Genetic algorithm optimization based infinite horizon LQ control for

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Genetic algorithm optimization based infinite horizon LQ control for injection molding batch processes with uncertainty xiaomin hu, Limin Wang, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b04921 • Publication Date (Web): 05 Dec 2018 Downloaded from http://pubs.acs.org on December 9, 2018

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Industrial & Engineering Chemistry Research

Genetic algorithm optimization based infinite horizon LQ control for injection molding batch processes with uncertainty Xiaomin Hu1, Limin Wang2,*, Furong Gao3

1. 2.

School of Science, Hangzhou Dianzi University, Hangzhou 310018, P. R. China

School of Mathematics and Statistics, Hainan Normal University, Haikou, 571158, P. R. China

3. Department of Chemical and Biomolecular Engineering, Hong Kong University of Science and Technology, Hong Kong

Corresponding author: Limin Wang Emails: [email protected]

Abstract: A new genetic algorithm (GA) based infinite horizon linear quadratic (IHLQ) control method is developed for injection molding batch processes under partial actuator faults and unknown disturbances. To obtain improved system performance, an extended state space model, which combines the process state and output error information, has been adopted, where extra tuning freedom are gained for the corresponding control system design by adjusting the extended weighting matrix. However, no specified rules are established for the choices of these weighting coefficients. To cope with such situations, GA is employed to optimize these weighting factors in this paper. Furthermore, the robust stability of the control system, which can be regarded as the process under uncertainty is addressed. Finally, the validity of the proposed approach is tested on the injection velocity regulation process. Key words：Genetic algorithm, infinite horizon linear quadratic control, partial actuator faults, extended state space model, injection molding

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1. Introduction As a vital role in the manufacturing industry, batch processes provide various necessary products in which may kinds are low-volume with high values, and the corresponding control strategies have made significant progress in the past decades1. Driven by the increasing economy and stricter product quality, there are challenges for the existing methods, and the corresponding research is still a hot topic. It is known that various uncertainties are inevitable in practice, such as load disturbances, frictions, module faults, and the system performance may deteriorate2. Among these disadvantages, the actuator faults affect the accurate execution of the controller signal, such that the effectiveness of the relevant control strategy may be discounted significantly. In a general way, there are actuator stuck, actuator outage and partial actuator fault3-4. Note that the whole system may be uncontrollable under the first two types of actuator faults, and it is not meaningful to investigate the further control design. Hence, the control system under partial actuator fault is what we need to focus on. For such issues, many researchers contribute to the development of the relevant control theories and applications5-6. As a promising control strategy in dealing with the process under system failures, fault-tolerant control (FTC) has drawn a lot of attention7. In8, the recent development of monitoring, optimization control and fault diagnosis theory based on data-driven models was addressed. In order to handle the control of discrete systems with delay, a relevant fault diagnosis and compensation problem was discussed by Zhao, et al9. As to large-scale systems, , an improved performance optimization of distributed model predictive control was provided by Zakharov, et al10. To cope with the control problem of batch processes under disturbance and partial actuator faults, a novel design of linear quadratic control strategy was proposed in11. In12, a robust two-dimensional hybrid composite FTC guaranteed cost controller was designed for the multiphase batch processes. Besides these relevant researches on FTC, the study about iterative learning control (ILC) is also an important

