Synthesis of Real-Time-Feedback-Based 2D Iterative Learning

Article pubs.acs.org/IECR

Synthesis of Real-Time-Feedback-Based 2D Iterative Learning Control−Model Predictive Control for Constrained Batch Processes with Unknown Input Nonlinearity Dewei Li,‡,¶ Yugeng Xi,‡ Jingyi Lu,¶ and Furong Gao*,¶ ‡

Department of Automation, Shanghai Jiao Tong University, Shanghai, China Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR

¶

ABSTRACT: Batch process is an important category of industrial processes. Recently, the combination of the real-time-feedback-based iterative learning control (ILC) and the model predictive controller (MPC) has demonstrated its advantage when applied to the batch process. In practical applications, the plants are always nonlinear and the model cannot be known exactly. Therefore, how to design such a control strategy for the constrained batch processes with unknown input nonlinearities and guarantee the convergence is interesting and valuable. Inspired by the dualmode MPC, this paper proposes a two-mode framework for the constrained ILC−MPC to solve this problem, which is constructed by a real-time-feedback-based strategy followed by a run-to-run strategy. Under the proposed framework, an invariant updating strategy based on the runto-run strategy is developed for constrained batch processes to act as the second mode, which describes the situation of infinite batch and gives an estimation on the upper bound of the influence from the nonlinearity. Then, a real-time-feedback-based ILC−MPC with a twodimensional (2D) model is designed for the considered batch process, which adopts the time-varying horizon, and is proven to be recursively feasible and convergent. The case studies verify the results of the paper.

1. INTRODUCTION As a well-known category of industrial processes, the batch process is common in practical applications, such as chemical processes,1 motion control,2 and even the biomedical process.3 For a batch process, the periodical character is its important property and the control goal is to steer the system output to track a given reference trajectory during each batch. Because of its importance for industries, many researchers have paid much attention to the control of batch processes. Therein, the iterative learning control (ILC) method, which was initially proposed by Arimoto et al.4 and Uchiyama et al.5 to deal with processes with repetitiveness, is a popular method. As discussed in the survey paper of Wang et al.,6 by making use of the periodical character, ILC provides the integrative action along the batch direction to erase or reduce the tracking error, which is similar to the repetitive control.7,8 For a practical batch process, it is unavoidably restricted by some constraints, such as the valve opening limit, etc. Meanwhile, optimal performance is also expected. This motivates researchers to introduce model predictive control (MPC) into ILC due to its capability of explicitly handling constraints and optimizing the control performance.9,10 As the batch process is a periodic process, to improve the tracking result and handle the constraints, the ILC−MPC can naturally be designed by extending the normal optimization ILC (NOILC)11 with the receding horizon optimization.12 In this design, the input © XXXX American Chemical Society

sequence of the next batch is calculated before its beginning based on the results of the current batch. The similar design also includes Chu et al.,13 where a different approach was adopted to handle input constraints. Since this kind of ILC−MPC calculates the control inputs at the beginning of each batch, we call it the run-to-run ILC−MPC, which is borrowed from the work of Lee et al.7 The main disadvantage of run-to-run ILC−MPC is that it cannot reduce the influences from some nonrepetitive factors within each batch since it is a feed-forward control in nature and the control actions have been determined at the beginning of each batch. This motivates the real-time-feedback-based ILC− MPC. Compared with the run-to-run ILC−MPC, the real-timefeedback-based ILC−MPC calculates the control input of each time instant within a batch according to the tracking errors of both the previous batch and the previous time instant. In the work of Lee et al.,14 a batch-MPC was proposed to integrate ILC with MPC for a batch reaction system. Lee et al.15 further extended this method to develop an integrated end product and a transient profile control technique. In addition, Lee et al.16 provided a more complete form to address the problem of Received: Revised: Accepted: Published: A

August 26, 2016 November 23, 2016 November 26, 2016 November 26, 2016 DOI: 10.1021/acs.iecr.6b03275 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research disturbances, stability, and constraints. In 2007, Shi et al.17 combined ILC with generalized predictive control (GPC) for which a 2D model was adopted. A multicycle prediction was applied and better control performance was obtained. Wang et al.18 applied the ILC−MPC to processes with a multiphase nature and constraints. Alternatively, Chin et al.19 presented a method to integrate ILC with MPC in a separate way, so that repetitive and nonrepetitive disturbances could be rejected separately. For the practical applications, the nonlinearity is another important issue, which may influence the control performance and even make the process unstable. In terms of the iterative control of nonlinear processes, many designs have been proposed in previous literature. In Cueli et al.,20 a run-to-run iterative MPC was designed, which linearized the nonlinear model and then proposed the design. Meanwhile, for the unconstrained case, Cueli et al.20 gave the stability analysis. Utilizing the linearization method20,21 can also give the iterative control design for the nonlinear model. In addition, Liu et al.22 adopted the fuzzy model of nonlinear processes to design the ILC−MPC. However, it is well-known that it is not an easy task to establish the nonlinear model for a practical application. Because of this factor, Chi et al.23 considered the process with unknown nonlinearities. An adaptive strategy was utilized and based on it a run-to-run ILC controller was developed. But the design in Chi et al.23 did not take the system constraints into account. Meanwhile, the adaptive strategy may be influenced by the external disturbance. In this paper, we focus on the batch process with unknown input nonlinearities. A real-time-feedback-based ILC−MPC is developed to handle the system constraints and optimize the control performance. First, we convert the nominal system model of the original controlled process into a 2D-model. To guarantee the convergence of the iterative control, inspired by the dual-mode control framework in MPC literature,9 we propose a two-mode framework of the real-time-feedbackbased ILC−MPC which is constructed by a real-time-feedbackbased strategy (the first mode) followed with a run-to-run strategy (the second mode). To approximate the output sequence of batch infinity, an invariant updating strategy of batch processes is developed for the second mode, which follows the run-to-run strategy and can ensure the constraints and give the estimation of the output sequence at batch ∞. In this way, the proposed ILC−MPC can optimize both the free control inputs (the first mode) and the invariant updating strategy (the second mode) to minimize the objective function. Furthermore, the proposed ILC−MPC adopts the time-varying horizon design to ensure the recursive feasibility. Since the output sequence of batch ∞ is considered into the objective function, the design can guarantee the convergence and optimize the control performance. This paper is organized as follows. Section 2 introduces the considered system model and its corresponding 2D model of nominal system. After that, section 2 also introduces some backgrounds about the ILC−MPC. In section 3, the invariant updating strategy of batch processes is developed and its characters are introduced in detail. The two-mode framework and the complete design of the proposed ILC−MPC are introduced in section 4. The proposed design is proved to be recursively feasible and convergent. Some further discussion about the proposed design is also given in section 4. The case studies of injection molding are used to verify the results of the paper in section 5.

