Iterative Learning Fault-Tolerant Control for Networked Batch

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Iterative Learning Fault-Tolerant Control for Networked Batch Processes with Multi-Rate Sampling and Quantization Effects Ming Gao, Li Sheng, Donghua Zhou, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b04609 • Publication Date (Web): 15 Feb 2017 Downloaded from http://pubs.acs.org on February 24, 2017

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Industrial & Engineering Chemistry Research

Iterative Learning Fault-Tolerant Control for Networked Batch Processes with Multi-Rate Sampling and Quantization Effects Ming Gaoa,c , Li Shenga , Donghua Zhoub Furong Gaoc,∗ a. College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, China b. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China c. Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

ABSTRACT: In this paper, the fault-tolerant control problem is investigated for a class of networked batch processes with actuator faults and external disturbances. A two-dimensional Fornasini-Marchesini (2D-FM) system with multi-rate sampling and quantization effects is introduced to model the networked batch processes, which may reflect the reality more closely. The aim of this paper is to design a dynamic output feedback controller such that the closed-loop system can achieve fault tolerance with the effect of actuator faults and satisfy the H∞ performance constraint for unknown external disturbances. By employing a combination of the Lyapunov stability analysis theory, lifting technique and logarithmic quantization method, a networked iterative learning fault-tolerant control (NILFTC) scheme is first proposed, and some sufficient conditions are established for the existence of the desired dynamic output feedback controller. Finally, an example is exploited to illustrate the effectiveness of the developed method.

1

INTRODUCTION

For several decades, the batch process control technology has been a hot research topic due to its successful applications in various domains such as chemical industry, bio-technology and material science.1−3 Since iterative learning control (ILC) exploits the repetitive nature of batch processes, it has gained increasing research attention and fruitful results have been available.4−13 For example, Xu et al.4 investigated the ILC algorithms incorporating a Smith predictor for the batch process with the effect of time delays in the frequency domain. Considering uncertain initializations and disturbances, Gao et al.5 proposed the optimal ILC algorithm for batch processes, whose effectiveness has been verified by the application to the injection molding control. In order to improve the robust convergence of ILC systems, Shi et al.6 investigated the problem of robust ILC for batch processes with model uncertainties. It is well known that faults in actuators and sensors are common in batch processes, which may cause unsatisfactory performance or even instability of the plant. Consequently, it is important to

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develop a fault-tolerant control (FTC) scheme to guarantee the performance of the system in spite of faults. Recently, some results have been reported for the fault-tolerant control of many practical systems.14−18 The FTC problems for batch processes have also been studied.19−23 By transforming the batch process into a two-dimensional Fornasini-Marchesini (2D-FM) model, Wang et al.20,21 proposed the iterative learning FTC scheme for batch processes with actuator/sensor faults. On the other hand, it is quite common in engineering practice that data transmission between system components is realized via networks. Therefore, it is necessary to consider the control problems of batch processes subject to network-induced phenomena. To the best of our knowledge, the related results for the control problem of networked batch processes are few, not to mention the case complicated further by faults and disturbances. With the development of communication technologies, the networked control systems (NCSs) have drawn significant research interest due to their advantages such as simple installation, low maintenance costs and flexible architecture.24,25 As is well known, most physical signals sometimes operate at different rates in complex systems.26 Zhang et al.27 pointed out that if the measurement transmission period is several times of the state updating period, then the transmitted information can be reduced, which is an effective way to save the communication burden in a networked environment. As such, the NCSs with multi-rate sampling have received remarkable attention and many research results have been reported.27−29 In terms of the quantization phenomenon in NCSs, lots of results have appeared on this topic since the original signal needs to be quantized before being transmitted owing to the limitation of communication bandwidth.30−33 For instance, Xiong el al.32 studied the ILC problem for a class of discrete-time systems with quantization and event-triggered scheme. With respect to the batch processes running in networked environment, it is important to analyze the effect that the quantization makes on the performance of batch processes and choose the appropriate sampling period to improve network transmission efficiency. Unfortunately, up to now, the ILC or FTC problem for batch processes with multi-rate sampling and quantization effects has not been investigated, although it has potential application in practical systems. Motivated by the preceding discussion, we aim to deal with the fault-tolerant control problem for a class of networked batch processes with actuator faults and unknown disturbances. The networked batch process is modeled by a 2D-FM model with multi-rate sampling and quantization effects. By means of the lifting technique, an augmented system model is constructed to facilitate the analysis of systems with multi-rate sampling. It is worth noting that there are mainly two types of quantization methods for the measured signal: logarithmic quantization and uniform quantization.30 The logarithmic quantization method is more preferable because it allows less information to be transmitted than uniform quantization method. Hence, the measured output is quantized by a logarithmic quantizer in this paper. By using the Lyapunov stability theory and the linear matrix inequality (LMI) approach, a networked iterative learning fault-tolerant control (NILFTC) scheme is proposed, which guarantees that the closed-loop system is stable with prescribed


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H∞ performance. The main contributions of this paper are outlined as follows. 1) The system model considered is quite comprehensive, where multi-rate sampling and quantization effects are taken into account in order to reflect the reality more closely. 2) The dynamic output feedback H∞ control problem is, for the first time, studied for the batch processes with consideration of the network-induced phenomena; 3) A theoretical framework for dealing with the problem of NILFTC is established for networked batch processes subject to actuator faults. The rest of the paper is organized as follows. In Section 2, some basic definitions and the fault-tolerant control problem are presented. The main results of this paper are stated in Section 3, and the comparisons between our results and previous results are made by some remarks. In Section 4, an illustrative example is given, and this paper is concluded in Section 5. Notations. In this paper, Rn and Rm×n denote the set of all real n-dimensional vectors and the set of all m × n real matrices, respectively. X T represents the transpose of a matrix X. For the symmetric matrices X and Y, X > Y (respectively, X ≥ Y ) is positive definite (respectively, positive semi-definite) matrix. X −1 describes the inverse of matrix X. diag{· · · } is a blockdiagonal matrix. I means the identity matrix. In symmetric block matrices, the notation ∗ is used to represent a term that is induced by symmetry. kxk is the norm of a matrix or vector. For a 2D qP N1 PN2 2 signal x(t, k), if kx(t, k)k2D , t=0 k=0 kx(t, k)k < ∞, for any integers N1 , N2 > 0, then x(t, k) is said to be in l2 space of all square summable function.

