Article pubs.acs.org/IECR
Fuzzy Iterative Learning Control for Batch Processes with Interval Time-Varying Delays Limin Wang,*,‡,† Chengjie Zhu,† Jingxian Yu,† Li Ping,† Ridong Zhang,*,§,∥ and Furong Gao∥ †
School of Information and Control Engineering, Liaoning Shihua University, Fushun, 113001, P. R. China HaiNan Normal University School of Mathematics and Statistics, HaiNan Normal University, Haikou, 571158, P. R. China § Key Lab for IOT and Information Fusion Technology of Zhejiang, Information and Control Institute, Hangzhou Dianzi University, Hangzhou 310018, P. R. China ∥ Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong ‡
ABSTRACT: In this paper, a T-S model-based fuzzy delay-rangedependent iterative learning control (ILC) scheme is developed for highly nonlinear batch processes with interval time-varying delays. The two-dimensional (2D) T-S time-delay model is constructed to remedy the disadvantage that the overall linear model cannot sufficiently describe the nonlinear batch process. Then, exploiting the repetitive nature of batch processes, a 2D fuzzy delay-rangedependent iterative learning control is designed. The delay-rangedependent stabilization problem and H∞ control are studied by using 2D Lyapunov function under a 2D system framework. At the same time, the controller gain design is given and its gain can be obtained in terms of LMIs. A water tank is taken as a simulation case to demonstrate the effectiveness of the proposed fuzzy iterative learning control scheme.
1. INTRODUCTION Batch processing technologies have developed greatly in recent decades due to the business of manufacture in the last 10 years. Advanced control methods of batch processes have attracted extensive attention for their great advantage in manufacturing low-volume products with high added value.1,2 Among the methods, iterative learning control (ILC) methods integrated with feedback control have been widely applied, which exploit the 2D nature that the system dynamics change along both batch and time direction.3−10 However, most of these existing results are based on linear models that are not sufficient enough to describe an actual nonlinear system. Especially, with many unique dynamic characteristics, it is more difficult to construct the linear models of highly nonlinear batch processes. On the other hand, because of the essential complexity and immature development, nonlinear control or sophisticated methods have not been the commonly used method although there are a few results.11−13 Nonlinear models always reveal the operation mechanism of a nonlinear process and are not difficult to obtain for many batch processes especially the chemical batch processes. If a method can solve the control problem of nonlinear batch processes, there will be a significant development for batch processes control theory. T-S fuzzy systems have been proven to be very useful for solving the control problem of nonlinear systems. Precisely represented by T-S fuzzy models with a series of linear models, many linear control methods can be applied to nonlinear processes.14−18 © 2017 American Chemical Society
Therefore, this paper will use the T-S fuzzy systems to deal with the nonlinear batch processes. The T-S fuzzy model has witnessed a rapid development in dealing with nonlinear systems. However, these methods cannot be used directly but need corresponding changes because they are designed for the one-dimension system, that is, the system dynamic only changes along the time direction. Time delay, which means that the future evolution of the system depends not only on its present state but also on a period of its history, is a common phenomenon in industrial systems. It is the main cause of system instability and weak performance. In view of this, study on the control of a time-delay system is of great importance. In the past decades, a lot of mature results are presented in terms of stability and feedback control of time-delay systems,19−22 among which the most typical one is time-delay relevant results.20−22 Time-delay relevance refers to the consideration of the magnitude of the delay when system stability and performance are investigated. Unfortunately, these results are all focusing on continuous systems, and the time delay is a constant or a value with zero as its lower bound. Until recently, researchers have found that the lower bound of a time delay may not be zero and this paves the way for the research on Received: Revised: Accepted: Published: 3993
November 30, 2016 March 18, 2017 March 21, 2017 March 22, 2017 DOI: 10.1021/acs.iecr.6b04637 Ind. Eng. Chem. Res. 2017, 56, 3993−4001
Article