Notation: 0 and I are the zero matrix and identity matrix with proper dimensions, respectively. diag(F) is a diagonal matrix with proper dimension and the diagonal element as F. a(k, t) is the variable a at time t of batch k and a(.|k, t) is the a(.) predicted at time t of batch k. ai is the ith element of vector a. a ⃗(k) is a vector composed by all the vector a(k, t) at batch k according to the time index t. a ⃗(k , tL+ 1|k , t ) is a vector composed by vectors from a(k, t + 1|k, t) to a(k, t + L|k, t) according to the time index. ||a||∞ is the infinite norm of vector a. For two integers r1 ≤ r2, I[r1, r2] = {r1, r1 + 1,···, r2}. ||x||2X = xTXx.

2. PROBLEM STATEMENT AND BACKGROUND 2.1. System description. Consider a batch process with the system model x(k , t + 1) = Ax(k , t ) + Bu(k , t ) + f (u(k , t ))

(1)

y(k , t ) = Cx(k , t )

(2)

where x(k, t) ∈ Rn is the measurable system state, u(k, t) ∈ Rm is the control input, y(k, t) ∈ Rl (l ≤ m) is the system output, and f(u(k, t)) is the unknown nonlinear dynamic related with the control input. When ignoring item f(u(k, t)), we can get the nominal model of system 1−2 as follows. x o(k , t + 1) = Ax o(k , t ) + Bu(k , t )

(3)

y o (k , t ) = Cx o(k , t )

(4)

The cycle time of each batch is fixed as L. The control goal of batch processes is to steer the system output to track the reference signal r(t) (t ∈ I[1, L]) and the sequence of r(t) is denoted as r ⃗ . In addition, for the batch process, the initial condition of each batch is reset as zero, i.e. x(k, 0) = 0 and u(k, i) = 0 (i < 0). Here, for the unknown nonlinear dynamic f(u(k, t)), we make the following assumption. Assumption 1 Nonlinear function f(.) is globally Lipschitz, that is, ∥f (u1) − f (u 2)∥ ≤ μ∥u1 − u 2∥

(5)

where μ < ∞ is the positive Lipschitz constant. Here, we consider a time-invariant system with unknown nonlinear input dynamics. For other model formulations, the results in the paper can also be applied by some extension or transformation, for example, the time-variant system with periodical operations.7 The considered input constraints of the batch process are given as below. u̲ i ≤ ui(k , t ) ≤ ui̅ ,

ui̅ > 0,

u̲ i ≤ 0

δui ≤ ui(k , t ) − ui(k , t − 1) ≤ δui ,

(6)

δui < 0,

δui > 0 (7)

dui ≤ ui(k , t ) − ui(k − 1, t ) ≤ dui , dui < 0, dui > 0 (8)

where i ∈ I[1, m]. Here, the output constraints (or state constraints) are not considered since the possible external disturbance and uncertainties of the controlled plant in practical applications always make the insurance of these constraints invalid. B

DOI: 10.1021/acs.iecr.6b03275 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research Considering system 1−2, we first transfer it as a 2D model to include system dynamics of both the time axis and the batch axis. With the following notations

Y (k + 1) = Sx(k , 0) + GU (k + 1)

where x(k, 0) = 0 is the initial state of each batch, Y (k + 1) = [y T (k + 1, 1), y T (k + 1, 2), ...,

dx(k , t ) = x(k , t ) − x(k , t − 1),

y T (k + 1, L)]T

du(k , t ) = u(k , t ) − u(k , t − 1), dr(k , t ) = r(k , t ) − r(k , t − 1),

U (k + 1) = [u T(k + 1, 0), u T(k + 1, 1), ...,

e(k , t ) = r(k , t ) − y(k , t ),

u T(k + 1, L − 1)]T

δx(k , t ) = dx(k , t ) − dx(k − 1, t ),

⎡ CA ⎤ ⎢ 2⎥ ⎢ CA ⎥ S=⎢ , ⋮ ⎥ ⎢ ⎥ ⎣⎢CAL ⎦⎥

δu(k , t ) = du(k , t ) − du(k − 1, t ), δr(k , t ) = dr(k , t ) − dr(k − 1, t ), δe(k , t ) = e(k , t ) − e(k − 1, t )

we can rewrite eq 1 as

− f (u(k , t − 1))

(9)

̃ (k , t ) z(k , t + 1) = Az̃ (k , t ) + Bδ̃ u(k , t ) + Dw

(10)

e(k , t ) = e(k − 1, t ) + Cz̃ (k , t )

(11)

l

C̃ = [0, I ]

min

ΔU (k + i)

and

∑ (∥e (⃗ k + i)∥Q2

+ ∥ΔU (k + i)∥2R )

i=1

s.t. (1) − (2) without f (u(k , t )), ΔU (k + i) ∈ Ωu

w(k) = f (u(k , t )) − f (u(k − 1, t )) − f (u(k , t − 1)) ⎡ A 0⎤ , + f (u(k − 1, t − 1)), Ã = ⎢ ⎣−CA I ⎥⎦

(16)

where Q and R are the weighting matrices, Ωu is the admissible sets of control inputs determined by constraints 6−8, l is the batch number within the predictive horizon. Solving the above optimization problem yields the control inputs of batch k + 1 which will be implemented during the next batch. This kind of design is called run-to-run ILC−MPC since it calculates the updating law before the beginning of a batch.12 Different from the run-to-run ILC−MPC, the real-timefeedback-based ILC−MPC considers not only the previous batch information but also the current batch information to reject the possible nonrepetitive disturbance within batch k. Then, the updating law du(k, t) at time t of batch k is determined by solving the following optimization problem at time t of batch k.

⎡ I ⎤ D̃ = ⎢ ⎥ ⎣− C ⎦

From the 2D model 10−11, the system state and control input can be achieved as x(k , t ) = x(k − 1, t ) + x(k , t − 1) − x(k − 1, t − 1) (12)

u(k , t ) = u(k − 1, t ) + u(k , t − 1) − u(k − 1, t − 1) + δu(k , t )

(15)

where ΔU(k + 1) = U(k + 1) − U(k) is the corresponding updating law. If matrix G is full-rank, the controlled batch process satisfies the “perfect tracking condition”. To calculate the updating law ΔU(k + 1), the optimization problem of ILC−MPC can be formulated as

where

+ δx(k , t )

⋯ 0⎤ ⎥ ⋱ ⋮⎥ ⋱ 0⎥ ⎥ ⋯ CB ⎦

e (⃗ k + 1) = e (⃗ k) − GΔU (k + 1)

