Robust Iterative Learning Fault-Tolerant Control for Multiphase Batch

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Robust iterative learning fault-tolerant control for multi-phase batch processes with uncertainties Limin Wang, Limei Sun, Jingxian Yu, Ridong Zhang, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00525 • Publication Date (Web): 12 Jul 2017 Downloaded from http://pubs.acs.org on July 13, 2017

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Industrial & Engineering Chemistry Research

Robust iterative learning fault-tolerant control for multi-phase batch processes with uncertainties Limin Wang1,2,*, Limei Sun2, Jingxian Yu2, Ridong Zhang3,4, Furong Gao4 1. School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, P. R. China; 2. College of Sciences, Liaoning Shihua University, Fushun, 113001, P. R. China; 3. Key Lab for IOT and Information Fusion Technology of Zhejiang, Information and Control Institute, Hangzhou Dianzi University, Hangzhou 310018, P. R. China 4. Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Hong Kong

Corresponding authors: Limin Wang

Emails: [email protected]

Abstract: In this paper, the iterative learning fault-tolerant control problem for multi-phase batch processes with uncertainty and actuator faults is studied. Firstly, making full use of the characteristics of two-time dimension (2D) feature and repetitiveness in batch processes and introducing the state error and output error between the adjacent batches, the established model is transformed into an equivalent 2D-Roesser switched system with different dimensions. Under the framework of the 2D system theory and by means of the average dwell time method, sufficient conditions ensuring the system to be 2D robustly stable along the time and batch directions and the minimum running time lower bound in each phase are given. Simultaneously, the designed updating law is figured out. In order to examine the control performance of the proposed method, the traditional reliable control method is also investigated in this paper. The batch process is regarded as a continuous system, in which only the fault-tolerant control along the time direction is considered. Finally, the injection modeling process is taken as an example, where the main parameters, namely the injection velocity and packing-holding pressure, are controlled in the filling and packing-holding phases. The simulation results show that the proposed iterative learning fault-tolerant control method is a better choice for the multi-phase batch processes with actuator faults. Key words: multi-phase batch processes; iterative learning fault-tolerant control; average dwell time method; actuator fault; different dimensional switched systems

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1. Introduction Widely applied in fine chemical engineering, food, beverage, medicine, microelectronics, polymer and other manufacturing industries in recent years, batch processes have become the first choice in manufacturing a variety of products with high additional value and the control issues of them have aroused great concerns among experts and scholars 1-2. As industrial technology is rapidly developing and manufacturing equipment is becoming increasingly automatic, massive and complicated, there may exist more chances of faults for equipment running for a long time. If the equipment fault happens and the reason and location are not detected or settled in time, equipment degradation and property loss and even endanger personal safety may result. It is quite necessary to keep equipment in steady operation under faulty conditions, which promotes the development of fault-tolerant control (FTC)

3-4

. FTC is classified into active FTC

5-7

and passive FTC

8-12

. The former requires controller’s parameter

changes and structure readjustments when system faults take place; whereas, the latter does not need the fault information and thus restructuring or system reconstruction are not needed. Due to the above merits, the latter once became a hot issue in research community. At present, FTC concerning continuous processes is basically mature

13-15

. However, with regard to batch processes, FTC studies start relatively late. In China, the first paper 16

. In the above paper, feedback control combined with

17-24

were applied in time direction and batch direction

on FTC in batch processes did not arise until 2006 iterative learning control based on the ideas in

respectively, where the batch process was converted into a two time-dimensional Fornasini-Marchesini (2D-FM) model and FTC law was designed. Subsequently, the study on FTC concerning batch processes has aroused extensive concerns of experts and scholars and resulted in innovative achievements

25-30

. For batch processes

with time delay and actuator faults, FTC that is dependent on time delay in a zone is considered based on 2D-FM


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models to ensure stable closed-loop systems under normal or faulty conditions. In fact, batch processes are multi-stage manufacturing processes in essence and its neighboring stages may differ in the control target due to their manufacturing purpose differences. In the injection molding process, for instance, a key variable to be controlled from the injection molding stage to the pressure packing stage is shifted from the injection velocity to the mold cavity pressure. Since the injection velocity may influence the mold cavity pressure, the occasion of switching becomes particularly important and the system is required to maintain stable after the switch. Besides, as running time is extended, chances for actuator faults in the equipment increase. Since actuator faults, if any, may result in breakdown in the whole manufacturing process, failure of the subsequent operation and tremendous loss and cost, it is quite important to study FTC in multistage batch processes under actuator faults. However, no references are offered for research on FTC of multistage batch processes nowadays，especially for multistage batch processes with different dimensions. It is necessary not only to keep the system in steady operation but also to maintain a shorter running time. It saves cost and enhances manufacturing efficiency by obtaining the expected results in a shorter time. The multistage characteristic of batch processes enables them to be deemed as switched systems, and theory on advanced control of switched systems can be borrowed for such processes, where the strategy of average dwelling time is a common way

31-32

. One of its great advantages is that it can enable scholars to find the

minimum running time of the whole system and even the minimum running time of each sub-system. It is without doubt that the most time-saving and best plan for the whole system is to determine the minimum running time of each stage. Based on this idea, a method of minimum running time control of multistage batch processes under normal situation was presented

33

, which ensures the minimum running time together with guaranteed

system performance. This paper is the first paper to determine the minimum running time for batch processes



