Adaptive neural network control for a two continuously stirred tank reactor with output constraints

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Adaptive neural network control for a two continuously stirred tank reactor with output constraints Dong-Juan Li

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S0925-2312(15)00518-4 http://dx.doi.org/10.1016/j.neucom.2015.04.049 NEUCOM15414

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Received date: 27 February 2015 Revised date: 14 April 2015 Accepted date: 20 April 2015 Cite this article as: Dong-Juan Li, Adaptive neural network control for a two continuously stirred tank reactor with output constraints, Neurocomputing, http: //dx.doi.org/10.1016/j.neucom.2015.04.049 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive neural network control for a two continuously stirred tank reactor with output constraints Dong-Juan Li School of Chemical and Environmental Engineering, Liaoning University of Technology, Jinzhou, Liaoning, 121001, China, E-mail: [email protected]

Abstract: An adaptive control scheme is studied for a class of continuous stirred tank reactors (CSTR). The considered reactors can be viewed as a class of MIMO systems with unknown functions and interconnections as well as the output constraints. These properties of the reactors will lead to a completed task for designing a stable control algorithm. To this end, several unknown functions are approximated based on the neural approximation, a novel recursive design method is used to remove the interconnection term, and Barrier Lyapunov function is introduced to avoid the violation of the output constraints. The stability of the proposed scheme is proved based on the Lyapunov analysis method. A simulation example for continuous stirred tank reactor is illustrated to verify the validity of the algorithm. Keywords: The neural networks, adaptive control, a two continuously stirred tank reactor, unknown nonlinear systems, Barrier Lyapunov function

1. Introduction Recently, the stability of uncertain systems has received much interest because a great deal of practical systems can be represented as uncertain systems. The control problem of uncertain systems becomes especially importance. Large numbers of design strategies with system modeling function approximation and big Data etc were studied in [1-16]. Specifically, it has received outstanding achievement in the area of adaptive control based on the neural or fuzzy approximation [17, 18]. Much advancement has been made for uncertain nonlinear systems. For example, some scholars proposed a series of adaptive neural network controls for nonlinear dynamics systems in the strict-feedback systems [19, 20], the pure-feedback systems [21, 22], MIMO block-triangular systems [23-25], and pure-feedback discrete-time systems [26-28]. Subsequently, multifarious adaptive neural control approaches were provided for different classes of nonlinear dynamic systems [29-36]. In addition, adaptive output feedback controller design was studied in [37-47] for nonlinear systems with unmeasured states. However, the approaches do not consider the constraint problem. When the output constraint appears in the systems, these approaches are difficult to ensure the stability. To this end, the output constraint control approach-based adaptive technique was designed in [48] for nonlinear parametric systems with the output constraint. In [49], the full state constraint was considered in the plant and an adaptive control technique was proposed to control a class of nonlinear systems in parametric output canonical form. In [50], an adaptive output feedback control was studied for nonlinear systems with output constraint. In real world, it is very significant for applying the theorem method to practical systems. In [51], an adaptive fuzzy control was developed to solve dynamic balance and motion for wheeled inverted pendulums with parametric and functional uncertainties and a systematic adaptation online mechanism is given to approximate the unknown parts. An adaptive neural approximation control was proposed in [52] for multiple mobile manipulators with considering time delays and input dead-zone. By designing a model reference neural control strategy with linear matrix inequalities and adaptive techniques, the control approach can guarantee that the tracking errors converge to zero whereas the coordination internal force errors remain bounded. In recent years, a much more attention has been allured for solving