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branch13-14. It is known that the repetitive nature exists in batch processes due to the fact that the same procedure will be executed repeatedly for manufacturing the specified products, and the system information of the previous batches may be useful for current batch15. Meanwhile, it is worth noticing that ILC approach is pure feedforward strategy and it is necessary to unite other feedback control methods to deal with uncertainties in practice16. There are many results in which ILC and other feedback control strategies are combined for batch processes17. In18, the ILC method was employed in the framework of model predictive control, then a two-dimensional controller was derived for the batch processes. For constrained batch process with unknown input nonlinearities, an ILC model predictive control was designed using a two-mode framework by Li, et al19. Jia, et al20 presented an improved control in which dynamic Rparameter is available. In21, an ILC strategy was put forward for the batch process under uncertainties. Based on the two-dimensional models, the control scheme in which dynamic matrix control and ILC are combined was investigated by Mo, et al22. In spite of these developments on the fault diagnosis and control theories, it is also a challenge to deal with the batch process with various uncertainties, disturbances and system failures, and many optimization approaches based on GA and its variants are recently adopted to enhance the relevant system performance further23-26. In order to solve the optimal problem of constrained batch process, the particle swarm optimization algorithm was employed in27. In28, an adaptive terminal ILC strategy with neural network is introduced for uncertain fed-batch processes. In29, a reliable optimal control approach which adopts ant colony optimization was presented. In30, a multi-objective optimization is further proposed for coke furnace systems. Among these improvements on the control strategies for batch processes, the adoption from the non-minimal state space model to the extended state space model31-38 is notable. Zhang et al39 proposed the first paper on extended non-minimal state space model predictive control and this work is very useful to provide a guidance for utilizing the improved model structure, the output error and the process state regulation with extra tuning. It is seen that the process

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performance is related to such tuning. If rules for the choices of these weighting coefficients can be offered, the ensemble control performance can be improved. To solve this problem, we propose a novel infinite horizon quadratic control approach in which GA is included for the selection of the coefficients. Meanwhile, the robust stability of the closed-loop system is also studied based on Lyapunov theory. Case studies on the injection velocity batch process under various uncertainties demonstrate the validity of the proposed IHLQ control method. The structure of the paper is shown as follows. The problem formulation is provided in section 2, and the GA based IHLQ control scheme in which the extended state space model is utilized is described in section 3. Section 4 illustrates the case studies on the control of the injection velocity. The conclusion is drawn in section 5.

2. Problem formulation Here a single-input single-output (SISO) batch process is considered for simplicity, and the relevant process model which is acquired by linearizing at its operation point is  x(k += 1) Ax(k ) + Bu (k − d )   y (k ) = Cx(k )

(1)

where x(k ) ∈ R n , y (k ) ∈ R , u (k ) ∈ R are the state, output and input. k is the current time instant and 0 < k ≤ L , and L is the end time point. { A, B , C} are system matrices of appropriate dimensions. d denotes dead time. Under partial actuator faults, the actual control action is

u F (k ) = α u (k )

(2)

where u F ( k ) is the actual control action, and u (k ) is the control calculated by the controller. α represents the actuator failure, and its value satisfies the following inequality.

0 0 denotes the partial actuator fault. 4


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By synthesizing Eqs.(1)~(2), the formula of the process with partial actuator faults is written as  x(k += 1) Ax(k ) + Bα u (k − d )   y (k ) = Cx(k )

(4)

The following controller will be designed to let the process output of the batch process track the reference value and maintain the acceptable performance under various uncertainties and partial actuator failures simultaneously.

3. GA based infinite horizon linear quadratic control 3.1 Extended state space model For Eq.(1), the differenced model can be acquired with the difference operator ∆ . ∆x(k + 1) = A∆x(k ) + B ∆u (k − d )   ∆y (k )= C ∆x(k )

(5)

In order to remove the dead time in Eq.(5), the state vector is further selected as

∆xm ( k ) = [ ∆x( k ) ∆u ( k − 1) ∆u ( k − 2)  ∆u ( k − d )]T

(6)

then we have

∆xm (k + 1) = Am ∆xm (k ) + Bm ∆u (k ) ∆y (k ) =Cm ∆xm (k ) where

A  0 0 Am =  0    0

B  0  0  0 1 0        0  0 1 0 

0 0 1

0 0 0

   0

0 0

Bm = [0 1 0  0]T Cm = [C 0 0  0] 0, 0 are zero vectors of corresponding dimensions. Define the output reference as yr (k ) , then the tracking error is calculated as

5


(7)


e= ( k ) y ( k ) − yr ( k )

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(8)