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using average dwell time method, however, the control problems of multistage batch processes with faults were not considered. In practical industrial processes, changes in process parameters and external factors may result in disturbance on the system. In other words, disturbance cannot be avoided, which may reduce product quality and influence manufacturing efficiency. Thus, how to restrain disturbance is a key issue for industrial processes. In this paper, the issue of iterative learning FTC for multistage batch processes under actuator gain faults, internal and external disturbances and running time is studied based on the receptiveness and multistage characteristics of batch processes. First of all, iterative learning reliable control (ILRC) law is designed based on the repetitive and 2D characteristics of batch processes to transform it into a 2D-Roesser switched system. Secondly, based on Lyapunov function and the average dwell time method, sufficient conditions for the switched system to show 2D robust stability for time and batch directions is given, methods of designing minimum running time for each stage are provided and the state feedback updating law is proposed as well. Besides, traditional reliable control (TRC) and ILRC are compared in this paper. Last but not least, ILRC is verified to show better performance through simulation examples of Injection modeling process.

2. Problem statement Based on the description in the introduction, a multistage batch process under disturbance is described as the following switched system:  x(t + 1, k ) = Aϑ (t ,k ) (t , k ) x(t , k ) + Bϑ (t , k ) u (t , k ) + wϑ (t , k ) (t , k )  ϑ (t ,k ) x(t , k )  y(t , k ) = C  x(0, k ) = x 0, k 

(1)

l n where t refers to time indicator, k is batch indicator, x(t , k ) ∈ R is the process state, y (t , k ) ∈ R is the system

output, u (t , k ) ∈ R m is the system input and ϑ (⋅, ⋅) : Z + × Z + → n = {1, 2,L , n} is the switching signal concerning not only time and batch but also the system state; for the i th system with Ai (t , k ) = Ai + ∆Ai (t , k ) , Ai , B i and


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C i are system state matrices, ∆Ai (t , k ) is the unknown uncertain system perturbation matrix, and

∆Ai (t , k ) = Di F i (t , k ) E i , F i (t , k ) F iT (t , k ) ≤ I i ,

{D , E } i

i

are constant matrices with proper dimensions, I i is

a unity matrix with proper dimensions, wi (t , k ) refers to the external unknown disturbance and x0.k refers to the initial condition of the system in the k th batch. In practice, disturbance is inevitable. If the uncertain system parameter perturbation depends upon time t only, the parameter perturbation will be fully repetitive; otherwise, it will be non- repetitive. Moreover, in practice, due to equipment’s longtime service and high-strength operation, system faults are inevitable. Generally, there are internal faults, actuator faults and sensor faults in a system. For actuator faults, the control input will not achieve its correct value and affect the system stability, which may even cause system failure under significant conditions. To ensure a stable and steady system operation, FTC for systems under actuator faults has become a hot topic and aroused concern among researchers. Under actuator faults, the system input u i (t , k ) hardly attains its expected value through the actuator. Since actuator faults are primarily classified into partial failure (gain fault), complete failure and stuck fault, different α i values are defined to express actuator faults. Here α i > 0 and α i = 0 indicate partial failure and complete failure respectively. Stuck fault may cause the system input to be a constant value and no control action can be exerted to the system, thus it can be deemed as a disturbance. In case of complete failure and stuck fault, the system is beyond control. Thus the partial failure is studied in this paper. Since the batch process is multistage and for the ith actuator fault α i , the system’s input signal is u i (t , k ) while u iF (t , k ) refers to the actuator fault signal. As a result, the faulty model is signified as follows: u iF (t , k ) = α i u i (t , k )

(i = 1, 2,...n)

(2)

0 < α ≤ αi ≤ αi

(i = 1, 2,...n)

(3)

i

where, α (α < 1) and α i (α i ≥ 1) are known variables. i

i

Define:

uiF = [u1 , u2iF ,..., umiF ]T

(4a)

α i = diag[α1i , α 2i ,..., α mi ]

(4b)

α i = diag[α 1i , α i2 ,..., α im ]

(4c)

iF



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α i = diag[α1i , α 2i ,..., α mi ]

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(4d)

Introduce the following notations and definitions

β i = diag[ β1i , β 2i ,..., β mi ]

(5a)

β 0i = diag[ β10i , β 20i ,..., β mi 0 ]

(5a)

其中

βi =

α i +αi 2

， β 0i =

α i −α i α i +αi

( for

i = 1, 2,...n)

(6)

From (4) and (6), there exists unknown α i such that

α i = (1 + α 0i ) β i

(7)

α 0i ≤ β 0i ≤ 1

(8)

and

where

α0i A diag[α 01i ,α02i ,...,α0i m ]

α0i A diag[ α01i , α02i ,..., α0i m ] Based on the above gain fault, the system in the ith stage is described as follows:  xi (t + 1, k ) = Ai xi (t , k ) + B iα i u i (t , k ) + wi (t , k )  i i i  y (t , k ) = C x (t , k )

(9)

The corresponding control target is to design different control laws concerning different stages and enable the output to track the given expected track yri (t ) as much as possible under the designed switched signals no matter whether system fails or not. The output error is defined as follows: ei (t , k ) A yri (t ) − y i (t , k )

(10)

As described before, the switch signal ϑ (⋅, ⋅) is pertinent to the system state, which is a switch condition in this paper. The condition is rendered to be ψ ϑ (t , k ) +1 ( x(t , k ) ) < 0 . Once this condition is fulfilled, the system will switch from one stage to another. As the control input in the system’s initial operation fails to take remarkable effects, the above switch condition will cost a long running time at the initial operation, where, in order to equal the switch time between the subsequent stages, the minimum switch time is defined as follows:


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Tki = min {t > Tki −1 ψ i ( x(t , k ) ) < 0} , Tk0 = 0 Process (1) has

(11)

n stage(s). Tki is the value at the i stage of the k batch, which is also called the switching

time or switching step ， ψ i ( x(t , k ) ) < 0

is the switching condition related to the system states.

