Robust boundary iterative learning control for a class of nonlinear hyperbolic systems with unmatched uncertainties and disturbance

Communicated by Dr. Yuan Yuan

Accepted Manuscript

Robust Boundary Iterative Learning Control for a Class of Nonlinear Hyperbolic Systems with Unmatched Uncertainties and Disturbance Chao He, Junmin Li PII: DOI: Reference:

S0925-2312(18)31081-6 https://doi.org/10.1016/j.neucom.2018.09.020 NEUCOM 19948

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

7 March 2018 21 July 2018 8 September 2018

Please cite this article as: Chao He, Junmin Li, Robust Boundary Iterative Learning Control for a Class of Nonlinear Hyperbolic Systems with Unmatched Uncertainties and Disturbance, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.09.020

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Robust Boundary Iterative Learning Control for a Class of Nonlinear Hyperbolic Systems with Unmatched Uncertainties and Disturbance Chao He, Junmin Li∗

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School of Mathematics and Statistics, Xidian University, Xi’an 710071, P.R. China

Abstract

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In this paper, the robust boundary iterative learning control for the output tracking and disturbance attenuation of the 2 × 2 nonlinear hyperbolic system is addressed. Since the measurement limitation, the control and measurement are implemented at the same boundary of the system and the disturbance is not necessary to be estimated, which makes the iterative learning control be easy in implementation and low in measurement cost. By using the characteristic method, the robust convergence with respect to iteration-varying uncertainties arising from initial states shift, external disturbances, model plants uncertainties and disturbed reference trajectories is analyzed without any model reduction, rigorously. It is shown that the robust convergence bound is continuously dependent on the bounds of the iteration-varying uncertainties. Furthermore, to implement the proposed iterative learning control, the actuator dynamic is considered, also. Finally, with the actuator dynamic, two examples are given to demonstrate the effectiveness of the proposed iterative learning control strategy for the 2 × 2 nonlinear hyperbolic system. Keywords: Disturbance rejection, iterative learning control, hyperbolic system, characteristic method 1. Introduction

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Iterative learning control is an effective method to improve the control performance by utilizing the previous 30 control information. Iterative learning control has the clear framework that the system is repeatable over a fixed time interval and the control is improved trial by trial. The primary iterative learning control is proposed in [1]. With several decades past, the iterative learning control 35 has been widely studied in theory and experimental applications [2, 3, 4, 5, 6]. Currently, the iterative learning control has achieved great development in the finite dimensional system fields, such as [7, 8, 9, 10, 11]. However, for the infinite dimensional system, the iterative learning 40 control is still located on the primary stage. Till now, there has been some works on the iterative learning control for the distributed parameter system. In the earlier work [12], based on the reduced order method, a learning control strategy has been presented for the con- 45 strained digital regulation of the linear hyperbolic system through the obtained ordinary differential equations. Since the infinite dimensional property of the distributed parameters governed by partial differential equations, the iterative learning laws have been proposed in the semi- 50 group form over infinite dimensional space [13]. From the actuator perspective, works [12, 13] both are the distributed control, where the actuator is distributed over the ∗ Corresponding

author Email address: [email protected] (Junmin Li)

Preprint submitted to Neurocomputing

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space domain. Except for the high cost in implementation, in some practical cases, the actuator can not be actuated in the specified domain. Therefore, the boundary iterative learning control should be more general and easy in implementation. For the boundary iterative learning control, based on the Lyapunov stability theory, [14] has proposed a control algorithm augmented with the learning controller to attenuate the unknown periodic disturbance on a transport string system. Similarly, [15] has presented a control strategy combined with the proportional derivative learning controller to attenuate the unknown periodical motion of a class of axially moving material system. The control schemes in [14, 15] are proposed to maintain the stability of the distributed parameter systems. While, the output tracking problem of the distributed parameter systems via the boundary iterative learning control is a quite challenge problem. In fact, in plenty of applications, the evolution motions of the systems can be modelled as many kinds of partial differential equations, in which the dynamics of the systems will become very complicated with different characteristics of the equations. Therefore, there is few works on the boundary iterative learning control for the distributed parameter systems. In recent years, the boundary iterative control for the tracking problem has been came out. In [16], without considering the transient dynamics, a P-type steady state boundary learning law for a class of nonlinear processes has been presented. With physical input output monotonicity, process stability assumption, the convergence has been guaranteed. Instead of considering the steady state, [17] has presented a D-type anticipatory September 20, 2018

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the disturbances dynamic governed by the finite dimensional equation only admits limited disturbances. When the system parameters of the disturbances is unknown, the proposed disturbance rejection control scheme for the linear 2 × 2 hyperbolic system will be invalid. Besides, comparing with the boundary disturbance rejection problem, the specified reference trajectory tracking for the output of the 2 × 2 hyperbolic system with uncertainties and disturbance at any given location remains to be further studied and becomes more challenging. Inspired by this limitations, the iterative learning control which needs little information of the 2 × 2 hyperbolic system will be an effective scheme to tackle the output tracking problem of the 2 × 2 hyperbolic system with unknown disturbances and modelling uncertainties. To our best of knowledge, there is no result on the robust boundary iterative learning control reported for the 2×2 hyperbolic system. Although the complex dynamics of the states of the system with the existence of the nonlinear uncertainties and time-varying disturbance make the tracking problem more complicated, the design of the iterative learning law and convergence analysis are urgent to be tackled with many potential applications. Inspired by discussions above, in this paper, an anticipatory robust learning control is proposed with the boundary measurement collocated with the control input, only. Through the characteristic curves method, the solution of the nonlinear 2 × 2 hyperbolic system with uncertainties and disturbances is transformed into the integral form. Without any model reduction, the fundamental convergence of the iterative learning control is guaranteed with iteration-dependent initial states shift. Based on this result, the robust convergence property of the proposed iterative learning control for the nonlinear 2×2 hyperbolic system with uncertainties subject to the iteration-dependent uncertainties arising from initial states shift, external disturbance, plant model and the reference trajectories is analyzed. Under the bounded assumption of the iterationvarying uncertainties, a robust error convergence bound which is continuously dependent on the the bounds of the nonrepetitive uncertainties is obtained. Then, based on the robust practical convergence property, the implementation and actuator dynamics are discussed, carefully. Finally, the proposed iterative learning law is applied to the outflow level control of the open-canal flows and downhole pressure fluctuation suppression in the managed pressure drilling system. Synthesizing the motivations, the proposed robust iterative learning control and robust convergence results, the main contributions can be summarized as follows:

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iterative learning control strategy for a class of inhomogeneous heat equations, where the completely dynamic of the115 heat equations has been considered by transforming the inhomogeneous heat equation into its integral form and exploiting the properties of the embedded Jacobi Theta functions. In addition to the parabolic equations, [18] has studied a class of distributed parameter systems through120 frequency domain design method. Although many elegant works have been done for the boundary iterative learning control of the distributed parameters systems, comparing with the plenty kinds of distributed parameter systems modelled with many different physical processes, the pro-125 posed boundary iterative learning control schemes above can not be used to deal with the tracking problem of the other kinds of distributed parameter systems, universally, since the huge difference of the characteristics of the dynamics of the distributed parameter systems. Especially130 for the nonlinear 2 × 2 hyperbolic system with uncertainties and disturbances which can be used to represent many kinds of processes [19, 20, 21, 22, 23, 24], the output tracking problem via the boundary iterative learning control, to our best of knowledge, is still a quite challenging problem135 with its complex dynamics, which is one of our motivations, in this paper. From the implementation perspective, the disturbances and modelling uncertainties are unavoidable in the practical plants. As a result, the robust control for the plants on140 the disturbance attenuation have attracted lost of attention. Many important results have been reported recently and references are therein [25, 26, 27]. Therefore, the disturbance attenuation for the nonlinear 2 × 2 hyperbolic system with uncertainties via boundary robust iterative145 learning control is another key of our motivations. In many applications, the disturbance is periodical since the inherent mechanism. For instance, in the managed pressure drilling system, the heave-induced periodical movement can lead to the downhole pressure variation. Therefore,150 it is necessary to maintain the downhole pressure around a specified set point with the unknown time-varying periodical disturbance [28, 29]. Furthermore, in [30], the boundary disturbance rejection for the linear 2 × 2 hyperbolic system whose boundary is coupled with disturbances155 is addressed. Based on the elegant full state feedback or an observer-based output feedback algorithm with the output measurement collocated with the actuator, the boundary disturbance modelled as a finite dimensional linear differential equation with known system matrix is estimated and160 the boundary condition at the opposite side of the actuator is regulated, asymptotically. In spite of the proposed feedback control scheme in [30] can be used to estimate the unknown time-varying disturbances and regulate the boundary conditions accurately, as the time goes to infinity. However, from the implement perspective, the accu-165 rate models are not available and the robust property of the control algorithm for the uncertainties must be guaranteed. Actually, for the 2 × 2 hyperbolic system the nonlinear modelling uncertainties will not be avoided. Besides,

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1. The boundary robust iterative learning control for the nonlinear 2 × 2 hyperbolic system with uncertainties and disturbances is proposed, for the first time, to tackle the output tracking problem at any given position with the measurement at the boundary collocated with the control input, only.

