ARTICLE pubs.acs.org/IECR
Optimal Structure of Learning-Type Set-Point in Various Set-PointRelated Indirect ILC Algorithms Youqing Wang,†,* Jianyong Tuo,† Zhong Zhao,† and Furong Gao‡,†,* † ‡
College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, China Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong ABSTRACT: According to the literature statistics, only less than 10% of reported iterative learning control (ILC) methods have been devoted to the indirect approach. Motivated by the full potential of research opportunities in this field, a number of studies on indirect ILC were proposed recently, where ILC-based P-type control and learning-type model predictive control (L-MPC) are two successful stories. All indirect ILC algorithms consist of two loops: an ILC in the outer loop and a local controller in the inner loop. The local controllers are, respectively, a P-type controller in the ILC-based P-type control and a model predictive control (MPC) in the L-MPC. Logically, this leads to the question of what type of ILC should be chosen respectively for the two above-mentioned indirect ILC methods. In this study, P-type ILC and anticipatory P-type (A-P-type) ILC are studied and compared, because they are typical and widely implemented. Based on mathematical analysis and simulation test, it has been proved that the A-P-type ILC should be used in the ILC-based P-type control and while the P-type ILC should be used in the L-MPC. Furthermore, an improved L-MPC with batch-varying learning gain was proposed to handle the trade-off between convergence rate and robustness performance. The simulation results on injection molding process and a nonlinear batch process validated the feasibility and effectiveness of the proposed algorithm.
1. INTRODUCTION Intelligent machines including industrial computers can be led from repeated training to reach superior performance of a specified task. Scholars and engineers have therefore developed iterative learning control (ILC) methods to formulate the learning procedure systematically. ILC was first presented in Japanese in 19781 and then was introduced in English in 1984.2 These contributions are widely considered to be the origins of ILC. Up to the present, ILC has become an ad hoc research topic and numbers of publications were achieved every year.39 After three decades’ development, ILC has been successfully implemented in industrial manipulators,4,10,11 chemical batch processes,1214 biomedical processes,7,15,16 and other processes.5,17,18 There are mainly two application modes for ILC. First, ILC is used to determine the control signal directly, and this kind of ILC is named direct ILC. Second, there is a local feedback controller in each batch and ILC is used to update some parameter settings of the local controller, so this kind is named indirect ILC. Compared with direct form, indirect ILC has some advantages. First, the existing process structure need no change; if there already exists a controller, only an ILC module is added in the outer loop to update some parameters of the existing controller and this ILC module could be easily moved at any time. Second, in most cases, indirect ILC has better robustness than the direct one; this is because direct ILC must have a feedforward term, which is sensitive to variations in batch direction, but a feedforward term is not necessary for the local controller of the indirect ILC. In addition, the idea of the indirect method is consistent with the developing trend of control engineering: stability and robustness are not the only requirements for control design, and an optimization scheme should be utilized r 2011 American Chemical Society
to improve the closed-loop control performance. According to the literature statistics in a recent survey,19 however, only less than 10% of the reported ILC methods were implemented in the indirect mode. The following two issues are essential for an indirect ILC: what algorithm is used to design the local controller, and which parameters of the local controller are adjusted by the ILC. Generally speaking, ILC could be used to adjust set-point,20 control gain,21 weight,22 and other parameters23,24 for the local controller. Particularly, an indirect ILC that updates the set-point for the local controller is termed as set-point-related (SPR) indirect ILC. Less than 20% of reported indirect ILC algorithms are SPR indirect ILC. For clarity, the block diagram of SPR indirect ILC was shown in Figure 1. Motivated by the full potential of research opportunities in this field, several SPR indirect ILC methods were proposed recently.7,25 In the literature,7 a novel combination of ILC and model predictive control (MPC), termed L-MPC, was proposed, where the local controller is MPC. It is valuable to point out that ILC and MPC have long been used together, in combinations such as BMPC,26 2D-GPILC,27 and MPILC.28 However, in each of these, MPC was used to design the updating law of ILC; therefore, these combinations belong to the direct ILC category. To our best knowledge, ref 7 is the first reported work on ILC-based MPC, or L-MPC. In the L-MPC framework, the set-point yr(t,k) for MPC could be different in various batches, and it is updated Received: September 8, 2010 Accepted: October 12, 2011 Revised: October 6, 2011 Published: October 12, 2011 13427
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2. SYSTEM DESCRIPTION Consider the following batch process, ( xðt þ 1, kÞ ¼ Axðt, kÞ þ Buðt, kÞ þ wðtÞ yðt, kÞ ¼ Cxðt, kÞ xð0, kÞ x0 ; t ¼ 0, 1, 3 3 3 T 1; k ¼ 1, 2, 3 3 3
Figure 1. Block diagram of set-point-related indirect ILC. The dotted line indicates the signal in the previous batch. Different line colors/ thicknesses denote various signal natures: black thick line indicates physical signal; blue thin line indicates command signal.