2

DEFINITIONS AND PRELIMINARIES

Consider the following class of batch processes with multi-rate sampled data: x(Ti+1 , k) = Ax(Ti , k) + Bu(Ti , k) + Dw(Ti , k) y(ti , k) = Cx(ti , k),

i = 0, 1, · · · , N

k = 1, 2, · · ·

(1) (2)

where k denotes the cycle, Ti and ti are the time index, x(Ti , k) ∈ Rn , y(ti , k) ∈ Rp and u(Ti , k) ∈ Rm represent the state, measured output and input of the process in the kth batch run, respectively. w(Ti , k) ∈ Rq represents the external disturbance belonging to l2 space. h , Ti+1 − Ti is the sampling period of (1), and the measurement period of (2) is denoted by bh , ti+1 − ti , in which h > 0 and b is a positive integer. The matrices A, B, C, D are constant matrices with appropriate dimensions.

2.1

Actuator Faults

Let uF j (Ti , k), j = 1, 2, · · · , m represent the signal from the failed actuator. Then the failure model is described by: uF (Ti , k) = Λu(Ti , k)



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Page 4 of 23

¯ j , j = 1, 2, · · · , m with λ and λ ¯ j being known where Λ = diag {λ1 , λ2 , · · · , λm } , 0 ≤ λj ≤ λj ≤ λ j scalars. Define H = diag {H1 , H2 , · · · , Hm } ,

H0 = diag {H01 , H02 , · · · , H0m }

(3)

¯ j )/2, H0j = (λ ¯ j − λ )/(λ ¯ j + λ ), j = 1, 2, · · · , m. Then, there exists some where Hj = (λj + λ j j unknown matrix Λ0 such that Λ = (I + Λ0 )H

(4)

|Λ0 | ≤ H0 ≤ I

(5)

and

where Λ0 = diag {λ01 , λ02 , · · · , λ0m } , |Λ0 | = diag {|λ01 |, |λ02 |, · · · , |λ0m |} with λ0j = (λj −Hj )/Hj , j = 1, 2 · · · , m. Hence, the multi-rate sampling batch processes with actuator faults can be modeled by:   x(Ti+1 , k) = Ax(Ti , k) + BΛu(Ti , k) + Dw(Ti , k) 

(6)

y(ti , k) = Cx(ti , k).

By applying the lifting technique26 on (1) and (2), a lifted system with a single time scale ti can be obtained: ΣBP :

  x(ti+1 , k) = A1 x(ti , k) + B1 Λ1 u ¯(ti , k) + D1 w(t ¯ i , k) 

(7)

y(ti , k) = C1 x(ti , k)

where A1 = Ab , C1 = C,

h i B1 = Ab−1 B Ab−2 B · · · AB B , h i D1 = Ab−1 D Ab−2 D · · · AD D ,

¯ 0 )H, ¯ Λ1 = (I + Λ

¯ 0| ≤ H ¯ 0, |Λ

Λ1 = diag{Λ, Λ, · · · , Λ}, {z } |

(8)

¯ 0 = diag{Λ0 , Λ0 , · · · , Λ0 }, Λ | {z }

b

b

¯ = diag{H, H, · · · , H }, H | {z }

H¯0 = diag{H0 , H0 , · · · , H0 }, | {z }

b

b

and 

 u(ti , k)

   u(ti + h, k) u ¯(ti , k) =  ..   .  u(ti + (b − 1)h, k)

   ,   



 w(ti , k)

   w(ti + h, k) w(t ¯ i , k) =  ..   .  w(ti + (b − 1)h, k)


   .   

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2.2

Iterative Learning Control

For the lifted system (7), an iterative learning fault-tolerant control (ILFTC) law of the following form is adopted: ΣILFTC :

 u ¯(ti , k) = u ¯(ti , k − 1) + r(ti , k) 

u ¯(ti , 0) = 0,

i = 0, 1, · · · , N,

(9) k = 1, 2, · · ·

where r(ti , k) is the updating law of the ILC. The purpose of this paper is to design the updating law r(ti , k) such that the output y(ti , k) of system (7) can track the desired trajectory yd (ti ) not only when all actuators are operational, but also in case of actuator failures. The tracking error e(ti , k) is defined by e(ti , k) = y(ti , k) − yd (ti ).

(10)

For presentation convenience, denote σ(f (ti , k)) = f (ti , k) − f (ti , k − 1).

(11)

In view of (7)−(11) and after some manipulations, we obtain σ(x(ti+1 , k)) = A1 σ(x(ti , k)) + B1 Λ1 r(ti , k) + D1 σ(w(t ¯ i , k))

(12)

e(ti+1 , k) = e(ti+1 , k − 1) + C1 A1 σ(x(ti , k)) + C1 B1 Λ1 r(ti , k) + C1 D1 σ(w(t ¯ i , k)).

(13)

and

By setting  x ¯(ti , k) = 

 σ(x(ti , k)) e(ti , k)

 ∈ Rn+p

 and y¯(ti , k) = 

 e(ti , k − 1) e(ti , k)

 ∈ R2p ,

an augmented model can be derived as follows:   ¯1 Λ1 r(ti , k) + D ¯ 1 σ(w(t  x ¯(ti+1 , k) = A¯1 x ¯(ti , k) + A¯2 x ¯(ti+1 , k − 1) + B ¯ i , k))    y¯(ti , k) = C¯1 x ¯(ti , k)      z¯(t , k) , e(t , k) = C¯ x i i 2 ¯(ti , k) where z¯(ti , k) represents the controlled output of (15) and         A1 0 0 0 B1 D1 ¯1 =  ¯1 =   , A¯2 =  , B , D , A¯1 =  C1 B1 C1 D1 C1 A1 0 0 I   −C1 I  , C¯2 = [0 I]. C¯1 =  0 I


(14)

(15)


