Optimal Iterative Learning Control Based on a Time-Parametrized

Article pubs.acs.org/IECR

Optimal Iterative Learning Control Based on a Time-Parametrized Linear Time-Varying Model for Batch Processes Zuhua Xu,† Jun Zhao,† Yi Yang,*,† Zhijiang Shao,† and Furong Gao†,‡ †

National Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China ‡ Department of Chemical & Biomolecular Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong ABSTRACT: In this paper, an optimal iterative learning control (ILC) algorithm based on a time-parametrized linear timevarying (LTV) model for batch processes is proposed. Utilizing the repetitive nature of batch processes, a time-parametrized LTV model is used to represent the nonlinear behavior, with its consistence and variance properties established. Furthermore, an optimal ILC algorithm based on the time-parametrized LTV model is developed, and its convergence property is analyzed. Simulations have demonstrated the effectiveness and excellent performance of the proposed method. However, developing a first principle model of batch processes is always costly and time-consuming; the accuracy of the first principle model is often not high enough for dynamic control. In the multimodel approach, the nonlinear system is represented by a combination of multiple linear systems. However, questions such as how many linear models are required used and which model is scheduled still remain open. For the neural network method, the main difficulty is the high cost in modeling and the inability to extrapolate. Compared to continuous processes, batch processes can be characterized by the frequent repetition of batch runs, which has never been fully utilized in the existing process modeling approaches. Applying the repetitive nature of batch processes, we develop a time-parametrized linear time-varying (LTV) model to represent the nonlinear behavior of batch processes, in which model coefficients are parametrized as nonlinear functions of the time index, such as the polynomial function and the cubic spline function. Compared with other modeling methods for batch processes, this method has the advantages of a simple structure and fewer parameters. Its consistence and variance properties are established in this paper. Furthermore, an optimal iterative learning control based on time-parametrized LTV model is proposed. The convergence property of the ILC algorithm is analyzed. This work is inspired by the linear parameter-varying (LPV) modeling approach16−18 widely used in continuous processes, which parametrizes the parameters of the process model as nonlinear functions of the scheduling variable. The existing LPV identification approaches mostly assume static dependence on the scheduling variable (depend only on the instantaneous value of the scheduling variable) and ARX structure. However, dynamic dependence on the scheduling variable and colored noise conditions are common for batch processes. Moreover, the

1. INTRODUCTION As a preferred choice for manufacturing low-volume and highvalue-added products, batch processes play an important role in modern industries, such as specialty chemicals, pharmaceutical products, and polymers.1 To exploit the repetitive nature of batch processes, iterative learning control (ILC) has been widely used to improve the tracking accuracy and ensure the product quality in batch processes.2,3 Recently, due to its superior performance and the capability of dealing with multivariate constrained problems, model-based iterative learning control algorithms have been widely studied. Lee et al.4 proposed the quadratic-criterion-based ILC (Q-ILC) algorithm for batch processes with deterministic, stochastic disturbances and noises. In order to reject the real-time disturbance effectively, Chin et al.5 presented a two-stage iterative learning control technique combined with real-time feedback for independent disturbance rejection. From a two-dimensional (2D) system point of view, Shi et al.6 developed single-cycle and multicycle generalized 2D model predictive iterative learning control schemes for batch processes. In order to improve the convergence rate, Chu et al.7 proposed two accelerated normoptimal iterative learning control algorithms using successive projection. Based on the internal model control (IMC) structure, Liu et al.8 presented an IMC-based ILC method for batch processes with uncertain time delay. To deal with the model uncertainty of batch processes, Xu et al.9 proposed two robust iterative learning control algorithms based on the worst-case and nominal performance indices. Due to the wide operating range of batch processes, process variables exhibit significant nonlinearities inherently in the process dynamics. The linear time-invariant models often fail to describe batch processes adequately. Consequently, the control strategies based upon linear models can lead to poor control performance. There are several different approaches to handle nonlinear problem for batch processes: (1) trajectory-linearizationbased ILC;10,11 (2) multiple-model-based ILC;12,13 (3) neuralnetwork-based ILC.14,15 The first approach is to linearize the first principle model of batch processes along the nominal trajectory. © 2013 American Chemical Society

Received: Revised: Accepted: Published: 6182

September 20, 2012 February 28, 2013 April 2, 2013 April 4, 2013 dx.doi.org/10.1021/ie302561t | Ind. Eng. Chem. Res. 2013, 52, 6182−6192