by using a P-type ILC, as shown below: yr ðt, kÞ ¼ yr ðt, k 1Þ þ Leðt, k 1Þ
ð1Þ
where L is the learning gain matrix. For simplicity, L could be designed as L ( diag{l1, 3 3 3 ,l1} with 0 < li < 1 (i = 1,..., m). In the literature,25 an ILC-based P-type controller was proposed, where a P-type controller works as the local controller. The set-point yr(t,k) for the local controller is updated by using an anticipatory P-type (A-P-type) ILC:20 yr ðt, kÞ ¼ yr ðt, k 1Þ þ Leðt þ 1, k 1Þ
ð2Þ
Please note the similarity and difference between eqs 1 and 2. It is very interesting and important to study the stability property of L-MPC and ILC-based P-type control if eqs 1 and 2 are switched. Based on 2-dimensional system formulation,29,30 it has been proved in Section 3 that the ILC-based P-type controller will be asymptotically unstable if the ILC module is changed to be P-type. In Section 4, the physical meaning of P-type ILC in L-MPC was explained based on the explicit solution of the MPC module. Using this explicit solution, one knows that the relative degree from the set-point to the tracking error is zero under a MPC. Therefore, P-type ILC is a suitable choice for L-MPC. To further study whether P-type or A-P-type ILC is a better choice for L-MPC, in Section 3, these two types of ILC methods were compared based on some simulation tests on the injection molding process and a numerical nonlinear batch process. If the ILC module is A-P-type, the closed-loop system under L-MPC is unstable in some situations. Therefore, P-type ILC is a better option for L-MPC. In the P-type ILC module, a learning gain should be designed; however, how to design this gain is still an open problem. It is well-known that there is a trade-off in designing the learning gain for L-MPC: a larger value of the learning gain can induce a faster convergence rate but worse robustness; a smaller value has opposite influences. To handle this trade-off, a batch-varying gain is proposed for L-MPC, where the learning gain is constant in each batch but can be changed from batch to batch. The simulation results on the injection molding process validated the feasibility and effectiveness of the proposed algorithm.
ð3Þ
where t denotes time; k denotes batch index; x(t,k)∈Rn, u(t,k)∈Rp, and y(t,k)∈Rm represent, respectively, the states, outputs, and inputs of the process at time t of the kth batch run; w(t)∈Rn denotes repetitive disturbances; {A, B, C} are the system matrices with appropriate dimensions; x0 is the identical initial condition for each batch. The control objective is to determine a control law such that the outputs tracks the given target, Y*(t), as closely as possible. The tracking error is defined as: eðt, kÞ ¼ ^ Y ðtÞ yðt, kÞ ð4Þ To achieve this objective, two indirect ILC algorithms, ILCbased P-type control and L-MPC, will be designed in Sections 3 and 4, respectively.
3. ILC-BASED P-TYPE CONTROL In the literature,25 a SPR indirect ILC was proposed, where the local controller is a P-type control as shown below, uðt, kÞ ¼ K½yr ðt, kÞ yðt, kÞ
ð5Þ