Tki −1 + 1, Tki  is called time interval of i (i = 1,2,...n) stage(s). (Ti n −1 , ki ), ϑ (Ti n −1 , ki )  refer to the values at the

end of the （n-1）stage of the ki batch under the switching signal ϑ (Ti n −1 , ki ) . Therefore, the switching sequence of the whole batch process is described as follows:

{

Σ = (T01 , k0 ), ϑ (T01 , k0 )  ,  (T02 , k0 ), ϑ (T02 , k0 )  , L , (T0n , k0 ), ϑ (T0n , k0 )  ,  (T11 , k1 ),ϑ (T11 , k1 )  , L (T1n , k1 ), ϑ (T1n , k1 )  , L , (Tk1 , kk ), ϑ (Tk1 , kk )  ,L ,  (Tkn , kk ), ϑ (Tkn , kk )  L

}

where, (Ti n , ki ), ϑ (Ti n , ki )  refers to a point of junction between the end of the previous batch and start of the next batch. In addition, different stages in a batch process are required to control different parameters. As a result, their dimensions may differ. The state relations between two stages at the moment of switch are expressed by the following formula 33, 34: x i +1 (Tki , k ) = Μ i x i (Tki , k )

(12)

where, Μ i ∈ R ni+1 ×ni is called the state transition matrix.

3 Traditional reliable control (TRC) There are two ideas in the design of advanced control strategy for batch processes. One is to deem the batch process as a continuous system and design the control law based on continuous system models; the other is to deem the batch process as a real batch process and design the control law based on the batch characteristics. In this paper, two methods are both applied to design the control law and will be analyze and compared for system control performance. Concerning a continuous process, the control law in the ith stage is designed as follows:

xei (t + 1, k ) = xei (t , k ) + ei (t , k ) =xei (t , k ) + yri (t ) − y i (t , k )

(13)

=xei (t , k ) + yri (t ) − C i x i (t , k ) where, xei (t , k ) refers to the state. By extending the models of (13) and (9), we obtain: X 1i (t + 1, k ) = A1i X 1i (t , k ) + B1iα i u (t , k ) + W1i (t , k )


(14)


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 x i (t , k )   Ai i i i where, X 1i (t , k ) =  i  , A1 = A1 + ∆A =  i  xe (t , k )   −C

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 wi (t , k )   Bi  0   ∆Ai (t , k ) 0  i i + B = W =      , 1  i 1 I  0 0 ,  yr (t )  0 

 i   i and ∆Ai =  ∆A (t , k ) 0  = D i ∆i E i =  D  ∆ i (t , k )  E i  0 0    0 

0 

For system (14), a state feedback control law is designed as follows: (15)

u i (t , k ) = K1i X 1i (t , k ) = K11i x i (t , k ) + K12i xei (t , k )

where, K1i (t ) =  K11i

K12i  ∈ R m×( n +l )

.

Thus, a closed-loop system equation is obtained as follows:

X 1i (t + 1, k ) = ( A1i + B1iα i K1i ) X 1i (t , k ) + W1i (t , k )

(16)

In order to analyze the stability of the above closed-loop system and the control law design, the following definitions and lemmas are introduced. At the same time, the lemmas are still applicable to the following Theorem 2 and 3. Definition 1: define t + k = F ≥ w = Ti w + kiw where t refers to total running time, N p ( F , w)

is the switch

time of the p sub-system in the time interval of ( w, F ) and Tp ( F , w ) is the total running time of the p sub-system in the interval of ( w, F ) . Here if Np (F, w) ≤

Tp ( F,w)

τp

holds for any given τ p > 0,

τ p will be called

the average dwelling time. Definition 2: in view of boundary conditions and external disturbances wi (t , k ) = 0 , it is defined

{

}

that χ l = sup X t , k : t + k = w, t , k ≥ 1 . Concerning the given w ≥ 0 and all F ≥ w , a positive definite matrix

γ exists, resulting in liml → F χ F ≤ ξγ ( F − w ) χ w . Thus, this closed-loop 2D system (16) is exponentially stable under the switch signal ϑ (⋅, ⋅) , where, • refers to Euclidean norm. Lemma 1: (Schur complement): assume that W , L and V are given matrices with proper dimensions,

where, W and V are positive definite symmetric matrices, the necessary and sufficient condition for LT VL − W < 0 to hold will be as follows:

 −W   L

 −V −1 LT  < 0 or  T −1  −V   L

L   0 exists, the necessary and sufficient condition will be as follows: M + ε −1 DDT + ε E T E < 0

Theorem 1: For given scalars θ1i > 1 and θ 2i > 1 , if there exist positive definite matrices S i ∈ R ( n + l )×( n +l ) ,

H i ∈ R m×( n + l ) and constants µi > 1 ε i > 0 , γ i > 0 such that the following inequality holds:  −S i   ∗  ∗   ∗  ∗   ∗ 