Industrial & Engineering Chemistry Research

Article

interval [1, N]. It is well-known that any continuous functions on a closed interval can be approximated as closely as desired by a polynomial function if the order of the polynomial is high enough. Hence, the coefficient functions can be given as

selection of the scheduling variable is difficult, especially for multi-input multi-output (MIMO) systems. The use of a timeparametrized LTV model can avoid these difficulties, as illustrated in section 2. The rest of this paper is organized as follows. In section 2, a time-parametrized LTV model is proposed to describe the nonlinear behavior of batch processes. In section 3, an optimal iterative learning control method based on the time-parametrized LTV model is proposed. Numerical examples are illustrated in section 4, and finally, the conclusions are drawn in section 5.

ai(t ) = ai1 + ai2t + ... + aimt m − 1

(8)

bi(t ) = bi1 + bi2t + ... + bimt m − 1

(9)

Here m is the order of the coefficient function. Other representations are also available, such as the cubic spline function,19 which is given as

2. TIME-PARAMETRIZED LTV MODEL FOR BATCH PROCESSES For simplicity, consider a single-input single-output plant, which performs repetitively a given task over a finite time duration, called a batch or cycle, as described by the following linear timevarying (LTV) model:

m

∑ aij |t − δj − 2|3

ai(t ) = ai1 + ai2t +

(10)

j=3

m

bi(t ) = bi1 + bi2t +

∑ bij |t − δj − 2|3 (11)

j=3

yk (t ) = G(q , t ) uk(t ) + H(q , t ) ek(t ) t = 1, ..., N ;

k = 1, 2, ...

where knots {δ1, δ2, ..., δm−2} are real numbers and satisfy (1)

1 ≤ δ1 , δ2 , ..., δm − 2 ≤ N

where t and k represent the discrete-time index and cycle/batch index, respectively; N is the time duration of each cycle; yk(t), uk(t), and ek(t) are the output, input, and noise at time t in the kth cycle, respectively; G(q,t)is a time-varying transfer function from uk(t) to yk(t); H(q,t) is a time-varying transfer function from ek(t) to yk(t); and q indicates the timewise unit forward-shift operator. One of the traits of batch processes is that they involve repetitive operations. For instance, errors due to bias in the input variable will repeat themselves in the subsequent batches. The same phenomenon can be observed for the disturbance vk(t) = H(q,t) ek(t). In batch processes, the disturbance repeats itself or exhibits a strong batchwise correlation. But at the same time, the disturbance commonly exhibits drifting behavior along the batch index and not just random fluctuations around a stationary mean.2,4,5 Such behavior can be reasonably represented as an integrated white-noise process along the batch index: vk(t ) = vk − 1(t ) + wk(t )

Remark 1: The time-parametrized LTV model can overcome the drawbacks of the LPV model as detailed in the Introduction, such as the static dependence problem and the scheduling variable selection problem. Moreover, this method can easily extend to multi-input multi-output (MIMO) batch processes. Remark 2: Due to the batchwise correlations of disturbance, we can omit the disturbance model by differencing the output and the input between batches. Moreover, it is unnecessary to estimate the disturbance model due to the use of the incremental model in ILC as shown in section 3. 2.1. Parameter Estimation. For polynomial or cubic spline parametrization, the parameter vector to be determined is θ = [a11, ..., a1m , ..., an1, ..., anm , b11, ..., b1m , ..., bn1, ..., bnm]T

θ = [a11, ..., a1m , ..., an1, ..., anm , b11, ..., b1m , ..., bn1, ..., bnm , δ1, ..., δm − 2]T

(14)

where wk(t) is a zero-mean independent and identically distributed (i.i.d.) sequence with respect to both k and t. By differencing the output and the input between two consecutive batches, the following input−output model for batch processes can be derived as follows:

For the linear time-varying OE model (eq 3), the one-step-ahead optimal predictor is20 yk̅̂ (t |θ ) =

where (4)

A(q , t ) = 1 + a1(t )q−1 + ... + an(t )q−n

(5)

B(q , t ) = b1(t )q−1 + ... + bn(t )q−n

(6)

yk̅ (t ) = yk (t ) − yk − 1(t ),

(7)

VK (θ ) =

uk̅ (t ) = uk(t ) − uk − 1(t )

B (q , t ) uk(t ) A (q , t ) ̅

(15)

Then the parameter vector θ can be determined by minimizing the loss function:

(3)