It has been proved that there exist an appropriate L such that system 3 under ILC-based P-type controller shown 5 and 2 is asymptotical stable. In the following, it will be proved that if the set-point is updated by using eq 1, then the closed-loop system is asymptotical unstable for all L. The stability of the closed-loop system will be analyzed in a 2-dimensional framework. Consider a Roesser’s system,31 " # # " #" A11 A12 xh ði, jÞ xh ði þ 1, jÞ ¼ ð6Þ A21 A22 xv ði, j þ 1Þ xv ði, jÞ where xh∈Rn1 is the horizontal state vector and xv∈Rn2 is the vertical state " vector.#The boundary condition of the Roesser’s xh ð0, jÞ , where xh(0,j) and xv(i,0) represent the system is xv ði, 0Þ horizontal and vertical boundary conditions, respectively. To study the stability for the ILC-based P-type controller, a lemma on the stability of the 2D system was first introduced. Lemma 131. The Roesser’s system 6 is asymptotically stable if and only if " # I z1 z1 1 A11 1 A12 1 1 dðz1 , z2 Þ ¼ det z1 I z1 2 A21 2 A22 1 6¼ 0, for alljz1 1 j e 1, jz2 j e 1:
ð7Þ
det(*) denotes the determinant of matrix *. In addition, it is always true that 1 1 dðz1 1 , z2 Þ ¼ detðI z2 A22 Þ 1 detfI z1 1 ½A11 þ A12 ðz2 I A22 Þ A21 g
ð8Þ
1 where d(z1 1 ,z2 ) is termed as the denominator polynomial for the
system. 13428
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For convenience, the following notation is introduced: δ0ðt, kÞ ¼ ^ 0ðt, kÞ 0ðt, k 1Þ, 0 ¼ x, y, u, 3 3 3
ð9Þ
where δ is the variation of 0 in the batch direction. Under the ILC-based P-type controller in 5 and 1 (P-type ILC), the system in 3 can be transformed to δxðt þ 1, kÞ ¼ Aδxðt, kÞ þ Bδu2 ðt, kÞ ¼ Aδxðt, kÞ þ BKδyr ðt, kÞ BKδyðt, kÞ ¼ ðA BKCÞδxðt, kÞ þ BKLeðt, k 1Þ
ð10Þ
and eðt, kÞ ¼ eðt, k 1Þ Cδxðt, kÞ
ð11Þ
Therefore, a 2D system can be obtained as follows: " # " #" # δxðt þ 1, kÞ A BKC BKL δxðt, kÞ ¼ eðt, kÞ C I eðt, k 1Þ ð12Þ According to 6, eq 12 is a typical Roesser’s model denoting xh(t + 1,k) = δx(t + 1,k) and xv(t,k + 1) = e(t,k). Analyzing the stability of the batch process 3 under the ILC-based P-type controller 5 and 1 is equivalent to analyzing the stability of the Roesser’s system 12. Its denominator polynomial is 1 1 dðz1 1 , z2 Þ ¼ detðI z2 IÞ
1 detfI z1 1 ½A BKC BKLðz2 I IÞ Cg
ð13Þ
1 1 If we set z1 2 = 1, then d1(z1 ,z2 ) = 0. According to Lemma 1,
the system 12 is asymptotically unstable. Because two kinds of ILC are studied in this paper, there are correspondingly two kinds of ILC-based P-type control: P-type ILC-based P-type control, which has a P-type ILC in the outer loop; A-P-type ILC-based P-type control, which has an A-P-type ILC in the outer loop. Let us analyze the differences between P-type and A-P-type ILC-based P-type control physically. Using 3 and 5, one knows that eðt þ 1, kÞ ¼ Y ðt þ 1Þ yðt þ 1, kÞ ¼ Y ðt þ 1Þ CAxðt, kÞ CBK½yr ðt, kÞ yðt, kÞ ð14Þ Taking the set-point, yr(t,k), as the input and the tracking error, e(t + 1,k), as the output, their dynamic relationship is termed as set-point influence dynamics for convenience. From 14, one knows that the relative degree of the set-point influence dynamics is one under ILC-based P-type control.32 Hence, the amplitude of e(t + 1,k) indicates the control performance of yr(t,k). Due to the repetitive nature of the system 3, e(t + 1,k 1) can be considered a estimation for e(t + 1,k); therefore, e(t + 1,k 1) was used in 2 to design yr(t,k). It has been proved, if the ILC module is P-type, the closedloop system under the ILC-based P-type controller is asymptotically unstable. In this section, it has been proved that P-type ILC cannot be used in ILC-based P-type control, but it does not mean P-type ILC cannot be used in L-MPC. In the sequel, we will explain why P-type ILC can be implemented in L-MPC.
Figure 2. 7 Block diagram of learning-type model predictive control (L-MPC). The solid arrow lines denote the real-time information; the dotted arrow lines denote the past information; components in the dashed frame comprise an iterative learning control (ILC).