Then, the 2D model of the batch process is given as

⎡ B ⎤ , B̃ = ⎢ ⎣−CB ⎥⎦

⎡ CB 0 ⎢ CAB CB ⎢ G=⎢ ⋮ ⋮ ⎢ − L 1 L ⎣CA B CA − 2 B

Then, due to the periodical character of reference r (⃗ k), the output errors can be given as

dx(k , t + 1) = Adx(k , t ) + Bdu(k , t ) + f (u(k , t ))

z(k , t ) = [ δx(k , t )T δe(k , t )T ]T ,

(14)

(13)

The above model is a modification to the 2D model in Shi et al.17 Here, the control goal is to steer e(k, t) to zero. 2.2. Background Technique. For a batch process, the ILC− MPC is generally designed to calculate the control law u(k, t) = u(k − 1, t) + Δu(k, t) with Δu(k, t) as the updating law. Note that Δu(k, t) is different from du(k, t). In previous literature, there are two schemes to calculate the updating law, that is, run-to-run scheme and real-time scheme. In the following, we will introduce them briefly. To simplify the presentation, here model 1−2 is used without the nonlinear part; that is, a linear time-invariant system is utilized. Along the batch direction, from model 1−2 without the nonlinear part, the outputs of the next batch of the batch process can be written as

min

δu(k , t ),..., δu(k , t + h − 1)

J (k , t )

s.t. (1) − (2) without input nonlinearity

(17)

where h is the control horizon, h−1

J(k , t ) =

∑ ∥e(k , t + i + 1|k , t )∥q2

+ ∥δu(k , t + i|k , t )∥2λ

i=0

and q and λ are the weighting matrices, respectively. Problem 17 only formulates the case with t + h ≤ L. If t + h > L, the problem will be correspondingly modified due to the zero initial states. According to 13, we can get the below relation between the updating laws of the run-to-run and real-time feedback ILC. C


Article

Industrial & Engineering Chemistry Research ⎡ I 0 ⋯ 0 ⎤⎡ δu(k , 0) ⎤ ⎥ ⎢ ⎥⎢ I I ⋱ ⋮ ⎥⎢ δu(k , 1) ⎥ ⎢ ΔU (k) = ⎥ ⎢ ⎢⋮ ⋱ 0 ⎥⎢ ⋮ ⎥ ⎢ ⎥ ⎣I ⋯ I ⎦⎢⎣ δu(k , T − 1)⎥⎦

by the iterative learning mechanism while the previous designs can only consider the dynamic of the current batch. By the two-mode framework 19, the corresponding ILC− MPC should be solved at each time instant of each batch to get the u(k, i|k, t) and g(k + s). As for this framework, the first mode is similar to the previous designs, such as in Shi et al.17 That is, the control inputs after the current time instant t within the current batch k should be online, obtained by solving an optimization problem. Compared with the first mode, the second mode is more complicated since the first mode is of finite horizon and the second mode is of infinite horizon. It is impossible for an ILC− MPC to calculate all the inputs of the second mode from batch k + 1 to ∞ as g(k + s, i) online. So we are more interested about the second mode in the following part of this section, and the design of the first mode should be introduced in section 4 along with the whole design of the proposed controller. 3.2. Invariant Updating Strategy of the Batch Processes. With consideration of the online implementation of the second mode in the framework 19, the resultant updating strategy g(k + s, i) should satisfy the following requirements:

(18)

Compared with the run-to-run ILC−MPC in 16, the control law of the constrained real-time-feedback-based ILC−MPC is timevarying within a batch and there is no analytical solution. This causes the difficulty of convergent analysis. Furthermore, the considered unknown dynamic f(u(k, t)) in model 1−2 makes it more difficult. Few works discuss how to guarantee the convergence and how to ensure the constraints. This motivates our work. Before the main results, we give the following lemma. Lemma 1 Consider a scalar a, if there are parameters α and ρ (1 > ρ α ≥ 0) such that a̲ ≤ a + 1 − ρ ≤ a ̅ and a̲ ≤ a ≤ a,̅ then a̲ ≤ a + ∑i = 0s α ρi ≤ a̅ (s = 0, 1, 2, ..., ∞). It is obvious that the a + ∑i s= 0 αρi in Lemma 1 increases (decreases) monotonically when α > 0 (α < 0). Hence, its proof can be easily achieved and is omitted here.

3. TWO-MODE FRAMEWORK FOR REAL-TIME-FEEDBACK-BASED 2D ILC−MPC 3.1. Two-Mode Framework for Batch Processes. In MPC literature, the dual-mode control is a common method to design a MPC with the guaranteed closed-loop stability.9 The main idea of the dual-mode control is to extend the finite control horizon of MPC to the infinite horizon. To make the online computation possible, according to the dual-mode control, the optimization problem of MPC at time k constructs the future control inputs after time k as some free control actions followed by a fixed (or designed online) terminal feedback control law. Thus, the dynamic of system at time infinity can be formulated. Hence, if the terminal feedback control law satisfies some conditions (see Mayne et al.9 for detail), the closed-loop system can be ensured stable. Similarly, for the ILC−MPC, if the dynamic of the controlled process at batch infinity can be described, the convergence and the corresponding outputs can be analyzed. Hence, inspired by the dual-mode control, we suggest the following two-mode framework for the real-time-feedback-based ILC−MPC of batch processes. ⎧ u(k , i) i ∈ [t , L − 1], ⎪ u(. ) = ⎨ u(k + s , i) + g (k + s , i) ⎪ ⎩ i ∈ [0, L − 1], s ∈ [1, ∞]

(R1) satisfy all the constraints from batch k + 1 to ∞ (R2) the output sequence at batch ∞ can be given analytically (R3) the g(k + s, i) can be calculated online Hence, we suggest the following run-to-run strategy for g(k + s, i) to simplify the online computation and propose the following run-to-run based updating strategy from batch k. ΔU (k + i + 1) = ρi ΔU (k + 1)

i ∈ I[0, ∞]

(20)

where 0 ≤ ρ < 1 is a given scalar. The strategy is an extension of the amplitude-decaying aggregations strategy (see details in Li et al.24,25) in the batch direction. For the updating strategy (20), 0 ≤ ρ < 1 means that ΔU(k + i + 1) will be zeros when i → ∞. Meanwhile, from model 1−2, the following yields Y (k + 1) = Sx(k , 0) + GU (k + 1) + GF ̅ (U (k + 1)) (21)

where F(U(k + 1)) = [f T(u(k + 1, 1)), f T(u(k + 1,2)), ..., f T(u(k + 1, L)) ]T, ⎡ C 0 ⎢ CA C ⎢ G̅ = ⎢ ⋮ ⋮ ⎢ L − 1 ⎣CA CAL − 2