B (q , t ) G (q , t ) = A (q , t )

(13)

or

(2)

yk̅ (t ) = G(q , t ) uk̅ (t ) + wk(t )

(12)

1 K

K

⎡1

N

⎤

t=1

⎥⎦

∑ ⎢ ∑ εk(t |θ)2 ⎥ k=1

⎢⎣ N

(16)

where εk(t |θ ) = yk̅ (t ) − yk̅̂ (t |θ )

Since the output error εk(t|θ) is nonlinear in the parameters of A(q,t), there exists no analytical solution to this minimization problem. Therefore, a numerical search algorithm is needed to find a minimum. There exist a large variety of numerical methods for optimization such as Gauss−Newton, Levenberg− Marquardt, Newton−Raphson, and others.20,21 Here, the Gauss−Newton method is derived for the minimization of VK(θ) in this work. Denote θ̂r as the estimate at iteration r. Assume that θ̂r is close to a local minimum. The output error

Here, eq 3 is referred to as the linear time-varying output error (OE) structure. In eqs 4 and 5, n is the order of the LTV OE model, ai(t) and bi(t) are time-varying coefficient functions. Due to the repetitive nature of batch processes, ai(t) and bi(t) can be parametrized as nonlinear functions of time index t on the 6183

dx.doi.org/10.1021/ie302561t | Ind. Eng. Chem. Res. 2013, 52, 6182−6192


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εk(t|θ) can be approximated by truncating high order terms of its Taylor expansion so that

The initial values of parameters aji and bji can be estimated by the following linear time-varying ARX model:

⎡∂ ⎤ εk(t |θ ) ≈ εk(t |θr̂ ) + ⎢ εk(t |θ )⎥ (θ − θr̂ ) ⎣ ∂θ ⎦ ̂

A(q , t ) yk̅ (t ) = B(q , t ) uk̅ (t ) + wk(t )

which leads to a least-squares (LS) problem. For parameters δj, it is common to let the knots be uniformly distributed in the range [1, N]. If the matrix ill-conditioned problem occurred in the optimization, the Levenberg−Marquardt method can be used to avoid it. 2.2. Order Selection. In the above identification algorithm, the order of LTV OE model and coefficient function (i.e., n and m) need to be determined. Overparametrization could lead to unnecessary moves by the controller. Hence, the minimum description length (MDL) criterion20 is used to determine n and m:

θr

= εk(t |θr̂ ) − ψk T(t , θr̂ )(θ − θr̂ )

(17)

where ⎡∂ ⎤T ⎡ ∂ ⎤T ψk(t , θ ) = −⎢ εk(t |θ )⎥ = ⎢ yk̅̂ (t |θ )⎥ ⎣ ∂θ ⎦ ⎣ ∂θ ⎦ (a column vector)

Under this approximation, the error is linear in the parameter. Minimize eq 16 with respect to θ and let the minimum point constitute the new parameter estimate θ̂r+1. Thus, it leads to the following Gauss−Newton algorithm: K

⎛ d log Nk ⎞ MDL = VK (θ )⎜1 + ⎟ Nk ⎠ ⎝

N

NK = K ·N

k=1 t=1 N

[∑ ∑ ψk(t , θr̂ ) εk(t |θr̂ )]

⎧ for polynomial function ⎪ 2n · m d=⎨ ⎪ ⎩(2n + 1) ·m − 2 for cubic spline function

(18)

k=1 t=1

where the scalar αr is used to control the step length so that VK(θr+1) ≤ VK(θr). Before applying the Gauss−Newton method to estimate parameters, we need also the gradient ψk(t,0) of the prediction yk̅̂ (t |θ ). Equation 15 can be rearranged as follows:

Here, NK is the number of estimation data, d is the number of estimated parameters, and the factor (1 + (d log Nk)/Nk)) is used to penalize too-complex model structures in view of the parsimony principle. 2.3. Theoretical Analysis. In this section, the limiting properties of the estimated parameters as the number of batch tends to infinity will be developed. Basic Assumptions. A1. The data {uk̅ (t), yk̅ (t)} are stationary processes. A2. The input is persistently exciting. A3. The model G(q,t) is a smooth function of the parameter vector θ. A4. The orders of the LTV model and coefficient functions are correctly determined. Theorem 1. Given the system (eqs 1−3) and assuming that assumptions A1−A4 hold, denote θ̂ = arg min VK(θ) as the parameter estimate that minimizes the output error loss function (eq 16) and Ĝ (q,t) as the corresponding transfer estimates. Denote G0(q,t) as the true transfer functions of the system and θ0 as the true parameter vector. Then 1. The output error model is consistent meaning that

yk̅̂ (t |θ ) + a1(t ) yk̅̂ (t − 1|θ ) + ... + an(t ) yk̅̂ (t − n|θ ) = b1(t ) uk̅ (t − 1) + ... + bn(t ) uk̅ (t − n)