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2. The proposed robust iterative learning control scheme is extended to the nonlinear 2 × 2 hyperbolic system with iteration-varying uncertainties arising from initial states shift, external disturbance, plant models215 and reference trajectory uncertainties, and a robust convergence bound continuously depending on the bounds of the iteration-varying uncertainties is also obtained. 3. Under the proposed learning control, the nonlinear220 uncertainties and the bounded unknown time-varying disturbance are attenuated without any estimation algorithm, which leads to a simple structure in control implementation. 225

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2. 2×2 Nonlinear Hyperbolic System with Disturbance

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Consider the following 2 × 2 repetitive hyperbolic sys-240 tem:

bi (1, t) = ui (t) ,

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ai (0, t) = ρbi (0, t) + vd (t) ,

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∂t ai (x, t) + ε1 (x) ∂x ai (x, t) = f1 ai , bi , t ,
∂t bi (x, t) − ε2 (x) ∂x bi (x, t) = f2 ai , bi , t ,

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yci (t) = αai (x∗ , t) + βbi (x∗ , t) ,

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where x ∈ [0, 1], t ∈ [0, T ] and i = 0, 1, 2, · · · denote the space variable, time variable and iteration index, respectively. Space varying coefficients ε1 (x) > 0 and ε2 (x) > 0 are measurable. Coefficient ρ 6= 0, vd (t) is the unknown time varying disturbance. ui (t) is the control input ac-255 tuated on the boundary of bi (1, t). Functions f1 and f2 representing the modelling uncertainties or nonlinear terms are continuously differentiable functions over domain R × R × R+ , with f1 (0, 0, t) = f2 (0, 0, t) = 0. Since i the limitation of measurement, the measurement ym (t) is260 i located on the same side of control u (t) at x = 1. The controlled output yci (t) with the constants α and β 6= 0 can not be measured directly, since the specified location x = x∗ , x∗ ∈ [0, 1].

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Remark 2. From the practical viewpoint, many physical processes are not performed over the infinite time domain. While, many physical processes modelled as 2 × 2 hyperbolic system evolute over the finite time domain, batch by batch. For instance, in the crystal production instruments, the crystal growth process [34] satisfying balance laws is performed, batch by batch. Also, the conservation law models of the production flows of the production lines in the manufacturing systems or the supply chains [36, 37], such as the semi-conductor production lines, are repetitive 2 × 2 hyperbolic systems, in some cases. Besides, for the the gas transport in the pipes [20, 21] and the fluid transport in canals of the chemical process [19] are both the repetitive 2 × 2 hyperbolic system, since the production of the chemical reactor is actuated, batch by batch. Therefore, the repetitive 2 × 2 hyperbolic system is widely existed and the boundary iterative learning control for this kind of systems with disturbances and unmatched uncertainties is necessary for the practical applications.

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The rest of the paper is organized as follows. In section 2, the characteristic curves method and the estimation of the solution are introduced. Based on the estimation of the solution of 2×2 hyperbolic system, the robust learning control law and robust convergence analysis are given in section 3. Furthermore, the implementation details of the230 iterative learning control and applications to outflow level control of the open-canal flows and the downhole pressure disturbance attenuation in managed pressure drilling are presented in section 4. Finally, section 5 concludes this 235 paper.

Remark 1. The 2 × 2 hyperbolic system is a basic equation in the description of some physical processes. The 2 × 2 hyperbolic system is widely used to model many physical processes which satisfy the balance laws. For instance, the Saint-Venant equations in the flows transport process [19], the isentropic equations in gas pipes transport process [20, 21], the telegrapher equations in the electrical transmission lines [31, 32], the balance law models of the fluid flow in elastic tubes [22, 23], the Aw-Rascle equations in road traffic [33, 34], the chemotaxis motion equation of the living microorganisms [35] and chromatography process [24], etc. All of the processes presented above can be transformed into the characteristic form as 2 × 2 hyperbolic system (1), by introducing the Riemann coordinates transform [24]. Therefore, the proposed boundary iterative learning control for the 2 × 2 hyperbolic system has many potential applications in many practical physical precesses.

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The control objective is to design a sequence of control signal {ui (t)} such that the controlled output yci (t) of (1) can track the specified trajectory yd (t): yci (t) → yd (t) , ∀t ∈ [0, T ] as i → ∞ over a fixed time period T > 0, i by using the measurable signal ym (t) only. Remark 3. It is noticed that the only available measurement data is ai (1, t) which is located at the same side with the control ui (t) and the control ui (t) is implemented at one boundary of the 2 × 2 hyperbolic system. In fact, because the 2 × 2 system is used to model many applications which is large in the space scale, the distributed control actuators and sensors implemented over the whole space domain is high in cost and even impossible with the tough working condition. Therefore, it is reasonable and more practical to consider the boundary iterative learning control with the available measurement at the same side x = 1. In another word, our proposed iterative learning control is easy in implementation and low in measurement

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∂t a∗ (x, t) + ε1 (x) ∂x a∗ (x, t) = f1 (a∗ , b∗ , t) , ∗

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∂t b (x, t) − ε2 (x) ∂x b (x, t) = f2 (a , b , t) ,

a∗ (0, t) = ρb∗ (0, t) + vd (t) ,

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Lemma R1. With the definition of two functions such that: R 1 dr 1 h1 (z) = z ε1dr , h (z) = , the solution a (x, t) of 2 (γ) z ε2 (γ) system (1) is of the form:

b∗ (1, t) = u∗ (t) , yd (t) = αa∗ (x∗ , t) + βb∗ (x∗ , t) , such that umin ≤ u∗ (t) ≤ umax and the desired states a∗ (x, t), b∗ (x, t) are bounded, with the initial states : a∗ (x, 0) = a0 (x), b∗ (x, 0) = b0 (x).

Assumption 2. (Lipschitz condition) For functions f1 and f2 , there exists a constant l > 0 such that |f1 (a1 , b1 , t) − f1 (a2 , b2 , t) | < l (|a1 − a2 | + |b1 − b2 |) and |f2 (a1 , b1 , t) − f2 (a2 , b2 , t) | < l (|a1 − a2 | + |b1 − b2 |), where a1 , a2 , b1 and b2 are the simplification forms of a1 (x, t), a2 (x, t), b1 (x, t) and b2 (x, t) defined on (x, t) ∈ [0, 1] × [0, T ] with | · | denoting the absolute value of a scalar variable.

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ai (x, t) =ρui (t − h1 (0) − h2 (0) + h1 (x)) Z t−h1 (0)+h1 (x)
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Remark 4. Actually, on the iterative learning control, it is impossible to keep the invariance of the initial states of the systems at every iteration, since the unavoidable initial states disturbances. Being similar to the assumption on the iteration-varying initial states of the discrete systems [38, 39], Assumption 3 is introduced to represent iteration-varying initial states shift of the distributed parameter systems (2 × 2 hyperbolic system), which leads to a robust iterative learning control algorithms. And, to our best of knowledge, the robustness of the iterative learning control with respect to the initial states shift, comparing with the important results in [16, 17, 18], is considered, for the first time, which makes the proposed iterative learning control more practical.

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Assumption 3. (Initial states resetting error) For every t−h1 (0)+h1 (x) iteration i, the initial states of the system (1) is located in a set such that ai (x, 0) ∈ Ba0 (ε0 ) , bi (x, 0) ∈ Bb0 (ε0 ) with + vd (t − h1 (0) + h1 (x)) , Bψ0 (ε0 ) = {ψ (x) |maxx∈[0,1] |ψ (x) − ψ0 (x) | < ε0 }, with a0 (x), b0 (x) and ψ0 (x) being the continuous functions320 with h1 (0) − h1 (x) < t ≤ h1 (0) + h2 (0) − h1 (x). defined on x ∈ [0, 1]. ε0 > 0 is a small constant.

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with t > h1 (0) + h2 (0) − h1 (x). And the solution bi (x, t) is of the form:
bi (x, t) = bi h−1 2 (h2 (x) − t) , 0 +

For system (1), the well-posedness can be obtained with 0 ≤ t ≤ h2 (x). through the characteristic method. The characteristic curves are defined as ∗

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ai (x, t) = ai h−1 1 (t + h1 (x)) , 0 +

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Based on the Riemann curves (3) and formula (4), the solution of hyperbolic system (1) has the following form.

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cost, comparing with the distributed control schemes for with t∗ , s∗ being the initial time and initial location of the other distributed parameter systems [12, 13]. 310 zr and zl , respectively, satisfying zr (t∗ , t∗ , s∗ ) = s∗ and zl (t∗ , t∗ , s∗ ) = s∗ . Before the presentation of the main results, some asAccording to the defined curves (3), we have the time sumptions on the 2 × 2 nonlinear hyperbolic system are relationship: necessary. Z t Z zr (t,t∗ ,s∗ ) dγ Assumption 1. (Existence of the desired output and indt = t − t∗ , = put) With known constants umin < umax , assume that, for ε1 (γ) t∗ s∗ (4) Z s∗ Z t the bounded desired output yd (t), there exists the unique dγ ∗ bounded desired input u∗ (t) satisfying dt = t − t . = zl (t,t∗ ,s∗ ) ε2 (γ) t∗

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case ii a)

case ii b)

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Figure 1: Three cases of the solution a (x, t) and b (x, t)