S i A1iT + H iT β i B1iT −S + ε D D + ε B β B ∗ i

i

iT

i

i

i 1

i2 0

iT 1

S i E iT

H iT β i

Si

0

0 0

0 0

−ε I i

i

∗

∗

−ε i I i

0

∗ ∗

∗ ∗

∗ ∗

−γ i I i ∗

θ1i ( S i − (γ i ) −1 S i S i )  S i 0   0 exists and the following

conditions hold: (1) The closed-loop system (39) of wi (t , k ) = 0 is stable. (2) Under zero initial condition, the controlled output Ζi (t , k ) meets:

Z i (t , k ) ≤ γ i wi (t , k )

(40)

The system will have robust H ∞ property indicator γ i . Lemma 4: Under zero input, if there is a functional V i (g) that meets the following conditions:

a) For ∀x (t , k ) ∈ R n1 + n2 , V i ( x (t , k )) ≥ 0 and V i ( x (t , k )) = 0 ⇒ x (t , k ) = 0 ; b) V i ( x (t , k )) → ∞,if x (t , k ) → ∞ c) Under any boundary condition, if a constant ρ i < 1 exists that satisfies the following:

∑

V i ( x (t , k )) < ρ i

i + j = I 0 + J 0 + k +1 I0 ≤i ≤ I0 + k J0 ≤ j ≤ J0 + k

∑

V i ( x (t , k )) , ∀I 0 > 0, J 0 > 0, k > 0

(41)

i + j = I0 + J0 + k I 0 ≤i ≤ I0 + k J 0 ≤ j ≤ J0 + k

The 2D Roesser system (39) will be stable and ρ i will be called the 2D robust convergence index that makes (39) hold. Theorem 2: Under repetitive disturbance, for the given 0 < β1i < 1, 0 < β 2i < 1 , if there exist a diagonal matrix

S i = diag{Shi , Svi } > 0 , matrix H i ∈ R m×( n1 + n2 ) and scalars ε ai , ε bi > 0 , such that the following matrix inequalities



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Page 14 of 32

hold:

 − S i ( β1i , β 2i ) S i A2iT + H iT β i B2iT  ∗ − S i + ε ai D2i D2iT + ε bi B2i β i 02 B2iT   ∗ ∗  ∗ ∗ 

S i E iT 0 −ε ai I i ∗

H iT β i   0  0, J 0 > 0, k > 0 , the following inequalities will hold according to (47):

Vhi ( xh ( I 0 + 1, J 0 + k )) + Vvi ( xv ( I 0 , J 0 + k + 1)) < ρ iV i ( x ( I 0 , J 0 + k )) M Vhi ( xh ( I 0 + k + 1, J 0 )) + Vvi ( xv ( I 0 + k , J 0 + 1)) < ρ iV i ( x ( I 0 + k , J 0 ))


(48a)

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The above inequalities are summated as follows:

∑

V i ( x (t , k ))

i + j = I 0 + J 0 + k +1 I0 ≤i ≤ I0 + k J0 ≤ j ≤ J0 + k

≤

∑

V i ( x (t , k )) + Vhi ( xh ( I 0 , J 0 + k + 1)) + Vvi ( xv ( I 0 + k + 1, J 0 ))

(48b)

i + j = I 0 + J 0 + k +1 I0 ≤i ≤ I0 + k J 0 ≤ j ≤ J0 + k

< ρi

∑

V i ( x (t , k ))

i + j = I 0 + J0 + k I 0 ≤i ≤ I0 + k J0 ≤ j ≤ J0 + k

Let kl − f +1 and kl refer to the initial batch and the end batch respectively and Nϑ ( w, F ) refer to switches of the switch signal in the time interval of [ w, F ] . The switch points are in the following forms:

T s , k  , T s +1 , k  , L , Tkp −1 , kk − f +1  , T p , kk − f +1  , Tks , kl − f + 2  , L , Tkp , kl  ,L  l   kl − f +1 l − f +1   kl − f +1 k − f +1   l− f +1   kl − f +1    l− f +2

(49)

where, Tklp− f +1 , kl − f +1  and Tkl − f +1 , kl − f +1  have the same meaning as the ending moment of the previous stage     and the beginning moment

of the

following stage.

mki −q 1 = Tkiq−1 + kq

Since

and

mkq = Tkq + kq ,

q = l − f + 1, l − f + 2, L , l − 1, l , the time interval of the k q batch in the i stage will be Tkiq−1 , Tkq  , where,   F ∈  mki −q 1 , mkq  , V ϑ (Tk , kl ) ( x (t , k ) ) and V   i

( )

ϑ  Tki 

−

 , kl  

( x (t , k ) )

refer to the i th stage and (i − 1) th stage of the kl

batch. From (48b), (50a) will be obtained:

∑V

ϑ (Tkil , kl )

t+k =F

( x (t , k ) ) < ( ρ i )

F − mki −l 1

∑

V

ϑ (Tkil , kl )

( x (t , k ) )

(50a)

( x (t , k ) )

(50b)

t + k = mki −l 1

At the switching moment mki −q 1 = Tkiq−1 + kq , from (42b), we have:

∑

t + k = mki −l 1

V

ϑ ( Tkil , kl )

( x (t , k ) ) ≤ µi ∑ (

t + k = mki −l 1

V

)

( )

 ϑ  Tkil 

−

 , kl  

−

Therefore, the following inequality will hold based on Definition 2:



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∑V

(

)

ϑ Tkil , kl

Page 16 of 32

( x (t , k ) )

t +k =F

< (ρi )

F − mki −l 1

∑

(

) x (t , k ) ( )

ϑ Tkli , kl

V

t + k = mki −l 1

≤ µi ( ρ i )