Industrial & Engineering Chemistry Research continuous systems with interval time-varying delays.23 For stability and control issues of 2D systems, results emerged around 2000.24−26 In 2010, issues of 2D systems with interval time-varying delays were conducted.27,28 Similar to continuous processes, batch processes are also affected by time delays. However, a limited number of results are achieved. In the frequency domain, the Smith predictor is known for improved control of time delay processes with a priori knowledge of the time delay, an ILC algorithm29 was presented for tracking performance improvement of ILC for batch processes with known and fixed time delay. Using internal model control (IMC), an ILC is proposed for batch processes with time delay mismatch.30 A reference shift algorithm was proposed using a double-loop ILC structure.31 Park et al.32 proposed an ILC using a holding mechanism for batch processes with time delay. However, how to find a control method for batch processes with time-varying delays is still an open issue. Recently, the feedback control combined with ILC has been adopted in batch processes to solve issues of stability and stabilization under interval time-delay. Typically, research on linear batch processes with time-varying delay varying in a range and the lower bound not restricted to be zero has been significantly studied.33−40 However, the models of most chemical processes are nonlinear with time-delay. How to design controllers to result in stable control is still a problem to be solved. On the basis of the above-mentioned facts, this paper contributes to a novel method to deal with the control problems of nonlinear batch processes with interval time-varying delays. With the iterative learning control strategy, the 2D T-S delayed fuzzy model is converted into an equivalent 2D delayed Rosser model. Combining feedback control with ILC, a 2D fuzzy ILC (FILC) law is designed and then the closed-loop 2D fuzzy system is established. Under 2D system framework and using a 2D Lyapunov function method, new delay-range-dependent stability criteria and stabilization conditions are derived in terms of linear matrix inequalities (LMIs), which depend on not only the difference between the upper and lower delay bounds but also the upper delay bound of the interval time-varying delay. At the same time, the controller’s gain matrices are given as well. The contribution lies in the fact that the nonlinear batch process is transformed into a 2D fuzzy system, and time delay is incorporated into the stability condition, through which sufficient conditions for the stability analysis and design of 2D fuzzy systems are provided. A highly nonlinear water tank is taken as an example to illustrate the feasibility and effectiveness of the proposed method. The paper is organized as follows. The 2D delayed T-S fuzzy model is established in section 2. In section 3, the main strategy of the proposed controller design is detailed. Compared with the traditional ILC strategy, the 2D fuzzy iterative learning control strategy proposed in this paper can accelerate the convergence speed by using the learning ability of the iterative learning control along the batch direction and suppress the time-varying delay by the feedback control in the batch system. Control application to a highly nonlinear water tank under repetitive and nonrepetitive disturbance is shown in section 4. Section 5 concludes the paper.
THEN ⎧ x(t + 1, k) = A x(t , k) + A x(t − d(t ), k) i id ⎪ + Bi u(t , k) + ω(t , k) ⎪ ⎨ y(t , k) = Cix(t , k) ⎪ ⎪ x(0, k) = x0, k ; 0 ≤ t ≤ T; k = 1, 2, ··· ⎩ i = 1, 2, ..., r ; j = 1, 2, ..., p .
(1)
where t denotes time, k is the cycle index, T is the duration of a batch; x0,k is the time-wise initial state of the kth batch, x(t,k) ∈ Rn, y(t,k) ∈ Rl, u(t,k) ∈ Rm, and ω(t,k) represent, respectively, the state, output, input, and unknown disturbances of the process at time t in the kth batch run; Mij is the fuzzy set, r is the number of the fuzzy rules; p is the number of the premise variables; Ai, Aid, Bi, Ci represent the corresponding matrix of system state, delayed state, input, and output, respectively, in rule i; z1(t,k), ..., zp(t,k) in this paper are known premise variables of the functions of the system state; the time-varying delay d(t) along horizontal direction satisfies dm ≤ d(t) ≤ dM, where dm and dM denote the lower and upper delay bounds, respectively. The part following the “IF” is called the premise part while that following the “THEN” is called the consequence part. Then the overall T-S fuzzy model can be described as follows: r ⎧ ⎪ x(t + 1, k) = ∑ hi(z(t , k))(Ai x(t , k) ⎪ i=1 ⎪ ⎪ + Aid x(t − d(t ), k) r ∑ ⎨ P − T − S − delay: ⎪ + ∑ hi(z(t , k))Bi u(t , k) + ω(t , k) ⎪ i=1 ⎪ ⎪ y (t ) = Cixk(t ) ⎩ k
(2)
where, hi(z(t , k)) =
wi(z(t , k)) , r ∑i = 1 wi(z(t , k)) p
wi(z(t , k)) =
∏ Mij(zj(t , k)),
Mij(zj(t , k))
j=1
is the grade of membership of zj(t,k) in Mij; for simplicity, hi(z(t,k)) is represented by hi(i = 1,2,...,r) as follows. ⎧ r ⎪∑ wi(z(t , k)) > 0, as ⎨ i = 1 , ⎪ ⎪ w (z(t , k)) ≥ 0, ⎩ i
⎧ r ⎪∑ hi(z(t , k)) = 1 so ⎨ i = 1 ⎪ ⎪ h (z(t , k)) ≥ 0 ⎩ i
For the 2D T-S delayed fuzzy model shown by eq 2, the control objective is to find a real time input u(t,k) to guarantee the outputs to track the set-point or expected profile.