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Proof. For the solution with positive slope a (zr (t, t∗ , s∗ ) , t)375 can be characterised as three cases, as shown in Figure 1. While, for the solution with negative slope b (zl (t, t∗ , s∗ ) , t) Remark 5. Although the nonlinear 2 × 2 hyperbolic syscan be classified into two parts. tem with uncertainties and disturbances has many potenCase i. Consider formula (4), we can find the coordiR x dγ tial applications and many physical processes ([20, 21], nate such that s∗ ε1 (γ) = t − 0, which is equivalent to ∗ 380 etc.) can be transformed into the standard form (1). Howh1 (s ) − h1 (x) = t. Since the defined function h1 (z) ever, to analyze the dynamics of the states of nonlinear is monotone decreasing with respect to z ∈ [0, 1], h1 (z) ∗ 2 × 2 hyperbolic system is not a easy task, since the comis invertible for all z ∈ [0, 1]. Then, s is obtained as −1 ∗ plicated coupled property of the two states ai (x, t) and s = h1 (t + h1 (x)). bi (x, t). By the characteristic curves method, the dynamic Synthesizing the solution on the characteristic curves (3), the hyperbolic system can be transformed as da (zr
(t, 0,385 of the nonlinear 2 × 2 hyperbolic system is transformed s∗ ) , t) /dt = f1 ai (zr (t, 0, s∗ ) , t) , bi (zr (t, 0, s∗ ) , t) , t with into a class of integral equations in different time domains without any model reduction, which is completely different initial state ai (zr (0, 0, s∗ ) , 0) = a (s∗ , 0). Therefore,
the −1 i i from the integral representation of the parabolic system in solution is of the form a (x, t) = a h1 (t + h1 (x)) , 0 +
Rt [17]. In fact, according to the integral representation, the −1 f ai , bi , s ds with 0 ≤ h1 (t + h1 (x)) ≤ 1 which 0 1 390 solution of the parabolic system is continuous with respect means that t ≤ h1 (0) − h1 (x). to space variable and time variable (t > 0), if the nonlinear Case ii. For t > h1 (0) − h1 (x), the solution ai (x, t) function is assumed continuous. While, from the Lemma can be divided into two cases: Case ii a) and Case ii b) 1, the nonlinear 2 × 2 hyperbolic equation may admit a as shown in Figure 1. The characteristic curves can be discontinuous solution, if the initial states are not contindivided into two periods. For the first part, we will obtain 395 uous. Therefore, the dynamic of the nonlinear 2 × 2 hythe time tr , when the curve touches theR boundary x = 0. x dγ perbolic system is completely different from the solution Considering the formula (4), we have 0 ε1 (γ) = t − tr . property of the parabolic system and will be more complex That is tr = t−h1 (0)+h1 (x). Being similar to Case i, the with the nonlinear uncertainties and time-varying disturhyperbolic system (1) over the curves zr (t, tr , 0) is of the bances, which leads to a different iterative learning law form da (zr (t, tr , 0) , t) /dt = f1 ai (zr (t, tr , 0) , t) , bi (zr (t,400 tr , 0) , convergence analysis method. and t) , t) with initial state a (zr (tr , tr , 0) , tr ) = a (0, tr ). FurIn Lemma 1, the ai (x, t) and bi (x, t) can be reprethermore, considering the boundary condition ai (0, t) = i sented as formulas (5)-(9). Furthermore, to derive the ρb (0, t) + vd (t) in (1), one has that convergence property of the iterative learning control, the following estimations of the errors defined as a ¯i (x, t) = Z t
i ∗ ¯bi (x, t) = bi (x, t) − b∗ (x, t) are i i i i 405 a (x, t) − a (x, t) and a (x, t) = ρb (0, tr ) + vd (tr ) + f1 a , b , s ds, (10) needed. tr Lemma 2. With the definition a ¯i (x, t), ¯bi (x, t) and ξ i (t) = Based on formula (10), the solution ai (x, t) can be sepi i ¯ max{ max |¯ a (x, t) |, max |b (x, t) |}, considering the hyarated into two cases according to the solution bi (x, t). x∈[0,1] x∈[0,1] R s∗ dγ Case ii a): From formula (4), one has x ε2 (γ) = t, perbolic system (1) and (2) under Assumptions 1-3, one which is equivalent to h2 (x) − h2 (s∗ ) = t. With the410 has the estimation such that ss   invertible property of h2 (z), the coordinate s∗ can be obtained as: s∗ = h−1 ξ i (t) ≤ kε0 + k max |¯ ui (t) | e2klt , 2 (h2 (x) − t). Then, the solution

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i i bi (x, t) = bi h−1 2 (h2 (x) − t) , 0 + 0 f2 a , b , s ds. According to 0 ≤ h−1 2 (h2 (x) − t) ≤ 1, 0 ≤ t ≤ h2 (x) is obtained. Then, consider (10), one has ai (x, t) = ρbi h−1 (h2 (0)

2 Rt R tr i i i i −tr ) , 0)+ρ 0 f2 a , b , s ds+ tr f1 a , b , s ds+vd (tr ) with tr ≤ h2 (0), which is equivalent to formula (6) with h1 (0) − h1 (x) < t ≤ h1 (0) + h2 (0) − h1 (x). Case ii b): In the last case, as shown in Figure 1, the characteristic curve will touch the boundary x = 1 after touching the boundary x = 0, therefore, it is necessary to get the touch time tl on the x = 1. From formula (4), we R1 have 0 ε2dγ (γ) = tr − tl , which is equivalent to tl = tr − h (0). Simultaneously, we have bi (x, t) = ui (t − h2 (x)) +
R 2t f ai , bi , s ds with t > h2 (x). Then, for tr = t − t−h2 (x) 2 h (0)+h1 (x) > h2 (0), one has bi (0, tr ) = ui (tr − h2 (0))+
R 1tr f ai , bi , s ds. Synthesizing formula (10), fortr −h2 (0) 2 mula (7) can be obtained.

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ACCEPTED MANUSCRIPT Rt where k = max{1, |ρ|} , u ¯i (t) = ui (t) − u∗ (t) and ε0 de|¯ ai (x, t) | ≤ ξ i (0) + 2l 0 ξ i (s) ds. For the second part: notes the resetting initial states error bound in Assumption440 h1 (0) − h1 (x) < t ≤ h1 (0) + h2 (0) − h1 (x), according to 3. formula (6), a ¯i (x, t) can be approximated as:

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Proof. According to Lemma 1, the ai (x, t) and bi (x, t) have different representation form with different x and t. Therefore, the estimation of ξ i (t) can be discussed in several cases as shown in Figure 2. a (x; t)

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Figure 2: Cases of the solution a ¯i (x, t) and ¯bi (x, t).

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Synthesizing cases b1 and b2 with formulas (11) and (12), one has the estimation of ¯bi (x, t) over t ∈ [0, T ]: 465 |¯bi (x, t) | ≤ ξ i (0) + max |¯ ui (t) | + 2l [0,t]

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In the following, the estimation of a ¯ (x, t) over t ∈ [0, T ], which can be divided into three cases as shown in Figure 2: case a1, case a2 and case a3. Firstly, for case a1: 0 ≤ t ≤ h2 (0), the solution of a (x, t) can be classified into two parts. The first part: 0 ≤ t ≤ h1 (0) − h1 (x), then we have the estimation as:

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ξ i (s) ds.

0

(15) Finally, consider the case a3: h1 (0) + h2 (0) ≤ t ≤ T , in this case, the estimation of solution can be estimated through formula (7): |¯ ai (x, t) | ≤|ρ|

(13)

i

435

Furthermore, we consider case a2: h2 (0) < t ≤ h1 (0)+ h2 (0), which can be classified into two or three parts. If h2 (0) < h1 (0), then case a2 can be classified into three parts as shown in Figure 2: part 1: 0 ≤ t ≤ h1 (0) − h1 (x); part 2: h1 (0) − h1 (x) < t ≤ h1 (0) + h2 (0) − h1 (x); part 3: h1 (0) + h2 (0) − h1 (x) < t ≤ h1 (0) + h2 (0). If h2 (0) ≥ h1 (0), then case a3 can be classified into two parts: part 1: h1 (0) − h1 (x) < t ≤ h1 (0) + h2 (0) − h1 (x); part 2: h1 (0) + h2 (0) − h1 (x) < t ≤ h1 (0). For more detail, we consider the estimation of solution with h2 (0) < h1 (0). The case h2 (0) ≥ h1 (0) can be obtained in the similar way. With h2 (0) < h1 (0), the part 1 can be estimated from formula (5) such that |¯ ai (x, t) | ≤ Rt i i ξ (0) + 2l 0 ξ (s) ds. Then, for part 2, according to (6), Rt one has |¯ ai (x, t) | ≤ |ρ|ξ i (0) + 2kl 0 ξ i (s) ds. Finally, by formula (7), part 3 satisfies |¯ ai (x, t) | ≤ |ρ| max |¯ ui (t) | + [0,t] R t−h (0)+h (x) Rt 2|ρ|l t−h11(0)−h21(0)+h1 (x) ξ i (s) ds+2l t−h1 (0)+h1 (x) ξ i (s) ds Rt ≤ |ρ| max |¯ ui (t) | + 2kl 0 ξ i (s) ds. Therefore, synthesizing [0,t]

Furthermore, we consider the case b2: h2 (0) < t ≤ T . From formula (9), one has

CE

430

ξ (s) ds

0

AN US

M

425

445

Firstly, consider the estimation of ¯b (x, t). It is divided into two parts as shown in Figure 2 case b1: 0 ≤ t ≤ h2 (0) and case b2: h2 (0) < t ≤ T . The case b1 can be classified into two parts. The first450 part: 0 ≤ t ≤ h2 (x). With the definition of ¯bi (x, t), from Lemma 1 and Assumption 2, we have |¯bi (x, t) | ≤ ξ i (0) +

Rt Rt i |f2 ai , bi , s −f2 (a∗ , b∗ , s) |ds ≤ ξ i (0)+l 0 |¯ a | + |¯bi ds 0 R t ≤ ξ i (0) + 2l 0 ξ i (s) ds. For the second part: h2 (x) < t ≤ 455 h2 (0), from formula (9) one has |¯bi (x, t) | ≤ max |¯ ui (t) | + [0,t] Rt 2l 0 ξ i (s) ds. Synthesizing the estimations of the two parts, one has the estimation of ˜bi (x, t) for case b1:

ED

420

ξ i (s) ds

0

x

x

ξ i (s) ds

0

with k = max{1, |ρ|}. Then, the estimation of the solution on case a1: 0 ≤ t ≤ h2 (0) is Z t |¯ ai (x, t) | ≤ kξ i (0) + 2kl ξ i (s) ds. (14)

case a2

h2 (0)

t−h1 (0)+h1 (x)

CR IP T

case b2

τ

Z

t−h1 (0)+h1 (x) Z t i

case a3

τ

i

Z

|¯ ui (t) |

t

ξ i (s) ds.

