking in the 189th trial is taken as another example in comparison with the tracking in the 200th trial. With regard to maximum magnir r (t)|, t ∈ [0, T ] is less than 4 (Fig.5), and |y200 (t)| tude, |y189 may be greater than 60. Using the proposed HOIM-based ILC, good tracking performance can also be achieved for the completely different tracking tasks.
Fig. 5 Output tracking in the 189th trial.
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Remark 8 In Fig.1, it is easy to see that the reference trajectory satisfying the HOIM is an oscillatory one. Fig.6 illustrates the desired trajectories in 200 trials at t = 0.22 and t = 0.67. Continuing from Remark 3, this HOIM’s characteristic equation
ries. Thus, more information from past trials are required by HOIM-based ILC to update the iteration-varying law in iteration domain.
C(z) z 2 − 2 cos(0.1)z + 1 = 0
In this paper, a new ILC scheme based on HOIM is developed for a class of iteration-varying reference trajectory tracking problem. Profiting from using classical Ztransformation in the discrete iteration domain, the nonrepetitiveness phenomenon is quantified in a simple and straightforward form for the first time by using a polynomial structure. The boundedness property and associated conditions of HOIM-based ILC are analyzed and presented clearly. Theoretical analysis and simulation results verify the effectiveness of the proposed ILC algorithm.
(31)
has a pair of conjugate roots on the unit circle. According to the limit theory of second-order difference equation, HOIM (29) leads to the neutral stability of the reference trajectory in the iteration domain with periodic oscillation. Regarding HOIM as Z-transformation in discrete iteration domain, we have the solution yir (t) = cos(0.1i)c1 (t) + sin(0.1i)c2 (t). Hence, there is a hidden periodicity in iteration domain N = 2π/0.1 ≈ 62.8, which is an irrational number but not an integer. In the presence of nonintegral periodicity, strict repetitiveness never appears in iteration domain, but oscillation does for about every 63 iterations. The phenomenon is strictly iteration varying unless N becomes an integer. Special HOIM examples Ha (w−1 ) and Hb (w−1 ) presented in Remark 3 are with strict repetitiveness because they have integral periodicity Na = 2 and Nb = 4 in iteration domain, respectively. The simple case is called partially iteration varying.
5 Conclusions
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Fig. 6 Reference trajectories generated by HOIM (29) in iteration domain at t = 0.22 and t = 0.67 (with circle).
Remark 9 Despite the linear HOIM with finite order that may evoke repetitiveness or oscillation for reference trajectory in the iteration domain, ILC (9) indeed benefits from the internal model comparing with conventional ILC. If an integral periodicity N exists for the iterationvarying tracking problem, classical ILC algorithm still take effect when it is used in batch mode, namely, we can update ILC scheme for every N iterations as a batch. The batchwise updating scheme must be based on partially iterationvarying HOIM. Another shortcoming is that it is in need of more memory, in proportion to the periodicity N , to store information in last batch (consists of N iterations). Consequently, the only way leading to convergence is to execute enough batches, not merely enough iterations. Comparatively, storage is not a tough problem for HOIM-based ILC except that the order of internal model is very high. In such circumstance, the internal model in reference trajectory is much more complicated than the one with low order, and can be described only by more past reference trajecto-
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[email protected].
Jianxin XU received the Bachelor’s degree from Zhejiang University, China, in 1982. He attended the University of Tokyo, Japan, where he received his Master’s and Ph.D. degrees in 1986 and 1989, respectively. All his degrees are in Electrical Engineering. He worked for one year in Hitachi Research Laboratory, Japan, one year in Ohio State University as a visiting scholar, and 6 months at Yale University as a visiting research fellow. In 1991, he joined the National University of Singapore, and is currently a professor in the Department of Electrical and Computer Engineering. His research interests include learning control, variable structure control, and machine learning. He has produced 130 peer reviewed journal papers, more than 230 peerrefereed conference papers, two monographs and three edited books published by Springer-Verlag, Kluwer Academic Press, and World Scientific. E-mail:
[email protected].
Zhongsheng HOU received his bachelor and master degrees in Applied Mathematics from Jilin University of Technology, China, in 1983 and 1988, respectively, and his Ph.D. degree in Control Theory from Northeastern University, China, in 1994. From 1988 to 1992, he was a lecturer with the Department of Applied Mathematics, Shenyang Polytechnic University, China. He was a postdoctoral fellow with Harbin Institute of Technology, China from 1995 to 1997, and a visiting scholar with Yale University, New Haven, CT, from 2002 to 2003. In 1997, he joined Beijing Jiaotong University, China, and is currently a professor with the Department of Automatic Control. His research interests cover the model free adaptive control, data-driven control, learning control, and intelligent transportation systems. Corresponding author of this paper. E-mail:
[email protected].