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2.3

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Logarithmic Quantization

Before transmitted to the controller, the measured signal y¯(ti , k) should be quantized in the network environment, and the quantizer h(·) is defined by h y˜(ti , k) = h(¯ y (ti , k)) = h1 (¯ y (1) (ti , k)) h2 (¯ y (2) (ti , k))

···

iT h2p (¯ y (2p) (ti , k))

where h(·) is supposed to be a logarithmic quantizer. For each hj (·)(1 ≤ j ≤ 2p), the set of quantization levels can be described as n (j) Uj = ±ul ,

(j)

ul

(j)

= ρlj u0 ,

o l = 0, ±1, ±2, . . . ∪ {0} ,

0 < ρj < 1,

(j)

u0 > 0,

where the parameter ρj represents the quantization density. The logarithmic quantizer hj (·) is modeled as

hj (¯ y (j) (ti , k)) =

 (j)   u ,   l

(j)

ul

(j)

≤ y¯(j) (ti , k) ≤ u ¯l

(16)

0, y¯(j) (ti , k) = 0     −h (−¯ y (j) (ti , k)), y¯(j) (ti , k) < 0 j

with (j)

ul

=

1 (j) u , 1 + δj l

(j)

u ¯l

=

1 (j) u , 1 − δj l

δj =

1 − ρj . 1 + ρj

According to (16), y˜(ti , k) can be expressed as y˜(ti , k) = h(¯ y (ti , k)) = (I + ∆(ti ))¯ y (ti , k) = Cˆ1 x ¯(ti , k)

(17)

where n o ∆(ti ) = diag ∆(1) (ti ), ∆(2) (ti ), · · · , ∆(2p) (ti )

Cˆ1 = (I + ∆(ti ))C¯1 ,

(18)

and |∆(j) (ti )| ≤ δj , which implies that 2

(∆(ti )) ≤ δ 2 I,

δ = diag {δ1 , δ2 , · · · , δ2p } .

(19)

Substituting (17) into (15), we can obtain the following multi-rate 2D-FM system with quantization effects:

Σ2D−BP :

  ¯1 Λ1 r(ti , k) + D ¯ 1 σ(w(t  x ¯(ti+1 , k) = A¯1 x ¯(ti , k) + A¯2 x ¯(ti+1 , k − 1) + B ¯ i , k))    y˜(ti , k) = Cˆ1 x ¯(ti , k)      z¯(t , k) = C¯ x i 2 ¯(ti , k)

(20)

To realize the fault-tolerant control of system (20), the 2D dynamic output feedback controller is designed as follows:  x ˆ(ti+1 , k) = F1 x ˆ(ti , k) + F2 x ˆ(ti+1 , k − 1) + G1 y˜(ti , k) + G2 y˜(ti+1 , k − 1) ΣC :  r(ti , k) = K1 x ˆ(ti , k) + K2 x ˆ(ti+1 , k − 1) + L1 y˜(ti , k) + L2 y˜(ti+1 , k − 1)


(21)

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where x ˆ(ti , k) ∈ Rn+p represents the state estimate, and {Fl , Gl , Kl , Ll }l=1,2 are the controller parameters to be determined. Remark 1. By employing the lifting technique, the model (15) for batch processes with multirate sampling has been obtained. By using the transformation described in (17), the logarithmic quantization effects can be transformed into sector bounded uncertainties. In case of ∆(ti ) = 0, i.e., there are no quantization effects in (17), it can be easily obtained y˜(ti , k) = y¯(ti , k). Comparing with the existing models of batch processes,20−23 the model (20) exhibits a distinguished feature by considering multi-rate sampling and quantization effects, and it is comprehensive to better reflect the networked environment.

2.4

Problem Formulation

£ T ¤T Letting x ˜(ti , k) = x ¯ (ti , k) x ˆT (ti , k) and combining (20), (21), we obtain the closed-loop system of the following form:  ˜ 1 σ(w(t x ˜(ti+1 , k) = A˜1 x ˜(ti , k) + A˜2 x ˜(ti+1 , k − 1) + D ¯ i , k))  where

 A˜1 =   ˜1 =  D

¯1 Λ1 L1 Cˆ1 A¯1 + B

¯ 1 Λ 1 K1 B

G1 Cˆ1 

F1

¯1 D 0

(22)

z¯(ti , k) = C˜2 x ˜(ti , k)

,

C˜2 =

h

C¯2

 ,

 A˜2 = 

¯1 Λ1 L2 Cˆ1 A¯2 + B

¯ 1 Λ 1 K2 B

G2 Cˆ1

F2

 ,

i 0

.

The closed-loop system (22) can be depicted by the block diagram in Figure 1, where the solid arrow lines stand for the flow of information from current cycles, while the dashed arrowed lines mean the flow of information from previous cycles. The signal y˜(ti , k) is obtained by quantizing the measured signal y¯(ti , k), and it is transmitted through the communication network. On the one hand, from the viewpoint of 2D system, the system (22) contains a 2D plant Σ2D−BP and a controller ΣC as shown by the dotted line block. On the other hand, from the viewpoint of batch process, the system (22) is made up of a plant ΣBP and an ILFTC law ΣILFTC as shown by the solid line block. The boundary conditions of the 2D system (22) are given by  x ˜(ti , 0) = x ˜ti ,0 , i = 0, 1, · · · 

x ˜(0, k) = x ˜0,k ,

k = 1, 2, · · · .

It should be pointed that, if both the actuator faults and signal quantization are not considered, ¯ and Cˆ1 = C¯1 , system (22) will be degenerated into the following system: i.e. Λ1 = H  ˜ 1 σ(w(t x ˜(ti+1 , k) = A˘1 x ˜(ti , k) + A˘2 x ˜(ti+1 , k − 1) + D ¯ i , k)) 

z¯(ti , k) = C˜2 x ˜(ti , k)


(23)


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Figure 1: Block diagram of the closed-loop system. where

 A˘1 = 

¯1 HL ¯ 1 C¯1 A¯1 + B

¯1 HK ¯ 1 B

G1 C¯1

F1

 ,

 A˘2 = 

¯1 HL ¯ 2 C¯1 A¯2 + B

¯1 HK ¯ 2 B

G2 C¯1

F2

 ,

˜ 1 , C˜2 are the same as defined in (22). and D To state the main results, the following definitions and lemmas are adopted. Definition 1.