concept of L-MPC is that MPC works as the local controller to regulate the glucose level and ILC is used to adjust the set-point for MPC. The block diagram of L-MPC is shown in Figure 2. The MPC module and its explicit solution are introduced in the following section. A. MPC Module and Its Explicit Solution. A brief overview of MPC is provided here; a more detailed overview of MPC can be found in textbooks, for example, ref 33. The deviations in this section are standard, and they were included to make this paper selfcontained. The key algorithmic components for MPC include: prediction model, cost function, and receding horizon optimization. Based on the system model in eq 3, the prediction model can be built. Given that the set-point for MPC is yr(t,k), the cost function could be chosen as Ω¼
M1
N
∑ ð^yðt þ ijt, kÞ yr ðt þ i, kÞÞ2 þ j∑¼ 0 αðΔuðt þ jjt, kÞÞ2 i¼1 ð15Þ
where the integers N and M (N g M) are referred to, respectively, as the prediction horizon and control horizon; ^y(t + i|t,k) denotes the prediction of y(t + i,k) based on the known information at time t of batch k; Δu(t + j|t,k)|j=0M1 denotes the possible control signal variations in the control horizon; the weight α adjusts the relative importance of input variation. Guidelines for choosing {N, M, α} can be found in the literature.33 The following optimization problem is solved to obtain the control signal: 1 Δuðt þ jjt, kÞjM j ¼ 0 ¼ arg min Ω Δu
ð16Þ
From 3, one gets that Y^ ðt þ 1jt, kÞ ¼ Ψxðt, kÞ þ ΦΔUðt, kÞ þ Γuðt 1, kÞ
ð17Þ
where 3 3 2 Δuðtjt, kÞ ^yðt þ 1jt, kÞ 7 6 6 7 6 Δuðt þ 1jt, kÞ 7 6 ^yðt þ 2jt, kÞ 7 7 7, ΔUðt, kÞ ¼ 6 ^ Y^ ðt þ 1jt, kÞ ¼ ^6 7 7 6 6 l l 5 5 4 4 Δuðt þ M 1jt, kÞ ^yðt þ Njt, kÞ 2
4. LEARNING-TYPE MODEL PREDICTIVE CONTROL (L-MPC) In the literature,7 L-MPC was proposed and implemented in the closed-loop glycemic control for type 1 diabetes mellitus. The
ð18Þ 13429
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CB 6 6 CB þ CAB 2 3 6 CA 6 l 6 M1 6 6 CA2 7 7 6 7 CAi B Ψ¼ ^6 ^6 6 l 7, Φ ¼ 6 i¼0 4 5 6 6 l CAN 6 6 N1 4 CAi B
∑
∑ i¼0
3 CB 7 6 6 CB þ CAB 7 7 6 Γ¼ ^6 l 7 7 6 N1 4 i 5 CA B 2
0 CB l
M2
∑
i¼0
CAi B l
N2
∑ CA B i¼0 i
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333 333 ⋱
0 0 l
333
CB
333
l
333
NM
∑
i¼0
3
CAi B
7 7 7 7 7 7 7, 7 7 7 7 7 5
ð19Þ
∑
i¼0
The cost function can be expressed as Ω ¼ ΔU T ðΦT Φ þ αIÞΔU þ 2ΔU T ΦT ½Ψxðt, kÞ þ Γuðt 1, kÞ YR ðt, kÞ þ ½Ψxðt, kÞ þ Γuðt 1, kÞ YR ðt, kÞT ½Ψxðt, kÞ þ Γuðt 1, kÞ YR ðt, kÞ ð20Þ where
2
3 " # yr ðt þ 1, kÞ yr ðt þ 1, kÞ 6 7 6 7 l YR ðt, kÞ ¼ ^4 ^ 5¼ Y̅ R ðt þ 1, kÞ yr ðt þ N, kÞ
differences, these two types of L-MPC will be implemented on the injection molding process and a nonlinear batch process in the sequel.
5. COMPARISON BETWEEN P-TYPE AND A-P-TYPE L-MPC ALGORITHMS To compare the P-type and A-P-type L-MPC methods, they were implemented on the injection molding process and a numerical nonlinear batch process. A. Injection Molding Process. Injection molding process is a typical batch process, which consists of three main phases: filling, packing/holding, and cooling.34 In the filling phase, injection velocity, a key process variable, should be controlled to follow a certain profile to ensure product quality. The injection velocity response to the proportional valve has been identified as an autoregressive model: ð1 1:582z1 þ 0:5916z2 Þ yðt, kÞ ¼ ð1:69z1 þ 1:419z2 Þuðt, kÞ þ wðt, kÞ