the control problems of continuous stirred tank reactors. In [53], an adaptive multilayer neural tracking control was designed for a class of general nonlinear systems and it is to establish an ideal of implicit feedback linearization control. A temperature controller based on adaptive fuzzy and feedback linearization was developed in [54] for a class of continuous stirred tank reactors. It can achieve that H∞ tracking control performance with a prescribed attenuation level and the effectiveness of the controller can been validated by applying this approach to a benchmark chemical reactor. In [55], the control problem of continuous stirred tank reactors with dead-zone input was first studied and an adaptive compensate controller was provided to avoid the effect of dead-zone input. For a two continuous stirred tank reactor, the problem of almost disturbance decoupling was solved in [56]. Based on this result, the neural network-based control for a two continuous stirred tank reactor with dead-zone input was studied in [57] and a novel recursive method was constructed to remove the interconnection term and the effect of dead-zone input. These approaches are not to consider the constraint problem. When there are the constraints in the practical systems, it must to ensure that the constraints are not violated. In [58], an output constraint controller was designed for continuous stirred tank reactor. However, these approaches can only to stabilize simple SISO systems and it is very difficult to control a two continuous stirred tank reactor. Based on the above presentations, this paper will try to solve the adaptive control problem for a two continuous stirred tank reactor with the output constraints. This reactor can be viewed as a class of MIMO systems and contain the interconnections. In [48-50], the constraint is considered in simple SISO systems and they are not used to control MIMO. Thus, compared with the previous works, the approach in this paper can be applied to a general class of systems. In addition, the unknown interconnections are included in the considered reactor and it will lead to the difficulty in the design. To design a stability controller, the neural networks are used to approximate the unknown functions and the adaptive compensatory terms are given to compensate for the unknown parameters of approximator. Based on Barrier Lyapunov function, it is proved that the control approach can guarantee the stability of the closed-loop system and the output constraint is not violated. The effectiveness of the proposed approach can be verified by a simulation example. 2. System Description and Preliminaries A two continuously stirred tank reactor process is given in [55, 56], which is shown in Fig. 1.

Fig. 1. Two continuous stirred tank reactor

An irreversible, exothermic reaction, 1 → 2 , occurs in two reactors. Cooling water is added to the cooling jackets around both reactors at flow rates q j1 and q j 2 , temperatures T j1 and T j 2 , respectively. Assume that the volume of the cooling jackets is V j1 = V j 2 = V j ; the volume of the reactor is V1 = V2 = V ; the flow of reactants are q0 = q2 = q and q1 = q + qR . The process is described by the following differential equations

q + qR qR  dC A1 q0 − ( Ea  dt = V C A 0 − V C A1 + V C A 2 − α C A1e   dC A 2 = q + qR C − q + qR C − α C e −( Ea RT2 ) A1 A2 A2  dt V V

RT1 )

(1)

q + qR qR  dT1 q0  dt = V T0 − V T1 + V T2  UA αλ − E RT  − C A1e ( a 1 ) − (T1 − Tj1 )  c c pV ρ ρ p   dT q q + qR αλ − E RT  2 = 0 T1 − T2 − C e ( a 2) V V ρ c p A2  dt  UA  − (T2 − Tj 2 )  c pV ρ 

(2)

 dT j1 q j1 = (Tj10 − Tj1 ) − ρ UA (T1 − Tj1 )  dt V j j c jV j    dT j 2 = q j 2 T − T − UA T − T ( j 20 j 2 ) ρ c V ( 2 j 2 )  dt Vj j j j 

(3)

Let x11 = C A2 − C Ad 2 , x12 = f 2 , x21 = T2 − T2d , x22 = T j 2 − T jd2 , x31 = T1 − T1d , x32 = T j1 − T jd1 . Then, the systems (1)-(3) can be changed to

 x
11 = g11 x12 , x
12 = g12 u1 , y1 = x11   x
21 = g 21 x22 + φ21 + Φx31 , x
22 = g 22 u2 + φ22 , y2 = x21  x
= g x + φ + Ψω , x
= g u + φ , y = x  31 31 32 31 32 32 3 32 3 31

(4)

where

g11 = 1, g12 = 1; g 21 = g 32 = u1 =

q j1 Vj

,Ψ=

( q + qR ) q0 V2

C A1 =

V q + qR

UA

φ32 =

UA

ρ c pV

C A 0 − f 4 , u2 = T j 20 − T jd20 , u3 = T j10 − T jd10

q + qR  d  x12 + V ( x11 + C A 2 ) 

(

(

− Ea R x21 +T2d

))  

q + qR d q + qR αλ T1 − x21 + T2d ) − ( ( x + CAd 2 ) V V ρ c p 11 ×e

φ22 =

, g 31 =

Vj

q0 q + qR ,Φ = , ω = T0 − T0d V V

+α ( x11 + C Ad 2 ) e

φ21 =

q j2

, g 22 =

ρ c pV

qj2 V q j1 Vj

(

(

− Ea R x21 +T2d

(T (T

d j 20

d j10

))

−

UA ( x + T d − T jd2 ) ρ c pV 21 2

− x22 − T jd2 ) +

− x32 − T jd1 ) +

UA

ρ j c jV j UA

ρ j c jV j

φ31 =

(x

21

+ T2d − x22 − T jd2 )