20

The closed-loop system (22) is called 2D-fault-tolerant system and the control

law (21) is called 2D-FTC law for the system (20) in the absence of disturbance, if lim x ˜(ti , k) = 0,

ti ,k→∞

(24)

for any bounded boundary conditions {˜ xti ,0 , x ˜0,k } and any admissible actuator faults. Definition 2.

34

The closed-loop system (23) is asymptotical stable in the absence of distur-

bance, if lim x ˜(ti , k) = 0,

ti ,k→∞

(25)

for any bounded boundary conditions {˜ xti ,0 , x ˜0,k }. The objective of this paper is to design the parameters {Fl , Gl , Kl , Ll }l=1,2 of the dynamic output feedback controller such that the 2D-FM system (22) satisfies the following conditions: (i) System (22) is a 2D-fault-tolerant system defined in Definition 1; (ii) With the effect of any disturbance σ(w(t ¯ i , k)) ∈ l2 , the controlled output z¯(ti , k) satisfies k¯ z (ti , k)k2D < γkσ(w(t ¯ i , k))k2D ,

(26)

under the zero boundary conditions, where γ > 0 is a given disturbance attenuation level. Lemma 1.35 Considering the following 2D-FM system ζ(i + 1, j + 1) = A1 ζ(i, j + 1) + A2 ζ(i + 1, j)


(27)

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where ζ(i, j) ∈ Rn is the state vector, and A1 , A2 are matrices of appropriate dimensions, if there exist P > Q > 0 such that  −

 P −Q

0

0

Q





+

AT1

P

AT2

h

i A1

0, the following inequality holds: U V + V T U T ≤ ²U U T + ²−1 V T V.

3

NILFTC SCHEME DESIGN

The following theorem presents a sufficient condition under which the system (22) achieves the desired fault-tolerant control and satisfies the H∞ performance constraint. Theorem 1. Let γ > 0 be given. The closed-loop system (22) is a 2D-fault-tolerant system and satisfies the H∞ performance requirement (26), if there exist positive definite matrices P > 0 ∈ R2(n+p)×2(n+p) , Q > 0 ∈ R2(n+p)×2(n+p) and matrices {Fl , Gl , Kl , Ll }l=1,2 such that the following matrix inequality holds:

          

 −P + Q

∗

∗

∗

∗

0

−Q

∗

∗

∗

0

0

−γ 2 I

∗

∗

C˜2

0

0

−I

∗

P A˜1

P A˜2

˜1 PD

0

−P

      < 0.    

(29)

Proof. We first prove the asymptotic stability of the system (22). Suppose σ(w(t ¯ i , k)) = 0 in system (22), and choose the following Lyanpunov function: V (ti , k − 1) = V1 (ti , k − 1) + V2 (ti , k − 1),

(30)

in which V1 (ti , k − 1) = x ˜T (ti , k − 1)(P − Q)˜ x(ti , k − 1), V2 (ti , k − 1) = x ˜T (ti , k − 1)Q˜ x(ti , k − 1).

(31)

The difference of V (ti , k − 1) along the trajectories of system (22) is given by ∆V (ti , k − 1) =∆V1 (ti , k − 1) + ∆V2 (ti , k − 1) =V1 (ti+1 , k) − V1 (ti , k) + V2 (ti+1 , k) − V2 (ti+1 , k − 1) =˜ xT (ti+1 , k)(P − Q)˜ x(ti+1 , k) − x ˜T (ti , k)(P − Q)˜ x(ti , k) +x ˜T (ti+1 , k)Q˜ x(ti+1 , k) − x ˜T (ti+1 , k − 1)Q˜ x(ti+1 , k − 1)  T   x ˜(ti , k) x ˜(ti , k)  Ψ , = x ˜(ti+1 , k − 1) x ˜(ti+1 , k − 1)


(32)


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Page 10 of 23

where  Ψ=

A˜T1 A˜T2

 P

h

A˜1

A˜2

i

 −

 P −Q

0

0

Q

.

(33)

According to (29), it is easy to verify that Ψ < 0. By using Lemma 1, it can be concluded that the closed-loop 2D system (22) is asymptotically stable when σ(w(t ¯ i , k)) = 0. Next, we discuss the H∞ performance under the zero boundary conditions and σ(w(t ¯ i , k)) 6= 0. Denote the performance index:

J(ti , k) = ∆V (ti , k − 1) + k¯ z (ti , k)k2 − γ 2 kσ(w(t ¯ i , k))k2 = η T (ti , k)Φη(ti , k),

(34)

where h η(ti , k) =

x ˜T (ti , k) x ˜T (ti+1 , k − 1) σ T (w(t ¯ i , k)

iT

and     Φ= 

A˜T1

  ˜T  A2  ˜T D





 h   P A˜1 

A˜2

˜1 D

i

  − 

 

P − Q − C˜2T C˜2

0

0

0

Q

0

0

0

γ2I

1

   

  . 

According to the matrix inequality (29) and Schur complement lemma36 , it is easy to obtain Φ < 0. Considering the zero boundary conditions, for any integers N1 , N2 > 0, we have N1 X N2 X

∆V (ti , k − 1) =

i=0 k=1

NX 2 −1

V1 (tN1 +1 , k) +

k=1

N1 X

V2 (ti , N2 ) + V1 (tN1 +1 , N2 ) ≥ 0.

(35)

i=1

Thus, k¯ z (ti , k)k22D − γ 2 kσ(w(t ¯ i , k))k22D ≤

N1 X N2 X ¡

∆V (ti , k − 1) + k¯ z (ti , k)k2 − γ 2 kσ(w(t ¯ i , k))k2

¢

i=0 k=1

≤

N1 X N2 X

J(ti , k) < 0,

(36)

i=0 k=1

which implies that k¯ z (ti , k)k2D ≤ γkσ(w(t ¯ i , k))k2D .