where w(t,k) was introduced to describe the unknown disturbances. The stability margin of the nominal model is narrow, as one of the poles located near the boundary of the united disk. Because only input and output variables are practically measurable, the state-space realization of 25 is shown below: 82 3 2 > yðt þ 1, kÞ 1:582 0:5916 > >6 6 7 > 7¼6 6 1 > yðt, kÞ 0 > 4 5 4 > > > > uðt, kÞ 0 0 > > > 2 3 2 3 > > > > 1:69 wðt, kÞ < 6 7 6 7 6 7 7 þ6 4 0 5uðt, kÞ þ 4 0 5 > > 0 1 > > > 3 2 > > > > yðt, kÞ > > 7 6 > 7 > yðt, kÞ ¼ ½ 1 0 0 6 > 4 yðt 1, kÞ 5 > > > : uðt 1, kÞ
ð21Þ
is the set-point sequence in the prediction horizon. Let (∂Ω)/(∂ΔU) = 0, one can get the solution of the optimization problem 16 as below: ΔU ðt, kÞ ¼ ðΦT Φ þ αIÞ1 ΦT ½Ψxðt, kÞ þ Γuðt 1, kÞ YR ðt, kÞ
ð22Þ
After a solution ΔU*(t,k) is obtained, only the first term Δu* (t|t,k) is implemented; therefore, one has 1 T T 3 3 3 0 ðΦ Φ þ αIÞ Φ ½Ψxðt, kÞ þ Γuðt 1, kÞ YR ðt, kÞ 1 2 3 ¼K ^ MPC uðt 1, kÞ þ KMPC xðt, kÞ þ KMPC YR ðt, kÞ 3, 1 1 2 ¼K ^ MPC uðt 1, kÞ þ KMPC xðt, kÞ þ KMPC yr ðt þ 1, kÞ
uðt, kÞ ¼ uðt 1, kÞ ½ I
32 3 1:419 yðt, kÞ 76 7 6 7 0 7 54 yðt 1, kÞ 5 0 uðt 1, kÞ
ð26Þ
0
3, 2 þ KMPC Y̅ R ðt þ 1, kÞ
ð23Þ B. Learning-Type Set-Point. From 3, 4, and 23, we see that
The initial value of the state is assumed to be zero for all k. In this paper, the target profile takes the following form: ( 15, 1 e t < 100 Y ðtÞ ¼ ð27Þ 30, 100 e t < 200 To evaluate the tracking performance, the following performance index was used,
eðt, kÞ ¼ Y ðtÞ yðt, kÞ ¼ Y ðtÞ CAxðt 1, kÞ CBuðt 1, kÞ 2 Þxðt 1, kÞ ¼ Y ðtÞ ðCA þ CBKMPC 3, 2 3, 1 1 CBKMPC uðt 2, kÞ CBKMPC yr ðt, kÞ Y̅ R ðt, kÞ CBKMPC
ð24Þ Taking yr(t,k) as the input and e(t,k) as the controlled output, the relative degree of the set-point influence dynamics in 24 is zero. Therefore, yr(t,k)’s predecessor, yr(t,k 1), and e(t,k)’s predecessor, e(t,k 1), could be used to design yr(t,k), as shown in 1. Because ILC in 1 is P-type, the combination of this ILC and a MPC is termed as P-type L-MPC. Similarly, the combination of ILC in 2 and a MPC is termed as A-P-type L-MPC. The dynamics in 24 demonstrates that P-type L-MPC is a more reasonable choice compared with A-P-type L-MPC. To further study their
ð25Þ
ATEðkÞ ¼ ^
200
∑ jeðt, kÞj=201
ð28Þ
t¼0
which is termed the average tracking error for the k th batch. For the MPC module, the prediction horizon and control horizon were fixed at 10 and 5, respectively. Hence, there are only two parameters need to design in the L-MPC: the learning gain L and the input variation weight α. First, the weight is fixed to α = 100, and three values for L (0.2, 0.5, and 0.8) were compared. Using 23, the gains for MPC are 1 2 ¼ 0:338, KMPC ¼ ½ 0:1217, 0:0649, 0:1556 KMPC 3 KMPC ¼ ½0:0057, 0:0136, 0:0156, 0:0128, 0:0089, 0:0057,
0:0029, 0:0004, 0:0020, 0:0041 13430
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Figure 5. ATE value comparison of L-MPC with three learning gains, where the disturbances are repetitive.
Figure 3. ATE comparison between P-type and A-P-type L-MPC methods with three values of the learning gain L, where the input variation weight is fixed at α = 100: (a) L = 0.2; (b) L = 0.5; (c) L = 0.8.
Figure 6. Output responses under L-MPC with α=100 and L = 0.5, where the disturbances are repetitive.
Figure 4. ATE comparison between P-type and A-P-type L-MPC methods with three values of the input variation weight α, where the learning gain is fixed at L = 0.5: (a) α = 10; (b) α = 100; (c) α = 1000.
First, it is assumed that there is no disturbance, that is, w(t,k) 0. The ATE values for two algorithms were compared under three cases: L = 0.2, 0.5, or 0.8. All results were shown in Figure 3. One can find that the P-type L-MPC has superior performance in all situations. Particularly from Figure 3 (b) and (c), one can see that the closed-loop system under A-Ptype L-MPC is unstable.