− ( Ea R ( x31 +T1d ) ) q0 d q + qR αλ T0 − x31 + T1d ) − C A1e ( V V ρ cp

−

(x

f1 =

q + qR q − E C A1 + R C A 2 − α C A1e ( a V V

f2 =

q + qR q − E C A1 − R C A 2 − α C A 2 e ( a V V

31

+ T1d − x32 − T jd1 )

RT1 )

RT2 )

qR UA x21 + T2d ) − ( ( x + T d − Tjd1 ) V ρ c pV 31 1

f3 =

q + qR q αλ − E T1 − R T2 − C e( a V V ρ c p A2 −α

Ea − E C A2 e ( a RT22

RT2 )

RT2 )

−

UA

ρ c pV

(T

2

− Tj 2 )

f4 =

q + qR  q + qR − E f1 −  +αe ( a V V 

RT2 )

  f2 

f3

{

In the system (4), the output variables yi , i = 1, 2,3 are constrained in the compact sets Ωi = yi yi ≤ kci

} where k

ci

is

a constant. The control objective is to design an adaptive NN controller for the system (4) such that all the signals in the closedloop system are bounded and the constraint of the system output is not violated. Because the system (4) contains the unknown functions, they can be used in the controller. Due to the approximation property of the neural networks, they have been used in the control problem and modelling of the nonlinear systems. The approximation property of the neural networks can be founded in [4]. Using the neural networks, unknown functions f ( y ) : R → R can be expressed as

f ( y ) = η *Tς ( y ) + ε ( y ) T

where η * is the optimal weight vector, ς ( y ) = ς 1 ( y ) ,  , ς N ( y )  are fuzzy basis function vector and ε ( y ) is the approximation error. Assumption 1 [17]: η * and ε ( y ) are bounded, i.e., η * ≤ η and ε ( X ) ≤ ε * . Remark 1: in [48-50, 58], several adaptive control schemes were designed for uncertain nonlinear systems to avoid the output constraint of the systems. However, these schemes are proposed for nonlinear SISO systems and it is very difficult to be applied in more complex MIMO feedback system. Specially, when the interconnection terms are included in the systems, the controller design is more complicated. Therefore, an adaptive neural control scheme guarantee that the output constraints are not violated for more complex nonlinear MIMO feedback systems. In the following design, it needs to use the Lemma.

：For any positive constant k , for all of e , exists e < k satisfying log k k− e 2

Lemma 1 [50]

2

2

<

e2 . k 2 − e2

3. Controller design and stability analysis In this section, stable controllers with adaptation laws are designed and the corresponding theorem is explained. The detailed design procedure is given in the following. Step 1: Define the tracking error z11 = x11 and z
11 is

z
11 = g11 x12

(5)

Design the virtual controller α11 as

α11 = − k11 z11

(6)

where k11 > 0 is the design parameter. Then, α11 is a function of x11 . Define z12 = x12 − α11 and by substituting (6) into (5), we have

z
11 = g11 ( − k11 z11 + z12 )

(7)

Consider the Lyapunov function candidate as follows 2

V11 =

kc 1 log 2 1 2 2 g11 kc1 − z11

(8)

where Γ1 = Γ1T > 0 is a design parameter. Its time derivative is

V
11 =

z11 z
11

(

g11 kc21 − z112

Basis on (7), (9) can be repressed as

)

(9)

k z2 z z V
1 = − 211 112 + 211 12 2 kc1 − z11 kc1 − z11

(10)

Step 2: Define the variable z12 = x12 − α11 and z
12 is z
12 = g12 u1 − α
11

where α
11 =

(11)

∂α11 x
11 . ∂x11

Define the unknown function

H1 ( χ1 ) = α
11 g12 The unknown function is approximated by the neural networks

H1 ( χ1 ) = η1∗T ς 1 ( χ1 ) + ε1 ( χ1 )

(12)

T

where χ1 = [ x11 , x12 ] ; η1∗ indicates the ideal constant weight, ε1 ( χ1 ) ≤ ε1∗ is approximation error and ε1∗ > 0 . Let ηˆ1 be the estimate of η1∗ and construct the actual controller as follows

u1 = − k12 z12 −

z11 + ηˆ1T ς 1 ( χ1 ) 2 k − z11 2 c1

(13)