(37)

The proof of this theorem is then complete. When there are no actuator faults and quantization effects in system (22), the following corollary can be directly derived for system (23) according to Theorem 1.

ACS Paragon 10Plus Environment

Page 11 of 23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Corollary 1.

Let γ > 0 be given.

The closed-loop system (23) is stable and satisfies

the constraint (26), if there exist positive definite matrices P > 0 ∈ R2(n+p)×2(n+p) , Q > 0 ∈ R2(n+p)×2(n+p) and matrices {Fl , Gl , Kl , Ll }l=1,2 such that  −P + Q ∗ ∗ ∗    0 −Q ∗ ∗   2  0 0 −γ I ∗    C˜2 0 0 −I  ˜1 P A˘1 P A˘2 P D 0

the following matrix inequality holds:  ∗   ∗    (38) ∗  < 0.   ∗   −P

Based on the above stability analysis, we deal with the dynamic output feedback H∞ control for system (23), this is, design the parameters of the controller (21) such that the system (23) is stable and satisfies the H∞ performance. Theorem 2. Let γ > 0 be given. The problem of dynamic output feedback H∞ control for system (23) is solved by the controller (21), if there exist positive definite matrices M > 0 ∈ ˜l, K ˜ l, L ˜ l }l=1,2 R(n+p)×(n+p) , N > 0 ∈ R(n+p)×(n+p) , W > 0 ∈ R2(n+p)×2(n+p) and matrices {F˜l , G such that the following LMI holds:  −Υ + W    0   Ξ1 =  0    Σ41  Σ51 where



Υ=

 N

I

I 

M

Σ52 = 

 < 0,

h Σ41 =

C¯2 N

 ∗

∗

∗

∗

−W

∗

∗

∗

0

−γ 2 I

∗

∗

0

0

−I

0

Σ52

Σ53

0

−Υ

C¯2

¯1 H ¯K ˜2 A¯2 N + B

¯1 H ¯L ˜ 2 C¯1 A¯2 + B

F˜2

˜ 2 C¯1 M A¯2 + G



i

Σ51 = 

, 

Σ53 = 

(39)

¯1 H ¯K ˜1 A¯1 N + B

¯1 H ¯L ˜ 1 C¯1 A¯1 + B

F˜1 

˜ 1 C¯1 M A¯1 + G



,

      < 0,    

¯1 D ¯1 MD

 ,

.

Moreover, if the LMI (39) is feasible, then the desired controller parameters are given by:  ˜l,  Ll = L       ˜ l − Ll C¯1 N )S −T ,  Kl = ( K  ˜l − M B ¯1 HL ¯ l ),  Gl = R−1 (G       ¯1 H ¯K ˜ l )S −T − Gl C¯1 N S −T , Fl = R−1 (F˜l − M A¯l N − M B

(40)

for l = 1, 2, in which nonsingular matrices S and R can be derived by the singular value decomposition of matrix I − N M due to N M + SRT = I. Proof. Let

 P =

 M

R

RT

Θ

,

 P −1 = 

 N

S

ST

Θ

,

 Ω=


 N

I

ST

0

,

(41)


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Page 12 of 23

where Θ is not relevant to the following discussion. In virtue of P −1 P = I, we have N M + SRT = I,

N R + SΘ = 0.

ˆ T = diag{ΩT , ΩT , I, I, ΩT }. Pre- and post-multiplying (38) by Ω ˆ T and Ω, ˆ respectively, Denote Ω we obtain

          

 −ΩT P Ω + ΩT QΩ

∗

∗

∗

∗

0

−ΩT QΩ

∗

∗

∗

0

0

−γ 2 I

∗

∗

C˜2 Ω

0

0

−I

∗

0

−ΩT P Ω

ΩT P A˘1 Ω

Moreover, it is easy to derive   N I , ΩT P Ω =  I M

˜1 ΩT P A˘2 Ω ΩT P D  ˜1 =  ΩT P D

¯1 D ¯1 MD

      < 0.    

 ,

and C˜2 Ω =

h

C¯2 N

(42)

C¯2

i .

(43)

Letting ˜ l = Ll , L

˜ l = Kl S T + Ll C¯1 N, K

˜l = M B ¯1 HL ¯ l + RGl , G

¯1 H ¯K ˜ l + RGl C¯1 N + RFl S T , F˜l = M A¯l N + M B for l = 1, 2, we have

 ΩT P A˘1 Ω = 

and

 ΩT P A˘2 Ω = 

¯1 H ¯K ˜1 A¯1 N + B

¯1 H ¯L ˜ 1 C¯1 A¯1 + B

F˜1

˜ 1 C¯1 M A¯1 + G

¯1 H ¯K ˜2 A¯1 N + B

¯1 H ¯L ˜ 2 C¯1 A¯1 + B

F˜2

˜ 2 C¯1 M A¯2 + G

 

(44)

 .

(45)

Noticing (43)-(45) and setting W = ΩT QΩ, it can be derived that (38) holds if and only if (39) holds under the condition (40). The proof of this theorem is complete. In what follows, the NILFTC scheme for system (22) is proposed, in which a sufficient condition is provided in order to design the parameters of the dynamic output feedback controller (21) such that system (20) can reach fault tolerance subject to quantization effects. Theorem 3. Let γ > 0 be given. The system (22) is a 2D-fault-tolerant system with the H∞ performance constraint (26), if there exist positive definite matrices M > 0 ∈ R(n+p)×(n+p) , ˜l, K ˜ l, L ˜ l }l=1,2 and positive scalars N > 0 ∈ R(n+p)×(n+p) , W > 0 ∈ R2(n+p)×2(n+p) , matrices {F˜l , G ε1 > 0, ε2 > 0, ε3 > 0, such that the following LMI holds:  e Ξ ∗ ∗ ∗   T  ρu1 −ε2 I ∗ ∗   −2 Ξ =  ε2 ςc1 0 −ε2 δ I ∗   T  ρu 2 0 0 −ε3 I  ε3 ςc2 0 0 0

 ∗ ∗ ∗ ∗ −ε3 δ −2 I


      < 0,    

(46)