Further, we will have by combining Eqs.(7)~(8)

e(k + 1) =e(k ) + Cm Am ∆xm (k ) + Cm Bm ∆u (k )

(9)

Here, an extended state vector is chosen to formulate the extended state space model

 ∆x (k )  z (k ) =  m   e( k ) 

(10)

then

z (k += 1) Az (k ) + B∆u (k )

(11)

0  Am  B  A = ; B  m  =  C m Am 1  Cm Bm 

(12)

where

0 in A is the zero vector of appropriate dimensions. 3.2 Controller design 3.2.1 Cost function The following is selected for the proposed IHLQ control strategy ∞

J =

∑ [ z (k ) Qz (k ) + ∆u T

T

(k ) R∆u (k )]

(13)

k =0

where Q and R are the weighting matrices for extended state vector and control increments, respectively. Here, Q is

Q = diag{q j x1 , q j x 2 , , q j xn , q j u1 , q j u 2 , , q j ud , q j e }

(14)

Remarks 2. By observing the form of Q in Eq.(14), we can see that q j x1 , q j x 2 ,  , q j xn are the weighting coefficients for state variable increments, q j u1 , q j u 2 ,  , q j ud are for the process input increments, and q j e is for the tracking error. Remarks 3. From Eq.(14), we find that q j x1 , q j x 2 ,  , q j xn affect the dynamic responses of the control system, and

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bigger q j x1 , q j x 2 ,  , q j xn will result in smoother response. Meanwhile, R in Eq.(13) is for control increments, and

q j u1 , q j u 2 ,  , q j ud will be selected as zeros for simplicity. Note that how to choose the values of q j x1 , q j x 2 ,  , q j xn is the optimization target for GA because there are no relevant guidelines. 3.2.2 GA optimization of the selection for Q As q j x1 , q j x 2 ,  , q j xn influence the dynamic responses of the process to a great extent, GA is utilized to find the values of q j x1 , q j x 2 ,  , q j xn . 3.2.2.1 Encoding approach In GA, the elements are obtained using binary encoding, and a ten-bit binary code is used as an example in the following.

1 0 0 111 0 111

(15)

The elements q j x1 , q j x 2 ,  , q j xn in Eq.(14) can be gained as

= q ji max( = q ji ) * b / 210 (i x1, x 2, , xn)

(16)

3.2.2.2 Target of Q optimization The following objective in which the process overshoot

ο (t ) and rise time tr (t ) are considered is introduced

for GA,

Min ο (t ) + tr (t )

(17)

Then the fitness is described as

Max 1/ [c + ο (t ) + tr (t )]

(18)

Here c is a fixed constant. 3.2.2.3 Operators (1) Selection operators Individuals are chosen from for reproduction according to its fitness, and note that bigger fitness value results

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in larger possibility to survive. Here the following Roulette wheel approach is utilized to obtain the selection possibility.

P(cl ) =

f (cl )

(19)

N

∑ f (c ) l

l =1

where P (cl ) is the chosen possibility for individual cl . f (cl ) is the fitness value for individual cl , and N is the amount of individuals. (2) Crossover operator pc and mutation operator pm After pc and pm are executed among individuals cu and cv , the offspring cu′ and cv′ can be generated. By implementing the above steps, the optimal Q is acquired. 3.2.3 Controller derivation The optimal control law of the IHLQ is as follows. Theorem 1. By minimizing the objective in Eq.(13), the control law is expressed in Eq.(20a), and K ∞ can be acquired using the Riccati Equation in Eq.(20b).