(16)

t−h1 (0)−h2 (0)

Synthesizing cases a1, a2 and a3 with formulas (14), (15) and (16), the uniform estimation of the solution a ¯i (x, t)

6

ACCEPTED MANUSCRIPT

over t ∈ [0, T ] is |¯ ai (x, t) | ≤ kξ i (0) + |ρ| max |¯ ui (t) | + 2kl [0,t]

ξ i (s) ds.

500

0

(17) Then, with the definition of ξ i (t) and estimations (13), (17), one has

[0,t]

Z

505

t

ξ i (s) ds.

(18)

With the iterative learning law (21), we have the following theorem.

0

CR IP T

ξ i (t) ≤ kξ i (0) + k max |¯ ui (t) | + 2kl

475

t

By using Gr¨ onwall’s inequality, from estimation (18), Theorem 1. Consider the 2 × 2 nonlinear hyperbolic syswe have 510 tem (1) with Assumptions 1-3, for the given bounds 1 =   |r|kε0 e2klT (|α| + 2l|β|h2 (x∗ )) + 2|r|lM (|α|h1 (x∗ ) + |β|× ξ i (t) ≤ kξ i (0) + k max |¯ ui (t) | e2klt , (19) h2 (x
∗ )) + Mu ) / (1 − |1 + rβ| − (|rα| + 2l|rβ|h2 (x∗ )) k× [0,t] e2klT + , 2 = (|α| + |β|) kε0 e2klT + (|α| + |β|) ke2klT 1 where ξ i (0) = max{ max |¯ ai (x, 0) |, max |¯bi (x, 0) |} ≤ with any  > 0 and uniform bound |ai (x, t) | ≤ M , |bi (x, t) | x∈[0,1] x∈[0,1] ∗ ∗ ∗ ε0 with Assumption 3. Thus, the estimation of ξ i (t) is515 ≤ M , ∀t ∈ [0, T ] and∗ |u0 (T − ∗h1 (x ) − h2 (x ))−ui (t) | ≤ Mu , ∀t ∈ [T − h1 (x ) − h2 (x ) , T ], the control u (t) and obtained. controlled output yci (t) satisfy

AN US

470

Z

38] where the output can be obtained, directly. From this viewpoint, the proposed iterative learning control (21) eliminates the limitation of the measurement of output. As shown in the augmented error (20) consisting of ai (1, t) and ui (t), all the signals can be obtained from previous iteration and measurement. Therefore, the proposed iterative learning control can be implemented. Besides, because the limited measurement ai (1, t) is used, the proposed iterative learning control is easy in implementation and low in measurement cost.

3. Robust Iterative Learning Control

max |ui (t) − u∗ (t) | ≤ 1 , max |yci (t) − yd (t) | ≤ 2 ,

t∈[0,T ]

AC

490

CE

PT

485

ED

M

480

495

t∈[0,T ]

With the limitation of the application environment,    the measurement sensor can not be implemented over the 1 ∗ as i > log () − log max |u (t) − u (t) | /log (|1+ t∈[0,T ] whole domain x ∈ [0, 1]. An implementable way is that
∗ 2klT the sensor is located at x = 1 in a collocated form with rβ| + (|rα| + 2l|rβ|h2 (x )) ke +1, if there exists learnthe control ui (t). Therefore, the controlled output yci (t)520 ing gain r in (21) such that at x = x∗ can not be available directly. In the following, an iterative learning control strategy by using the avail|1 + rβ| + (|rα| + 2l|rβ|h2 (x∗ )) ke2klT < 1. (22) i (t) = ai (1, t) is proposed to supable measurement ym Proof. Firstly, from the Lemma 1, one has press the disturbance without estimation of the unknown disturbance. Z t+h1 (x∗ ) Before the presentation of the iterative learning control
i ∗ i ∗ a (x , t) = a (1, t + h (x )) − f1 ai , bi , s ds, 1 scheme, we define the following error variable. t (23)
with t ∈ [0, T − h1 (x∗ )]. i i ∗ ∗ i E (t) = αa (1, t + h1 (x ) + h2 (x )) + βu (t) (20) Z t − yd (t + h2 (x∗ )) ,
i ∗ i ∗ b (x , t) = u (t − h (x )) + f2 ai , bi , s ds, 2 ∗ ∗ with t ∈ [0, T − h1 (x ) − h2 (x )]. t−h2 (x∗ ) (24) Then, the iterative learning control law is proposed as: with t ∈ [h2 (x∗ ) , T ]. Based on the formulas (23) and (24), the defined error variable E i (t) over t ∈ [0, T − h1 (x∗ ) −
 i i Sat u (t) + rE (t) ,  525 h2 (x∗ )] can be transformed as   ∗ ∗  t ∈ [0, T − h1 (x ) − h2 (x )] , ui+1 (t) = (21) i  E i (t) =αai (1, t + h1 (x∗ ) + h2 (x∗ )) + βui (t) u (T − h1 (x∗ ) − h2 (x∗ )) ,    ∗ ∗ − yd (t + h2 (x∗ )) t ∈ (T − h1 (x ) − h2 (x ) , T ] ,
= αai (x, t + h2 (x∗ )) + βbi (x∗ , t + h2 (x∗ )) where Sat (·) is the saturation function defined as : Sat (u) Z t+h1 (x∗ )+h2 (x∗ )
= u, if umin < u < umax ; Sat (u) = umax , if u ≥ umax ; (25) +α f1 ai , bi , s ds Sat (u) = umin , if u ≤ umin . r is the learning gain. t+h2 (x∗ ) Z t+h2 (x∗ )
Remark 6. Since the limited measurement ai (1, t) at the − β f2 ai , bi , s ds i controlled boundary, the output yc (t) is unavailable, which t is different from the previous results in [16, 17, 18, 39, − yd (t + h2 (x∗ )) . 7

ACCEPTED MANUSCRIPT

Substituting the estimation of ξ i (t) in Lemma 2 into (29), we get

From the Assumption 1, the specified output is yd (t+ h2 (x∗ )) = αa∗ (x∗ , t + h2 (x∗ )) + βb∗ (x∗ , t + h2 (x∗ )). Simultaneously, being similar to the formula (24), one has

|ui+1 (t) | ≤|1 + rβ||¯ ui (t) | + |rα|ke2klT max |¯ ui (t) | t∈[0,T ]

¯bi (x∗ , t) =¯ ui (t − h2 (x∗ )) Z t

f2 ai , bi , s − f2 (a∗ , b∗ , s) ds, +

2klT

+ |rα|kε0 e

+ 2|rβ|lh2 (x∗ ) ke2klT max |¯ ui (t) |

t−h2 (x∗ )

t∈[0,T ]

(26)

∗

+ 2|rβ|lM h2 (x )

with t ∈ [h2 (x∗ ) , T ]. Therefore, the E i (t) in formula (25) can be further derived as E i (t) =α¯ ai (x, t + h2 (x∗ )) + β u ¯i (t) Z t+h2 (x∗ )

f2 ai , bi , s − f2 (a∗ , b∗ , s) ds +β

+ 2|rα|lM h1 (x∗ ) + Mu .

545

t

−β

Z

t+h1 (x∗ )+h2 (x∗ )

t+h2 (x∗ ) t+h2 (x∗ )

f2

t


f1 ai , bi , s ds

550


a , b , s ds. i

i

(27) Based on the robust iterative learning control law (21) and formula (27), the error u ¯i+1 (t), ∀t ∈ [0, T − h1 (x∗ ) − ∗ h2 (x )] can be approximated as: |¯ u

∗

i

i

= max |α¯ ai (x∗ , t) + β¯bi (x∗ , t) | t∈[0,T ]

ED

= |¯ ui (t) + rβ u ¯i (t) + rα¯ ai (x∗ , t + h2 (x∗ )) Z t+h2 (x∗ )

+ rβ f2 ai , bi , s − f2 (a∗ , b∗ , s) ds t

− rβ

Z

t+h1 (x∗ )+h2 (x∗ )

t+h2 (x∗ ) t+h2 (x∗ )

f2

t


f1 ai , bi , s ds


a , b , s ds|.

PT

+ rα

i

i

555

(28) According to Assumption 2 and the saturated iterative learning law, from the estimation in Lemma 2, we can conclude that the solution ai (x, t) and bi (x, t) are bounded such that |ai (x, t) | ≤ M and |bi (x, t) | ≤ M with M > 0560 for all i = 1, 2, · · · . Also, from the iterative learning control law (21), for t ∈ (T − h1 (x∗ ) − h2 (x∗ ) , T ], we have |¯ ui+1 (t) | ≤ Mu with Mu > 0. Then, synthesizing this inequality, u ¯i+1 (t) can be further estimated as

AC

540

i+1

|¯ u

565

i

i

≤ (|α| + |β|) max |ξ i (t) |

∗

∗

(t) | ≤|1 + rβ||¯ u (t) | + |rα||¯ a (x , t + h2 (x )) | + 2|rβ|lh2 (x∗ ) max |ξ i (t) | t∈[0,T ]

+ 2|rβ|lM h2 (x∗ ) + 2|rα|lM h1 (x∗ )

≤ (|α| + |β|) kε0 esklT + (|α| + |β|) ke2klT max |¯ ui (t) | t∈[0,T ]

≤ (|α| + |β|) kε0 e as i >

CE

535

max |yd (t) − yci (t) |

t∈[0,T ]

(t) | ≤ |u (t) − u (t) + rE (t) |

Z

t∈[0,T ]

µ = |1 + rβ| + (|α| + 2|β|lh2 (x∗ )) |r|ke2klT and d = (|α|+ 2|β|lh2 (x∗ )) |r|kε0 e2klT +2 (|α|h1 (x∗ ) + |β|h2 (x∗ )) |r|lM + d with i > 1. Mu . Therefore, if µ < 1, φi ≤ µi−1 φ1 + 1−µ d For any  > 0, given the prescribed bound 1 = 1−µ + , 1 log()−log φ ( ) d then, φi ≤ µi−1 φ1 + 1−µ ≤ 1 as i > + 1. log(µ) i Furthermore, for the controlled output yc (t), one has

t∈[0,T ]

M

i+1

(30) Then, we have φi+1 = µφi +d, where φi = max |¯ ui (t) |,

AN US

+α

Z

CR IP T

530

+ 2|rβ|lh2 (x∗ ) kε0 e2klT

570

+ Mu . (29) 8

log()−log (φ1 ) log(µ)

sklT

+ (|α| + |β|) ke

2klT

1 = 2 , (31)

+ 1.

