From Two-Dimensional Linear Quadratic Optimal Control to Iterative

Ind. Eng. Chem. Res. 2006, 45, 4617-4628

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From Two-Dimensional Linear Quadratic Optimal Control to Iterative Learning Control. Paper 2. Iterative Learning Controls for Batch Processes Jia Shi,†,‡ Furong Gao,*,† and Tie-Jun Wu‡ Department of Chemical Engineering, Hong Kong UniVersity of Science & Technology, Clear Water Bay, Kowloon, Hong Kong, and Institute of Intelligent Systems and Decision Making, Zhejiang UniVersity, Hangzhou, 310027, PRC

On the basis of the two-dimensional linear quadratic (2DLQ) optimal control developed in the first paper of this series, systematical design and analysis of optimal iterative learning control (ILC) schemes are developed in this paper. The structure analysis indicates that the resulting optimal ILC laws are the combination of 2D ILC and 2D state feedback control optimized in a 2D sense. The proposed 2D based design methods not only result in a unified design of these two types of control but also allow easy extension of 2D information for better/balanced performance in terms of both time and batches. To avoid state measurement/estimation, a state-space realization based on input-output information is presented. The applicability and effectiveness of the proposed schemes are illustrated with velocity control in injection molding. 1. Introduction Iterative learning control (ILC) is an intelligent control motivated to mimic human learning. For a process performing a given task repetitively or cyclically, ILC can improve the control accuracy iteratively by updating the control signal of each new cycle. After its original introduction in 19841 for industrial robot manipulation, ILC has been developed as an effective control technique widely used in many industrial processes with repetitive natures. Nowadays, the ILC technique covers the original open-loop feed-forward control to the recent feedback and feed-forward combined control.2 The conventional ILC schemes, such as P-type ILC, are designed to be model-free. They are essentially timewise open-loop feed-forward controls where the control inputs of the current cycle are computed from the control inputs and control errors of the previous cycles. This type of ILC control cannot guarantee the control performance within the cycle. The combination of timewise feedback control with cyclewise ILC can not only guarantee the timewise control performance but also improve the cyclewise convergence and robustness. With feedback control included, the design of such combined ILC is usually model-dependent and more complicated than the conventional ILC. In most existing methods,3-5 the feedback controller and feed-forward ILC were separately designed, resulting in difficulties for control performance analysis and optimization. A repetitive process can be viewed as a two-dimensional (2D) system with respect to the time and cycle axes, where the dynamical behavior over the time axis is mainly determined by the process dynamics, whereas the cyclewise dynamics is governed by the repeatability of the process. In a 2D system view, a perfect repeating process has no cyclewise dynamics. Different from a general 2D system,6 a repetitive process has timewise dynamics over a finite time duration. In a 2D system view, the timewise open-loop feed-forward ILC is essentially a cyclewise feedback control with cyclewise integral action; the * To whom correspondence should be addressed. Telephone: +8522358 7139. Fax: +852-2358 0054. E-mail: [email protected]. † Hong Kong University of Science & Technology. ‡ Zhejiang University.

combination of feedback control and feed-forward ILC leads naturally to 2D feedback control with cyclewise integral action. Similar to the integral action of a 1D system, the cyclewise integral action can eliminate the steady-state control error along cycle index, which explains why ILC can improve the control accuracy from cycle to cycle. On the basis of this understanding, the design and analysis of ILC systems can be considered and solved in the 2D system frame. Research based on 2D control theory began to emerge in 1990. Geng et al.7 proposed to describe an ILC system as a 2D system for design and analysis. The proposed ILC scheme, however, is a timewise open-loop feed-forward ILC. On the basis of a 2D Roesser model, Kurek and Zaremba8 and Fang and Chow9 developed feedback and feed-forward combined ILC schemes for deterministic repetitive processes. On the basis of the robust stability of a 2D Roesser system, Shi et al.10 proposed an integrated design method for uncertain batch processes. These designs are all convergence analysis based, resulting in control performance that may be conservative. The ILC proposed by Amann et al.11 is an optimal ILC in terms of one cycle’s performance. Extension to that, Owens and Amann12 proposed to take the predictive control performance over several cycles into consideration, but the resulting ILC law was noncausal. For a class of repetitive 2D processes, two-dimensional linear quadratic (2DLQ) optimal control schemes have been developed in the first paper of this series. The objective of this paper is to propose optimal ILC schemes within the framework of 2DLQ optimal control. First, an ILC system is represented by an equivalent 2D Fornasini-Marchesini (2D-FM) model. On the basis of this, the similarities of control performance between the ILC system and the equivalent 2D-FM model are presented. Using these similarities, single-batch LQ (SB-LQ), multibatch LQ (MB-LQ), and batchwise receding horizon LQ (BW-RHLQ) optimal ILC problems are defined and solved in the framework of 2DLQ optimal control. As the batchwise (robust) convergences of the resulting ILC systems are equivalent to the cyclewise (robust) stabilities of the equivalent 2D-FM models, the cyclewise (robust) stability analysis methods developed in the first paper of this series can be directly used for the batchwise (robust) convergence analysis of the resulting ILC system. The structure analysis indicates that the resulting ILC schemes essentially consist of two types of controls: one is a

10.1021/ie051298a CCC: $33.50 © 2006 American Chemical Society Published on Web 05/26/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006

Figure 1. Block diagrams of a control system: (a) SB-LQ optimal ILC scheme; (b) BW-RHLQ optimal ILC scheme.

Figure 2. Block diagrams of the SB-LQ optimal ILC scheme with extension of the 2D state information.