(19)

where t is the current time and g(k + s, i) is the updating strategy after batch k. In the framework of 19, the future control actions after time instant t of batch k are divided into two parts: the future control inputs within the current batch k (named the first mode) and a updating strategy from batch k + 1 to ∞ (named the second mode). Compared with the previous real-time-feedback-based ILC−MPC designs, which can be seen as only having the first mode of the two-mode framework, the design following the twomode framework can take both the dynamic of the current batch and the future evolution procedure along the batch direction into account. Therefore, the convergence can be designed to be ensured, and the dynamic of the current batch can be improved

⋯ 0⎤ ⎥ ⋱ ⋮⎥ ⋱ 0⎥ ⎥ ⋯ C⎦

Similarly to 15, we can obtain e ⃗(k + 1) = e ⃗(k) − GΔU (k + 1) − G̅ {F(U (k + 1)) − F(U (k))}

(22)

Then, the difference between the output and reference at batch ∞ will be D


Article

Industrial & Engineering Chemistry Research ∞

∑ Gρi ΔU (k + 1)

e (⃗ k + ∞) = e (⃗ k) −

i=0

∞

−

GΔU (k + 1) 1−ρ

e ⃗ ( k + ∞ ) − e ⃗( k ) +

= −G̅ {F(U (k + ∞)) − F(U (k))}

∑ G̅ {F(U (k + i + 1)) − F(U (k + i))}

Strategy (20) can yield

i=0

∞

GΔU (k + 1) = e (⃗ k) − 1−ρ

U (k + ∞ ) = U (k ) +

∞

−

i=0

∑ G̅ {F(U (k + i + 1)) − F(U (k + i))}

= U (k ) +

i=0

(23)

borrowing the concept of positive invariant set in,26 we name the updating strategy 20 which satisfies conditions 24−26 as an invariant updating strategy of the batch process. Although the output sequence at batch ∞ is not a determined one, it satisfies all the requirements of the second mode in 19. For the evolution procedure under the updating strategy 20, we can give the following proposition. Proposition 2. For a batch process with the nominal model of system 1−2 controlled by strategy 20, if ρiΔU(1) (i ∈ I[0,∞]) makes

u̲ i ≤ ui(k , t ) ≤ ui̅ Δui(k + 1, t ) ≤ ui̅ 1−ρ

(24)

δui ≤ ui(k , t ) − ui(k , t − 1) ≤ δui , δui ≤ ui(k , t ) − ui(k , t − 1) Δui(k + 1, t ) − Δui(k + 1, t − 1) + ≤ δui 1−ρ

(25)

dui ≤ Δui(k + 1, t ) ≤ dui

(26)

e ⃗ T(0)Qe ⃗(0) > e ⃗ T(∞)Qe ⃗(∞)

then

where i ∈ I[1, m], t ∈ I[0, L − 1], and Y(k) and U(k) are the output sequence and control input sequence at batch k, respectively. Additionally, the output error at batch ∞ will satisfy

e ⃗ T(0)Qe (0) > e ⃗ T(1)Qe (1) ⃗ ⃗

Proof. Consider i = 0. For a nominal model of system 1−2, since

GΔU (k + 1) e ⃗ ( k + ∞ ) − e ⃗( k ) + 1−ρ ≤

e (⃗ ∞) = e (0) − ⃗

μ∥G̅ ∥∥ΔU (k + 1)∥ 1−ρ

(27)

e ⃗ T(0)Qe ⃗(0) > e ⃗ T(∞)Qe ⃗(∞)

dui ≤ Δui(k + 1, t ) ≤ du i j

we have j

means dui ≤ ρ dui ≤ ρ Δui(k + 1, t ) ≤ ρ du i ≤ du i(j ≥ 0). That is, condition 26 can ensure updating strategy 20 always satisfies the constraint 8 from batch k. For condition 24, we get

2 2 | e (0) ⃗ ∥Q > ∥ e (0) ⃗ ∥Q +

GΔU (1) 1−ρ

2

Q

GΔU (1) − 2 e ⃗ T(0)Q 1−ρ

j−1

ui(k + j , t ) = ui(k , t ) +

GΔU (1) 1−ρ

and

Proof. For condition 26, since 0 ≤ ρ < 1 and dui < 0, j

ΔU (k + 1) 1−ρ

Hence, from Assumption 1, condition 27 holds, which completes the proof.□ Proposition 1 reflects that if conditions 24−26 hold by the updating strategy 20, it can steer the output sequence of the batch process to a sequence and ensure the output error at batch ∞ will GΔU (k + 1) belong to the neighbor region of e ⃗(k) − 1 − ρ . So,

Then, for system 1−2 subjected to constraints 6−8, we give the following proposition. Proposition 1. Consider a batch process with model 1−2 subject to constraints 6−8. If there exists a ΔU(k + 1) of the updating strategy (20) such that the following conditions are satisfied, then constraints 6−8 are always ensured from batch k.

u̲ i ≤ ui(k , t ) +

∑ ρi ΔU (k + 1)

∑ ρs Δui(k + 1, t )(j > 0)

That is,

s=0

Since 0 ≤ ρ < 1, from Lemma 1, if conditions 24 hold, constraints 8 can be ensured from batch k. Similarly to condition 24, we can use Lemma 1 to prove that condition 25 ensures constraints 7 from batch k, respectively. The detail is omitted here. As for condition 27, from eq 23, we can get

0>

∥GΔU (1)∥Q2 2

(1 − ρ)

− 2 e ⃗ T(0)Q

GΔU (1) 1−ρ

From 1 − ρ > 0, we have 0>

GΔU (k + 1) e ⃗ ( k + ∞ ) = e ⃗( k ) − 1−ρ

∥GΔU (1)∥Q2 1−ρ

− 2 e ⃗ T(0)QGΔU (1)

= e (0) − GΔU (1) yields Meanwhile, e (1) ⃗ ⃗

− G̅ {F(U (k + ∞)) − F(U (k))}

e (1) ⃗

That is, E

2 Q

= e (0) ⃗

2 Q

+ GΔU (1)

2 Q

− 2 e ⃗ T(0)QGΔU (1)


Article

Industrial & Engineering Chemistry Research Thus, from 0>

1 1−ρ

And, the output error sequence will be steered to

> 1, we get

∥GΔU (1)∥Q2 1−ρ

e ⃗(∞|k , t ) = e ⃗(k|k , t ) −

T

− 2 e ⃗ (0)QGΔU (1)

− G̅ {F(U (k + ∞)) − F(U (k))} ⎡ e ⃗(k , t |k , t ) ⎤ 1 ⎥ − GΔU (k + 1) =⎢ ⎢ e ⃗(k , L |k , t )⎥ 1−ρ ⎣ ⎦ t+1

> ∥GΔU (1)∥Q2 − 2 e ⃗ T(0)QGΔU (1) 2 This yields ∥ e ⃗(0)∥Q2 > ∥ e (1) ⃗ ∥Q .□

− G̅ {F(U (k + ∞)) − F(U (k))} ⎡ e ⃗(k , t |k , t ) ⎤ ⎡ 0 1 ⎥+⎢ =⎢ ⎢ e ⃗(k , L |k − 1, t )⎥ ⎢⎣ St̃ z(k , t ) + Gt̃ δU (k , ⎣ ⎦ t+1