(19)

By differentiating the equation with respect to aij and bij respectively, we get A (q , t )

A (q , t )

∂yk̅̂ (t |θ ) ∂aij

∂yk̅̂ (t |θ ) ∂bij

= −t j − 1yk̅̂ (t − i|θ )

(20)

= t j − 1uk̅ (t − i)

(21)

The gradient is thus obtained by filtering −t j − 1yk̅̂ (t − i|θ ) or t u̅k(t − i) through the filter 1/A(q,t). A similar result can be obtained for the cubic spline function: j−1

A(q , t )

A (q , t )

A (q , t )

∂yk̅̂ (t |θ ) ∂aij

∂yk̅̂ (t |θ ) ∂bi

j

∂yk̅̂ (t |θ ) ∂δj

⎧− t j − 1y ̂ (t − i|θ ) j≤2 ⎪ k̅ =⎨ ⎪− (t − δj − 2)3 y ̅̂ (t − i|θ ) j > 2 ⎩ k

⎧ t j − 1u (t − i) j≤2 ⎪ k̅ =⎨ ⎪(t − δj − 2)3 uk̅ (t − i) j > 2 ⎩

Ĝ (q , t ) → G0(q , t )

= 3(t − δj)

∑ [ai

j+2

(22)

(27)

dist

K (θ ̂ − θ0) ⎯⎯⎯⎯⎯→ (0, P) K →∞

(28)

where

(23)

⎡N ⎤−1 T ⎢ P = λ ∑ E[ψk(t , θ0) ψk (t , θ0)]⎥ ⎢⎣ t = 1 ⎥⎦

yk̅̂ (t − i|θ )

i=1

− bij + 2uk̅ (t − i|θ )]

for t = 1, ..., N as K → ∞

2. The parameter estimate θ̂ follows an asymptotic Gaussian distribution

n 2

(26)

where

θr̂ + 1 = θr̂ + αr[∑ ∑ ψk(t , θr̂ ) ψkT(t , θr̂ )]−1 K

(25)

(29)

λ is the variance of noise wk(t). Proof. 1. Denote the model error at the time t of each cycle

(24)

Remark 3: In order to obtain an accurate estimate in the optimization scheme, a good initial estimate should be provided.

ΔG(q , t ) = G0(q , t ) − Ĝ (q , t ) 6184

(30)



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Then, if K → ∞, we have 1 N

VK (θ) → V∞(θ) =

1 = N 1 = N

Since wk(t) is a zero-mean i.i.d. sequence N

P0 = lim E[KV K′ (θ0)T V K′ (θ0)]

∑ [E(yk̅ (t ) − yk̅̂ (t |θ))2 ]

K →∞ N

t=1

=

N 2

∑ E[ΔG(q , t ) uk̅ (t ) + wk(t )] t=1 2

2

∑ [ΔG(q , t )] E[uk̅ (t )] +

E[wk2(t )]

⎡N ⎤−1 T P = λ⎢∑ E[ψk(t , θ0) ψk (t , θ0)]⎥ ⎢⎣ t = 1 ⎥⎦

t=1

The last equality in eq 31 holds because wk(t) is a zero-mean i.i.d. sequence and uncorrelated with all past data. Provided the model order is correct and the minimization finds the global minimum, we have for t = 1, ..., N as K → ∞

Remark 4: For the linear time-invariant model of continuous processes, the limiting properties of the parameter estimates are well studied by Ljung20 and Soderstrom.21 For the timeparametrized LTV model of batch processes, we can prove its limiting properties of the parameter estimates using a similar method. Remark 5: If wk(t) is not a zero-mean i.i.d. sequence, Theorem 1 can still hold under the additional assumption that the data are collected from an open-loop experiment, and only eq 29 needs to be revised.