2D ILC and the other is a 2D state feedback control. Both of these two controls use the 2D information of the process to ensure the optimal control performance. The 2D based design methods not only give a unified design of these two types of controls but also allow easy extension of 2D state information for better/balanced control performance. As the 2DLQ optimal controls proposed in the first paper are based on a state-space model, an input-output based state-space realization is given

in this paper to ensure the wide applicability of the proposed schemes. The feasibility and effectiveness of the proposed optimal ILC scheme are illustrated with velocity control of injection molding. The organization of this paper is as follows: The mathematical description of the ILC system and its equivalent 2D-FM representation are given in section 2. On the basis of the equivalent 2D-FM model, in section 3, SB-LQ, MB-LQ, and

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4619

Figure 3. Output responses (case 1): (a-d) SB-LQ optimal ILC scheme; (e-h) BW-RHLQ optimal ILC scheme. Table 1. Similarities Between the Control Performance of the ILC System and 2D-FM System ∑2D-P1 2D-FM system ∑2D-P1

ILC system

timewise (robust) stability cyclewise (robust) stability cyclewise (robust) H∞ performance

timewise (robust) stability batchwise (robust) convergence batchwise (robust) disturbance rejection

BW-RHLQ optimal ILC problems are defined and solved in the framework of 2DLQ optimal control. Batchwise robustness and convergence of the resulting control systems are analyzed in section 4. System structure analysis is given in section 5, together with an input-output based realization of the control schemes. To improve control performance, a technique to extend 2D state information is proposed in section 6. The application to

injection velocity control illustrates the effectiveness of the proposed schemes in section 7. Conclusions are drawn in section 8. For consistency, the same notations of the first paper of this series are used in this paper. 2. ILC System and Equivalent 2D Representation 2.1. ILC System. A batch process, repetitively performing a given task over a period of time (called a batch/cycle), can be described by the following time-varying state-space model:

{

}

x(t + 1,k) ) A(t)x(t,k) + B(t)u(t,k) y(t,k) ) C(t)x(t,k) x(0,k) ) x0,k, t ) 0, 1, 2, ..., T; k ) 1, 2, ...

∑BP1:

(1)

4620


By defining the state variable as

X(t,k) )

(

δk(x(t,k)) e(t,k)

)

(6)

and setting tracking error, e(t,k) as the controlled output, the combination of eqs 4 and 5 can be expressed in matrix form

∑2D-P1:

[

X(t + 1,k) ) A1(t)X(t,k) + A2(t)X(t + 1,k - 1) + B(t)r(t,k) e(t,k) ) C(t)X(t,k) (7)

where

A1(t) )

(

Figure 4. Batchwise tracking errors of different control schemes (case 1).

where t is the time, k is the batch index, x(t,k) ∈ Rn, u(t,k) ∈ Rm, and y(t,k) ∈ Rl represent, respectively, the state, input, and output of the process at time t in the kth batch run, {A(t),B(t),C(t)} are the time-varying matrices with appropriate dimensions, and x0,k is the timewise initial state of the kth batch run. As the initial state of each batch can usually be reset in many applications, it is assumed in this paper that x0,k ≡ x0. For the above batch process, introduce an ILC law with the form:

∑ILC:

u(t,k) ) u(t,k - 1) + r(t,k)

(2)

u(t,0) ) 0, t ) 0, 1, 2, ..., T where u(t,0) is the initial profile of iteration and r(t,k) ∈ Rm is referred as the updating law of the ILC. The objective for ILC design is to determine ILC law ∑ILC (or updating law r(t,k)) such that the output of the process tracks a given trajectory, yr(t), as accurately as possible. Different updating laws result in different ILC schemes, depending on what and how the available information of the process is used. For example, for the conventional P-type ILC scheme,2 the updating law is proportionally determined by the tracking error of the last batch only, i.e., r(t,k + 1) ) L(yr(t) - y(t,k)), where L is called the learning rate of the ILC scheme. 2.2. Equivalent 2D Representation. It is noticed from the description of ILC system (eqs 1 and 2) that all variables of the control system are the functions of both time t and batch index k; the process ∑BP1 determines the dynamics of the control system along the time axis, whereas the ILC law introduces batchwise dynamics. In other words, the ILC system has 2D dynamical characteristics. Define

δt(f(t,k)) ) f(t,k) - f(t - 1,k), δk(f(t,k)) ) f(t,k) - f(t,k - 1) (3)

( ) )

(

Model ∑2D-P1, called the equiValent 2D-FM model of the ILC system, is essentially an FM-type 2D system,6 where r(t,k) is the input of the system. With the assumption that x(t,0) ≡ 0, the boundary condition of the above 2D model can be determined as follows:

X(0,k) )

(

)

( )

0 0 , X(t,0) ) , yr(0) - C(0)x0 yr(t) t ) 0, 1, 2, ..., T; k ) 1, 2, ... (9)

Definition 6 indicates clearly the equivalence of control performance between the ILC system and its equivalent 2DFM model, as listed in Table 1. The importance of building the equivalent 2D-FM model for an ILC system is that the ILC problem can be treated in the framework of 2D systems. 3. ILC Schemes Based on 2DLQ Optimal Control In most existing designs of ILC, only batchwise convergence of the control system is considered, resulting in open-loop feedforward control, that may not guarantee timewise control performance which is found in ref 10 to also be important for batchwise convergence. Treating an ILC problem from a 2D system viewpoint allows both the timewise and batchwise control performance of the ILC system to be considered and optimized simultaneously. In this section, based on the equivalent 2D-FM model of the ILC system, the 2DLQ optimal control methods developed in the first paper of this series will be applied for the design and analysis of the ILC system. Before the main results are provided, the following problems are defined first. Definition 3.1. Single-Batch LQ (SB-LQ) Optimal ILC Problem. For batch process ∑BP1, design the ILC law (eq 2) such that the following single-batch quadratic performance index is minimized:

Jk+1(r([0:T - 1],k + 1)) ) T-1

It is derived from model 1 and the ILC law (eq 2) that

δk(x(t + 1,k)) ) A(t)δk(x(t,k)) + B(t)r(t,k)

)

A(t) 0 0 0 , A2(t) ) , 0 I -C(t + 1)A(t) 0 B(t) , C(t) ) (0 I ) (8) B(t) ) -C(t + 1)B(t)

(4)

e(t + 1,k) ) yr(t + 1) - y(t + 1,k) ) e(t + 1,k - 1) C(t + 1)δk(x(t + 1,k)) ) e(t + 1,k - 1) C(t + 1)A(t)δk(x(t,k)) - C(t + 1)B(t)r(t,k) (5)

(XT(t + 1,k + 1)Q(t + 1)X(t + 1,k + 1) + ∑ t)0 rT(t,k + 1)R(t)r(t,k + 1)) (10)

where Q(‚) and R(‚), respectively, are the semipositive definite (SPD) and positive definite (PD) matrices and variable X(‚,‚) is defined by eq 6.


Figure 5. Output responses (case 2): (a-d) SB-LQ optimal ILC scheme; (e-h) BW-RHLQ optimal ILC scheme.

Definition 3.2. Multibatch LQ (MB-LQ) Optimal ILC Problem. For batch process ∑BP1, design the ILC law (eq 2) such that the following multibatch quadratic performance index is minimized:

Definition 3.3. Batchwise Receding Horizon LQ (BWRHLQ) Optimal ILC Problem: For batch process ∑BP1, design the ILC law (eq 2) such that the following multibatch quadratic performance index is minimized:

J[1:N](r([0:T - 1],[1:N])) )

J[k+1:k+N](r([0:T - 1],[k + 1:k + N])) )

N T-1

∑ ∑ i)1 t)0

N T-1

(XT(t + 1,i)Q(t + 1,i)X(t + 1,i) +

∑ ∑(XT(t + 1,k + i)Q(t + 1,i)X(t + 1,k + i) + i)1 t)0

rT(t,i)R(t,i)r(t,i)) (11)

rT(t,k + i)R(t,i)r(t,k + i)) (12)

where Q(‚,‚) and R(‚,‚) are SPD and PD matrices, variable X(‚,‚) is defined by eq 6, and integer N > 0 is defined as the batchwise optimization horizon in this paper.

where Q(‚,‚) and R(‚,‚) are SPD and PD matrices, variable X(‚,‚) is defined by eq 6, and integer N > 0 is defined as the batchwise optimization horizon in this paper.

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Theorem 3.2. MB-LQ Optimal ILC Scheme: For the MBLQ optimal ILC problem proposed in definition 3.2, the optimal updating laws over the optimization horizon are defined as follows (i) for i ) 1, 2, ..., N - 1,

r* (t,i) ) -K(t,i)

(

)

X(|0t ,i) , t ) 0, 1, ..., T - 1 X(|t+1 T ,i - 1)

(19)

(ii) for i ) N,

r* (t,N) ) -K(t,N)


Remark 3.1. From the definition of the state variable X(‚,‚)s see eq 6sit clear that not only tracking error within the batch but also the changes of the state and control from batch to batch are included and considered in the performance indices (eq 1012). This allows for the batchwise and cyclewise control performance to be adjusted and balanced by selecting weighting matrices Q(‚) (or Q(‚,‚)) and R(‚) (or R(‚,‚)) properly. As the timewise boundary conditions, X(0,k), are the same, the SB-LQ, MB-LQ, and BW-RHLQ optimal ILC problems proposed in the above are essentially, respectively, SC-2DLQ, MC-2DLQ, and CW-RHLQ (here, C stands for cycle) optimal control problems with respect to the equivalent 2D-FM model ∑2D-P1. Therefore, these problems can be solved in the framework of the 2DLQ optimal controls developed in the first paper. By applying theorem 3.2, theorem 3.3, and theorem 3.4 of the first paper, the following optimal ILC schemes are obtained. Theorem 3.1. SB-LQ Optimal ILC Scheme. For the SBLQ optimal ILC problem proposed in definition 3.1, the optimal updating law for the (k + 1)th batch is

r* (t,k + 1) ) -K(t)

(

)

X(t,k + 1) , t ) 0, 1, ..., T - 1 (13) X(|t+1 T ,k)

where the time-varying matrix K(t) is determined by the following backward recursive algorithm: Algorithm 1

S(T) ) Q(T)

(

)

Q(t) 0 0(T-t)n×(T-t)n + P(t) 0

(16)

-1 T

(R(t) + B h (t)S(t + 1)B h (t)) B h (t)S(t + 1)A h (t) (17) where t ) 0, 1, ..., T - 1, and

A h (t) )

(

)

where the time-varying matrices K([0:T - 1],[1:N]) are designed by the following backward recursive algorithm: Algorithm 2 (i) Step 1. For i ) N, the matrices K([0:T - 1],N) are determined by algorithm 1 proposed in theorem 3.1, where the matrices {P(t),Q(t),R(t),S(t),K(t)} are, respectively, replaced by matrices {P(t,N),Q(t,N),R(t,N),S(t,N),K(t,N)}. (ii) Step 2. For i ) 1, 2, ..., N - 1, the matrices K([0:T 1],i) are determined by the following backward recursive algorithm:

S(T,i) )

(

)

0Tn×Tn 0 + P(0,i + 1) Q(T,i) 0

(

A1(t) A2(t) 0 B(t) h (t) ) 0 I(T-t-1)n , B 0 0 (T-t-1)n×m

)

(18)

(21)

P(t,i) ) A ˜ T(t)(S(t + 1,i) - S(t + 1,i)B ˜ (t)(R(t,i) + ˜ (t))-1B ˜ T(t)S(t + 1,i))A ˜ (t) (22) B ˜ T(t)S(t + 1,i)B

(

)

0tn×tn 0 0 Q(t,i) 0 + P(t,i) S(t,i) ) 0 0(T-t)n×(T-t)n 0 0

(23)

K(t,i) ) ˜ (t))-1B ˜ T(t)S(t + 1,i)A ˜ (t) (24) (R(t,i) + B ˜ T(t)S(t + 1,i)B where the initial matrix P(0,N) is determined in step 1, t ) 0, 1, ..., T - 1, and

(

Itn 0 A ˜ (t) ) 0 0

0 In A1(t) 0

0 0 A2(t) 0

)

0 0 , 0 I(T-t-1)n

( )

0tn×m 0n×m B ˜ (t) ) B(t) 0(T-t-1)n×m

(25)

Theorem 3.3. BW-RHLQ Optimal ILC Scheme. For the BW-RHLQ optimal ILC problem proposed in definition 3.3, the optimal updating law for the (k + 1)th batch is

r* (t,k + 1) ) -K(t,1)

K(t) ) T

)

X(t,N) , t ) 0, 1, ..., T - 1 X(|t+1 T ,N - 1) (20)

(14)

h (t)(R(t) + P(t) ) A h T(t)(S(t + 1) - S(t + 1)B T B h (t)S(t + 1)B h (t))-1B h T(t)S(t + 1))A h (t) (15) S(t) )

(

(

)

X(|0t ,k + 1) , t ) 0, 1, ..., T - 1 X(|t+1 T ,k) (26)

where the time-varying matrices K([0:T - 1],1) are determined by algorithm 2 proposed in theorem 3.2. Remark 3.2. From the viewpoint of a 2D system, optimal updating laws 13, 19, 20, and 26 are all time-varying 2D state feedback controllers for the equivalent 2D-FM model ∑2D-P1.


Figure 7. Output responses (case 3): (a-d) SB-LQ optimal ILC scheme; (e-h) BW-RHLQ optimal ILC scheme.

As the system matrices of 2D model ∑2D-P1 are time-dependent only, all the resulting feedback gain matrices K(‚) or K(‚,‚) are cycle-independent and can be computed offline. It is also noted that the feedback gain matrices of control law 26 in any batch are the same as the feedback gain matrices of control law 20 in the first batck if the same batchwise optimization horizons are used in the performance indices 11 and 12. From the viewpoint of ILC schemes for a batch process, optimal updating laws 13, 19, 20, and 26 are all computed from the state and output information obtained in the past times or past batches; in other words, all the schemes are causal. Remark 3.3. The MB-LQ optimal ILC scheme is suitable for batch processes with certain and known batch runs. Larger optimization horizons can lead to heavy computational burdens

for the design. Furthermore, the number of the batches may not be predetermined in many applications. Therefore, an SBLQ or a BW-RHLQ optimal ILC scheme is more applicable for design and implementation. In the following, the discussion will focus mainly on these two schemes. Remark 3.4. If weighting matrices Q(‚) (or Q(‚,‚)) are fixed, matrices R(‚) (or R(‚,‚)) can be used as a knob to balance the timewise control performance and batchwise robust convergence. Smaller values of R(‚) (or R(‚,‚)) allow larger control changes from batch to batch; this may lead to a better tracking performance within a batch but worse batchwise robust stability. Note, from definition 6, if weighting matrices Q(‚) (or Q(‚,‚)) are selected as the following diagonal matrices

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Q(‚) )

(

Q1(‚) 0 Q2(‚) 0

)

or Q(‚,‚) )

(

Q1(‚,‚) 0 Q2(‚,‚) 0

)

(27)

where both Q1(‚) ∈n×n (or Q1(‚,‚) ∈ Rn×n) and Q2(‚) ∈ Rl×l (or Q2(‚,‚) ∈ Rl×l) are SPD matrices, then the relative values of the matrices Q1(‚) (or Q1(‚,‚)) and Q2(‚) (or Q2(‚,‚)) can also be specified to balance the tracking performance and batchwise robust stability. In addition, when Q1(‚) ≡ 0, the SB-LQ optimal ILC scheme is reduced to the norm-optimal ILC scheme proposed by Amann et al..11 4. Batchwise Convergence and Robustness Analysis It is clear from definition 8 that a deterministic batch process results in a deterministic equivalent 2D-FM model. According to Table 1, the batchwise convergence of the ILC system corresponds to the cyclewise stability of the 2D-FM model. Therefore, the methods of cyclewise stability analysis for a 2D system developed in the first paper of this series can be used for the batchwise convergence analysis of the ILC system. Now, consider an uncertain batch process described by the following state-space model:

∑

BP2: x(t + 1,k) ) (A(t) + ∆A(t))x(t,k) + (B(t) + ∆B(t))u(t,k) + w(t,k) y(t,k) ) C(t)x(t,k)