Remark 7. Consider the presentation of Theorem 1, the designed learning parameter r will be selected to satisfy the convergence condition (22). When the uncertainties vanishing, the Lipschitz coefficient l = 0, the convergence condition is reformulated as: |1 + rβ| + k|rα| < 1 with k = max{1, |ρ|}. From this inequality, there always exists a learning gain r, if the coefficients α, β and ρ in the 2 × 2 hyperbolic system (1) satisfy |β/ (kα) | > 1. With the learning gain r chosen as |1 + rβ| + k|rα| < 1 and the continuous property of the right side of the convergence condition (22) with respect to l, there always exists a bound l∗ > 0 such that the convergence condition (22) is satisfied over l ∈ [0, l∗ ). Therefore, the proposed iterative learning control law (21) can attenuate the uncertainties and disturbances as l ∈ [0, l∗ ). In Theorem 1, we observe that the tracking error is convergent to a bounded set around 0. However, the convergence bound can be further reduced. In addition, in practical cases, the nonrepetitive uncertainties are unavoidable. As the presentation of [38, 39] where a extended contraction mapping method and sufficient or necessary

ACCEPTED MANUSCRIPT

585

590

Remark 8. Assumption 4 proposed in this paper is similar to the assumption in [38, 39]. Actually, in practical processes, the nominal parameters and trajectory can be obtained through modelling and measurement. And iteration-dependent uncertainties are unavoidable since the inaccurate modelling, but the uncertainties limited into a635 bounded set around the nominal parameters is reasonable. Actually, the parameters ε1 (x) and ε2 (x) have the same property. When the solution of every iteration is continuous (2 × 2 hyperbolic system may include a discontinuous solution [40]), the convergence result on the uncertainties of ε1 (x) and ε2 (x) is similar to the result of the uncertainties in Assumption 4. Since the limitation of space, for simplicity, we will perform the convergence analysis with Assumption 4. With the existence of the iteration-dependent uncertainties in Assumption 4, following the same proof procedure of Lemma 2, the estimation of the a ¯i (x, t) and ¯bi (x, t) can be modified as the following lemma.



max ∗

t∈[0,T −h1 (x )−h2 (x∗ )]

 |¯ u1 (t)|

+ 1, with ¯1 , ¯2 as i > log(δ) defined as (41) and (42), respectively, and the learning gain is designed as r = −θ/β with θ ∈ (0, 1]. Proof. With the augmented error E i (t) defined as (32), being similar to (27), one has E i (t) =α¯ ai (x, t + h2 (x∗ )) + β u ¯i (t) Z t+h2 (x∗ )

+β f2i ai , bi , s − f2 (a∗ , b∗ , s) ds t

+α

Z

Z

t+h1 (x∗ )+h2 (x∗ )

t+h2

(x∗ )

t+h2 (x∗ )

t


f1i ai , bi , s ds


f2i ai , bi , s ds

− y¯di (t + h2 (x∗ )) .

(33) Then, based on (33) and the iterative learning control law (21), for t ∈ [0, T − h1 (x∗ ) − h2 (x∗ )], we have

Lemma 3. Consider the nonlinear 2×2 hyperbolic system (1) with iteration-varying uncertainties under Assumptions 1-4, one has the estimation of ξ i (t) = max{ max |¯ ai (x, t) |,

|¯ ui+1 (t) | ≤ |1 + rβ||¯ ui (t) | + |rα||¯ ai (x∗ , t + h2 (x∗ )) |

x∈[0,1]

615

log(¯ )−log

−β

AC

610

PT

605

CE

600

ED

M

595

CR IP T

580

convergence condition are derived for the discrete-time dyFurthermore, the limitation of coefficients |β/ (kα) | > namical system, the nonrepetitive uncertainties arise from 1 can be further released as |αρ/β| < 1. Then, we have the initial states shift, external disturbances, plant mod-620 the following Theorem. elling uncertainties and reference trajectory uncertainties. Then, according to the definition, the augment error In Theorem 1, the only initial states shift is addressed. E i (t) is written as Therefore, to implement the proposed iterative learning control law, the iteration-dependent external disturbance, E i (t) = αai (1, t + h1 (x∗ ) + h2 (x∗ )) + βui (t) plant model uncertainties and reference trajectory distur(32) bance will be further addressed. − ydi (t + h2 (x∗ )) , Then, we have the following assumption. with t ∈ [0, T − h1 (x∗ ) − h2 (x∗ )], where α, β are the nomAssumption 4. For the iteration-dependent coefficients inal coefficients. αi , β i and ρi , the iteration-dependent external disturbance vdi , the trajectory with uncertainties ydi (t) and
the625 Theorem 2. Consider the nonlinear 2 × 2 hyperbolic sysiteration-dependent modelling uncertainties f i ai , bi , t sat- tem (1) with iteration-dependent uncertainties under Assumptions 1-4 and the iterative learning control law (21). isfied with Assumption 2, there exist the nominal α, β, ρ, If the coefficients |αρ/β| < 1, then for any given ¯ > 0 and vd (t), yd (t) such that δ ∈ [|αρ/β|, 1), there always exists a constant l∗ ≥ 0 such 630 that, for l ∈ [0, l∗ ], the control and output satisfy |¯ αi | ≤ Mα , |¯ ρi | ≤ Mρ , |β¯i | ≤ Mβ , |¯ vdi | ≤ Mv , |¯ ydi | ≤ My ,

i |fm ai , bi , t − fm (a, b, t) | ≤ l |ai − a| + |bi − b| + Mf , max |ui (t) − u∗ (t) | ≤ ¯1 , ∗ )−h (xast )] t∈[0,T −h (x 1 2 with m = 1, 2, α ¯ i = αi − α, β¯i = β i − β, ρ¯ = ρi − ρ, i i i max |yci (t) − yd (t) | ≤ ¯2 , v¯d = vd − vd , y¯d = ydi − yd and some constants Mα ≥ 0, t∈[0,T −h1 (x∗ )−h2 (x∗ )] Mρ ≥ 0, Mβ ≥ 0, Mv ≥ 0, My ≥ 0 and Mf ≥ 0.

AN US

575

max |¯bi (x, t) |} such that

+ 2|rβ|lh2 (x∗ ) max ξ i (t)

x∈[0,1]

t∈[0,T ]

+ |rβ|h2 (x∗ ) Mf

ξ i (t) ≤ kξ i (0) +k max |¯ ui (t) |

+ 2|rβ|lM h2 (x∗ )

x∈[0,t]

+ 2|rα|lM h1 (x∗ )

+Mρ λρ + Mf λf + Mv ) e2klt ,

+ |r|My .

¯ u + 2lh2 (0) M with |ui (t) | ≤ M ¯ u and where λρ = M + M λf = h1 (0) + h2 (0) + |ρ|h2 (0). 9

(34)

ACCEPTED MANUSCRIPT

640

In the following, with a constant defined as ∆ = h1 (0)+ h2 (0) − h1 (x∗ ) − h2 (x∗ ), a ¯i (x∗ , t + h2 (x∗ )) will be estimated. For t ∈ [0, T ], from the solution representation in Lemma 1, the a ¯i (x∗ , t + h2 (x∗ )) can be classified three cases. Case 1, if h1 (0) − h1 (x∗ ) − h2 (x∗ ) ≥ 0, according to Lemma 1, one has

For t ∈ [0, T − h1 (x∗ ) − h2 (x∗ )], substituting (35) into (28), it can be obtained that |¯ ui+1 (t) | ≤ |1 + rβ||¯ ui (t) | + |rαρ||¯ ui (t − ∆) | + 2l|r| [|α| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0))

+|β|h2 (x∗ )] max ξ i (t) + |rα| (kε0 + Mv ) t∈[0,T ]

+ [|rβ|h2 (x ) + |rα| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0))] Mf
¯ u Mρ + |rα| M + 2lh2 (0) M + M + |r|My + 2|rβ|lM h2 (x∗ ) + 2|rα|lM h1 (x∗ ) .

0

645

∗

∗

with t ∈ [0, h1 (0) − h1 (x ) − h2 (x )]. Case 2, t ∈ (h1 (0) − h1 (x∗ ) − h2 (x∗ ) , ∆], from Lemma 1, we have

655

a ¯i (x∗ , t + h2 (x∗ ))
∗ ∗ = ρ¯bi h−1 2 (h1 (0) + h2 (0) − h1 (x ) − h2 (x ) − t) , 0
∗ ∗ + ρ¯i bi h−1 2 (h1 (0) + h2 (0) − h1 (x ) − h2 (x ) − t) , 0 Z t−h1 (0)+h1 (x∗ )+h2 (x∗ )

+ρ f2i ai , bi , s − f2 (a∗ , b∗ , s) ds + ρ¯ + +

t−h1 (0)+h1 (x∗ )+h2 (x∗ )

0 t+h2 (x∗ )

Z

t−h1 (0)+h1 (x∗ )+h2 (x∗ ) i v¯d (t − h1 (0) + h1 (x∗ )

f2i


a , b , s ds i

|¯ ui+1 (t) | ≤ |1 + rβ||¯ ui (t) | + |rαρ||¯ ui (t − ∆) |

i

+ 2l|r| [|α| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0)) +|β|h2 (x∗ )]

∗

∗

+ +2l|r| [|α| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0)) +|β|h2 (x∗ )] Mξ .