∆u (k ) = − R −1 BT [ I + K ∞ BR −1 BT ]−1 K ∞ Az (k ) (20a)

K∞ = AT [ I + K ∞ BR −1 BT ]−1 K ∞ A + Q AT K ∞ A − AT K ∞ B( R + BT K ∞ B) −1 BT K ∞ A + Q =

(20b)

Proof： Here, a finite horizon problem is introduced firstly. k f −1

J =

∑ [ z (k ) Qz (k ) + ∆u T

k = k0

T

(k ) R∆u (k )] + z (k f )T Qz (k f )

where k ∈ [k0 , k f ] . On the basis of the minimum principle of Pontryagin, we have the Hamiltonian for Eq.(11) as

8


(21)

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[ z (k ) Qz (k ) + ∆u (k ) R∆u (k )] + pk +1[ Az (k ) + B∆u (k )] Ηk = T

T

T

(22)

where pk +1 is the Lagrange multiplier . The necessary conditions for the control are

∂Η k = 2Qz (k ) + AT pk +1 ∂z (k ) pk f = 2Q f z (k f )

pk =

The following equation is gained by letting

(23)

∂Η k =0. ∂∆u (k )

∂Η k =2 R∆u (k ) + BT pk +1 =0 ∂∆u (k )

(24)

∂ 2Η k = 2 R is positive definite, hence the It is worth noticing that Η k is quadratic for ∆u (k ) and ∂∆u 2 (k ) optimal solution for minimizing Η k can be acquired by solving Eq.(24).

1 ∆u (k ) = − R −1 BT pk +1 2

(25)

We suppose that

pk = 2 H k ,k f z (k )

(26)

then the following formula is obtained by synthesizing Eq.(11), Eqs.(25)~(26).

1 = pk +1 2 H k +1,k f ( Az (k ) − BR −1 BT pk +1 ) 2

(27a)

Eq.(27a) is equivalent to

pk += [ I + H k +1,k f BR −1 BT ]−1 2 H k +1,k f Az (k ) 1

(27b)

By Combining Eq.(25) and Eq.(27b), the optimal control law in Eq.(20a) is gained.

∆u (k ) = − R −1 BT [ I + H k +1,k f BR −1 BT ]−1 H k +1,k f Az (k ) .

(28)

From Eqs.(23) and (27b), the following formula is derived.

= pk 2Qz (k ) + AT pk +1 = 2[ AT ( I + H k +1,k f BR −1 BT ) −1 H k +1,k f A + Q]z (k )

9


(29)


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Uniting Eqs.(26) and (29), we have

H k ,k f = AT [ I + H k +1,k f BR −1 BT ]−1 H k +1,k f A + Q = AT H k +1,k f A − AT H k +1,k f B( R + BT H k +1,k f B) −1 BT H k +1,k f A + Q

(30)

For Eqs.(28) and (30), if k f goes to ∞ , then the proposed IHLQ control is described by Eqs.(20). □ Note that the proposed IHLQ controller design is a robust controller design. Here a robust stability criterion, which is suitable for the conditions that desired set-point tracking or disturbance rejection under unknown actuator failures and uncertain process parameters can be obtained for the control system, is presented in this section. In the following, the corresponding robustness criterion is given. Theorem 2. Assume that the proposed IHLQ control scheme is based on the process model in Eq.(1) and serves as the controller for the processes with partial actuator failures in Eq.(4), then the following condition is tenable. 2 σ max (∆A) < −σ max ( A − BK s ) + σ max ( A − BK s ) +

λmin (W ) λmax ( P)

(31)

where σ max ( χ ) is the maximum singular value ， λmin ( χ ) is the minimum eigenvalue and λmax ( χ ) is the maximum eigenvalue of χ . Define two symmetric positive matrices P and W which satisfy

( A − BK s )T P( A − BK s ) − P = −W

(32)

where

 0  0 0  ∆A =    0 0 

0

0

 0

Bα − B

0 0

0 0

0 0





 0  0  



 

   



0



0 0   0 CBα − CB

0  0 0     0 0 

= K s R −1 BT [ I + K ∞ BR −1 BT ]−1 K ∞ A

0 in ∆A is the zero matrix with dimension n × n . Then robust stability can be satisfied under the proposed IHLQ control approach. 10


(33)

(34)

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Proof. By combining Eqs.(20a) and (34), it is obvious that the control law is

∆u (k ) = − K s z (k )

(35)