{

(28)

where w(t,k) represents the disturbance, possibly resulting from the unmodeled dynamics or external disturbances, {A(t),B(t),C(t)} are the nominal system matrices with appropriate dimensions, and {∆A(t),∆B(t)} denote the uncertain parameter mismatches and/or perturbations at time t with the structure as follows:

∆A(t) ) E1(t)∆1(t)F1(t), ∆B(t) ) E2(t)∆2(t)F2(t)

(29)

where matrices {Ei(t),Fi(t)}i)1,2 are known matrices with appropriate dimensions, characterizing the structures of the parameter mismatches and/or perturbations. {∆i(t)}i)1,2 are unknown matrices satisfying the following norm-bounded conditions:

∆iT(t)∆i(t) < I, i ) 1, 2; ∀ t ) 0, 1, 2, ..., T

(30)

For the above uncertain batch process and ILC law (eq 2), by using the same manipulation used in subsection 2.2, the equivalent 2D-FM model can be derived as follows

∑

{

2D-P2:

X(t + 1,k) ) (A1(t) + ∆A1(t))X(t,k) + A2(t)X(t + 1,k - 1) + (B(t) + ∆B(t))r(t,k) + D(t)δk(w(t,k)) e(t,k) ) C(t)X(t,k) (31)

where state variable X(t,k) is defined by eq 6, {A1(t),A2(t),B(t),C(t),} are nominal system matrices given by eq 8, and the rest of the parameters are defined as follows:

(

)

I -C(t + 1) E (t) ∆A(t) 0 ∆A1(t) ) ) 1 ∆ (t) -C(t + 1)E1(t) 1 -C(t + 1) ∆A(t) 0 (F1(t) 0 ) D(t) )

(

∆B(t) )

(

) (

)

) (

)

E (t) ∆B(t) ) 2 ∆ (t)F2(t) -C(t + 1)E2(t) 2 -C(t + 1)∆B(t) (32)

Clearly, model ∑2D-P2 is a 2D-FM system with parameter uncertainties and disturbance, and the similarities of control performance between the ILC system and the 2D model ∑2D-P2 listed in Table 1 still hold. Therefore, the batchwise robust convergence and robust disturbance rejection performance of the optimal ILC system can be analyzed by using the robust analysis methods developed for the 2D system in the first paper of this series. 5. System Structure Analysis In an ILC sense, the structure analysis of the resulting control system is important not only for investigating the control performances but also for the implementation of the control schemes. 5.1. State Feedback Realization. As mentioned before, both the SB-LQ optimal ILC law (eq 13) and the BW-RHLQ optimal ILC law (eq 26) are essentially time-varying 2D state feedback controllers for the equivalent 2D-FM model. From definition 6, the two optimal ILC laws can be rewritten as follows by rearranging the columns of gain matrix K: (i) SB-LQ optimal ILC scheme

( )

δk(x(t,k + 1)) δ (x(|t+1,k)) u(t,k + 1) ) u(t,k) - K′(t) k T e(t,k + 1) e(|t+1 T ,k)

( )

(33)

(ii) BW-RHLQ optimal ILC scheme

δk(x(|0t ,k + 1)) δ (x(|t+1,k)) u(t,k + 1) ) u(t,k) - K′(t,1) k 0 T e(|t ,k + 1) e(|t+1 T ,k)

(34)

u(t,k + 1) ) uilc(t,k + 1) + ufbc(t,k + 1)

(35)

Now, let K′(t) ) (K′1(t) K′2(t) K′3(t) K′4(t)) and K′(t,1) ) (K′1(t,1) K′2(t,1) K′3(t,1) K′4(t,1)). The above ILC laws can be decomposed as

where uilc(t,k + 1) and ufbc(t,k + 1), respectively, result from a 2D ILC law and a 2D state feedback control law formulated as follows: (i) SB-LQ optimal ILC scheme

∑2D-ILC:

uilc(t,k + 1) ) uilc(t,k) - K′3(t)e(t,k + 1) -

∑2D-FBC:

ufbc(t,k + 1) ) -K′1(t)x(t,k + 1) -

K′4(t)e(|t+1 T ,k) (36)


K′2(t)x(|t+1 T ,k) (37)


∑2D-ILC:

uilc(t,k + 1) ) uilc(t,k) - K′3(t,1)e(|0t ,k + 1) K′4(t,1)e(|t+1 T ,k) (38)

Σ2D-FBC: ufbc(t,k + 1) ) -K′1(t,1)x(|0t ,k + 1) -

only, the state feedback controller can be easily realized in applications. Note that the process with input-output time delay can also be handled by this realization. 6. Extension of 2D State Information

K′2(t,1)x(|t+1 T ,k)

(39)

From the above decomposition, it can be seen that the output of 2D ILC law ∑2D-ILC is updated based on the historical tracking errors of the current batch and all tracking errors of the last batch over the remaining time, leading to the improvement of control performance not only from batch to batch but also over time, whereas 2D state feedback law ∑2D-FBC, depending on the state information of the current batch over the past time and the last batch over the remain time, guarantees both timewise and batchwise (robust) stability. The 2D based design framework gives a united design of these two types of control laws. With the available state information, both control laws are physically realizable. The difference between these two schemes is that all historical information of the current batch is used in the BW-RHLQ optimal ILC scheme, whereas only the current information of the current batch is used in the SB-LQ optimal ILC scheme. The system structures for these two schemes are shown in Figure 1, where the triangular blocks represent the proportional units. 5.2. Input-Output Based State-Space Realization. Note that optimal laws 37 and 39 require state information. In many applications, state measurement or estimation is not practical. To ensure wide applicability of the proposed methods, an inputoutput based state-space realization will be developed for the widely used controlled auto-regressive moving-average (CARMA) input-output model. Assume that the following CARMA model is used to describe a single-input single-output (SISO) batch process:

y(t + 1,k) ) a1y(t,k) + a2y(t - 1,k) + ‚‚‚ + anyy(t - ny + 1,k) + b1u(t,k) + b2u(t - 1,k) + ‚‚‚ + bnuu(t - nu + 1,k) + w(t,k) (40) where u(t,k), y(t,k), and w(t,k) are, respectively, the input, output, and unknown disturbance of the process. With the following definition of the state