(37) Synthesizing Lemma 3, formula (37) is reformulated as

ED

Case 3, t ∈ (∆, T − h1 (x ) − h2 (x )], derived from formula (7), a ¯i (x∗ , t + h2 (x∗ )) is of the form.

+ 650

t−∆ t+h2 (x∗ )

i

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

|¯ ui (t) |

(38)

660

where

i

t+h1 (x∗ )+h2 (x∗ )−h1 (0) i v¯d (t − h1 (0) + h1 (x∗ )

µ ¯ = |1 + rβ| + |rαρ| + kλ (l, x∗ ) e2klT



f1i ai , bi , s − f1 (a∗ , b∗ , s) ds + h2 (x∗ )) .


d¯(ε0 , l, x∗ , Mρ , Mf , My , Mv ) = |rα| + λ (l, x∗ ) e2klT kε0

i

+ [|rβ|h2 (x∗ ) + |rα| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0))  
¯u +λ (l, x∗ ) λf e2klT Mf + |rα| M + 2lh2 (0) M + M  +λ (l, x∗ ) λρ e2klT Mρ + |r|My
+ |rα| + λ (l, x∗ ) e2klT Mv + 2|rβ|lM h2 (x∗ )

|¯ ai (x∗ , t + h2 (x∗ )) | ≤ |ρ||¯ ui (t − ∆) |

+ 2l (h1 (0) − h1 (x∗ ) + |ρ|h2 (0)) max ξ i (t)

+ 2|rα|lM h1 (x∗ ) + λ (l, x∗ ) Mξ .

t∈[0,T ]

+ (h1 (0) − h1 (x∗ ) + |ρ|h2 (0)) Mf
¯ u Mρ + M + 2lh2 (0) M + M

(39)

with λ (l, x∗ ) = 2l|r| [|α| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0)) +|β|h2 (x∗ )] and

¯u Due to Assumptions 1-4 and the bound |u (t) | ≤ M which can be obtained from control (21), synthesizing cases 1, 2 and 3, we have the following estimation

+ kε0 + Mv .

max

|¯ ui+1 (t) |

+ d¯(ε0 , l, x∗ , Mρ , Mf , My , Mv ) ,


a , b , s ds

AC

+

Z

PT

+ ρ¯

f2i

≤µ ¯

CE

i

t−∆ t−∆+h2 (0)

Z

max

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

a ¯i (x∗ , t + h2 (x∗ )) = ρ¯ ui (t − ∆) + ρ¯i ui (t − ∆) Z t−∆+h2 (0)

+ρ f2i ai , bi , s − f2 (a∗ , b∗ , s) ds

ξ i (t) + |rα| (kε0 + Mv )

+ |r|My + 2|rβ|lM h2 (x∗ ) + 2|rα|lM h1 (x∗ )



f1i ai , bi , s − f1 (a∗ , b∗ , s) ds + h2 (x∗ )) .

max

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

+ [|rβ|h2 (x∗ ) + |rα| (h1 (0) − h1 (x∗ ) + |ρ|h2 (0))] Mf
¯ u Mρ + |rα| M + 2lh2 (0) M + M

M

i

(36) As a result of the boundedness of u ¯i (t) with the saturated iterative learning control law (21), the boundedness of ξ i (t) can be obtained from Lemma 2 such that |ξ i (t) | ≤ Mξ . Then, formula (36) can be further written as

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∗ ∗ a ¯i (x∗ , t + h2 (x∗ )) = a ¯i h−1 1 (t + h1 (x ) + h2 (x )) , 0 Z t+h2 (x∗ )

+ f1i ai , bi , s − f1 (a∗ , b∗ , s) ds,

∗

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(40) In formula (39), given l = 0, then the coefficient µ ¯= |1 + rβ| + |rαρ|. Therefore, for any given δ ∈ (|αρ/β|, 1), there always exists the learning r = −θ/β with θ ∈ (0, 1)

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max

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

≤ and

|¯ ui (t) |

d¯(ε0 , l, x∗ , Mρ , Mf , My , Mv ) + ¯ = ¯1 1−δ

(41)

705

max

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

|yci

with x∗ > 0. Then, ∆ = h1 (0)+h2 (0)−h1 (x∗ )−h2 (x∗ ) > 0. By using the Lemma 1, we have, for t > ∆,

(t) − yd (t) |

b∗ (x∗ , t) = −

≤ (ε0 + ¯1 ) (|α| + |β|) ke2klT + (Mα + Mβ ) M = ¯2 , (42)    max

t∈[0,T −h (x∗ )−h2 (x∗ )]

1 as i > log(δ) tracking error estimated as

|¯ u1 (t)|

1 αρ ∗ ∗ b (x , t − ∆) + yd (t) β β

and b∗ (x∗ , t) = b (t), for t ∈ [0, ∆). This system is a continuous time difference equation [41], which is stable as |αρ/β| < 1. Actually, if |αρ/β| > 1, for t ∈ [n∆, (n + 1) ∆) with n = 1, 2, · · · , we have

+ 1, with the

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|αρ/β| > 1, then for most given desired output yd (t), the desired state a∗ (x, t) and b∗ (x, t) will become very large, rapidly, which is not practical for the engineering problem. Then, we give an example with l = 0:  ∂t a∗ (x, t) + ε1 (x) ∂x a∗ (x, t) = 0     ∂ b∗ (x, t) − ε (x) ∂ b∗ (x, t) = 0 t 2 x ∗ ∗  a (0, t) = ρb (0, t)    yd (t) = αa∗ (x∗ , t) + βb∗ (x∗ , t)

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such that |1 + rβ| + |rαρ| < δ. Since the continuous prop-700 erty of l in µ ¯, there always exists a constant l∗ > 0 such that, for all l ∈ [0, l∗ ], µ ¯ ≤ δ. Furthermore, since the inequality (38), being similar to the proof in Theorem 1. For any ¯ > 0, we have

|yci (t) − yd (t) | ≤ |α¯ ai (x∗ , t) + β¯bi (x∗ , t) +α ¯ i ai (x∗ , t) + β¯i bi (x∗ , t) |

b∗ (x∗ , t) =

n  n−1 X  αρ i yd (t − i∆) αρ − b (t − n∆)+ . − β β β i=0

≤ (|α| + |β|) max ξ i (t) + (Mα + Mβ ) M. 710 Therefore, |b∗ (x∗ , t) | will become very large, dramatically, s∈[0,t] i Pn−1  yd (t−i∆) (43) if i=0 − αρ is bounded. Therefore, the deβ β

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Remark 9. In Theorem 2, when |αρ/β| < 1, the robust convergence bound of the proposed iterative learning control (21) for the nonrepetitive 2 × 2 hyperbolic system is 715 analyzed. As the presentation in [38, 39], the nonrepetitive uncertainties (iteration-dependent uncertainties) arising from initial states shift, external disturbances, plants model and reference trajectory are considered in Theorem 2. With a bounded iteration-dependent uncertainties assumption, Assumption 4, from the Theorem 2, the tracking error convergence bound is continuously dependent on the bounds of nonrepetitive uncertainties, which is consistent with the conclusions of [38, 39]. From this perspective, Theorem 2 extends the results of [38, 39] to the 2 × 2 nonlinear hyperbolic systems with nonrepetitive uncertainties. Furthermore, it is shown that if the iteration-varying un-720 certainties αi → α, β i → β, ρi → ρ, ydi → yd , vdi → vd , initial states ε0 → 0, nonlinear uncertainties l → 0 and Mf → 0 as i → ∞, the perfect tracking error can be achieved such that

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t∈[0,T −h1 (x )−h2 (x∗ )]

Corollary 1. Consider the 2 × 2 hyperbolic system with iteration-dependent uncertainties under Assumptions 1-4 and iterative learning control law (21). If |αρ/β| =1, for h

αρ |, 1 any ¯ > 0, 4κ > 0, given a constant δ ∈ | κβ

|yci (t) − yd (t) | = 0,

with

∗

κ = 1 + 4κ, there exists l > 0 such that, for l ∈ (0, l∗ ], max

t∈[0,T −h1 (x∗ )−h2 (xast )]

max

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

|ui (t) − u∗ (t) | ≤ ¯1 ,

|yci (t) − yd (t) | ≤ ¯3 ,

with ¯1 > 0 defined as (41), ¯3 = (ε0 + ¯1 ) (|α| + |κβ|) k × e2klT + (Mα + 4κ|β| + Mβ ) M as

i>

log (¯ ) − log



max

t∈[0,T −h1 (x∗ )−h2 (x∗ )]

log (δ)

 |¯ u (t) | 1

+ 1,

θ and the learning gain is designed as r = − κβ with θ ∈ [0, 1].

as i → ∞. 695

sired state b∗ (x∗ , t) does not exist in practical cases. Thus, in this paper, we consider coefficients satisfying |αρ/β| ≤ 1.

Remark 10. In Theorem 1, the learning gain does exist, if the coefficients satisfy |kα/β| < 1. Furthermore, in Theorem 2, this condition is relaxed into |αρ/β| < 1. When725 |αρ/β| > 1, the tracking problem of the 2 × 2 hyperbolic system will no be well-defined. Actually, if the coefficients 11

Proof. As the proof procedure of Theorem 2, we consider an augmented parameter κβ as the nominal parameter. It αρ is easily seen that | κβ | < 1. From Assumption 4, the iteration-varying coefficient β i satisfies |β i − β| ≤ Mβ .