(32)

which implies that Ĝ (q , t ) → G0(q , t )

for t = 1, ..., N

(33)

As the model order is correct and model parameters are uniquely determined (identifiable), eq 33 is equivalent to θ → θ0 as K → ∞. 2. Since the estimate θ̂ is a minimum point of VK(θ), we have V′K(θ̂) = 0 A Taylor series expansion of VK′ (θ̂) around θ0 retaining the first two terms gives:

3. OPTIMAL ITERATIVE LEARNING CONTROL ALGORITHM FOR BATCH PROCESSES In this section, an optimal iterative learning control based on time-parametrized LTV model will be derived. The convergence property of the ILC algorithm is analyzed. 3.1. Derivation of the ILC Law. Consider the following time-parametrized LTV model of batch processes

0 = V K′ (θ )̂ T ≈ V K′ (θ0)T + V K″ (θ0)(θ ̂ − θ0) ≈ V K′ (θ0)T ″ (θ0)(θ ̂ − θ0) + V∞

(34)

The second approximation follows since V″K(θ0) → V″∞(θ0) with probability 1 as K → ∞. Since θ̂ convergences to θ0 as K tends to infinity, for large K the estimation error θ̂ − θ0 can be written as ″ (θ0)]−1 [ K V K′ (θ0)T ] K (θ ̂ − θ0) ≈ −[V∞

yk (t ) =

2 V K′ (θ) = − KN

(35)

N

∑ ∑ εk(t , θ)

ψkT(t ,

θ)

k=1 t=1

⎡ ⎤ ∂2 2 V K″ (θ) = ∑ ∑ ⎢ψk(t , θ) ψkT(t , θ) + εk(t , θ) 2 εk(t , θ)⎥ ⎦ KN k = 1 t = 1 ⎣ ∂θ K

N

(36)

V K′ (θ0) = − ″ (θ0) = V∞

2 KN

2 N

A(q , t ) = 1 + a1(t )q−1 + ... + an(t )q−n

(42)

B(q , t ) = b1(t )q−1 + ... + bn(t )q−n

(43)

vk(t ) = vk − 1(t ) + wk(t )

(44)

xk(t + 1) = A(t + 1) xk(t ) + B(t + 1) uk(t )

N

k=1 t=1

where

N

∑ E[ψk(t , θ0) ψkT(t , θ0)] t=1

⎡0 ⎢ ⎢1 ⎢ A (t ) = ⎢ 0 ⎢ ⎢⋮ ⎢ ⎣0

(37)

Since wk(t) and ψk(t,θ0) are uncorrelated, the following result can be obtained from lemmas B.3 and B.4 in ref 19. dist

K (θ ̂ − θ0) ⎯→ ⎯ (0, P)

(38)

with

0 ··· 0 − an(t ) ⎤ ⎥ 0 ··· 0 − an − 1(t ) ⎥ ⎥ 1 ··· 0 − an − 2(t )⎥ , ⎥ ⋮ ⋱ ⋮ ⋮ ⎥ ⎥ 0 ··· 1 − a1(t ) ⎦

C = [0 0 ··· 0 1], −1

″ (θ0)] P0[V∞ ″ (θ0)] P = [V∞

P0 = lim E[KV K′ (θ0) V K′ (θ0)]

xk(0) = x0

⎡ b (t ) ⎤ ⎢n ⎥ ⎢bn − 1(t )⎥ ⎢ ⎥ B(t ) = ⎢⋮ ⎥ ⎢ ⎥ ⎢ b2 (t ) ⎥ ⎢ ⎥ ⎣b1(t ) ⎦ (46)

Denote

T

K →∞

(45)

yk (t ) = Cxk(t ) + vk(t )

∑ ∑ wk(t ) ψkT(t , θ0)

−1

(41)

Here, ai(t) and bi(t) are nonlinear functions of time index t, given by eqs 8−12. Then, a state-space description can be derived as follows:

Since ψk(t, θ) and ∂2/∂θ2[εk(t,θ)] depends on the data up to time t − 1, they will be independent of wk(t) ≡ εk(t,θ0). Thus K

B (q , t ) uk(t ) + vk(t ) A (q , t )

where

By definition, we have K

(40)

Therefore, the covariance matrix is given by

N

(31)

[ΔG(q , t )]2 → 0

4λ ∑ E[ψk(t , θ0) ψkT(t , θ0)] N2 t = 1

yk = [ ykT (1) ykT (2) ... ykT (N )]T

(39) 6185

(47)



Article

uk = [ ukT(1) ukT(2) ... ukT(N − 1)]T

(48)

wk = [ wkT(1) wkT(2) ... wkT(N )]T

(49)