F − mki −l 1

∑

V

( )

ϑ  Tkil 

−

 , kl  

( x (t , k ) )

t + k = mki −l 1

(50c)

M ≤ Π (µp ) i

Tp ( F , w )

τp

p =1

Tp ( F , w)

× Π (ρ p ) i

p =1

Tp ( F , w )

1 i   = Π  ( µ p )τ p ( ρ p )  p =1   1

It is known that

(µ ) (ρ ) < 1 p

p

τp

(

 ϑ  Tl1− f +1 

)

−

 , kl − f +1  

( x (t , k ) )

t +k =w

∑V

(

ϑ Tl1− f +1 , kl − f +1

) x (t , k ) ( )

t +k =w

1   and let γ = max p∈n  ( µ p )τ p ( ρ p )  . Thus, the following will be drawn:  

∑V

t +k =F

∑

V

(

ϑ Tkil , kl

)

ϑ T ( x (t , k ) ) ≤ ξγ F − w ∑ V (

1 l − f +1 , kl − f +1

)

( x (t , k ) )

(51)

t +k =w

ϑ Ti , k It is concluded that as long as the switch signal meets (44), convergence of V ( kl l ) will be obtained. The

closed-loop 2D system (39) will be 2D-fault-tolerant with 2D-convergence index not more than

ρ i = max( β1i , β 2i ) . Theorem 3: Under non-repetitive disturbance, for scalars 0 < β1i < 1, 0 < β 2i < 1 , if there exist a positive definite i i i i i i matrix S = diag{Sh , Sv } > 0 , a matrix H i ∈ R m× ( n1 + n2 ) , scalars ε a , ε b > 0 and γ > 0 make the following

inequalities hold:

 − S i ( β1i , β 2i ) S i A2iT + H i β i B2i T  i ∗ − S + ε ai D2i D2iT + ε bi B2i β 0i 2 B2i T   ∗ ∗  ∗ ∗   ∗ ∗  ∗ ∗ 

S i E iT

H iT β i

S i G iT

0

0 0

−ε ai I i ∗

−ε bi I i

0 0 0

∗ ∗

∗ ∗

−γ i I i ∗

0      0 and assume that all boundary conditions are equal to 0, the following inequations will be drawn: N1 N2

N1 N2

∑∑∆V ( x(t, k)) < ∑∑V (x (t +1, k)) +V (x (t, k +1)) −V (x (t, k)) −V (x (t, k)) i

t =0 k =1

i h

i v

h

v

i h

h

i v

v

t =0 k =1

N1

N1

k =1

t =0

(56)

=∑Vhi ( xh (N1, k)) + ∑Vvi ( xv (t, N2 )) ≥0 Therefore, N1

N2

∑∑ ((γ t = 0 k =1

) VGi iT Gi ( x (t , k ) ) − γ iVIii ( wi (t , k )))

i −1

(57)

N1 N 2

≤ ∑∑  ∆V i ( x (t , k ) ) + (γ i ) −1VGi iT Gi ( x (t , k ) ) −γ iVIii ( wi (t , k ))  t = 0 k =1 N1

N2

= ∑∑ J k (t , k ) < 0 t = 0 k =1

In other words,

Z i (t , k ) ≤ γ i wi (t , k ) . This completes the proof.

5. Algorithm design The goal of the paper is to design ILRC law to robustly stabilize system (39) of arbitrary boundary and repetitive disturbances with the minimum horizontal（t） and vertical（k）convergence indices under normal and permitted faults. For non-repetitive disturbances, system (39) should have the 2D robust H∞ index upper bound as small as possible. The following will consider both cases, i.e., repetitive and non-repetitive disturbances. In order to ensure that the designed ILRC system with wi (t , k ) = 0 has sound convergence in the orientations of t and k, it is necessary to optimize system’s “horizontal (t)” and “vertical (k)” convergence index in the controller design. In view of the sophistication of nonlinear optimization algorithm, the following weighting optimization algorithm may be applied in practice: Case 1: wi (t , k ) = 0



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Page 18 of 32

Algorithm 1: weighting optimization algorithm:

Given the weighting factor 0 ≤ λ i ≤ 1 , the following optimization issue is solved:

( ρ i )* = Minimize ρi i i i S , H , ε a ,b

Subject to

 − S i (1 − λ i ) + λ i ρ i，λ i + (1 − λ i ) ρ i  S i A2iT + H iT β i B2i T     ∗ − S i + ε ai D2i D2iT + ε bi B2i β i 02 B2i T  ∗ ∗   ∗ ∗ 

S i E iT 0 −ε ai I i ∗

H iT β i   0  0 S i = diag{Shi , Svi }

ε ai , ε bi > 0 If the optimization issue has a feasible solution and ρ * < 1 , the iterative updating law can be designed with K i = H i ( S i ) −1 .

It is noted that λ i reflects the relevant importance of the horizontal and vertical convergence indices 19. λ i >0.5 indicates that the horizontal convergence is more important while λ i 0

ε ai , ε bi > 0 If the above optimization issue has a feasible solution, the iterative updating law may be designed.