From the procedures in Eq.(6)~(12) and the following formula, we can find that Eq.(4) is associated with Eq.(1) through the following.

z (k + 1)= ( A + ∆A) z (k ) + B∆u (k )

(36)

Synthesizing Eqs.(35)~(36), we have

z (k + 1)= ( A − BK s ) z (k ) + ∆Az (k )

(37)

then the Lyapunov function between time instant k and k + 1 is

∆V ( z (k = )) V ( z (k + 1)) − V ( z (k ))

(38)

= z T (k + 1) Pz (k + 1) − z T (k ) Pz (k ) Through combining Eq.(37) with Eq.(38), the following holds.

∆V ( z (k )) = z T (k )( A − BK s )T P ( A − BK s ) z (k ) + z T (k )( A − BK s )T P∆Az (k ) + z T (k )∆AT P ( A − BK s ) z (k )

(39)

+ z T (k )∆AT P∆Az (k ) − z T (k ) Pz (k ) By uniting the first and last term in Eq.(39) with Eq.(32), the following formula is derived.

z T (k )[( A − BK s )T P ( A − BK s ) − P ]z (k ) ≤ −λmin (W ) z (k )

2

(40)

The following inequality is obtained from the second and third term in Eq.(39)

z T (k )( A − BK s )T P∆Az (k ) + z T (k )∆AT P ( A − BK s ) z (k ) ≤ 2σ max ( A − BK s )λmax ( P) ∆A z (k )

(41)

2

From the fourth term in Eq.(39), we have

z T (k )∆AT P∆Az (k ) ≤ λmax ( P ) ∆A

2

z (k )

2

(42)

Through synthesizing Eqs.(40)~(42), we can gain the following formula.

∆V ( z (k )) ≤ z (k ) (−λmin (W ) + 2σ max ( A − BK s )λmax ( P ) ∆A + λmax ( P ) ∆A ) 2

2

11


(43)


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If the following is tenable 2 −σ max ( A − BK s ) − σ max ( A − BK s ) +

< −σ max ( A − BK s ) + σ

2 max

λmin (W ) < ∆A λmax ( P)

(44)

λ (W ) ( A − BK s ) + min λmax ( P)

here the first and last term satisfy

0 −λmin (W ) + 2σ max ( A − BK s )λmax ( P) ∆A + λmax ( P) ∆A = 2

(45)

then ∆V ( z (k )) < 0 , which implies that robust stability is ensured. By observing Eq.(44), we can easily see that its first inequality is always satisfied due to the fact that ∆A ≥ 0 and the left hand side is always negative. Finally, Eq.(44) is simplified as 2 ( A − BK s ) + ∆A = σ max (∆A) < −σ max ( A − BK s ) + σ max

λmin (W ) λmax ( P)

(46)

□

4. Case studies 4.1 Injection molding process [44] In Fig.1, the schematic diagram and the process flow are shown. Under batch mode, the goal of the process is to convert plastic granules into certain products, and there are three main stages during the whole process, i.e., filling, packing/holding and cooling. During the filling, plastic will be melt and sent to the mold cavity. After the mold cavity is totally filled, packing/holding will start. In the packing/holding stage, additional material is put into the mold cavity for compensating for the shrinkage. Then the cooling stage starts, and the material will be cooled to be rigid enough and ejected subsequently. Note that plastication is also happened simultaneously, which results in the fact that the polymer is melt. After the material becomes rigid sufficiently, the current cycle is completed and the next cycle is under preparation.

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Fig.1. Injection molding machine.

In the total injection molding process, the injection velocity affects the product quality and needs to be regulated at the reference value to gain more benefits. Refer to [4], the process model for the injection velocity to the value is

1.582 −0.5916  1  = x(k + 1)  x(k ) +   u (k )  0  0   1 = y (k ) [1.69 1.419] x(k ), 0 ≤ k ≤ 100

(47)

The set-point of the injection velocity regulation process is

= yr (k ) 10 (for 1 ≤ k ≤ 50) = yr (k ) 20 (for 51 ≤ k ≤ 100)

(48)

To test the proposed IHLQ control, the conventional IHLQ control [41] in which only the extended state space model is utilized is adopted as the comparison. Note that the elements in the weighting matrix Q in the proposed approach is obtained by GA optimization. Here different cases of partial actuator failures are considered. Meanwhile, the random white noise sequence with standard deviation 0.2 is added as the unknown disturbances to evaluate the control performance of both schemes further. The control parameters are listed in Table 1 and Table 2.