Page 13 of 23

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where



 Ξ1

∗

∗

   e= Ξ  ε1 ρTB −ε1 I  < 0, ∗   −2 ¯ ςV 0 −ε1 H 0 h iT h ˜T ρB = 0 0 0 0 B , ς = V1 V 1 iT h ˜T H ¯ , ρu1 = 0 0 0 0 U1T 0 L 1 iT h ¯ ˜T H , ρu2 = 0 0 0 0 U2T 0 L 2 h i h ¯K ˜1 H ¯L ˜ 1 C¯1 , V2 = H ¯K ˜2 V1 = H h iT h ˜T H ¯B ¯T G ˜ T , U2 = L ˜T H ¯B ¯T U1 = L 1 1 1 2 1

(47) i 0 0 0 , h i ~1 0 0 0 0 0 0 , ςc1 = C h i ~1 0 0 0 0 0 , ςc2 = 0 C i ¯L ˜ 2 C¯1 , H iT h i ˜T , C ~ 1 = 0 C¯ G . 2 1 V2

Moreover, if the LMI (46) is solvable, then the desired controller parameters are given by (40). ¯ by Λ1 = (I + Λ ¯ 0 )H, ¯ Σ51 Proof. When there exist actuator faults in system (22), replacing H in Theorem 2 can be written as follows:   ¯1 (I + Λ ¯ 0 )H ¯K ˜ 1 A¯1 + B ¯1 (I + Λ ¯ 0 )H ¯L ˜ 1 C¯1 A¯1 N + B   ˜ 1 C¯1 F˜1 M A¯1 + G   h i ¯1 B ¯0 H Λ ¯K ˜1 H ¯L ˜ 1 C¯1 . =Σ51 +  0

(48)

Similarly, it follows from Σ52 in Theorem 2 that   ¯1 (I + Λ ¯ 0 )H ¯K ˜ 2 A¯2 + B ¯1 (I + Λ ¯ 0 )H ¯L ˜ 2 C¯1 A¯2 N + B   ˜ 2 C¯1 F˜2 M A¯2 + G   h i ¯1 B ¯0 H Λ ¯K ˜2 H ¯L ˜ 2 C¯1 . =Σ52 +  0 ˜1 = Setting B

h

iT

¯T B 1

0

h , V1 =

obtained that

¯K ˜1 H

¯L ˜ 1 C¯1 H

i

h and V2 =

      Ξ2 , Ξ1 +     

∗

0

0

0

0

0

0

˜1 Λ ¯ 0 V1 B

˜1 Λ ¯ 0 V2 B V1T

V2

i , it can be

0 0

∗

∗

∗

∗

∗

  ∗    0 ∗ ∗    0 0 ∗   0 0 0 



0

     0   h  ¯  = Ξ1 +  0  Λ V1 0      0    ˜1 B

¯L ˜ 2 C¯1 H

 0





¯K ˜2 H

(49)

  T  V i  2  0 + 0    0  0

    h ¯  Λ0 0 0    


0

0

˜T B 1

i .

(50)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 23

For presentation convenience, define h ρB =

0 0 0 h

ςV =

V1

V2

0

iT

˜T B 1

0 0

,

i .

0

(51)

By using Lemma 2 and considering the condition (8), one has T¯ ¯ Ξ2 ≤ Ξ1 + ε1 ρB ρTB + ε−1 1 ςV Λ0 Λ0 ςV , T ¯2 ≤ Ξ1 + ε1 ρB ρTB + ε−1 1 ςV H0 ςV .

(52)

e < 0 in (47) holds. In view of Schur complement lemma, Ξ2 < 0 holds if Ξ Next, considering the quantization effects of the output shown in (17), and replacing C¯1 by Cˆ1 in Theorem 2, the following transformations can be obtained:     ~1 = where C

h

¯1 H ¯K ˜1 A¯1 N + B

¯1 H ¯L ˜ 1 (I + ∆(ti ))C¯1 A¯1 + B

F˜1

˜ 1 (I + ∆(ti ))C¯1 M A¯1 + G

¯1 H ¯K ˜2 A¯2 N + B

¯1 H ¯L ˜ 2 (I + ∆(ti ))C¯1 A¯2 + B

F˜2

˜ 2 (I + ∆(ti ))C¯1 M A¯2 + G

0 C¯1

 ~1,  = Σ51 + U1 ∆(ti )C

(53)

 ~1  = Σ52 + U2 ∆(ti )C

(54)

i .

Furthermore, V1 and V2 in (51) can be replaced by h i ¯L ˜ 1 ∆(ti )C ~1, ¯K ˜1 H ¯L ˜ 1 (I + ∆(ti ))C¯1 = V1 + H H

(55)

and h

¯K ˜2 H

¯L ˜ 2 (I + ∆(ti ))C¯1 H

Noting that (53)-(56) and in view of  0    0    0   e Ξ3 , Ξ +  0   ~1  U1 ∆(ti )C    0  ¯L ˜ 1 ∆(ti )C ~1 H

i

¯L ˜ 2 ∆(ti )C ~1. = V2 + H

(56)

Lemma 2, it follows that  ∗

∗

∗

∗

∗

0

∗

∗

∗

∗

0

0 ∗

∗

∗

0

0

0 ∗

∗

~1 U2 ∆(ti )C

0

0

0 ∗

0

0

0

0

0

¯L ˜ 2 ∆(ti )C ~1 H

0

0

0

0

∗

  ∗    ∗    ∗    ∗    ∗   0

e + ρu ∆(ti )ςc + ςcT ∆(ti )ρTu + ρu ∆(ti )ςc + ςcT ∆(ti )ρTu =Ξ 1 1 2 2 2 1 2 2 e + ε−1 ρu ρT + ε2 ς T (∆(ti ))2 ςc + ε−1 ρu ρT + ε3 ς T (∆(ti ))2 ςc . ≤Ξ c1 c2 1 u1 1 2 u2 2 2 3 Considering (19), we obtain e + ε−1 ρu ρTu1 + ε2 δ 2 ςcT ςc + ε−1 ρu ρT + ε3 δ 2 ς T ςc . Ξ3 ≤Ξ c2 2 1 1 2 u2 2 3 1


(57)