6 Simulation example An injection molding process is taken as a case and actuator gain faults are taken into account. At the stage of injection, the model form the valve opening to the injection velocity is as follows 33,34:

IV 8.687 z −1 − 5.617 z −2 = VO 1 − 0.9291( ± ∆11 ) z −1 − 0.03191( ± ∆12 ) z −2 and the model form the valve opening to the nozzle pressure is NP 0.1054 = IV 1 − z −1

(60a)

(60b)

Here ∆1i ( i ∈ {1, 2} ) is the corresponding uncertainty with ∆1i ≤ 0.002 , which is obtained based on open loop data. Let

x11 (t , k ) A IV (t , k ), x12 (t , k ) A 0.03191( ± ∆12 ) IV (t − 1, k ) − 5.617VO(t − 1, k ), 1 3

1

(60c)

1

x (t , k ) A NP(t , k ), u (t , k ) A VO (t , k ), y (t , k ) A IV (t , k ). Based on (60a)- (60b), we have

   0.9291 1 0    8.687   1   + ∆A1 (t , k )  x1 (t , k ) +  −5.617  α u1 (t , k ) x ( t + 1, k ) = 0.03191 0 0          0.1054 0 1     0        y1 (t , k ) = [1 0 0] x1 (t , k ) 


(60d)


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 20 of 32

0.002δ (t , k ) 0 0  where, ∆A (t , k ) =  0.001δ (t , k ) 0 0  , δ (t , k ) refers to a random variable in [0,1] ,  0 0 0  1

Similarly, during the packing holding, the model is

NP 171.8 z −1 − 156.8 z −2 = VO 1 − 1.317( ± ∆12 ) z −1 + 0.3259( ± ∆ 22 ) z −2

(60e)

Define

x12 (t , k ) A NP (t , k ), x22 (t , k ) A −0.3259( ± ∆ 22 ) NP(t − 1, k ) − 156.8VO(t − 1, k ), u 2 (t , k ) A VO(t , k ), y 2 (t , k ) A NP(t , k ).

(60f)

Based on (60e)和 (60f), we have

 2   1.317 1  171.8  2 +∆A2 (t , k )  x 2 (t , k ) +   x (t + 1, k ) =     u (t , k )  −156.8    −0.3259 0    2 2  y (t , k ) = [1 0] x (t , k )

(60g)

 0.02δ (t , k ) 0  where ∆ i2 ( i ∈ {1, 2} ) is the uncertainty with ∆ i2 ≤ 0.02 , then ∆A2 (t , k )=  .  0.01δ (t , k ) 0  The switch condition is [0 0 1]x1 (t , k ) ≥ 350 , signifying that once the nozzle pressure is higher than 350, the process will switch from the injection stage to the pressure packing stage. In order to evaluate the tracking performance, the following index is introduced. Tk

DT ( k ) A

∑ e(t , k )

(61)

t =1

In this paper, assume that the unknown actuator fault α i exists with 0.8 = α i ≤ α i ≤ α i = 1 , and it will be shown by Formula (6) that β i = 0.9 and β 0i = 0.1 . At the first stage and the second stage, according to Theorem 1 and the above control algorithm 2, the minimum running time lower bound, updating law and H ∞ performance index of each stage can be easily obtained. Different results by comparing between TRC and ILRC

are shown in Table 1. It is seen that under actuator faults and no matter which stage is considered, the lower bound of the minimum running time and H ∞ performance index of TRC are both better than those of ILRC, which will be illustrated later.

Figure 1 shows the tracking performance of all batches using (61). It is illustrated that TRC is better than ILRC for the first several batches. However, from the 10th batch to the occurrence of fault, TRC has little performance change, whereas, ILRC improves significant. When the fault occurs, performance of TRC deteriorates a little and remains unchanged as that under such fault. For ILRC, although performance deteriorates at first, the tracking performance improves greatly after several batches, just like the condition without faults. This further shows the


Page 21 of 32

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effectiveness of the proposed method. The same conclusion applies to the minimum running time of the first stage, as shown in Figure 2. In the 50th batch, the fault occurs and Figure 3 shows the performance of batch 49, batch 50, batch 51 and batch 100. For batch 49, the proposed method manages to track the set-point but TRC fails. For batch 50, both methods fail to track the set point, but in the end, the ILRC manages to track the set-point, while, for TRC, no improvement is seen. It is observed in Figure 3a that ILRC attains the expected output from the 49th batch while TRC has the same tracking effects in every batch without batch improvement. Although the system’s control performance in ILRC algorithm clearly deteriorates in a fault, it quickly recovers and even attains the original level after undergoing several batches and ILRC has better output tracking performance than TRC. It is observed in Figure 3b and 3c that the output response suffers from minor effects of the fault in TRC; whereas, outstanding change happens in ILRC. Also, after running in several batches, as shown in Figure 3d, ILRC overcomes the fault and the system’s output is enabled to fully attain its expected set-point and realizes a perfect tracking. Overall, ILRC has improved output response than TRC.

7 Conclusion In this paper, FTC is studied for multistage batch processes under actuator faults and disturbance. Based on 2D system theory and using multiple Lyapunov function, an equivalent 2D switched system is obtained and a hybrid ILRC law is proposed to give sufficient conditions for exponentially stability indices of both time and batch axes. Meanwhile, the lower bound of the minimum running time of each stage is calculated using the average dwell time method. The effectiveness of the proposed method is illustrated in comparison with traditional reliable control on an injection molding process. Simulation results show that ILC is quicker in terms of convergence and better in tracking performance.

Acknowledgments This work is supported by National Natural Science Foundation of China under Grant (61673147, 61433005) and Guangdong Innovative and Entrepreneurial Research Team Program (2013G076).

References (1) Bonvin, D.; Srinivsan, B.; Hunkeler, D. Control and optimization of batch processes. IEEE Contr. Syst. Mag. 2006, 26(6), 43−45.