3. FUZZY ITERATIVE LEARNING CONTROLLER DESIGN As is described in the introduction, batch processes have a unique repetitiveness that facilitates the wide use of ILC. In view of this, ILC is adopted here. However, feedback control is also considered. Under the proposed control strategy, the error variables in terms of time and batch are introduced to form an equivalent 2D model, which serves for transforming the design of ILC into an updating control law. In this paper, the control goal has considered reforming the tracking control into the H∞ stabilizing control issue of the fuzzy systems. For the batch
2. ESTABLISHMENT OF THE 2D T-S FUZZY MODEL A nonlinear batch process performing a given task in a batch can be described by a series of discrete T-S fuzzy rules with interval time-varying delays and is shown as follows: Rule i: IF z1(t,k) is Mi1 and zj(t,k) is Mij, ..., zp(t,k) is Mip 3994
DOI: 10.1021/acs.iecr.6b04637 Ind. Eng. Chem. Res. 2017, 56, 3993−4001
Article
Industrial & Engineering Chemistry Research process ∑P−T−S−delay that is described by eq 2, design an ILC law with the form in eq 3 to achieve the control objective: ⎧ ⎪ u(t , k ) = u(t , k − 1) + r (t , k ) :⎨ ⎪ f − ilc ⎩ u(t , 0) = 0
Therefore, an equivalent 2D system description of the batch processes eq 2 together with two augmented state variables of the tracking error information can be expressed by r ⎧ ⎡ δ(x(t + 1, k))⎤ ⎡ δ(x(t , k)) ⎤ ⎪⎢ ⎥ ⎥ = ∑ h A̅ ⎢ ⎪ ⎢⎣ δ(x(t + 1, k))⎥⎦ i = 1 i i ⎢⎣ e(t + 1, k − 1)⎥⎦ ⎪ r ⎤ ⎡ δ(x(t − d(t ), k)) ⎪ ⎥ ⎪ + ∑ hiA̅ di ⎢⎢ ⎪ ⎣ e(t + 1, k − 1 − h(k − 1))⎥⎦ i=1 r ⎨ ∑ 2D − T − S − delay: ⎪ + ∑ hiBi̅ r(t , k) + Cw ̅ ̅ (t , k) ⎪ i=1 ⎪ ⎪ ⎡ δ(x(t , k)) ⎤ ⎪ z(t , k)Ae(t + 1, k − 1) = D̅ ⎢ ⎥ ⎪ ⎢⎣ e(t + 1, k − 1)⎥⎦ ⎩
∑
(3)
where r(t,k) is the updating law to be determined in the current cycle k. u(t,0) denotes the initial value of the iteration that is generally set to zero for implementation. From eq 3, it is known that when the updating law r(t,k) is obtained, the ILC law can be designed. The objective of the ILC design is then transformed into determining the updating law r(t,k) such that y(t,k) tracks the given profile yr(t,k). To achieve this goal, define the system state error in adjacent cycle as eq 4 and the output tracking error as eq 5: δ(x(t , k)) = x(t , k) − x(t , k − 1)
(4)
e(t + 1, k) = yr (t + 1, k) − y(t + 1, k)
(5)
where h(k − 1) satisfies hm ≤ h(k − 1) ≤ hM with hm and hM ⎡ Ai 0 ⎤ denoting the lower and upper delay bounds, A̅ i = ⎢−CA ⎥, ⎣ i I⎦ ⎡ Bi ⎤ ⎡ Aid 0 ⎤ ⎡I⎤ A̅ id = ⎢−CA ⎥, Bi̅ = ⎢⎣−CBi ⎥⎦, C̅ = ⎣C ⎦, D̅ = [0 1]. Here ⎣ id 0 ⎦ e(t + 1,k − 1 − h(k − 1)) does not exist. In order for eqs 6 and 7 to be formulated into eq 8, that is, the Rosser model with time delay, the output error variable is introduced artificially. However, h(k − 1) can be viewed as the lagged value through ⎡ Aid 0 ⎤ batch orientation. From A̅ id = ⎢−CA ⎥, it is seen that ⎣ id 0 ⎦