Then, the system dynamics in eq 45 can be written equivalently as the N × N-dimensional lifted system yk = G·uk + vk + y0 (51)

vk = [ vkT(1) vkT(2) ... vkT(N )]T

(50)

where vk = vk − 1 + wk

⎡CB(1) ⎢ ⎢CA(2) B(1) ⎢ ⎢CA(3) A(2) B(1) G=⎢ ⎢⋮ ⎢ N−2 ⎢C ∏ A(N − i) B(1) ⎢⎣ i = 0

(52)

⎤ ⎥ ⎥ CB(2) ⎥ CA(3) B(2) CB(3) ⎥ ⎥ ⋮ ⋮ ⋱ ⎥ ⎥ N−3 N−4 C ∏ A(N − i) B(2) C ∏ A(N − i) B(3) ... CB(N )⎥ ⎥⎦ i=0 i=0

(53)

N−1

y0 = [[CA(1)]T [CA(2) A(1)]T ... [C ∏ A(N − i)]T ]T x0

(54)

i=0

Let ek = r − yk represent the output error, where r is the desired output trajectory. Then the following transition model for tracking error trajectory can be derived as e k = e k − 1 − GΔuk − wk

Algorithm. Step 0. Carry out the identification experiment and estimate model parameters. Step 1. Compute model matrix G and select controller parameters Q and R. Step 2. Set batch index k = 0 and initialize U0. Step 3. Apply Uk to the system, measure output Yk, and compute tracking error ek. Step 4. Calculate control actions for next batch using the following control law

(55)

where Δuk = uk − uk−1 is the difference of the control input between batches. The ILC design can now be formulated to find a control law such that the system output has the asymptotic convergence property, i.e., ek → 0 as k → ∞. There are different choices of designs to solve this ILC problem. The quadratic (norm) optimal formulation is adopted here due to its superior performance and the capability of dealing with multivariate constrained problems. At each batch, the following quadratic performance index is minimized to obtain the input vector: min Δuk

1 E{e Tk Qe k + Δu Tk RΔuk } 2

Uk + 1 = Uk + He k ,

H = (GTQG + R)−1GTQ

Step 5. Set k = k + 1 and go back to step 3. The proposed optimal ILC scheme is called LTV-ILC and is illustrated in Figure 1.

(56)

where Q and R are positive definite matrices. The weighting matrices Q and R affect the contributions of the tracking errors and control changes. A large weight on the input change will lead to more conservative adjustments and slower convergence. Since wk(t) is a zero-mean i.i.d. sequence, the optimization problem (eq 56) is equivalent to the following formulation: min Δuk

1 {(e k − 1 − GΔuk )T Q(e k − 1 − GΔuk ) + Δu Tk RΔuk } 2

Therefore, we can obtain the following ILC algorithm by solving a standard least-squares problem: uk = uk − 1 + He k − 1

Figure 1. Schematic of the proposed LTV-ILC system.

(57)

3.2. Convergence Analysis. In this section, we analyze the convergence property of the optimal ILC algorithm based on the time-parametrized LTV model for the deterministic case (wk = 0) under the following assumptions: B1. G is a perfect model of a batch process and has a full row− rank. B2. Q and R are positive definite.

where H = (GTQG + R)−1GTQ

(58)

Hence, an optimal iterative learning control based on the timeparametrized LTV model for batch processes is outlined as follows. 6186



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Theorem 2. Under assumptions B1 and B2 and wk = 0 ∀ k, system (eq 55) converges to the origin under the optimal ILC algorithm (eqs 57 and 58), i.e., Δuk → 0 and ek → 0 as k → ∞. Proof: Let

By discretizing the process model (eq 59), we have yk (t ) =

Vk = min Jk (Δuk )

b1(t )q−1 1 + a1(t )q−1

uk(t ) + vk(t )

(61)

where

Δuk

where

a1(t ) = −exp(−ts/T (t ))

1 T T {e k Qe k + Δu k RΔuk } 2 Hence, we have 1 Vk ≤ Jk (Δuk ) = e Tk − 1Qe k − 1 Δuk = 0 2 1 T = Vk − 1 − Δu k − 1RΔuk − 1 2 Therefore, sequence {Vk} is nonincreasing and bounded from below by 0. Consequently, sequence {Vk} converges. Then, it follows that 1 lim Δu Tk − 1RΔuk − 1 ≤ lim (Vk − 1 − Vk ) k →∞ 2 k →∞

b1(t ) = K (t )(1 − exp(−ts/T (t )))