Substituting (12) and (13) into (11), z
12 can be repressed as

  z z
12 = g12  −k12 z12 − 2 11 2 + η1T ς 1 ( χ1 ) − ε1 ( χ1 )  (14) kc1 − z11   Consider the Lyapunov function

V12 = V11 +

1 2 1 T −1 z12 + η1 Γ1 η1 2 g12 2

(15)

Using (14), we have

V
12 = V
11 + η1T Γ1−1η
ˆ1   z + z12  − k12 z12 − 2 11 2 + η1T ς1 ( χ1 ) − ε1 ( χ1 )  (16) kc1 − z11   Design the adaptation laws as follows

η
ˆ1 = Γ1  −ς 1 ( χ1 ) z12 − ζ 1ηˆ1 

(17)

where ζ 1 > 0 is the design constant. Substituting (10), (17) into (16) yields

k z2 V
12 = − 2 11 112 − k12 z122 − z12ε1 ( χ1 ) − ζ 1η1Tηˆ1 (18) kc1 − z11 Utilizing the Young’s inequality, we obtain * 2

z 2 ( ε1 ) − z12ε1 ( χ1 ) ≤ 12 + 2 2

(19)

−ζ 1η1Tηˆ1 = −ζ 1η1T (η1 + η1∗ ) 2

≤ −ζ 1 η1 + ζ 1 η1 η1∗ ≤−

ζ 1 η1 2

2

+

ζ 1η12 2

Substituting (19), (20) into (18) leads to

(20)

2

ζ η k z2 V
12 ≤ − 2 11 112 − k12∗ z122 − 1 1 kc1 − z11 2 ζ 1η12

+ L1

(21)

* 2 1

(ε ) +

1 and k12 is chosen to satisfy k12∗ = k12 − > 0 . 2 2 2 Step 3: Define the variable z21 = x21 and z
21 is where L1 =

z
21 = g 21 x22 + φ21 + Φx31

(22)

Define the unknown function

H 2 ( χ 2 ) = −φ21 / g 21

(23) T

According to the definition of φ21 , we have χ 2 = [ x11 , x21 ] . Using the neural networks, the unknown function H 2 ( χ 2 ) can be repressed as

H 2 ( χ 2 ) = η 2∗T ς 2 ( χ 2 ) + ε 2 ( χ 2 )

(24)

where η 2∗ indicates the ideal constant weight, let ηˆ2 be the estimate of η 2∗ and η2 = ηˆ2 − η 2∗ . The virtual control input is designed as

1 z21 2 2 kc22 − z21

α 21 = −k21 z21 + ηˆ2T ς 2 ( χ 2 ) −

(25)

where k 21 > 0 is the design parameter. Define z22 = x22 − α 21 and using (24)-(25), we have z
21 = g 21  z22 + η2T ς 2 ( χ 2 ) − k21 z21

−

1 z21 − ε 2 ( χ 2 ) + Φx31 2 2 kc22 − z 21

(26)

Choose the adaptation law as follows

 z ς (χ )  η
ˆ2 = Γ 2  − 212 2 22 − ζ 2ηˆ2   kc − z21 

(27)

2

where Γ 2 = ΓT2 > 0 is the adaptive gain matrix. Consider the Lyapunov function candidate as follows 2

V21 = V12 +

kc 1 1 log 2 2 2 + η2T Γ −2 1η2 2 g 21 kc2 − z21 2

(28)

Its time derivative as

V
21 = V
12 +

z21 z
21

(

)

2 kc22 − z21 g 21

+ η2T Γ 2−1η
ˆ2

(29)

Using (26) and (27), (29) can be repressed as

z ε (χ ) z21 V
21 ≤ V
12 − 212 2 22 − 2 2 kc2 − z21 2 kc2 − z21 2

(

+

)

2

− k21

z Φx z21 z22 + 2 21 2 31 − ζ 2η2Tηˆ2 2 kc22 − z 21 kc2 − z21 g 21

(

)

2 z21 2 k − z21 2 c2

(30)

Utilizing Young’s inequality, we obtain

−

z21ε 2 ( χ 2 ) 2 kc22 − z21

≤

2 z21

(

2 2 kc22 − z21

)

2

+

* 2 2

(ε ) 2

(31)

−ζ 2η2Tηˆ2 ≤ −

ζ 2 η2

2

+

ζ 2η22

(32)

2 2 Substituting (21), (31) and (32) into (30) leads to

2 ζ η j k z2 k z2 j V
21 ≤ − 2 11 112 − 2 21 212 − k12∗ z122 − ∑ kc1 − z11 kc2 − z21 2 j =1