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4. Implementation of the Robust Iterative Learning Control

4.2. Outflow Level Control of the Open-Canal Flow We consider a channel constructed by two spillways which the left spillway is fixed while the right one is a adjustable spillway. If assumes that [19]: 1) the channel is horizontal, 2) the channel is prismatic with a constant rectangular section and a unit width. Then, the Saint Venant equation with friction effect can be represented by: ∂t H + ∂x Q = 0,   2 gH 2 Q2 Q + + Cf 2 = 0, ∂t Q + ∂x H 2 H

(45)

where Cf denotes the friction coefficient, g is the acceleration of gravity, H (x, t) and Q (x, t) are the flow level and flow rate at (x, t) ∈ [0, L] × [0, T ] with L being the length of the open-canal, respectively. In practical cases, the inflow rate Q (0, t) at x = 0 is disturbed by the upstream unknown disturbance inflow rate qd (t). With the existence of the disturbance of inflow rate Q (0, t), the inflow rate Q (0, t) and outflow level H (L, t) will
not be main¯ H ¯ (L) with u (t) = u tained at the specified set point Q, ¯. Therefore, it is necessary to control the outflow spillway high u (t) to maintain the specified outflow level H (L, t) with the only measurement H (L, t) at x = L. The flow level is controlled through the change of the outflow rate Q (L, t) by adjusting the height of the spillway u (t). Then, boundary condition is 3

¯ d (t) , Q (L, t) = γ (H (L, t) − u (t)) 2 (46) Q (0, t) = Q+q

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To maintain the specified outflow level H (L, t), we can develop
a linear model around the specified set point ¯ H ¯ (x) . With the boundary condition Q (0, t) = Q ¯ and Q, u (t) = u ¯, the steady states satisfy

4.1. Actuator Dynamic In practical implementation, the control signal ui (t) can not be actuated directly with the influence of the actuator dynamic. For instance, in the outflow level control, the spillway is driven by a electric motor system. Also, in the suppression of pressure fluctuations in the managed pressure drilling, the control signal top-side flow velocity is driven by a choke system. Therefore, it is necessary to discuss the influence of the actuator dynamic. According to [42, 43], we model the actuator dynamic as

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Remark 11. Comparing with the disturbance rejection control for the 2 × 2 hyperbolic systems [30], the proposed 790 iterative learning control can deal with the output tracking at any location x∗ ∈ [0, 1], while, the algorithm in [30] is proposed to handle boundary disturbance at given boundary x∗ = 0. Furthermore, a state feedback and observerbased output feedback for the linear 2 × 2 hyperbolic sys795 tem [30] are presented to attenuate the disturbances whose system matrix is known. However, when the dynamic of the disturbance is unknown and the 2 × 2 hyperbolic system is modelled with unknown nonlinear uncertainties or unknown parameters uncertainties, the proposed scheme in [30] will be invalid. While, the proposed iterative learning control scheme can accommodate the uncertainties and the robust convergence can be guaranteed. It is shown from Theorem 1, Theorem 2 and Corollary 1 that the robust convergence can be guaranteed with the unknown800 nonlinear uncertainties whose Lipschitz constants are limited within a given bound l∗ ≥ 0. While, for the linear 2 × 2 hyperbolic system [30], the feedback-based disturbance attenuation is proposed for any known linear terms. The reason inducing this difference is the difference be-805 tween feedback and feedforward strategies [5]. Therefore, the iterative learning control combined with the feedback control can synthesize the advantages of these two methods, which will induce new iterative learning control algorithms. 810

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iterative learning control (21). λ > 0 is a constant denoting the response speed of the actuator. In the following, we will discuss the iterative learning control performance with different λ. In the following applications, the outflow level control of the open-canal flow is the controlled output yc (t) with x = x∗ = 1. While, for the pressure fluctuation suppression of the managed pressure drilling, the downhole pressure fluctuation is equivalent to the controlled output yc (t) at x = x∗ = 0 with measurement ym (t) at x = 1.

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Thus, the new nominal parameter κβ has the estimation: |β i − κβ| ≤ Mβ + 4κ|β|, which satisfies Assumption 4.780 Therefore, as the presentation of Theorem 2, the error convergence bounds at |αρ/β| = 1 are obtained. A larger 4κ will lead to a faster convergence rate δ. While, a larger convergence bound ¯3 will be derived. On the contrary, a smaller 4κ will result in a slower convergence rate and a785 smaller convergence bound. Therefore, the choice of 4κ, a tradeoff between convergence rate and convergence bound can be done by simulation.

¯ = 0, ∂x Q ¯ (x) = ∂x H ¯ Q (0) = Q,

¯ 2 /H ¯ 2 (x) −Cf Q 2 ¯ ¯ ¯ 2 (x) , g H (x) − Q /H


2 ¯ (L) = u ¯ 3. H ¯ + γ −1 Q

(47)


dˆ ui From (47), we can see that if Cf = 0, there exists the = λ ui (t) − u ˆi (t) . (44)815 ¯ ¯ satisfying dt constant steady state H p for anyQ¯ constant Q and p ¯ Q i ¯ + ¯ > 0 and ¯ − g H ¯ < 0. where u ˆ (t) is the control driven by the actuator dynamic fluvial condition: gH H H and ui (t) is the control signal generated from the robust Then, for small enough Cf > 0, because the solution of 12

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¯ (L) . Finally, the adjustable spillway posiH (L, t) − H ¯ tion ui (t) can be of the form:ui (t) = H (L, t) − γ −1 Q+ 2   p
3 ¯ (L) ¯ (L) + ¯Q¯ H (L, t) − H gH . uib (t) + H(L) To illustrate the effectiveness of robust boundary iterative learning control (21), the parameters of system ¯ = 3m, Q ¯ = 10m3 /s, the length of (45) are given as H the channel L = 10m and the characteristic constant of the spillway γ = 10, the constant spillway vertical position u ¯ = 2m, the acceleration of gravity g = 9.81m/s2 . ˜ + ∂x q˜ = 0, ∂t h 850 The disturbance qd (t) can be any bounded time-varying  function. For instance, in this example, the disturbance ¯2  ¯ Q ˜ + 2 Q ∂x q˜ ¯ ∂t q˜ + g H (x) − ¯ ∂x h is chosen as qd (t) = 0, t ∈ [0, 1); qd (t) = (t − 1) /5, ¯ (x) (48) H (x) H t ∈ [1, 6); qd (t) = 1, t ∈ [6, 8); qd (t) = −t/8 + 2, t ∈ ¯ ¯2 Q Q [8, 12); qd (t) = 0.5, t ∈ [12, 25] . The initial conditions ˜ = −2Cf ¯ 2 q˜ + 2Cf ¯ 3 h. H (x) H (x) 855 are given as Q (x, 0) = 10, H (x, 0) = 3. The learning gain is chosen as r = 0.9. Then, with the proposed roConsider system (48), introduce the transform bust iterative learning control law (21), the outflow level ¯ is maintained as shown in Figure 3 without H i (L, t) = H  q ¯ the initial state shift. Furthermore, with the existence of Q ˜ (xL, t) , ¯ (xL) − gH h a (x, t) = q˜ (xL, t) + 860 initial resetting errors H i (x, 0) = 3 + 0.02 sin (i) cos (x), ¯ (xL) H i q Q (x, 0) = 10 + 0.02 cos (i) sin (x), the robust convergence ¯  Q ˜ ¯ property can still be guaranteed within a small bound, as b (x, t) = q˜ (xL, t) − g H (xL) + ¯ h (xL, t) , H (xL) shown in Figures 4 and 5, which is consistent with the ro(49) bust convergence with respect to iteration-varying initial the linearised model (48) can be reformulated as 865 states shift in Theorems 1 and 2. As discussed in section 4.1, consider the implemen  q tation of the control algorithm (21), the control actua¯ Q 1 ¯ (xL) ∂x a (x, t) + g H ∂t a (x, t) + tor dynamic as shown in (44) is studied with different λ. ¯ (xL) L H To show the property clearly, the iteration-varying initial = f1 (Cf , a (x, t) , b (x, t) , x) , shift is settled as 0 such that H i (x, 0) = 3 and (50)870 states q  i ¯ 1 Q Q (x, 0) = 10. The λ is given as λ = 10, λ = 1 and ¯ (xL) − ∂t b (x, t) − gH ∂x b (x, ) ¯ λ = 0.03 to perform the proposed iterative learning control L H (xL) scheme, respectively, and the results are collected in Fig= f2 (Cf , a (x, t) , b (x, t) , x) , ures 4-6. As shown in Figure 4, the convergence rate is dewhere f1 (Cf , a (x, t) , b (x, t) , x) and f1 (Cf , a (x, t) , b (x, t)875 , generated as the actuator coefficient λ decreasing. When λ = 10, the robust convergence is still guaranteed with x) are bilinear functions with respect to Cf , a (x, t), b (x, t) a small error bound which can be find in Figure 5, also. and f1 (0, a, b, x) = f2 (0, a, b, x) = 0. ˜ However, for the λ = 1 and λ = 0.03, the iterative learnFrom the transform (49), to maintain h (L, t) = 0 is ing control will be invalid with a large error bounds as equivalent to let a (1, t) = b (1, t). Furthermore, the bound880 shown in Figure 4. For more detail, comparing λ = 1 ary condition can be converted into with λ = 0.03, in Figure 5, a smaller error bound is observed than λ = 0.03. The main reason of this difference p p ¯ (0) − ¯Q¯ gH ¯ is shown in Figure 6, where the λ = 0.03 case owns an 2 g H (0) H(0) a (0, t) = − p b (0, t) + q (t) , p d obvious larger difference to the control without actuator ¯ ¯ ¯ (0) + ¯Q ¯ (0) + ¯Q gH gH H(0) H(0) 885 dynamic than λ = 1 case. While, there almost no difference between the λ = 10 and the control without actuator b (1, t) = ub (t) , dynamic. y (t) = a (1, t) − ub (t) .