Jk (Δuk ) =

To obtain a time-parametrized LTV model, the identification experiment is carried out by a GBN (generalized binary noise) signal23 with an average switch time of 5 samples. In this example, the polynomial function is used to parametrize model coefficients ai(t) and bi(t). The order of the LTV model and the polynomial function are selected by minimizing the MDL criterion: n = 1 and m = 6. The design parameters of the ILC controller based on the LTV model are chosen as follows: Q = I and R = 10I. To investigate the performance of the proposed iterative learning control strategy, it is compared with iterative learning control based on the linear time-invariant (LTI) model, in which the first-order linear OE model is identified. In the simulation, we first consider the deterministic case in order to clearly compare the convergence rate of two ILC algorithms. Figures 2 and 3 plot the output and input of ILC based on the LTV model, while Figures 4 and 5 show the output and input of

= 0 ⇒ lim Δu Tk − 1RΔuk − 1 = 0 k →∞

It implies that Δuk → 0 as k → ∞. Since lim (e k − 1 − e k ) = lim GΔuk = 0

k →∞

k →∞

we obtain ek → e∞ as k → ∞. Assume that e∞ ≠ 0. Then, the gradient of J∞ at Δu∞ = 0 is ∇J∞(Δu∞)|Δu∞=0 = −GTQe∞. Since by assumption G has a full row−rank, we have ∇J∞(Δu∞)|Δu∞=0 ≠ 0. This contradicts the result that Δu∞ = 0 is the optimal solution, hence proves ek → 0 as k → ∞. Remark 6 :The above-mentioned theorem is also established by Lee et al.4 and Amann et al.22

4. NUMERICAL ILLUSTRATIONS The performance of the proposed iterative learning control algorithm based on the time-parametrized LTV model is illustrated through two numerical examples. 4.1. Example 1. In the first example, it is assumed that the batch process is described by the following continuous-time linear time-varying model with sampling period ts = 1: yk (t ) =

K (t ) uk(t ) + vk(t ) T (t )s + 1

Figure 2. Output response of ILC based on LTV model (deterministic case).

(59)

ILC based on the LTI model. The dotted lines represent the data of the first run, the dashed lines denote the results of the second run, and the solid lines plot the results of the third run. Figure 6 compares ∥ek∥Q values of these two ILC algorithms. It is clearly shown that ILC based on the time-parametrized LTV model converges much faster than the one based on linear time-invariant model. Figure 7−11 show the results of the numerical simulation with the stochastic case, which lead to the same conclusion. It must be noted that, due to the existence of the stochastic case, the control error cannot converge to zero. 4.2. Example 2. In the second example, we consider the temperature control of a fed-batch reactor where the irreversible, exothermic liquid-phase reaction A → B takes place.24 In this case, an initial amount of material is placed in the reactor, the

where T (t ) = 0.001t 2 + 3,

K (t ) = −0.03t 2 + 1.7t + 5

vk(t ) = vk − 1(t ) + 0.1nk (t ),

v0(t ) = 0.5 cos(0.04πt )

and nk(t) is a zero-mean Gaussian i.i.d. sequence with standard deviation 1. In this case, the desired reference trajectory is ⎧ 0.4t t ⎪ ⎪6 t r (t ) = ⎨ ⎪ 6 − 0.2(t − 25) t ⎪ t ⎩3

∈ [0, 15] ∈ [15, 25] ∈ [25, 40] ∈ [40, 50]

(60) 6187



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Figure 3. Input response of ILC based on LTV model (deterministic case).

Figure 6. Q-norm of error vs number of run (deterministic case).

Figure 4. Output response of ILC based on LTI model (deterministic case).

Figure 7. Output response of ILC based on LTV model (stochastic case).

Figure 5. Input response of ILC based on LTI model (deterministic case).

Figure 8. Input response of ILC based on LTV model (stochastic case). 6188



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equations describe the dynamics of the fed-batch reactor system: ⎧ dVR =F ⎪ ⎪ dt ⎪ dz F (z 0 − z ) ⎪ − kz ⎨ dt = VR ⎪ ⎪ UA(TR − TJ) ⎪ dTR = F(T0 − TR ) − λkz − ⎪ dt ρCpVR VR MCp ⎩ k = α e−E / RTR ,

A = 4VR /DR

(62)

where VR is the reaction volume, TR is the reactor temperature, z is the reactant concentration, and TJ is the jacket temperature. It is assumed that the reactor has a cooling jacket whose temperature is directly manipulated. Zero-mean i.i.d. noise with standard deviation 0.2 is assumed to corrupt TR. The initial values of the states are VR(0) = 200 ft3, z(0) = 0, and TR(0) = 90 °F. The model parameters are listed in Table 1.