Another advantage of designing an ILC system based on equivalent 2D-FM model ∑2D-P1 or ∑2D-P2 is that the 2D state information of the 2D model can be flexibly extended to be included in a performance index for better batchwise and/or timewise performance. For example, to eliminate the timewise steady-state tracking error, the integral of the tracking error can be included in the performance index by introducing the integral of the tracking error as an extended state of the 2D system. Let the state to be extended be defined by the following general 2D dynamic model

∑E:

xe(t + 1,k) ) A1exe(t,k) + A2exe(t + 1,k - 1) + B1ee(t,k) + B2ee(t + 1,k - 1) (44)

where xe(t,k) is the extended state dynamically driven by tracking error, e(t,k), and {A1e,A2e,B1e,B2e} are the parameter matrices to be specified according to the characteristic of the extended states. For example, a state representing the timewise integral of the tracking error can be created by simply specifying A1e ) B1e ) I, A2e ) B2e ) 0. The following parameter specifications can result in proportional-integral-derivative (PID) information of the tracking error:

( ) ()

0 0 0 I A1e ) 0 I 0 , B1e ) I , A2e ) 0, B2e ) 0 -I 0 0 I

The combination of the extended model ∑E with the equivalent 2D-FM model ∑2D-P1 leads to an augmented 2D-FM model:

∑2D-EP1:

{

X ˜ (t + 1,k) ) A ˜ 1(t)X ˜ (t,k) + A ˜ 2(t)X ˜ (t + 1,k - 1) + B ˜ (t)r(t,k) z(t,k) ) C ˜ (t)X ˜ (t,k) (46)

where the augmented state variable is

( )

x(t,k) ) (y(t,k) ... y(t - ny + 1,k) u(t - 1,k) ... u(t - nu +

δk(x(t,k)) X ˜ (t,k) ) xe(t,k) e(t,k)

T

1,k)) (41) a state-space realization of model 40 can be obtained as

(

ΣBP3:

{

x(t + 1,k) ) Ax(t,k) + Bu(t,k) + Dw(t,k) (42) y(t,k) ) Cx(t,k)

where

a1 1 ·· · A) 0 0 0 ·· · 0

a2 0 ·· · 0 0 0 ·· · 0

··· ··· ··· ··· ··· ··· ··· ···

any b2 0 0 ·· ·· · · 0 0 0 0 0 1 ·· ·· · · 0 0

··· ··· ··· ··· ··· ··· ··· ···

bnu-1 bnu

) ()

b1 0 0 0 ·· ·· ·· · · · 0 0 , B) 0 , 1 0 0 0 0 0 ·· ·· ·· · · · 0 1 0 C ) [1 0 · · · 0 0 · · · 0 ] (43)

As the state variable consists of input and output information

(45)

(47)

z(t,k) is the output variable determined by the specification of output matrix C ˜ (t), and the other parameters are constructed as follows:

(

)

(

)

A(t) 0 0 0 0 0 A1e B1e , A ˜ 2(t) ) 0 A2e B2e , A ˜ 1(t) ) 0 0 0 I -C(t + 1,k) 0 0 B(t) (48) B ˜ (t) ) 0 -C(t + 1)B(t)

(

)

On the basis of the augmented state variable, X ˜ (t,k), new performance indices involving new objectives can be defined, resulting in new SB-LQ and BW-RHLQ optimal ILC laws as follows: (i) SB-LQ optimal ILC scheme

4626


( )


(49)

7.1. CARMA Model and Input-Output Based State-Space Realization. On the basis of the open-loop test and analysis, the model of the injection velocity response to the hydraulic control valve opening has been identified as the following CARMA model with parameter uncertainties:

e(t,k + 1) e(|t+1 T ,k) : u (t,k + 1) ) u (t,k) K ˜ (t) 2D-ILC ilc ilc 1 xe(t,k + 1) xe(|t+1 T ,k)

(50)

x(t,k + 1) x(|t+1 T ,k)

(51)

(4.084 + ∆b2)u(t - 3,k) + w(t,k) (55)

(52)

where u(t,k) is the input, i.e., the valve opening, y(t,k) is the output, i.e., injection velocity, w(t,k) represents the unknown disturbance, possibly resulting from unmodeled dynamics and/ or external disturbances, and {∆a1,∆a2,∆b1,∆b2} are unknown parameters, indicating the parameter mismatch and/or disturbances satisfying the following norm-bounded conditions

∑

(

∑

˜ 2(t) 2D-FBC: ufbc(t,k + 1) ) -K

)

y(t + 1,k) ) (1.319 + ∆a1)y(t,k) + (-0.350 + ∆a2)y(t - 1,k) + (5.782 + ∆b1)u(t - 2,k) +



∑2D-ILC:

∑

uilc(t,k + 1) )