where yr(t + 1,k) denotes the set-point profile (or the desired output trajectory). Then, we have δ(x(t + 1, k)) = x(t + 1, k) − x(t + 1, k − 1) r
=
r
+
although e(t + 1,k − 1 − h(k − 1)) exists, they are the same since eq 8 can be reformulated in to eqs 6 or 7. For hm and hM, they are chosen artificially. Utilizing the parallel distributed compensation (PDC) method, the updating law r(t,k) based on the tracking error is designed as follows. Rule i:
∑ hiAiδ(x(t , k)) + ∑ hiA d δ(x(t − d(t ), k)) i
i=1 r
i=1
∑ hiBi r(t , k) + w̅(t , k)
(6)
i=1
where
⎡ δ(x(t , k)) ⎤ ⎥ r (t , k ) = K i ⎢ ⎢⎣ e(t + 1, k − 1)⎥⎦
w̅ (t , k) = w(t , k) + ω(t , k) − ω(t , k − 1), w(t , k) r
=
∑ δ(hi)[Aix(t , k − 1) + Bi u(t , k − 1) i=1
For simplicity, let hi stand for hi(z(t,k)). Here δ(hi) is expressed as δ(hi) = hi(z(t,k)) − hi(z(t,k − 1). Since the function hi is related to system states and under the condition that the system is with repetitive or without disturbances, the states at the same time instant of different batches will be equal under the control law. This means that δ(hi) = 0, that is, w(t,k) will converge to zero. For systems with nonrepetitive disturbances, this may not be true, that is, δ(hi) ≠ 0. Under this condition, w(t,k) that consists of the state and input information on the system will be viewed as the external disturbance. For the output tracking error, a special condition is considered, Ci = C, (i = 1,2,...,r), that is, a common C matrix condition is considered. A common C matrix case can be easily obtained by augmenting the outputs of the system with integrators and using the augmented states as a new set of outputs.29 Then, from eqs 4, 5 and 7, the following is derived
r
∑ 2D − T − S − ilc
⎡
⎤ ⎥ ⎢⎣ e(t + 1, k − 1)⎥⎦
∑ hiK i⎢ i=1
δ(x(t , k))
(10)
⎡ δ(x(t + 1, k))⎤ ⎡ xh(t + 1, k)⎤ ⎥, ⎥=⎢ xt̅ ′, k = ⎢ ⎢⎣ e(t + 1, k) ⎥⎦ ⎢⎣ xv(t , k + 1)⎥⎦ ⎡ δ(x(t , k)) ⎤ ⎡ xh(t , k)⎤ ⎥, ⎥=⎢ xt̅ , k = ⎢ ⎢⎣ e(t + 1, k − 1)⎥⎦ ⎢⎣ xv(t , k) ⎥⎦ ⎡ xh(x(t − d(t ), k)) ⎤ ⎥ xd̅ (t , k) = ⎢ ⎢⎣ xv(t + 1, k − 1 − h(k − 1))⎥⎦
, where the 2D Rosser closed-loop system eq 11 is obtained from eq 8 and 10. r r r ⎧ ⎪ xt̅ ′, k = ∑ ∑ hihjÃij xt̅ , k + ∑ hiA̅ id xd̅ (t , k) ⎪ i=1 j=1 i=1 ⎨ ∑ Cw ( t , k ) + ̅ ̅ 2D − T − S − delay − C : ⎪ ⎪ z ( t , k ) e (t + 1, k − 1) = Dx = ̅ t̅ , k ⎩
r
= e(t + 1, k − 1) − C(∑ hiAi δ(x(t , k)) + w̅ (t , k) i=1 r
∑ hiBir(t , k) + ∑ hiA diδ(x(t − d(t ), k))) i=1
: r (t , k ) =
Let
e(t + 1, k) = yr (t + 1) − y(t + 1, k)
+
(9)
Here, Ki is the control gain to be calculated. Then the overall 2D T-S fuzzy updating law is
+ Aid x(t − d(t ), k − 1)]
r
(8)
(11)
i=1
with à ij = A̅ i + B̅ iKj(i,j ≤ r).