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(2) Zhang, R.; Lu, R.; Xue, A.; Gao, F. New minmax linear quadratic fault-tolerant tracking control for batch processes. IEEE Trans. Autom. Control 2016, 61, 3045-3051. (3) Zhang, H.; Guan, Z. H.; Feng, G. Reliable dissipative control for stochastic impulsive systems. Automatica 2008, 44(4), 1004−1010. (4) Zhang, R.; Lu, J.; Qu, H.; Gao, F. State space model predictive fault-tolerant control for batch processes with partial actuator failure. J. Process Control 2014, 24(5), 613-620. (5) Wu, D.; Song, J.; Shen, Y.; Ji, Z. Active fault-tolerant linear parameter varying control for the pitch actuator of wind turbines. Nonlinear Dynam. 2017, 87(1), 475-487. (6) Qin, L.; He, X.; Yan, R.; Zhou, D. Active Fault-Tolerant Control for a Quadrotor with Sensor Faults. J. Intell. & Robot. Syst. 2017, 1-19. (7) He, X.; Wang, Z.; Qin, L.; Zhou, D. Active Fault-Tolerant Control for an Internet-Based Networked Three-Tank System. IEEE Trans. on Control Syst. Technol. 2016, 24(6), 2150-2157. (8) Yang, G. H.; Ye, D. Reliable H ∞ control of linear systems with adaptive mechanism. IEEE Trans. Autom. Control 2010, 55(1), 242−247.

(9) Zhang, R.; Gao, F. Improved infinite horizon LQ tracking control for injection molding process against partial actuator failures. Comput. Chem. Eng. 2015, 80, 130-139. (10) Boskovic, J. D.; Mehra, R. K. A decentralized fault-tolerant control system for accommodation of faults in higher-order flight control actuators. IEEE Trans. on Control Syst. Technol. 2010, 18(5), 1103−1115. (11) Mao, Z. H.; Jiang, B.; Shi, P. Fault-tolerant control for a class of nonlinear sampled-data systems via a Euler approximate observer. Automatica 2010, 46(11), 1852−1859. (12) Zhang, R.; Gan, L.; Lu, J.; Gao, F. New design of state space linear quadratic fault-tolerant tracking control for batch processes with partial actuator failure. Ind. Eng. Chem. Res. 2013, 52(46), 16294-16300. (13) Bin, J. Fault-tolerant control for a class of non-linear systems with dead-zone. Int. J. Syst. Sci. 2015, 47(7), 1689−1699. (14) Polycarpou, M. M. Fault accommodation of a class of multivariable nonlinear dynamical systems using a learning approach. IEEE Trans. Autom. Control 2001, 46(5), 736−742. (15) Kabore, P.; Wang, H. Design of fault diagnosis filters and fault tolerant control for a class of nonlinear systems. IEEE Trans. Autom. Control 2001, 46(11), 1805−1810. (16) Wang, Y. Q.; Shi, J.; Zhou, D. H.; Gao, F. Iterative learning fault-tolerant control for batch processes. Ind. Eng. Chem. Res. 2006, 45, 9050−9060.


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(17) Shi, J.; Gao, F.; Wu, T. J. A robust iterative learning control design for batch processes with uncertain perturbation and initialization. AICHE J. 2006, 52(6), 2171-2187. (18) 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(8), 907-924. (19) Shi, J.; Gao, F.; Wu, T. J. From two-dimensional linear quadratic optimal control to iterative learning control. Ind. Eng. Chem. Res. 2006, 45(13), 4603-4616.

(20) Shi, J.; Gao, F.; Wu, T. J. Iterative learning controls for batch processes. Ind. Eng. Chem. Res. 2006, 45(13), 4617-4628. (21) Shi, J.; Gao, F.; Wu, T. J. Single-cycle and multi-cycle generalized 2D model predictive iterative learning control (2D-GPILC) schemes for batch processes. J. Process Control 2007, 17(9), 715-727. (22) Wang, L.; Zhu, C.; Yu, J.; Li, P.; Zhang, R.; Gao, F. Fuzzy iterative learning control for batch processes with interval time-varying delays. Ind. Eng. Chem. Res. 2017, 56(14), 3993-4001. (23) Wang, L.; Mo, S.; Zhou, D.; Gao, F.; Chen, X. Delay-range-dependent guaranteed cost control for batch processes with state delay. AICHE J. 2013, 59(6), 2033–2045. (24) Wang, L.; Mo, S.; Qu, H.; Zhou, D.; Gao, F. H_infinity design of 2D controller for batch processes with uncertainties and interval time-varying delays. Control Eng. Pract. 2013, 21, 1321-1333. (25) Wang, L.; Mo, S.; Zhou, D.; Gao, F.; Chen, X. Robust delay dependent iterative learning fault-tolerant control for batch processes with state delay and actuator faults. J. Process Control 2012, 22(7), 1273−1287. (26) Wang, L.; Mo, S.; Zhou, D.; Gao, F. Delay-range-dependent method to iterative learning fault-tolerant guaranteed cost control for batch processes. Ind. Eng. Chem. Res. 2013, 52 (7), 2661–2671. (27) Wang, Y. Q.; Zhou, D. H.; Gao, F. R. Iterative learning reliable control of batch processes with sensor faults. Chem. Eng. Sci. 2008, 63, 1039-1051. (28) Zhang, R.; Jin, Q.; Gao, F. Design of state space linear quadratic tracking control using GA optimization for batch processes with partial actuator failure. J. Process Control 2015, 26, 102-114. (29) Zhang, R.; Lu, R.; Xue, A.; Gao, F. Predictive Functional Control for Linear Systems under Partial Actuator Faults and Application on an Injection Molding Batch Process. Ind. Eng. Chem. Res. 2014, 53(2), 723-731. (30) Zhang, R.; Zou, H.; Xue, A.; Gao, F. GA based predictive functional control for batch processes under actuator faults. Chemom. Intell. Lab. Syst. 2014, 137(1), 67-73. (31) Zhao, X. D.; Zhang, L. X.; Shi, P.; Liu, M. Stability and stabilization of switched linear systems with