4. SYNTHESIS OF 2D REAL-TIME-FEEDBACK-BASED ILC−MPC 4.1. Design of 2D Real-Time-Feedback-Based ILC− MPC. According to the two-mode framework 19, we choose the invariant updating strategy 20 with a given parameter ρ as the second mode. For the first mode, since the system model is known, the control inputs and corresponding outputs within the current batch can be formulated which is similar to that in previous works (e.g., Shi et al.17). Adopting the 2D model 10−11 and using 13 as the control law of 2D ILC−MPC and δ u(k, t) as the updating law, the output error sequence and input sequence of the first mode in 19 predicted at time t of batch k can be given by the following equations. e (⃗ k ,

L t + 1|k ,

L t + 1)

t ) = e (⃗ k − 1,

+ Gt̃ δU (k , ≔ e ⃗ o(k ,

L t + 1|k ,

⎡ 0 +⎢ ̃ ⎢⎣ Et w⃗(k ,

+ Et̃ w⃗(k ,

t ) + Et̃ w⃗(k ,

L−1 ) t

min

L t + 1),

s.t. U (k ,

L−1 ) t

∈ Ωu , (24) − (26)

(32)

J(k , t ) = qo∥ e ⃗ o(k|k , t )∥2 + q∥ e ⃗(∞|k , t )∥2 L−t−1

⋯ 0 ⎤ ⎥ ̃ ̃ CB ⋱ ⋮ ⎥ ⎥, ⋮ ⋱ 0 ⎥ ̃ ̃ L − t − 2 B̃ ⋯ CB ̃ ̃⎥⎦ CA ⎡CD ̃ ̃ 0 ⋯ 0 ⎤ ⎥ ⎢ ̃ ̃ ̃ ̃ ̃ ⎢CAD CD ⋱ ⋮ ⎥ ̃ Et = ⎢ ⎥ ⋮ ⋱ 0 ⎥ ⎢⋮ ⎢⎣CA ̃ ̃ L − t − 1D̃ CA ̃ ̃ L − t − 2 D̃ ⋯ CD ̃ ̃ ⎥⎦ ⎡CB ̃ ̃ ⎢ ̃ ̃ ̃ ⎢CAB Gt̃ = ⎢ ⎢⋮ ⎢⎣CA ̃ ̃ L − t − 1B̃

J (k , t )

where Ωu is the admissible set on U (k , tL − 1) determined by constraints 6−8, qo, q, and λ are the weighting parameters and qo < q,

(28)

= [δu(k , t )T ,..., δu(k , L − 1)T ]T

⎡CA ̃ ̃ ⎤ ⎥ ⎢ ̃ ̃2 ⎥ ⎢CA St̃ = ⎢ ⎥, ⎥ ⎢⋮ ⎢⎣CA ̃ ̃ L − t ⎥⎦

(31)

Then, according to Proposition 1, we can formulate the following optimization problem.

where δU (k ,

⎤ GΔU (k + 1) ⎥− 1−ρ

L ⎦ t + 1)⎥

− G̅ {F(U (k + ∞)) − F(U (k))}

δU (k , tL − 1), ΔU (k + 1)

L t + 1)

⎤ ⎥

L−1 )⎥⎦ t

− G̅ {F(U (k + ∞)) − F(U (k))} ⎡ ⎤ 0 ⎥ ≔ e ⃗ o(∞|k , t ) + ⎢ L ̃ ⎢⎣ Et w⃗(k , t + 1)⎥⎦

+ St̃ z(k , t )

L−1 ) t

GΔU (k + 1) 1−ρ

0

+

∑

λ∥Δu(k , t + i)∥2 +

i=0

λ∥ΔU (k + 1)∥2 1 − ρ2

(33)

In the problem 32, the δU (k , tL − 1) and ΔU(k + 1) correspond to the first mode and the second mode of 19, respectively. Although the constraints of the optimization problem 32 are linear, its objective function is not a quadratic function because of the unknown input linearity in the system model. Hence, we choose the upper bound of the function as the objective function of ILC−MPC. From Assumption 1, similarly to 27, we can get L−t−1

And at time t of batch k, the error sequence e ⃗(k − 1, known. In addition, we also have ΔU (k ,

L−1 ) t

L t + 1)

J(k , t ) ≤ qo∥ e ⃗ o(k|k , t )∥2 + q∥ e ⃗ o(∞|k , t )∥2 +

is

λ∥δu(k , t + i)∥2

i=0

+

⎡ I 0 0⎤ ⎡I ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢⋮⎥Δu(k , t − 1) + ⎢⋮ ⋱ 0 ⎥δU (k ,tL − 1 ) ⎢⎣ I ⎥⎦ ⎢⎣ I ⋯ I ⎥⎦

λ∥ΔU (k + 1)∥2 + q∥G̅ {F(U (k + ∞)) − F(U (k))}∥2 1 − ρ2

⎡ 0 +q ⎢ ⎢⎣ Et̃ w⃗(k ,

(29)

⎤ ⎥ L ) ⎦ t+1 ⎥

2

L−t−1

≤ qo∥ e ⃗ o(k|k , t )∥2 + q∥ e ⃗ o(∞|k , t )∥2 +

where ΔU (k ,

∑

∑

λ∥δu(k , t + i)∥2

i=0

L−1 ) t

= U (k ,

L−1 ) t

− U (k − 1,

L−1 ) t

+

and Δu(k, t − 1) = u(k, t − 1) − u(k − 1, t − 1) is known at time t of batch k. According to eq 28 and 29, we can formulate the system constraints 6−8 as the inequalities about the input sequences. For the second mode of 19, according to 20, the control input sequences are U (k + i ) = U (k ) + ρ

i−1

ΔU (k + 1) (i > 0)

+

λ∥ΔU (k + 1)∥2 + q∥Et̃ ∥2 ∥w⃗(k , 1 − ρ2

L 2 t + 1)∥

qμ2 ∥G̅ ∥2 ∥ΔU (k + 1)∥2 (1 − ρ)2

(34)

From w(k, t) = f(u(k, t)) − f(u(k, t − 1)) − (f(u(k − 1, t)) − f(u(k − 1, t − 1))) and Assumption 1, we can get || w(k , t )|| ≤ μ || δ u(k , t )|| + μ || δ u(k − 1, t )||

(30)