(7) 3995
DOI: 10.1021/acs.iecr.6b04637 Ind. Eng. Chem. Res. 2017, 56, 3993−4001
Article
Industrial & Engineering Chemistry Research ⎡ Ψ̅ ⎢ 1 ⎢ ⎢∗ ⎢∗ ⎢ ⎢∗ ⎢ ⎣∗
The boundary conditions are defined by xh(i , j) = ρij ,
∀ 0 ≤ j < γ1 , −dM ≤ i ≤ 0
xv(i , j) = σij ,
∀ 0 ≤ i < γ2 , −hM ≤ j ≤ 0
ρ00 = σ00
(12)
and the controller gains are designed with Ki = YiL , Kj = YjL−1, where Ψ̅1 = −L + M1 + DS + S − X , ⎡(dM − dm)I ⎤ 0 ⎥, D=⎢ ⎢⎣ 0 (hM − hm)I ⎥⎦ Gij̅ =
κ1
T
t = κ0
(13)
xd̅ T(t , k)
xMT (t , k)]T
(17)
5
V h(xt̅ h, k) = Σ Vnh(xt̅ h, k)
t = κ0
n=i
V1h(xt̅ h, k) = xt̅ h, kTP hxt̅ h, k
V 2h(xt̅ h, k) = V3h(xt̅ h, k) = V 4h(xt̅ h, k) =
i−1
Σ xr̅ h, Tk Q hxr̅ h, k
r = i − d(i) i−1
Σ xr̅ h, kM hxr̅ h, k
r = i − dM −d m
i−1
Σ xr̅ h, Tk Q hxr̅ h, k
Σ
s =−dM r = i + s −d m
V5h(xt̅ h, k) = dM Σ
0⎤ ⎥ ∈ R(n + l) × (n + l) Xv⎦
i−1
Σ xr̅ h, Tk Ghxr̅ h, k
s =−dM r = i + s
5
V v(xt̅ v, k) = Σ Vnv(xt̅ v, k) n=1
V1v(xt̅ v, k) = xt̅ v, Tk Pvxt̅ v, k
LA̅ i T + Yi TBi̅ T LA̅ i T + Yi TBi̅ T − L ⎤ ⎥ ⎥ LA̅ dTi LA̅ dTi −S 0 ⎥ ⎥ 0 for any w̅ (t,k) (1) the resulting closed-loop system with w̅ (t,k) = 0 is asymptotically stable; (2) with the zero boundary conditions, and for any disturbance w̅ ∈ l2D−T−S−delay−C−2e, if the controlled output satisfies
κ1
L
(i = 1, 2, ..., r )
The following definitions and lemmas are given for the closedloop control law design. Definition 1.41 The2D T-S fuzzy closed-loop system ∑2D−T−S−delay−C with w̅ (t,k) = 0 is called 2D stabilizable if the system state satisfies lim || xt̅ , k || = 0 for any boundary t , k →∞
|| z ||2D − T − S − delay − C − 2e < γ || w̅ ||2D − T − S − delay − C − 2e
LGij̅ + H̅ ij LGij̅ + H̅ ij − L ⎤ ⎥ ⎥ T T LA̅ di LA̅ di −S 0 ⎥ ⎥