(47) is continuously dependent on Cf , equation (47) has a unique solution for all boundary condition Q (0, t) =
2 ¯ and H ¯ (L) = u ¯ 3 satisfying fluvial condition: Q ¯ + γ −1 Q q p ¯ ¯ Q Q ¯ (L) < 0. > 0 and H(L) g H ¯(L) + H(L) − gH ¯ ¯
¯ (x) , Q ¯ with the defini-845 Then, around the set point H ˜ (x, t) = H (x, t)− H ¯ (x) and q˜ (x, t) = Q (x, t)− Q, ¯ tion of h the linearised model of system (45) can be written as

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(51) where the specified trajectory is specified as yd (t) = 0 and the coefficients are α = 1 and β = −1. From the output y (t), we can see that the outflow level control problem is890 equivalent to the system (1) with the controlled output yc (t) at x = 1 with x∗ = 1. Because the controlled output y i (t) at x = p 1. Then, ¯ (L) × the defined variable E i (t) is E i (t) = y i (t) = 2 g H 895

13

4.3. Pressure Fluctuations Suppression in Managed Pressure Drilling In the drilling operations, to avoid collapsed of the well, or influx of fluids from the surrounding rock formations, it is necessary control the pressure at the bottom of the well carefully. In managed pressure drilling, the annulus is sealed and the mud fluid is controlled by a choke to control the annular pressure, precisely. During the drilling

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(53)

p¯t (x, t) = −

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where x ∈ [0, L], t are the space and time variable, respectively. Besides, p (x, t) is the pressure, q (x, t) denotes the volumetric flow, ρd is the density of the mud, b is the bulk modulus of the mud, F1 is the friction coefficient, g is the acceleration of gravity, A1 is the cross sectional area of the annulus, A2 is the cross sectional area of the drill bit, vd (t) is the velocity disturbance. In physical perspective, the system (52) is controlled through a choke by changing the pressure. Therefore, it is reasonable to treat pL (t) as the control input. What’s more, the top-side volumetric flow qL (t) = q (L, t) is measured. The control objective is designing control pL (t) to maintain the downhole pressure at the set point psp : p (0, t) = psp without estimation of the unknown disturbance vd (t). Furthermore, to shift the state p (0, t) around set point psp to the origin, introduce the transform of variable p¯ (x, t) = p (x, t) − psp + ρd gx. Then, the system can be transformed as

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Figure 5: Outflow level H i (L, t).

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2.05 2 1.95 1.9 1.85

b qx (x, t) , A1 A1 F1 qt (x, t) = − p¯x (x, t) − q (x, t) , ρd ρd p¯ (L, t) = pL (t) − psp + ρd gL

p (L, t) = pL (t)

2

2.75

(52)

b qx (x, t) , A1 A1 F1 qt (x, t) = − px (x, t) − q (x, t) − A1 g, ρd ρd q (0, t) = −A2 vd (t) , pt (x, t) = −

3

0

operation, the heave motion of the rig is compensated to eliminate the influence of the heave. However, after the drilling operation, when the pump stopped and the compensated mechanism becomes invalid since the string is disconnected from the heave compensation mechanism. Then the drilling string moves with the heave motion of the floating rig, which generates the pressure fluctuations in the bottom of the well. For more details, please see the works on the managed pressure drilling [30]. Then, from [30], the model for the annular pressure and flow in a drilling well is given as

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Then, being similar to the transform (49), we introduce the transform of variable as   1 A1 u ¯ (x, t) = q (x, t) + √ p¯ (x, t) , 2 bρd   (54) 1 A1 √ v¯ (x, t) = q (x, t) − p¯ (x, t) 2 bρd and define the new variables

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Figure 6: Actual control action: spillway vertical position

ui

LF1 √ z

u (z, t) = u ¯ (zL, t) e 2

(t).

14

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, v (z, t) = v¯ (zL, t) e

LF − √ 1 z 2

. (55)

bρd

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With transforms (54) and (55), system (52) can be transformed into the standard form (1):

1

0.5

s

1 F1 √LFbρ1 z b uz (z, t) − e d v (z, t) , L ρd 2ρd s b F1 − √lFbρ1 z 1 d u (z, t) , vz (z, t) − e vt (z, t) = L ρd 2ρd ut (z, t) = −

0

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v (1, t) = U (t) ,

Figure 7: Downhole pressure pi (0, t).

with z ∈ [0, 1]. For the measurement, from the transforms (54) and (55), the measurement can be obtained as 1

 LF1 √ A1 (pL (t) − psp + ρd gL) e 2 bρd qL (t) + √ bρd (57) and the actual control law 1 2



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-1

√ LF1 √ bρd bρd qL (t) − 2 U (t) e 2 bρd + psp − ρd gL. A1 A1 (58) From (57), we can conclude that ym (t) = u (1, t) is measurable. Furthermore, consider the control objective p¯ (0, t) = 0, with the transforms (54) and (55), the control objective is equivalent to design the control v (1, t) such that u (0, t) = v (0, t), which means that yd (t) = 0 and the controlled output yc (t) = u (0, t) − v (0, t), which is equivalent to the system (1) with the controlled output yc (t) at the x = 0 with x∗ = 0. Then, based on the robust iterative learning control (21) with the E i (t) = uip (1, t + h1 (0) + h2 (0)) − U i (t) and h1 (0) = h2 (0) = L ρbd , the controller U i+1 (t) is obtained. Equivalently, the actual control pi+1 L (t) is obtained. In order to illustrate the effectiveness of the iterative learning control in the pressure fluctuations suppression without any estimation of the unknown disturbance vd (t), we choose the parameters b = 7317×105 Pa, A1 = 0.024m2 , A2 = 0.02m2 , ρd = 1250kg/m3 , F1 = 10kg/m3 , g = 9.81m/s2 , L = 3000m, psp = 500 × 105 Pa and vd (t) is given as vd (t) = sin(t). The iterative learning control proposed as (21) with the measurable variable q (L, t) is used. The learning gain is chosen as r = 0.6. The initial states are given as p (x, 0) = 500 × 105 (Pa), q (x, 0) = 0(m3 /s). Then, we obtain the results in Figure 7. From Figure 7, the iterative learning control is effective to regulate the downhole pressure p (0, t) without estimation of the unknown disturbance, which is consensus with the conclusions in Theorem 2. Furthermore, consider the actuator dynamic as (44) with λ = 0.5, λ = 0.1 and λ = 0.03. Figure 8 shows the downhole pressure pi (0, t) with λ = 0.5. Although there exists an obvious lag between the actuator dynamic pL (t) =

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It has been shown that the robust convergence bound is continuously dependent on the bounds of the iterationvarying uncertainties. Finally, with the actuator dynamics, two examples, Saint-Venant equation and managed pressure drilling system, have been performed to illustrate the effectiveness of the proposed iterative learning control for the 2 × 2 nonlinear system. In the near future, the iterative learning control for more physical systems governed by the other distributed parameter systems can be further studied.

We would like to express our gratitude to the editor and anonymous reviewers for their hard work and suggestions to improve the quality of the presentation. This work was supported by the Ph.D. Programs Foundation of Ministry of Education of China under Grant 20130203110021 and National Natural Science Foundation of China under Grant No. 61573013.

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Figure 11: Pressure p5 (x, t) without control actuator.

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Figure 12: Volumetric flow q 5 (x, t) with the downhole disturbance vd (t)

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and control signal as shown in Figure 10, the performance is acceptable and there exists a small amplitude of the transient performance. While, for the λ = 0.1 a poor per1020 formance is achieved in Figure 9, since the degeneration of the actuator dynamic comparing with the ideal control signal without actuator dynamic as shown in Figure 10. From Figure 10, when λ = 0.03, there is a large dif1025 ference between the actual control signal p (L, t) and the control signal p (L, t) without actuator dynamic. Therefore, the control performance can not be guaranteed any more. Furthermore, from Figure 11 we can observer that1030 the downhole pressure pi (0, t) is convergent to the specified pressure psp = 500 × 105 (Pa) with i = 5. Also, the boundary disturbance vd (t) on the downhole volumetric flow q i (0, t) is shown in Figure 12. 1035

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In this paper, the robust boundary iterative learning1040 control for the 2 × 2 nonlinear hyperbolic system with uncertainties and disturbance has been addressed. Without any disturbance estimation, the control and measurement have been implemented at the same boundary of1045 the system. The robust convergence of the proposed iterative learning control has been analyzed with respect to iteration-varying uncertainties from the initial states shift, external disturbance, plant model uncertainties and dis-1050 turbed reference trajectory by the characteristic method. 16

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Chao He received the B. degree, M.S. degree in applied mathematics from Xidian University Xi’an, China, in 2011 and 2014, respectively. He is a Ph. D candidate with the School of Mathematics and Statistics, Xidian University, Xi'an, China, since 2014. His current research interests include distributed parameter systems, adaptive control and learning control.

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Junmin Li received the Bachelor Science Degree and Master Science Degree in Applied Mathematics from Xidian University, in 1987 and 1990, respectively, and the Ph. D. degree in Systems Engineering from Xi’an Jiaotong University, Xi’an, China, in 1997. He has been a Professor with Xidian University since 2002. He has authored and / or coauthored more than 160 referred technical papers. His current research interests include robust and adaptive control, optimal control, iterative learning control, hybrid systems and networked control systems.