This yields F


Article

Industrial & Engineering Chemistry Research

considered to accelerate the convergence. Hence, this design is reasonable. Furthermore, we can notice that the control horizon of eq 36 is varying with the time in batch k. Remark 2. The online computational burden is important for the real implementation of the real-time-feedback-based ILC− MPC. For the proposed design, the optimization problem is a QP problem, the computational burden of which is mainly dependent on the number of the online optimization variables. As for the number of the online optimization variables of Algorithm 1, we can see that the maximum value is 2 mL and the minimum value is mL + m. Compared with the previous designs, such as Lee et al.,14 in which the number of optimization variables is determined by the length of control horizon, the computational burden of Algorithm 1 is heavier. Compared with the design of Shi et al.,17 the computational burden of Algorithm 1 is similar to that with two cycles and lighter than that with multicycles. In addition, in order to reduce the online computation, the matrices Ẽ t can be calculated in advance. 4.2. Convergence Analysis. For the convergence of Algorithm 1, we can give the following theorem. Theorem 1. Consider a batch process with model 1−2 subjected to 6−8 controlled by the ILC−MPC in Algorithm 1. If Assumption 1 holds and Problem 36 is feasible at time t of batch k, then Algorithm 1 is always feasible and reaches the convergence. Proof. For the considered batch process, since Problem 36 in Algorithm 1 is feasible at time t of batch k, we can assume the optimal solution is

J(k , t ) ≤ qo∥ e ⃗ o(k|k , t )∥2 + q∥ e ⃗ o(∞|k , t )∥2 L−t−1

∑

+

λ∥ΔU (k + 1)∥2 1 − ρ2

λ∥δu(k , t + i)∥2 +

i=0

+ qμ2 ∥Et̃ ∥2 (∥δU (k , +

L−1 2 )∥ t 2

+ ∥δU (k − 1,

L−2 2 t − 1 )∥ )

qμ2 ∥G̅ ∥2 ∥ΔU (k + 1)∥ (1 − ρ)2

≔ J (̂ k , t )

(35)

Notice that, at batch k, the δU (k − 1, eq 18, we rewrite J(̂ k, t) as

L−2 t − 1 ) is

determined. So by

J (̂ k , t ) = qo∥ e ⃗ o(k|k , t )∥2 + q∥ e ⃗ o(∞|k , t )∥2 L−t−1

+

∑

λ∥δu(k , t + i)∥2 +

i=0

+ qμ2 ∥Et̃ ∥2 ∥δU (k , +

λ∥ΔU (k + 1)∥2 1 − ρ2

L−1 2 )∥ t

qμ2 ∥G̅ ∥2 ∥Λ∥2 ∥δU (k + 1,

L−1 2 0 )∥

(1 − ρ)2

where ⎡ I 0 ⋯ 0⎤ ⎢ ⎥ I I ⋱ ⋮⎥ Λ=⎢ ⎢⋮ ⋱ 0⎥ ⎢ ⎥ ⎣I ⋯ I⎦

Σ*(t ) = {δU *(k ,

Then, we can give the optimization problem of ILC−MPC as min

δU (k , tL − 1), δU (k + 1, 0L − 1)

s.t. U (k ,

L−1 ) t

t ), δU *(k + 1,

L−1 0 |k ,

t )}

which satisfies all the constraints and corresponds to J*̂ (k, t). If L − 1 > t ≥ 0, at time t + 1, we have a solution as

J (̂ k , t )

∈ Ωu , (24) − (26)

L−1 |k , t

Σ(t + 1) = {δU *(k ,

(36)

L−1 t + 1 |k ,

t ), δU *(k + 1,

L−1 0 |k ,

t )}

which is obtained by removing δu*(k, t|k, t) from the solution at time t. By the feasibility of Σ*(t) at time t, Σ(t + 1) is also feasible at time t + 1. In J(̂ k, t + 1), since the uncertainty of time t in the item

which is a standard quadratic programming (QP) problem and can be efficiently solved by many QP solvers. The complete algorithm of the 2D real-time-feedback-based ILC−MPC is given below. Algorithm 1 (2D real-time-feedback-based ILC-MPC) Step 1. Choose parameter ρ ∈ [0, 1). Step 2. At time t of batch k, solve the optimization problem 36 to achieve the δu(k, t). Step 3. Calculate u(k, t) according to eq 13 and act it on the controlled system. Step 4. Record the control input u(k, t), y(k, t + 1), and x(k, t + 1). Step 5. Set t = t + 1, return to Step 2. Remark 1. For the objective function of Algorithm 1, besides the input sequences, the objective function includes both the nominal error sequence at the current batch and the error sequence at batch ∞ rather than all the future error sequences. The reason is that the control goal of ILC is to erase the errors between the output sequence and reference as soon as possible, that is, the aim of ILC is to make the convergent sequence is as close as possible to the reference. The convergent sequence is composed by two parts, that is, the part of error sequences of the nominal model of model (1) and the part resulted by the unknown nonlinearity in model (1). Hence, to minimize the objective function can minimize both the nominal error sequence and the uncertainty caused by the nonlinearity, which is to make the convergent error sequence as small as possible. Meanwhile, the nominal error sequence at the current batch is also

q∥ e ⃗ o(∞|k , t )∥2 + qμ2 ∥Et̃ ∥2 ∥δU (k ,

L−1 2 )∥ t

does not exist, it is easy to get that q∥ e ⃗ o(∞|k , t )∥2 + qμ2 ∥Et̃ ∥2 ∥δU (k ,

L−1 |k , t

t )∥2

≥ q∥ e ⃗ o(∞|k , t + 1)∥2 + qμ2 ∥Et̃ + 1∥2 ∥δU (k ,

L−1 t + 1 |k ,

t + 1)∥2

Meanwhile, the item of δ u*(k, t|k, t) in J*̂ (k, t) is erased in J(̂ k, t + 1) and qo∥ e ⃗ o(k|k , t )∥2 is unchanged. So, Σ(t + 1) leads to J*̂ (k, t) ≥ J(̂ k, t + 1). Therefore, when L − 1 > t ≥ 0, we conclude the optimal J(̂ k, t + 1) must satisfy J*̂ (k, t + 1) ≤ J(̂ k, t + 1) ≤ J*̂ (k, t). If t = L − 1, batch k + 1 will begin at next time instant. For the new batch, we have a solution as Σ(k + 1) = {δ U*(k + 1, L−1 0 |k, L − 1), ρ δ U*(k + 1, L−1 0 |k, L − 1)}. According to the optimization theory, we can easily know that qo∥ e ⃗ o(k|k , 0)∥2 > qo∥ e ⃗ o(∞|k , 0)∥2 L−1 Otherwise, the {δ U*(k+1, L−1 0 |k, L − 1), ρ δ U*(k + 1, 0 |k, L − 1) }will be zero. Therefore, we can get

qo∥ e ⃗ o(k|k , L − 1)∥2 > qo∥ e ⃗ o(k + 1|k + 1, 0)∥2

from Proposion 2. Note that, Σ(k + 1) at batch k + 1 is just the second mode of ILC−MPC at batch k. This yields G


Article

Industrial & Engineering Chemistry Research ∥δU *(k + 1, + =

L−1 0 |k ,

ρ2 ∥δU *(k + 1, 1 − ρ2

∥δU *(k + 1,

example of section 5. But a bigger ρ also causes the weighting parameter of input variations to be larger which may lead to the convergent process being very slow. So for a given application, the ρ should be chosen to trade off between the robustness and the convergent speed.