( )

e(|0t ,k + 1) e(|t+1,k) ˜ 1(t,1) T0 uilc(t,k) - K xe(|t ,k + 1) xe(|t+1 T ,k)

˜ 2(t,1) 2D-FBC: ufbc(t,k + 1) ) -K

(

x(|0t ,k + 1) x(|t+1 T ,k)

)

|∆R| e 0.05|R|, R ∈ {a1,a2,b1,b2} (53)

(54)

where {K ˜ 1(t),K ˜ 2(t)} and {K ˜ 1(t,1),K ˜ 2(t,1)} are designed by algorithm 1 and algorithm 2, respectively. Let K ˜ 2(t) ) (K′1(t) K′2(t)) and K ˜ 1(t) ) (K′3(t) K′4(t) K′5(t) K′6(t)); then, the block diagram of the SB-LQ optimal ILC scheme with the extended state information used is shown in Figure 2. Numerous ways of extending state information can be easily incorporated in the proposed scheme for control performance enhancement at the cost of a somewhat more complex control structure. 7. Application Illustration Injection modeling, a cyclic process, consists of three main stages: filling, packing/holding, and cooling. For each stage, the key variables of the process should be controlled to follow certain profiles to ensure the product quality. During the filling stage, injection velocity, a key variable in determining the mechanical and surface properties of the part, is controlled by a hydraulic valve. The complexity and uncertainty of the hydraulic system coupled with the nature of the materials impart to the batch process some uncertainties and significant disturbances.13 A conventional ILC scheme, such as P-type ILC, cannot provide satisfactory control performance within the batch. On the basis of the equivalent 2D Roesser representation of an ILC system, Shi et al.10 proposed a robust design method for a feedback and feed-forward combined ILC scheme to guarantee the robust control performance along both the time and batch indices. However, only convergence of the control system was taken into account in the designs, resulting in a conservative controller. On the basis of the state-space model, Gao et al.13 proposed using a norm-optimal ILC scheme with respect to the control performance within one batch to improve the timewise and batchwise control performance. The resulting control law, however, depends on state information that may not be directly measured. In this section, the proposed optimal ILC schemes will be applied to injection velocity control, described by an CARMA model with parameter uncertainties, to demonstrate the feasibility and effectiveness of the proposed schemes.

(56)

where {a1,a2,b1,b2} denote the corresponding nominal parameters. To ensure the implementation of the control laws, the inputoutput based state-space realization proposed in subsection 5.2 is conducted as follows. Chose the state variable as

x(t,k) ) (y(t,k) y(t - 1,k) u(t - 1,k) u(t - 2,k) u(t - 3,k) )T (57) Then the state-space realization of model 55 is as follows:

∑BP3:

{

x(t + 1,k) ) (A + ∆A)x(t,k) + Bu(t,k) + Dw(t,k) (58) y(t,k) ) Cx(t,k)

(

) ()

where

1.319 1 A) 0 0 0

-0.345 0 0 0 0

0 0 0 1 0

5.782 0 0 0 1

4.084 0 , B) 0 0 0

0 0 1 , 0 0

()

1 0 C ) (1 0 0 0 0 ), D ) 0 0 0

(

∆A is an uncertain parameter with the structure

0.066 0 ∆A ) 0 0 0

0.0173 0 0 0 0

0 0 0 0 0

0.289 0 0 0 0

)

0.204 0 ∆ 0 0 0

(59)

(60)

Clearly, condition 56 is equivalent to following norm-bounded condition:

∆T∆ e I

(61)

7.2. Design Parameters and Results. To illustrate and compare the effectiveness of the SB-LQ optimal ILC scheme and the BW-RHLQ optimal ILC scheme, three cases are conducted.


Case 1. Note that model ∑BP3 is a simplified version of model ∑BP2 and that model ∑2D-P2 can be viewed as the equivalent 2D-FM model of ILC of process ∑BP3. On the basis of this 2D model, the following performance indices are selected for designs of SB-LQ and BW-RHLQ optimal ILC schemes. (i) SB-LQ performance index

T-1

(X (t + 1,k + 1)QX(t + 1,k + 1) + ∑ t)0 T

r (t,k + 1)Rr(t,k + 1)) (62) T

(ii) BW-RHLQ performance index

J[k+1:k+N](r([0:T - 1],[k + 1:k + N])) ) N T-1

(XT(t + 1,k + i)Q(i)X(t + 1,k + i) + ∑ ∑ i)1 t)0 rT(t,k + i)R(i)r(t,k + i)) (63) where

( )

( )

0 0 0 0 , R ) 10, Q(i) ) 2i-1 × , 0 1 0 1 R(i) ) 2i-1 × 10, T ) 100, N ) 3 (64)

In this case, it is assumed that ∆ ) diag{-1,1,1,1,1} and w(t,k) ) 0. The system responses to a given trajectory under the SBLQ optimal ILC scheme and the BW-RHLQ optimal ILC scheme are compared in Figure 3. It is clear that both control schemes guarantee the robust stability against model mismatch and that the control performance is improved from batch to batch. The sums of the batchwise tracking errors, shown in Figure 4, clearly indicate that the BW-RHLQ optimal ILC scheme provides better control performance and faster convergence. Case 2. In case 1, with model mismatch and constraints on the batchwise input change, noticeable steady-state tracking errors can be found in the output responses of the first few batches, as shown in Figure 3a, b, e, and f. There are two solutions to this problem. One is to choose a smaller value of R which will lead to better control performance in the first batch; however, this may be at the cost of batchwise robustness. The second solution is to extend a time integral of the tracking error as a new state in the equivalent 2D-FM model and define performance indices as discussed in section 6. This extension is tested in case 2 simulations. On the basis of the process nominal model, it is assumed that the augmented equivalent 2D-FM model is formulated by eq 46, where the extended state variable xe(t,k) is the timewise integral of the tracking error determined by model ΣE with parameters A1e ) B1e ) 1 and A2e ) B2e ) 0. To eliminate the tracking error along the time index, the following performance indices are used in this case: (i) SB-LQ performance index T-1