+ L2 +

where L2 = L1 +

z Φx z21 z22 + 2 21 2 31 2 kc22 − z 21 kc2 − z21 g 21

ζ 2η 22

(

+

2

(33)

)

* 2 2

(ε )

. 2 2 Step 4: Define the variable z22 = x22 − α 21 and z
22 is z
22 = g 22 u2 + φ22 − α
21

(34)

Then, α 21 is a function of χ 2 and ηˆ2 . According to the definition of α 21 , α
21 is

α
21 =

∂α 21 ∂α ∂α ∂α x12 + 21 ( x22 +φ21 ) + 21 ηˆ
2 + 21 Φx31 (35) ∂x11 ∂x21 ∂ηˆ2 ∂x21

Define the unknown function

 ∂α H 3 ( χ 3 ) = − φ22 + 21 x12 ∂x11  +

 ∂α 21 ∂α ( x22 +φ21 ) + ˆ21 η
ˆ2  / g 22 ∂x21 ∂η 2 

where χ 3 = [ x11 , x12 , x21 , x22 ,ηˆ2 ] . The unknown function is approximated by the neural networks

H 3 ( χ3 ) = η3∗T ς 3 ( χ 3 ) + ε 3 ( χ 3 )

(36)

where η3∗ indicates the ideal constant weight, ε 3 ( χ 3 ) ≤ ε 3∗ is approximation error and ε 3∗ > 0 . Let ηˆ3 be the estimate of η3∗ , construct the actual controller as follows u2 = − k22 z22 −

z21 + ηˆ3T ς 3 ( χ 3 ) 2 kc22 − z21

(37)

where k 22 > 0 is the design parameter. Using (36) and (37), we have

  z z
22 = g 22  − k22 z22 − 2 21 2 + η3T ς 3 ( χ 3 ) − ε 3 ( χ 3 )  kc2 − z21   −

∂α 21 Φ x31 ∂x21

(38)

Consider the Lyapunov function

V22 = V21 +

1 2 1 T −1 z22 + η3 Γ3 η3 2 g 22 2

(39)

Its time derivative is as follows

1 V
22 = V
21 + z22 z
22 + η3T Γ3−1η
ˆ3 g 22

(40)

Design the adaptation law as follows

η
ˆ3 = Γ3  −ς 3 ( χ 3 ) z22 − ζ 3ηˆ3 

(41)

Utilizing Young’s inequality, we obtain

−

z22ε 3 ( χ 3 ) 2 kc22 − z21

* 2 3

2 z22

≤

(

2 2 kc22 − z21

−ζ 3η3Tηˆ3 ≤ −

ζ 3 η3

)

2

(ε ) +

2

+

2

(42)

ζ 3η32

(43) 2 2 Using (33), (38), (41),(42) and (43), (40) can be repressed as

k z2 k z2 V
22 ≤ − 211 112 − 2 21 212 − k12∗ z122 kc1 − z11 kc2 − z21

ζ j η j

3

−k z − ∑ ∗ 2 22 22

z21Φx31

(

+ L3

2

j =1

+

2

)

2 kc22 − z 21 g 21

ζ 3η32

−

z22 ∂α 21 Φx31 g 22 ∂x21

(44)

* 2 3

(ε ) +

1 ∗ and k22 is chosen to satisfy k22 = k22 − > 0 . 2 2 2
Step 5: Define the variable z31 = x31 and z31 is where L3 = L2 +

z
31 = g31 x32 + φ31 + Ψω

(45)

Define the unknown function

(

)

 z22 kc23 − z312 ∂α 21 H 4 ( χ 4 ) = − (φ31 + Ψω ) / g31 − Φ g 22 ∂x21  

(k

)

  (46) z Φ 21 2  kc22 − z21 g 21  The unknown function is approximated by the neural networks

+

2 c3

2 − z31

(

)

H 4 ( χ 4 ) = η 4∗T ς 4 ( χ 4 ) + ε 4 ( χ 4 )

(47)

T

where χ 4 = [ x11 , x12 , x21 , x22 , x31 ,ηˆ2 ] ; η4∗ indicates the ideal constant weight, ε 4 ( χ 4 ) ≤ ε 4∗ is approximation error and

ε 4∗ > 0 . Let ηˆ4 be the estimate of η4∗ . Define z32 = x32 − α 31 and α 31 is the virtual control input as