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Table 1. Parameters for GA

Parameters

GA

N

20

Pc

0.8

Pm

0.05

Table 2. Control parameters

Parameters

Proposed

Conventional

Q (case 1)

diag (13.92, 0,1)

diag (5, 0,1)

Q (case 2)

diag (44.28, 0,1)

diag (5, 0,1)

Q (case 3)

diag (109.87, 0,1)

diag (5, 0,1)

Q (case 4)

diag (21.77, 0,1)

diag (5, 0,1)

Q (case 5)

diag (24.35, 0,1)

diag (5, 0,1)

Q (case 6)

diag (27.64, 0,1)

diag (5, 0,1)

R

0.1

0.1

4.2 Constant failure with unknown disturbances To mimic practical situation, three cases under different constant actuator failures are studied as follows Case 1. α = 0.75 Case 2. α = 0.5 Case 3. α = 0.25 The responses of both strategies are shown in Figs.2-4. It shows that the proposed IHLQ control scheme gives better ensemble performance. In Figs.2a~4a, it is obvious that the responses of the conventional method deteriorate gradually and significantly with the increase of the actuator faults. However, the responses of the proposed method are satisfactory with smaller overshoot and oscillations. The responses of the input increment signals in Figs.2b~4b also indicate the improved control performance under the proposed method further, since the corresponding responses are smoother. 14


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25

20

Output

15

10

5 Set-point GAIHLQ IHLQ

0

-5

0

20

60

40

80

100

k

Fig.2a. Output responses for case 1. 3 GAIHLQ IHLQ

2.5 2

Input

1.5 1 0.5 0 -0.5 -1

0

20

40

60

80

100

k

Fig.2b. Input increment signals for case 1.

25

20

15

Output

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


10


0

-5

0

20

40

60

80

k

Fig.3a. Output responses for case 2.

15


100


3 GAIHLQ IHLQ

2.5 2 1.5

Input

1 0.5 0 -0.5 -1 -1.5

0

20

60

40

80

100

k

Fig.3b. Input increment signals for case 2. 30 25

Output

20 15 10 5 Set-point GAIHLQ IHLQ

0 -5

0

20

40

60

80

100

k


3 2

Input

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 0 -1 -2 -3

0

20

80

60

40 k


4.3 Time-varying failure with unknown disturbances

16


100

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Here, three cases of time-varying faults are utilized to test the two approaches further, and the three cases are as follows. Case 4. = α 0.6 + 0.01sin(k ) Case 5. = α 0.6 + 0.1sin(k ) Case 6. = α 0.6 + 0.4sin(k ) In Figs.5~7, the responses of both methods are presented. In Figs.5a~7a, it is obvious that the overshoot and oscillations for the conventional approach are larger, which proves the improved control performance under the proposed strategy. Meanwhile, the input increment responses under the proposed control are smoother, which is shown in Figs.5b~7b.

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Fig.5a. Output responses for case 4.

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Fig.5b. Input increment signals for case 4. 25

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5. Conclusion To cope with the control of industrial processes under partial actuator faults and unknown disturbances, an improved GA based IHLQ control is developed. By utilizing GA to optimize the weighting coefficients, better ensemble control is anticipated. Case studies on the injection molding process under various partial actuator failures and unknown disturbances demonstrate the validity of the proposed IHLQ control approach.

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Supporting Information The Supporting Information is available free of charge on ACS Publications website at http://pubs.acs.org/.

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Genetic algorithm optimization based infinite horizon LQ control for

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