L − 1)∥2

L−1 0 |k , 2

L−1 0 |k ,

L − 1)∥2

L − 1)∥2

5. CASE STUDY Injection molding is a typical batch process. The injection velocity is a key variable in the filling stage and is controlled by manipulating the opening of a hydraulic valve. The injection velocity response to the proportional valve away from an operation point can be modeled as Li et al.8 1.69z + 1.419 p(z ) = 2 z − 1.582z + 0.5916 whose state-space representation is

1−ρ

Notice that, Ẽ 0 = G¯ Λ. That means, ||Ẽ 0|| ≤ || G¯|| ||Λ||. Therefore, we can get L−1 0 |k , 2 2

∥E0̃ ∥2 ∥δU *(k + 1, + ≤

L − 1)∥2

∥G̅ ∥2 ∥Λ∥ ρ ∥δU *(k + 1, 1 − ρ2

∥G̅ ∥2 ∥Λ∥2 ∥δU *(k + 1, 1 − ρ2

L−1 0 |k ,

L−1 0 |k ,

L − 1)∥2

L − 1)∥2 (37)

⎡1.582 − 0.5916 ⎤ ⎡1 ⎤ ⎡1⎤ x(t + 1) = ⎢ ⎥x(t ) + ⎢⎣ ⎥⎦v(t ) + ⎢⎣ ⎥⎦w(t ), ⎣ 1 0 1 0 ⎦

Also since the uncertainty of time L − 1 of batch k in the item q∥ e ⃗ o(∞|k , L − 1)∥2 + qμ2 ∥EL̃ − 1∥2 ∥δU (k ,

L−1 2 L − 1)∥

y(t ) = [1.69 1.419 ]x(t )

does not exist and the item of δ u*(k, L − 1|k, L − 1) in J*̂ (k, L− 1) is erased in J(̂ k + 1, 0), we can conclude that J*̂ (k + 1, 0) ≤ J(̂ k + 1, 0) ≤ J*̂ (k, L − 1). The above analysis illustrates that the value of cost function of Problem 36 will decrease along the time direction and the batch direction until convergence, and the problem is always feasible if it is feasible at time t of batch k. This completes the proof. 4.3. Further Discussions. For a batch process, the external disturbance includes the periodic one with cycle time L and the nonrepetitive one. The model including disturbance can be

where w(t) is the disturbance. Here, the considered input is like

v(t ) = 2 sin(u(t )) and Figure 1 shows the nonlinearity. The considered constraint is |u(t)| ≤ 2. For this input function, we can rewrite it as v(t ) = u(t ) + 2 sin(u(t )) − u(t )

and then μ = 3.

x(k , t + 1) = Ax(k , t ) + Bu(k , t ) + f (u(k , t )) + d1(k , t ), y(k , t ) = Cx(k , t ) + d 2(k , t )

where d1(k, t) and d2(k, t) are the disturbance. The output sequence is Y (k + 1) = Sx(0) + GU (k + 1) + GF ̅ (U (k + 1)) ⃗ ⃗ + Gwd1(k) + d 2(k)

(38)

e ⃗(k + 1) = r (⃗ k) − GU (k + 1) − G̅ {F(U (k + 1)) − F(U (k))} − Gwd1⃗ (k) − d 2⃗ (k)

(39)

Figure 1. Input nonlinearity.

where Gw is a fixed matrix determined by A and C according the system model. From eq 39, if the disturbance d1(k, t) and d2(k, t) are periodic with cycle time L, the effect is same to that caused by the change of reference. Thus, for the periodic disturbance with cycle time L, the proposed 2D ILC−MPC can reduce the influence to the smallest one allowed by constraints after several batches. Meanwhile, by investigating the 2D model (10), we can find if the disturbance d1(k, t) is constant, the disturbance item will not appear in model 10. That implies that the influence of constant disturbance d1(k, t) can be erased within one batch. For parameter ρ of Algorithm 1, we can see that if the ρ is close to 1, then the final sequence of the second mode by the current optimized control sequence is more similar to the sequence of batch ∞. This can improve the robustness. In practical applications, there are always some uncertain parameters, for example the uncertainty of the μ. This may influence the control performance. The improved robustness will be illustrated by the

The process is requested to track a given trajectory shown in Figure 2 with the dash-dot line. For the ILC−MPC, the parameters are qo = 1, q = 100, λ = 1. Case 1: with different ρ. To verify the proposed algorithm, we first choose ρ = 0.5. The control results are shown in Figure 2. To save the space, we here just give the figures of batch 1, 10, 20, and 40. The evolution of tracking error is shown in Figure 4, which illustrates that the tracking error is smaller and smaller along with the batch, and the output errors are erased or reduced by iterative learning. The corresponding inputs are shown in Figure 3, and they reflect that the input constraints are ensured. But from the figures, we can also see that the convergent process is a little slow. The control result of batch 20 when ρ = 0.1 is chosen is shown in Figure 6. Meanwhile, Figure 5 illustrates the evolution of tracking errors. From these figures, we can see that the convergent process when ρ = 0.1 is quicker than that when ρ = H


Article


Figure 2. Output sequences in Case 1 (reference with dash line and outputs with solid line).

Figure 3. Input sequences in Case 1.

first batch is influenced greatly as shown in Figure 7. But due to the iterative learning, the response is improved greatly along with batches. At batch 40, the influence is almost erased. By combining with Figure 8, we can say that the proposed design can guarantee the convergence under the periodic disturbance. Case 3: The Robustness. To investigate the robustness under the different ρ, we assume that the v(t) = 2 sin(0.1u(t)) is used to design the controller and the real value of v(t) = 2 sin(2u(t)) is 6. By choosing ρ = 0.1, 0.5, and 0.9, respectively, the

0.5 and the tracking error at batch 20 is almost erased as shown in Figure 6. Case 2: With Disturbance. To reflect the capability of tackling the disturbance for the proposed design with ρ = 0.5, we choose w(k, t) = 0.3 + 0.01mod(t, 10), where mod(t, 10) is the remainder of t divided by 10 and is a periodic disturbance. The control results of batch 1, 4, and 40 are shown in Figure 7 and the evolution of tracking errors is illustrated in Figure 8. Since the disturbance is unknown and time-varying, the response of the I


Article


Figure 4. Evolution of tracking errors in Case 1 when ρ = 0.5.