Figure 9. Output response of ILC based on LTI model (stochastic case).

Table 1. Parameters for Case Study 2 parameter

value

units

mass density (ρ) molecular weight (M) activation energy (E) universal gas const (R) heat capacity (Cp) heat of reaction (λ) overall heat transfer coeff (U) reaction vessel diam (DR) fresh feed flow rate (F) reactant concn of fresh feed (z0) temp of fresh feed (T0) preexponential factor (α)

50 50 30 000 1.986 0.75 −20 000 100 5 20 1 90 4.354 × 1011

lb/ft3 lb/lb-mol Btu/lb-mol Btu/lb-mol−1 °R−1 Btu/lb·°F Btu/lb-mol Btu/h·°F·ft2 ft ft3/h mol °F

Figure 10. Input response of ILC based on LTI model (stochastic case).

In this case, the desired reference trajectory of the reactor temperature is ⎧ 90 + 10t t ⎪ ⎪120 t r (t ) = ⎨ ⎪120 + 10(t − 5) t ⎪ t ⎩150

∈ [0, 3] ∈ [3, 5] ∈ [5, 8] ∈ [8, 10]

(63)

In this example, the sampling period is selected as ts = 0.1 h, and the identification experiment is performed by a GBN signal with an average switch time of 6 samples. Here, the cubic spline function is used to parametrize model coefficients ai(t) and bi(t). The order of the LTV model and the order of cubic spline function are selected by minimizing the MDL criterion: n = 2 and m = 5. The design parameters of ILC controller based on LTV model are chosen as follows: Q = I and R = 0.02I. To illustrate the performance of the proposed iterative learning control strategy, it is compared with the iterative learning control strategy based on the LTI model, in which the second-order linear OE model is identified. In the simulation, we first consider the deterministic case. Figures 12 and 13 plot the output and input of ILC based on the LTV model, while Figures 14 and 15 show the output and

Figure 11. Q-norm of error vs number of run (stochastic case).

liquid is heated to the desired temperature, and then additional feed of fresh reactant is gradually added to the vessel. The nonlinearity is mainly due to variable volume. The following 6189



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Figure 12. Output response of ILC based on LTV model (deterministic case).

Figure 15. Input response of ILC based on LTI model (deterministic case).

Figure 13. Input response of ILC based on LTV model (deterministic case).

Figure 16. Q-norm of error vs number of run (deterministic case).

Figure 14. Output response of ILC based on LTI model (deterministic case).

Figure 17. Output response of ILC based on LTV model (stochastic case). 6190



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input of ILC based on the LTI model. The dotted lines represent the data of the first run, the dashed lines denote the results of the

Figure 21. Q-norm of error vs number of run (stochastic case).

second run, and the solid lines plot the results of the third run. Figure 16 compares ∥ek∥Q values of these two ILC algorithms. It again clearly proves that ILC based on the time-parametrized LTV model gives a better performance than the one based on the linear model, in terms of convergence rate and final control error. Figures 17−21 show the results of the numerical simulation with the stochastic case, with the same result.

Figure 18. Input response of ILC based on LTV model (stochastic case).

5. CONCLUSION In this paper, an optimal iterative learning control algorithm based on a time-parametrized LTV model for batch processes is proposed. In process identification, a time-parametrized LTV model is used to describe batch processes and its consistence and variance distribution properties are established. Furthermore, an optimal iterative learning control based on the time-parametrized LTV model is proposed. Illustrative examples were performed and clearly demonstrated the effectiveness and merits of the proposed control scheme.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Figure 19. Output response of ILC based on LTI model (stochastic case).

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study is financially supported by the National Science Foundation of China (Nos. 61273145, 61273146, 60934007), the 973 Program of China (No. 2012CB720503), and the Foundation of Key Laboratory of Advanced Control and Optimization for Chemical Processes. We acknowledge their support.

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REFERENCES

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Figure 20. Input response of ILC based on LTI model (stochastic case). 6191



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Optimal Iterative Learning Control Based on a Time-Parametrized

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