Jk(r([0:T - 1],k + 1)) )

N T-1

(X ˜ T(t + 1,k + i)Q(i)X ˜ (t + 1,k + i) + ∑ ∑ i)1 t)0 rT(t,k + i)R(i)r(t,k + i)) (66) where

Jk+1(r([0:T - 1],k + 1)) )

Q)

J[k+1:k+N](r([0:T - 1],[k + 1:k + N])) )

(X ˜ T(t + 1,k + 1)QX ˜ (t + 1,k + ∑ t)0 1) + rT(t,k + 1)Rr(t,k + 1)) (65)

(ii) BW-RHLQ performance index

( )

( )

0 0 0 0 0 Q ) 0 1 0 , R ) 10, Q(i) ) 2i-1 × 0 1 0 0 1 0 0 i-1 R(i) ) 2 × 10, T ) 100,

0 0 , 1 N ) 3 (67)

On the basis of these new performance indices and the same process model used in case 1, the SB-LQ optimal ILC scheme and the BW-RHLQ optimal ILC scheme are applied. The control results are shown in Figure 5, where it can be seen that the tracking errors of even the first cycle reduce gradually to zero over time. The batchwise tracking errors, shown in Figure 6, indicate also that BW-RHLQ optimal control is better. Case 3. In the previous cases, it has been assumed that w(t,k) ≡ 0. To illustrate the robustness of the proposed control schemes against unmodeled dynamics and/or external disturbances, now, w(t,k) is taken as a random variable distributed within the interval [-0.2,0.2]. The performance indices and control schemes remain as the same as those used in case 2. The output responses and batchwise tracking errors respectively shown in Figures 7 and 8 indicate good robustness of the proposed control schemes.


In all the above cases, it is found that the performance of the BW-RHLQ optimal ILC scheme, multicycle receding horizon control, is superior to that of the SB-LQ optimal ILC scheme, single-cycle optimal control. This paper is the first in proposing causal multicycle optimal ILC for batch processes. 8. Conclusions On the basis of 2DLQ optimal control developed in the first paper of this series, systematical design and analysis of optimal ILC schemes for a batch process have been developed in this paper. The structure analysis indicates that the resulting ILC laws are the combination of two types of controls: 2D ILC and 2D state feedback control. The proposed 2D based design methods not only give a unified design of these two types of control but also allow an easy extension of the 2D state

4628


information for the better/balanced control performance over time and batches. To avoid state measurement/estimation, an input-output based state-space realization of an CARMA model is also given. The feasibility and effectiveness of the proposed optimal ILC schemes have been illustrated with injection velocity control under different cases. Acknowledgment This work is supported in part by the Hong Kong Research Grant Council under Project No. 601104. Literature Cited (1) Arimoto, S.; Kawamura, S.; Miyazaki, F. Bettering operation of robots by learning. J. Rob. Syst. 1984, 1 (2), 123-140. (2) Xu, J.-X.; Bien, Z. Z. The frontiers of iterative learning control. IteratiVe Learning Control: Analysis, Design, Integration and Application; Kluwer Academic Publishers: Boston/Dordrecht/London, 1998; 9-35. (3) Moon, J.-H.; Doh, T.-Y.; Chung, M. J. A robust approach to iterative learning control design for uncertain systems. Automatica 1998, 34 (8), 1001-1004. (4) Tayebi, A.; Zaremba, M. B. Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition. IEEE Trans. Autom. Control 2003, 48 (1), 101-106. (5) Gorinevsky, D. Loop shaping for iterative control of batch processes. IEEE Control Syst. Mag. 2002, 55-65.

(6) Kaczorek, T. Two-dimensional linear system; Springer: Berlin, 1985. (7) Geng, Z.; Carroll, R.; Xie, J. Two-dimensional model and algorithm analysis for a class of iterative learning control systems. Int. J. Control 1990, 52 (4), 833-862. (8) Kurek, J. E.; Zaremba, M. B. Iterative learning control synthesis based on 2-D system theory. IEEE Trans. Autom. Control 1993, 38 (1), 121-125. (9) Fang, Y.; Chow, T. W. S. 2-D analysis for iterative learning controller for discrete-time systems with variable initial conditions. IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 2003, 50 (5), 722-727. (10) Shi, J.; Gao, F.; 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, 907-924. (11) Amann, N.; Owens, D. H.; Rogers, E. Iterative learning control using optimal feedback and feedforward actions. Int. J. Control 1996, 65 (2), 277-293. (12) Owens, D. H.; Amann, N.; Rogers, E.; French, M. Analysis of linear iterative learning control schemes - a 2D system/repetitive processes approach. Multidim. Syst. Signal Process. 2000, 11, 125-177. (13) Gao, F.; Yang, Y.; Shao, C. Robust iterative learning control with applications to injection molding process. Chem. Eng. Sci. 2001, 56, 70257034.

ReceiVed for reView November 22, 2005 ReVised manuscript receiVed March 23, 2006 Accepted March 29, 2006 IE051298A

From Two-Dimensional Linear Quadratic Optimal Control to Iterative

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