Then, L is fixed at 0.5, three values for the weight, α = 10, 100, or 1000, were compared. Please note that the gains for MPC are related with α; however, these gains were omitted in this paper for space reasons. All ATE values were shown in Figure 4. The P-type L-MPC is better than the A-P-type L-MPC in all situations. Furthermore, the closed-loop system under the A-P-type L-MPC is unstable. Now, it has been confirmed that the P-type ILC is a better choice for the L-MPC compared with the A-P-type ILC. In the sequel, we will focus on the P-type L-MPC and study how to design the learning gain. It is assumed that the unknown disturbance w(t,k) is repetitive, for example wðt, kÞ ¼ 0:2sinðt=10Þ
ð29Þ
The ATE values under L-MPC with three learning gains (L = 0.2, 0.5, or 0.8) were compared in Figure 5. From Figure 5, one finds that a larger learning gain induce a faster convergence rate. Particularly when L = 0.5, the output responses in Batches 1, 10, 13431
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Figure 8. Mean ATE value comparison for three learning gains using Monte Carlo method, where 100 groups of simulations were run for each case and the nonrepetitive disturbances are uniformly distributed within [0.4 0.4]. Figure 7. ATE comparison between P-type and A-P-type L-MPC methods on a nonlinear batch process with three values of the learning gain L: (a) L = 0.2; (b) L = 0.5; (c) L = 0.8.
and 100 were shown in Figure 6. One can clearly see how the tracking performance improved in the batch direction. In the sequel, the robustness of the L-MPC with respect to nonrepetitive disturbances was studied under different learning gains. Similarly to the previous section, the weight is fixed at α = 100, and three values for the learning gain (L = 0.2, 0.5, or 0.8) were compared. It is assumed that w(t,k) denotes white noises uniformly distributed within [-0.4 0.4]. Obviously, the closedloop system is a stochastic process due to the nonrepetitive disturbances, so Monte Carlo method35 was used to compare these three learning gains. For each gain, 100 groups of Monte Carlo simulations were done. Given ATEi(k) is the ATE value for batch k in the ith simulation, then the mean ATE can be defined as MATE ¼
100
∑ ATEi ðkÞ=100
ð30Þ
i¼1
The mean ATE values for three learning gains were compared in Figure 7, which illustrates that the tracking performance when L = 0.2 is the best. It is clear that a smaller learning gain has better robustness to nonrepetitive disturbances. B. Nonlinear Batch Process. In this section, two kinds of L-MPC are compared on the following nonlinear batch process:
To simplify the design procedure, only the linear part, {A,B, C}, is used to design a linear MPC controller. For the MPC module, the prediction horizon, control horizon, and input variation weight are fixed at 10, 5, and 10, respectively. Using 23, the gains for MPC are 1 2 ¼ 0:7096, KMPC ¼ ½ 0:0522, 0:0905 KMPC 3 KMPC ¼ ½0:0710, 0:0529, 0:0019, 0:0381, 0:0506,
0:0476, 0:0359, 0:0238, 0:0169, 0:0165 The ATE values for two algorithms were compared under three cases: L = 0.2, 0.5, or 0.8. All results were shown in Figure 8. One can find that the P-type L-MPC has superior performance in all situations. Remark 1. In all cases introduced in Section 3, P-type L-MPC has superior performance compared with A-P-type method. On the other hand, A-P-type ILC should be used in the ILC-based P-type control. The reason for the difference can be found from equations in 14 and 24: the set-point influence dynamics in 14 has relative degree one and while that in 24 has relative degree zero. Remark 2. All conclusions presented in Sections 2, 3, 4, and 5 are based on an assumption that the controlled system is delayfree. If there is a time delay, td, in the input variable (i.e., u(t, k) in 3 is replaced by u(ttd, k)), then the P-type and A-P-type ILCbased set-points should be modified to the following form: P-type : yr ðt, kÞ ¼ yr ðt, k 1Þ þ Leðt þ td , k 1Þ A-P-type : yr ðt, kÞ ¼ yr ðt, k 1Þ þ Leðt þ td þ 1, k 1Þ
ð32Þ Strictly speaking, both of them are A-P-type ILC.
In this study, the target profile is Y*(t) 0. The ATE index defined in 28 is also used to evaluate the tracking performance.
6. L-MPC WITH BATCH-VARYING GAIN According to the simulation results in the previous section, it is very difficult to find a good constant learning gain: a larger value indicates faster convergence rate but worse robustness; a smaller value has opposite influences. A batch-varying learning 13432
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Figure 9. ATE value comparison for L-MPC with three various learning gains and/or batch-varying gain.
gain might be a promising solution: in the beginning, a larger value of the learning gain should be used to improve the convergence rate; if the tracking performance can be improved evidently, a larger value should be used; after the tracking performance becomes stable, a smaller value should be used to improve the robustness. According to this principle, the learning gain in 1 was proposed as below. First, the following index was introduced for convenience: Fðk 1Þ ¼ ^
ATEðk 2Þ ATEðk 1Þ ATEðk 2Þ
Figure 10. Mean ATE value comparison for L-MPC with four various gains, where 100 groups of Monte Carlo simulations were run for each case and the nonrepetitive disturbances are uniformly distributed within [0.4 0.4].