Page 15 of 23

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According to Schur complement lemma, Ξ3 < 0 is satisfied if Ξ < 0 in (46) holds. Then, the proof of Theorem 3 is completed. Remark 2. The fault-tolerant control problems for batch processes have been discussed based on the iterative learning method.20−22 However, the results in the existing literature are fairly conservative in practical applications such as FTC of networked batch processes. Owing to the increasingly wide utilization of the networks, more and more control processes are executed in a networked environment. The NILFTC scheme proposed in this paper can effectively solve the FTC problem for the networked batch processes with multi-rate sampling and quantization effects. It is worth mentioning that, when both the multi-rate sampling and quantization effects are not taken into account, the condition in Theorem 3 will degenerate to the result in Wang et al.20 Therefore, our results extend and improve the previous results. When the signal quantization is not considered, the system (22) degenerates into the the following form:  ~1x ~2x ˜ 1 σ(w(t x ˜(ti+1 , k) = A ˜(ti , k) + A ˜(ti+1 , k − 1) + D ¯ i , k)) 

(58)

z¯(ti , k) = C˜2 x ˜(ti , k)

where  ~1 =  A

¯1 Λ1 L1 C¯1 A¯1 + B

¯ 1 Λ 1 K1 B

G1 C¯1

F1

 ,

 ~2 =  A

¯1 Λ1 L2 C¯1 A¯2 + B

¯ 1 Λ 1 K2 B

G2 C¯1

F2

 ,

˜ 1 and C˜2 are defined as in (22). Then, the following corollary can be easily derived from Theorem D 3. Corollary 2. Let γ > 0 be given. The system (58) is a 2D-fault-tolerant system with the H∞ performance constraint (26), if there exist positive definite matrices M > 0 ∈ R(n+p)×(n+p) , ˜l, K ˜ l, L ˜ l }l=1,2 and a positive N > 0 ∈ R(n+p)×(n+p) , W > 0 ∈ R2(n+p)×2(n+p) , matrices {F˜l , G scalar ε1 > 0 such that the following linear matrix inequality holds:   Ξ1 ∗ ∗    e= Ξ  ε1 ρTB −ε1 I  < 0, ∗   ¯ −2 ςV 0 −ε1 H 0

(59)

where Ξ1 , ρB and ςV are the same as defined in (47). Moreover, if the LMI (59) is solvable, then the desired controller parameters are given by (40). Remark 3. In Theorem 2 and Corollary 2, the dynamic output feedback controller is designed for the H∞ control and fault-tolerant control problems of multi-rate sampling batch processes, respectively. To the best of the authors’ knowledge, this is the first of few attempts to consider the control problem of batch processes with multi-rate sampling. The conditions presented in Theorem 2 and Corollary 2 can be viewed as an extension of the results in the existing literature6,20 to the multi-rate sampling case.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4

Page 16 of 23

ILLUSTRATION

In this section, a numerical example is given to demonstrate the usefulness of the proposed NILFTC scheme for the batch processes which cover the network-induced phenomena. Injection molding is a typical batch process, and consists of three main phases: filling, packing and cooling.5 And the nozzle pressure is a key process variable in the packing stage.6,21 . In order to guarantee the product quality, the nozzle pressure should be controlled to follow a desired profile. The nozzle pressure dynamics can be described by the following model       1.607 1 1.2390    x(Ti , k) +   u(Ti , k) + w(Ti , k),  x(Ti+1 , k) =  −0.6086 0 −0.9282     y(ti , k) = [1 0]x(ti , k), i = 0, 1, · · · , 50, k = 1, 2, · · · 50

(60)

where w(Ti , k) represents the unknown disturbance. In this example, the initial conditions of each cycle are fixed to be x(0, k) = [300, 0]T . The disturbance w(Ti , k) is assumed to be 0.05 sin(k) and the desired trajectory yd (ti ) is set to be 400 in all cases. Case I. FTC for the batch processes with multi-rate sampling Choose b = 2, and it means that the state signal of system (60) is updating with period h and the measurement output is sampled with period 2h. Set the H∞ performance level γ = 4.2 and the sampling period h = 1. The actuator fault is supposed to be Λ = 0.8 + 0.1 sin(0.01Ti ), then it can be derived that H = 0.8 and H0 = 0.125 from (4) and (5). By solving LMI (59) in Corollary 2, the parameters of the desired dynamic output feedback controller are shown in Table I. Table 1: The parameters of the controller (21) in Case I

Ll Kl

Gl

Fl

l=1  3.0109  −0.8011  −0.0072  −0.0001  0.0011    4.1842  −0.0397  −0.0000    −0.0266  0.0002

 −0.8011



−0.8847



0.0688 −0.0004 0.0732



0.0003 

−0.0009

  −4.1562   0.0295 0.0001

 0.0001

  −0.0012   −0.0031 −0.0062 0.6017

l=2  0.5127  −0.4424  −0.9608  0.8288  −0.0029    −0.2376  0.0029  0.0000    0.0004  −0.0000

 −0.4424



−0.5606



0.0184 −0.9061 0.0092 −0.4586  533.6757   18.1537   0.5038 −0.0000 0.0002 0.0001

 × 10−3



0.0001

  −0.0069   −0.0056

According to Corollary 2, it is concluded that the system (60) can reach fault tolerance. When the actuator is in normal operations, that is, Λ = 1, the output response trajectories of the system (60) are shown in Figure 2, which implies that the tracking performance improves in the cycle


450

400

400

i

Output y(ti,k)

450

350

300

250 60

350

300

250 60

50

40 20

Cycle (k)

50

40

40

40

30

30

20

20 0

20

10 0

0

Cycle (k)

Step (i)

10 0

Step (i)

Figure 2: Output responses of system (60) with- Figure 3: Output responses of system (60) with out faults in Case I.

faults in Case I.