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mode-dependent average dwell time. IEEE Trans. Autom. Control 2012, 57(7), 1809−1815. (32) Lu, Q.; Zhang, L.; Karimi, H. R.; Shi, Y. H∞ control for asynchronously switched linear parameter-varying systems with mode-dependent average dwell time. IET Control Theor. Appl. 2013, 7(5), 673−683. (33) Wang, L. M.; Zhou, D. H. Average dwell time-based optimal iterative learning control for multi-phase batch processes. J. Process Control 2016, 40, 1−12. (34) Wang, Y. Q.; Zhou, D. H.; Gao, F. Iterative learning model predictive control for multi-phase batch processes. J. Process Control 2008, 18, 543−557.

Table 1 Comparative result between TRC and ILRC TRC


ILRC

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Page 25 of 32

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


Updating law

Minimum running time at the first stage Minimum running time at the second stage Stage 1: H ∞ performance index Stage II: H ∞ performance index

K11 = [−0.2024 − 0.1514 − 0.0063 0.0595]

K11 = [ −0.0962 − 0.1042 − 0.0028 0.0563]

K 21 = [−0.0105 − 0.0073 0.0012]

K 21 = [−0.0082 − 0.0061 0.0029]

µ1 = 4.1824e + 04, ρ1 = 0.9334

( µ1 = 5.3104e+03, ρ1 = 0.9131

 µ  (τ 1a )∗ =  − 1  = (90.0568)=90  ρ1 

 µ  (τ 1a )∗ =  − 1  = (94.3509)=94  ρ1 

µ 2 = 4.5154e+03, ρ 2 = 0.9811

µ 2 = 3.3893e+03, ρ 2 = 0.9189

 µ  (τ 2a )∗ =  − 2  = (92.4432) = 92  ρ2 

 µ  (τ 2a )∗ =  − 2  = (96.1051) = 96  ρ2 

γ 1 =0.6046

γ 1 =14.7249

γ 2 = 0.6132

γ 2 =12.4801



5000 TRC model ILRC model

4500 4000 3500 3000 DT(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

Page 26 of 32

2500 2000 1500 1000 500 0

0

10

20

30

40 50 60 Batch number

70

80

Figure 1. Tracking performance comparison between TRC and ILRC


90

100

Page 27 of 32

220 TRC model ILRC model

200

180 Switching step

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


160

140

120

100

80

0

10

20

30

40 50 60 Batch number

70

80

Figure 2. Switching time comparison between TRC and ILRC


90

100


IV(mm/s)

40 30 Injection Velocity Nozzle Pressure Set-point for IV Set-point for NP

20 10 0

0

20

40

60

80

100

120

140

160

350 300 250 200 150 100 50

NP/bar

TRC 50

0 180

50 40 30 Injection Velocity Nozzle Pressure Set-point for IV Set-point for NP

20 10 0

0

20

40

60

80

100

120

140

160

step Figure 3a Output response comparison between TRC and ILRC in the 49th batch


350 300 250 200 150 100 50 0 180

NP/bar

step ILRC

IV(mm/s)

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 28 of 32

Page 29 of 32

IV(mm/s)


20 10 0

0

20

40

60

80

100

120

140

160

350 300 250 200 150 100 50

NP/bar

TRC 50

0 180


20 10 0

0

20

40

60

80

100

120

140

160

step Figure 3b Output response comparison between TRC and ILRC in the 50th batch


350 300 250 200 150 100 50 0 180

NP/bar

step ILRC

IV(mm/s)

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



IV(mm/s)


20 10 0

0

20

40

60

80

100

120

140

160

350 300 250 200 150 100 50

NP/bar

TRC 50

0 180


20 10 0

0

20

40

60

80

100

120

140

160

step Figure 3c Output response comparison between TRC and ILRC in the 51th batch


350 300 250 200 150 100 50 0 180

NP/bar

step ILRC

IV(mm/s)

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 30 of 32

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IV(mm/s)


20 10 0

0

20

40

60

80

100

120

140

160

350 300 250 200 150 100 50

NP/bar

TRC 50

0 180


20 10 0

0

20

40

60

80

100

120

140

160

step Figure 3d Output response comparison between TRC and ILRC in the 100th batch


350 300 250 200 150 100 50 0 180

NP/bar

step ILRC

IV(mm/s)

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



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 32 of 32

TOC

Iterative learning FTC

y(t + 1, k − 1)

u(t , k − 1) Memory

y yr (t + 1)

— e (t + 1) k −1

⊗

FTC

δ ( x(t , k ))

r (t , k )

⊗

u (t , k )

∑

2D − switch

y(t , k )

x(t , k )

Memor —

x(t , k − 1)

⊗ ∑


2 D − Roesser − switch

Batch process

Robust Iterative Learning Fault-Tolerant Control for Multiphase Batch

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