ð33Þ
Obviously, F(k 1) indicates the relative improvement of the ATE values in batch k 1, termed as improvement coefficient. A larger value of F(k 1) means a more rapid improvement and particularly F(k 1) = 1 indicates a perfect improvement. Then the learning gain is determined by the following algorithm. Batch-Varying Gain Algorithm (1) In batches 1 and 2, the learning gain is fixed at L = 0.8. (2) The learning gain for batch k (k g 3) is 8 > < 0:8, if 0:3 < Fðk 1Þ e 1 LðkÞ ¼ 0:5, if 0:1 < Fðk 1Þ e 0:3 ð34Þ > : 0:2, if Fðk 1Þ e 0:1 where demarcation points 0.3 and 0.1 are determined based on experience and simulation. First, the unknown disturbances are assumed to be repetitive as shown in 29. The ATE values in four scenarios, L-MPC with constant learning gains (L = 0.2, 0.5, or 0.8) or with batch-varying gain, are compared in Figure 9. Furthermore, nonrepetitive disturbance is also considered. The same with Section 3.C, white noises uniformly distributed within [0.4 0.4] are included and 100 groups of Monte Carlo simulations are done. The mean ATE values in four scenarios are compared in Figure 10. In sum, L-MPC with batch-varying gain has the smallest mean ATE values with nonrepetitive disturbances and has comparable convergence rate to that under L-MPC with L = 0.5 when disturbances are repetitive. As a result of balance, L-MPC with batch-varying gain might be the best choice.
Figure 11. An example of function L(k) = F(F(k 1)).
One possible shortcoming of the proposed batch-varying gain is that the relationship between L(k) and F(k 1) is discontinuous. If a continuous function L(k) = F(F(k 1)) is used, the learning gain can be updated continuously from batch to batch. One knows that the range for F is (∞, 1]. A reasonable F( 3 ) should be nondecreasing on (∞, 1]. One possible shape of F( 3 ) was shown in Figure 11. This direction will be addressed in the future.
5. CONCLUSIONS It has been proved the optimal structures of ILC are different for ILC-based P-type controller and/or L-MPC, even though both of them belong to SPR indirect ILC: A-P-type ILC is required for the former algorithm and P-type ILC is required for the latter one. In addition, the improved L-MPC with a batch-varying gain was proposed to balance the convergence rate and robustness performance. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected] (Y.W.); kefgao@ ust.hk (F.G.). 13433
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’ ACKNOWLEDGMENT This work is supported by the Fundamental Research Funds for the Central Universities (ZZ1012), Doctoral Fund of Ministry of Education of China (20100010120011), National 863 Project of China under Grant 2009AA04Z135, and National Nature Science Foundation of China under Grant 61074081 and 60974065. ’ REFERENCES (1) Uchiyama, M. Formulation of high-speed motion pattern of a mechanical arm by trial (in Japanese). Trans. SICE (Soc. Instrum. Contr. Eng.) 1978, 14, 706–712. (2) Arimoto, S.; Kawamura, S.; Miyazaki, F. Bettering operation of dynamic systems by learning: A new control theory for servomechanism or mechatronic system. In 23rd Conference on Decision and Control, Las Vegas, Nevada, 1984; pp 10641069. (3) Chu, B.; Owens, D. H. Iterative learning control for constrained linear systems. Int. J. Control 2010, 83 (7), 1397–1413. (4) Freeman, C. T.; Lewin, P. L.; Rogers, E.; Ratcliffe, J. D. Iterative learning control applied to a gantry robot and conveyor system. Trans. Inst. Meas. Control 2010, 32 (3), 251–264. (5) Liu, T.; Gao, F.; Wang, Y. IMC-based iterative learning control for batch processes with uncertain time delay. J. Process Control 2010, 20 (2), 173–180. (6) Meng, D.; Jia, Y.; Du, J.; Yu, F. Robust iterative learning control design for uncertain time-delay systems based on a performance index. IET Control Theory Appl. 2010, 4 (5), 759–772. (7) Wang, Y.; Zisser, H.; Dassau, E.; Jovanovic, L.; Doyle, F. J., III Model predictive control with learning-type set-point: Application to artificial pancreatic beta-cell. AIChE J. 2010, 56 (6), 1510–1518. (8) Yan, J. W.; Hou, Z. S. Iterative learning control based freeway ramp metering with iteration-varying parameter. Int. J. Innovative Comput. Inf. Control 2010, 6 (3A), 875–884. (9) Zhang, Y. W.; Fan, Y. P.; Zhang, P. C. Combining kernel partial least-squares modeling and iterative learning control for the batch-tobatch optimization of constrained nonlinear processes. Ind. Eng. Chem. Res. 2010, 49 (16), 7470–7477. (10) Chen, Y.; Moore, K. L.; Yu, J.; Zhang, T. Iterative learning control and repetitive control in hard disk drive industry—A tutorial. Int. J. Adapt. Control Signal Process. 