Figure 5. Evolution of tracking errors in Case 1 when ρ = 0.1.

Figure 7. Output sequences in Case 2 with disturbance (reference with dash-dot line and outputs/inputs with solid line).

Figure 6. Output sequence of batch 20 in Case 1 with ρ = 0.1.

evolution of tracking errors are illustrated in Figure 9. From the figure, we can find that when ρ = 0.1, the convergence is not guaranteed. In fact, after batch 50, the tracking error becomes larger and larger with the batches. At the same time, the increase of ρ reflects the improved robustness. So combining with the results in the above figures, we can see that although the large ρ may cause the slow convergence, it can improve the robustness. This verifies the analysis in section 4.3.

Figure 8. Evolution of tracking errors in Case 2 with disturbance and ρ = 0.5.

6. CONCLUSION The real-time-feedback-based 2D ILC−MPC for constrained batch processes with unknown input nonlinearities is studied in this paper. The two-mode framework of the ILC−MPC is proposed, which is formulated as a sequence of free control inputs followed by an invariant updating strategy. The invariant updating strategy is designed according to the run-to-run

scheme, and the upper bound of influence caused by the unknown input nonlinearity is estimated for the second mode. On the basis of these, the real-time-feedback-based 2D ILC− MPC is developed, for which the time-varying horizon is adopted. For the proposed ILC−MPC, the convergence can be guaranteed and constraints can be satisfied. These results are verified by the case studies. The proposed two-mode framework J


Article


(10) Qin, S. J.; Badgwell, T. A. A survey of industrial model predictive control technology. Control engineering practice 2003, 11, 733−764. (11) Amann, N.; Owens, D. H.; Rogers, E. Predictive optimal iterative learning control. Int. J. Control 1998, 69, 203−226. (12) Xu, Z.; Zhao, J.; Yang, Y.; Shao, Z.; Gao, F. Optimal Iterative Learning Control Based on a Time-Parametrized Linear Time-Varying Model for Batch Processes. Ind. Eng. Chem. Res. 2013, 52, 6182−6192. (13) Chu, B.; Owens, D. H. Iterative learning control for constrained linear systems. Int. J. Control 2010, 83, 1397−1413. (14) Lee, K. S.; Chin, I.-S.; Lee, H. J.; Lee, J. H. Model predictive control technique combined with iterative learning for batch processes. AIChE J. 1999, 45, 2175−2187. (15) Lee, K. S.; Lee, J. H. Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables. J. Process Control 2003, 13, 607− 621. (16) Lee, J. H.; Lee, K. S.; Kim, W. C. Model-based iterative learning control with a quadratic criterion for time-varying linear systems. Automatica 2000, 36, 641−657. (17) Shi, J.; Gao, F.; Wu, T.-J. Single-cycle and multi-cycle generalized 2D model predictive iterative learning control (2D-GPILC) schemes for batch processes. J. Process Control 2007, 17, 715−727. (18) Wang, Y.; Zhou, D.; Gao, F. Iterative learning model predictive control for multi-phase batch processes. J. Process Control 2008, 18, 543−557. (19) Chin, I.; Qin, S. J.; Lee, K. S.; Cho, M. A two-stage iterative learning control technique combined with real-time feedback for independent disturbance rejection. Automatica 2004, 40, 1913−1922. (20) Cueli, J. R.; Bordons, C. Iterative nonlinear model predictive control. Stability, robustness and applications. Control Engineering Practice 2008, 16, 1023−1034. (21) Liu, T.; Wang, Y. A synthetic approach for robust constrained iterative learning control of piecewise affine batch processes. Automatica 2012, 48, 2762−2775. (22) Liu, X.; Kong, X. Nonlinear fuzzy model predictive iterative learning control for drum-type boiler−turbine system. J. Process Control 2013, 23, 1023−1040. (23) Chi, R.; Hou, Z.; Huang, B.; Jin, S. A unified data-driven design framework of optimality-based generalized iterative learning control. Comput. Chem. Eng. 2015, 77, 10−23. (24) Li, D.; Xi, Y. Quality guaranteed aggregation based model predictive control and stability analysis. Science in China Series F: Information Sciences 2009, 52, 1145−1156. (25) Li, D.; Xi, Y.; Lin, Z. An improved design of aggregation-based model predictive control. Systems & Control Letters 2013, 62, 1082− 1089. (26) Blanchini, F. Set invariance in control. Automatica 1999, 35, 1747−1767.

Figure 9. Evolution of tracking errors in Case 3 with different ρ (ρ = 0.1 with dash line, ρ = 0.5 with line marked by star, and ρ = 0.9 with solid line).

and the invariant updating strategy can also be used to design other ILC−MPCs for batch processes with guaranteed convergence and performance.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Furong Gao: 0000-0002-5900-1353 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 61333009, 61374110, 61521063, 61590924, 61573239), and Hong Kong Research Grant Council under Project No. GRF612512.

■

REFERENCES

(1) Gao, F.; Yang, Y.; Shao, C. Robust iterative learning control with applications to injection molding process. Chem. Eng. Sci. 2001, 56, 7025−7034. (2) Hakvoort, W.; Aarts, R.; van Dijk, J.; Jonker, J. Lifted system iterative learning control applied to an industrial robot. Control Engineering Practice 2008, 16, 377−391. (3) Wang, Y.; Dassau, E.; Doyle, F. J. Closed-loop control of artificial pancreatic-cell in type 1 diabetes mellitus using model predictive iterative learning control. IEEE Trans. Biomed. Eng. 2010, 57, 211−219. (4) Arimoto, S.; Kawamura, S.; Miyazaki, F. Bettering operation of dynamic systems by learning: A new control theory for servomechanism or mechatronics systems. IEEE Proc. on Decision and Control. 1984, 23, 1064−1069. (5) Uchiyama, M. Formulation of high-speed motion pattern of a mechanical arm by trial. Trans. SICE Soc. Instrum. Control Eng. 1978, 14, 706−702. (6) Wang, Y.; Gao, F.; Doyle, F. J., III Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control 2009, 19, 1589−1600. (7) Lee, J. H.; Natarajan, S.; Lee, K. S. A model-based predictive control approach to repetitive control of continuous processes with periodic operations. J. Process Control 2001, 11, 195−207. (8) Li, D.; Gao, F.; Xi, Y. Separated design of robust model predictive control for LPV systems with periodic disturbance. J. Process Control 2014, 24, 250−260. (9) Mayne, D. Q.; Rawlings, J. B.; Rao, C. V.; Scokaert, P. O. M. Constrained model predictive control: Stability and optimality. Automatica 2000, 36, 789−814. K


Synthesis of Real-Time-Feedback-Based 2D Iterative Learning

Recommend Documents