35

45 40

30

35 25 30 DT(k)

20 DT

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Output y(t ,k)

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15

25 20 15

10 10 5

0

5

0

10

20

30

40

0

50

0

10

20

Cycle (k)

30

40

50

Cycle (k)

Figure 4: Tracking performances DT(k) without Figure 5: Tracking performances DT(k) with faults in Case I.

faults in Case I.

direction. The actuator fault is supposed to occur after the 20th cycle. In the presence of faults, Figure 3 depicts the output responses of the system (60) by using the designed controller (21). It can be seen that the NILFTC can effectively handle the actuator faults. The following performance index DT (k) is used to evaluate the the tracking performance of the system (60) v u 50 uX DT (k) = t e2 (ti , k),

k = 1, 2, · · · , 50.

(61)

i=1

The values of DT (k) without and with faults are presented in Figures 4 and 5, respectively, and the smaller DT (k) implies the better tracking performance in the kth cycle. From Figure 5, it can be found that the tracking performance still keeps at a good level even if the actuator faults occur when k ≥ 20, which illustrates the effectiveness of the designed controller. Case II. FTC for the batch processes with quantization effects In this case, we will study the influence of quantization effects on the performance of networked



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batch processes. Choose b = 1 and h = 1, then the model (60) reduces to the single-rate system. Our objective is to design the fault-tolerant dynamic output feedback controller in the form of (21). The actuator fault is assumed to be Λ = 0.9 + 0.045sin(0.1Ti ), and it can be obtained that H = 0.9, H0 = 0.05 from (4) and (5). Set the H∞ performance γ = 2.2 and the quantization density ρ = 0.9048, i.e., δ = 0.05. By using the Matlab LMI toolbox, a set of feasible solutions to LMI (46) is derived in Table II. Table 2: The parameters of the controller (21) in Case II

Ll Kl Gl

Fl

l=1 h i 1.3728 −1.4173 h i 0.1508 −0.0095 −0.0013   3.2871 −3.4715      −0.2139 0.3204    0.0075 −0.0128  0.7598 −0.0411 −0.0075    −0.0731 0.0262 −0.0010  0.0029 −0.0014 0.0005

    

l=2 h −0.0188 h −0.0004  0.0528    0.0948  −0.0111  0.0004    −0.0016  0.0002

i −0.8339

i

−0.0036 −0.0008  −4.0338   −5.5108   0.6608 0.0077 −0.0126 0.0014



0.0026

  0.0032   −0.0007

According to Theorem 3, the system (60) is a 2D-fault-tolerant system with disturbance attenuation level γ. The actuator fault is supposed to occur after the 15th cycle in this case. Figure 6 shows the output responses of the system (60), which indicates that output responses converge rapidly in each cycle when the actuator faults occur. By choosing different quantization densities ρ = 1 (δ = 0) and ρ = 0.7857 (δ = 0.12), the control results are depicted in Figures 7 and 8, respectively, which imply that the fault tolerance has been reached under the proposed NILFTC scheme. Moreover, the tracking performances DT (k) with different quantization parameters δ are shown in Figure 9. Remark 4. From Figure 9, it can be observed that the tracking performance of system (60) becomes worse with the increase of δ (the decrease of ρ). The basic idea of quantization is to reduce the transmitted information by choosing the appropriate quantization density ρ for the purpose of saving network resources in the networked environment.30 In practical engineering, if the network resource constraint is of more interest, then the quantization density ρ should be set a bit smaller. However, if we are more concerned about system performance, then ρ can be chosen a bit bigger. Therefore, the quantization density ρ should be properly fine-tuned in the practical application to ensure a trade-off between engineering performance requirements and network resources saving via quantization.


Page 19 of 23

420

450

Output y(t ,k)

400

i

380

i

Output y(t ,k)

400

360 340

350

320 300 60

300 60 50

40

40

10 0

30

20

20 0

Cycle (k)

50

40

40 30

20

Cycle (k)

Step (i)

20 0

10 0

Step (i)

Figure 6: Output responses of system (60) with Figure 7: Output responses of system (60) with δ = 0.05 in Case II.

δ = 0 in Case II.

20 δ=0 δ=0.05 δ=0.12

18 450

16 14

400

12 DT(k)

i

Output y(t ,k)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


350

10 8 6

300 60

4 50

40

40 30

20

Cycle (k)

2

20 0

10 0

0

Step (i)

0

10

20

30

40

50

Cycle (k)

Figure 8: Output responses of system (60) with Figure 9: Tracking performances DT(k) with difδ = 0.12 in Case II.

ferent δ in Case II.



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5

CONCLUSIONS

In this paper, we have addressed FTC issue for a class of networked batch processes with actuator faults, multi-rate sampling and quantization effects. The system under consideration is new in the sense which covers the network-induced phenomena and can be executed in the networked environment. By utilizing the lifting technique, logarithmic quantizer and linear matrix inequality theory, the fault-tolerant dynamic output feedback controller has been obtained for a class of networked batch processes with actuator faults. Moreover, an illustrative example has been provided to demonstrate the validity of the proposed FTC scheme. Some future research topics include further extensions of the obtained results to batch processes with other networked-induced phenomena such as missing measurements, and the investigation on the FTC for networked batch processes in the case of sensor faults.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Telephone: +852-23587139. Fax: +852-23580054. Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (Nos. 61573377, 61433005, 61403420) and Project for the Applied Basic Research of Qingdao (Nos. 16-5-1-3-jch, 16-8-3-1-zhc).

References (1) Korovessi, E.; Linninger, A. A. Batch Processes; CRC/Taylor and Francis, Boca Raton, FL, 2006. (2) Lee, J. H.; Lee, K. S. Iterative learning control applied to batch processes: An overview. Control Eng. Pract. 2007, 15, 1306-1318. (3) Lv, Z. M.; Jiang, Q. C.; Yan, X. F. Batch process monitoring based on multisubspace multiway principal component analysis and time-series Bayesian inference. Ind. Eng. Chem. Res. 2014, 53, 6457-6466. (4) Xu, J. X.; Hu, Q. P.; Lee, T. H.; Yamamoto, S. Iterative learning control with Smith time delay compensator for batch processes. J. Process Control 2001, 11, 321-328. (5) Gao, F.; Yang, Y.; Shao, C. Robust iterative learning control with applications to injection molding process. Chem. Eng. Sci. 2001, 56, 7025-7034.


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Iterative Learning Fault-Tolerant Control for Networked Batch

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