2008, 22 (4), 325–343. (11) Tayebi, A.; Islam, S. Adaptive iterative learning control for robot manipulators: Experimental results. Control Eng. Practice 2006, 14 (7), 843–851. (12) Xiong, Z. H.; Zhang, J. Product quality trajectory tracking in batch processes using iterative learning control based on time-varying perturbation models. Ind. Eng. Chem. Res. 2003, 42 (2), 6802–6814. (13) 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 (7), 607–621. (14) Tang, A.; Xiao, D.; Mao, Z. Z. Batch-to-batch iterative learning control of a batch polymerization process based on online sequential extreme learning machine. Ind. Eng. Chem. Res. 2009, 48 (24), 11108–11114. (15) Huang, Y. C.; Chan, M.; Hsin, Y. P.; Ko, C. C. Use of PID and iterative learning controls on improving intra-oral hydranlic loading system of dental implants. JSME Int. J., Ser. C 2003, 46, 1449–1455. (16) Dou, H.; Tan, K. K.; Lee, T. H.; Zhou, Z. Iterative learning feedback control of human limbs via functional electrical stimulation. Control Eng. Pract. 1999, 7, 315–325. (17) Hou, Z.; Xu, J. X.; Zhong, H. Freeway traffic control using iterative learning control-based ramp metering and speed signaling. IEEE Trans. Veh. Technol. 2007, 56 (2), 466–477. (18) Tan, K. K.; Zhao, S.; Chua, K. Y.; Ho, W. K.; Tan, W. W. Iterative learning approach toward closed-loop automatic tuning of PID controllers. Ind. Eng. Chem. Res. 2006, 45 (12), 4093–4100.
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(19) Wang, Y. Q.; Gao, F. R.; Doyle, F. J., III Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control 2009, 19 (10), 1589–1600. (20) Wu, J. H.; Ding, H. Reference adjustment for a high-acceleration and high-precision platform via A-type of iterative learning control. Proc. Inst. Mech. Eng., Part J 2007, 221 (I5), 781–789. (21) Xu, J. X.; Yan, R. Iterative learning control design without a priori knowledge of the control direction. Automatica 2004, 40 (10), 1803–1809. (22) Jiang, P.; Bamforth, L. C. A.; Feng, Z.; Baruch, J. E. F.; Chen, Y. Indirect iterative learning control for a discrete visual servo without a camera-robot model. IEEE Trans. Syst. Man Cybern. Part B 2007, 37 (4), 863–876. (23) Bone, G. M. A novel iterative learning control formulation of generalized predictive control. Automatica 1995, 31 (10), 1483–1487. (24) Tayebi, A.; Chien, C. J. A unified adaptive iterative learning control framework for uncertain nonlinear systems. IEEE Trans. Autom. Control 2007, 52 (10), 1907–1913. (25) Wang, Y. Q.; Doyle III, F. J. Stability analysis for set-pointrelated indirect iterative learning control. In Proceedings of 48th IEEE Conference on Decision and Control, 2009; pp 57025707. (26) Lee, K. S.; Chin, I. S.; Lee, H. J. Model predictive control technique combined with iterative learning for batch processes. AIChE J. 1999, 45, 2175–2187. (27) Shi, J.; Gao, F. R.; 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. (28) Wang, Y.; Dassau, E.; Doyle, F. J., III Closed-loop control of artificial pancreatic beta-cell in type 1 diabetes mellitus using model predictive iterative learning control. IEEE Trans. Biomed. Eng. 2010, 57 (2), 211–219. (29) Shi, J.; Gao, F. R.; Wu, T.-J. Integrated design and structure analysis of robust iterative learning control system based on a twodimensional model. Ind. Eng. Chem. Res. 2005, 44, 8095–8105. (30) Shi, J.; Gao, F. R.; Wu, T. J. Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system. J. Process Control 2005, 15 (8), 907–924. (31) Kaczorek, T., Two-dimensional linear system.: Springer-Verlag: Berlin, 1985. (32) Sun, M.; Wang, D. Initial shift issues on discrete-time iterative learning control with system relative degree. IEEE Trans. Autom. Control 2003, 48 (1), 144–148. (33) Camacho, E. F.; Bordons, C., Model Predictive Control in the Process Industry; Springer-Verlag: London, 1995. (34) Wang, Y.; Shi, J.; Zhou, D.; Gao, F. Iterative learning faulttolerant control for batch processes. Ind. Eng. Chem. Res. 2006, 45 (26), 9050–9060. (35) Fishman, G. S. Monte Carlo: Concepts, Algorithms, and Applications; Springer-Verlag: New York, 1996.
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