α 31 = −k31 z31 + ηˆ4T ς 4 ( χ 4 ) −

1 z31 2 2 kc23 − z31

(48)

where k31 > 0 is design parameter. Then, substituting (47) and (48) into (45), z
31 is

z
31 = g31 [ z32 − k31 z31 −

1 z31 + η4T ς 4 ( χ 4 ) − ε 4 ( χ 4 ) 2 2 kc23 − z31

Choose adaptive law as follow

 z31ς 4 ( χ 4 )

 − ζ 4ηˆ4  (50)  k − z  Consider the Lyapunov function candidate as follows

η
ˆ4 = Γ 4  −

2 c3

2 31

2

V31 = V22 +

kc 1 1 log 2 3 2 + η4T Γ 4−1η4 2 g31 kc3 − z31 2

Its time derivative as

(51)

+

2  z22 kc23 − z312 ∂α kc2 − z31 21 Φ− 2 3 2 z21Φ  g 22 ∂x21  kc2 − z21 g 21 

(

)

(

(

)

)

(49)

V
31 = V
22 +

z31 z
31

(

)

2 kc23 − z31 g31

+ η4T Γ 4−1η
ˆ4

(52)

Basis on(49) and (50), (52) can be repressed as

z z − z ε ( χ ) z ∂α 21 z2 V
31 = V
22 + 31 32 2 31 24 4 + 22 Φ − k31 2 31 2 g 22 ∂x21 kc3 − z31 kc3 − z31

(

−

)

z21Φ

(

2 c2

k −z

2 21

)

2 z31 1 2 k 2 − z2 c3 31

−

(

g 21

)

2

− η4T ζ 4ηˆ4

(53)

Utilizing Young’s inequality, we obtain

−

z31ε 4 ( χ 4 ) 2 kc23 − z31

* 2 4

2 z31

≤

(

2 2 kc23 − z31

−ζ 4η4Tηˆ4 ≤ −

ζ 4 η4

2

2

2

ζ 4η 42

+

2

)

(ε ) +

(54)

(55)

2

Using (44), we have 3 2 k z2 V
31 ≤ −∑ 2 i1 i1 2 − ∑ ki∗2 zi22 i =1 kci − zi1 i =1

4

−∑

ζ j η j

2

+ L4 +

2

j =1

where L4 = L3 +

ζ 4η 42

z31 z32 2 kc23 − z31

(56)

* 2 4

(ε ) +

. 2 2 Step 6: Define the variable z32 = x32 − α 31 and z
32 is z
32 = g32 u3 + φ32 − α
31

(57)

Define

H 5 ( χ5 ) = −

1 [φ32 − α
31 ] g32

(58)

Using the neural networks, the unknown functions H 5 ( χ 5 ) can be approximated as

H 5 ( χ 5 ) = η5∗T ς 5 ( χ 5 ) + ε 5 ( χ5 )

(59)

T

where, χ 5 = [ x11 , x12 , x21 , x22 , x31 , x32 ,ηˆ4 ] , η5∗ indicates the ideal constant weight, ε 5 ( χ 5 ) ≤ ε 5∗ is approximation error and ε 5∗ > 0 . Let ηˆ5 be the estimate of η5∗ , construct the actual controller as follows

u3 = −k32 z32 −

z31 + ηˆ5T ς 5 ( χ 5 ) k − z312 2 c3

(60)

where k 22 > 0 is the design parameter. Using (36) and (37), we have

  z z
32 = g32  −k32 z32 − 2 31 2 + η5T ς 5 ( χ 5 ) − ε 5 ( χ5 )  (61) kc3 − z31   Consider the Lyapunov function

V32 = V31 + Its time derivative as

1 2 1 T −1 z32 + η5 Γ5 η5 2 g 32 2

(62)

1 V
32 = V
31 + z32 z
32 + η5T Γ5−1η
ˆ5 g32

(63)

Design the adaptation laws as follows

η
ˆ5 = Γ5  −ς 5 ( χ5 ) z32 − ζ 5ηˆ5 

(64)

Utilizing Young’s inequality, we obtain

−

z32ε 5 ( χ 5 ) 2 kc23 − z31

2 z32

≤

(

2 2 kc23 − z31

−ζ 5η5Tηˆ5 ≤ −

ζ 5 η5

2

+

)

2

+

* 2 5

(ε )

(65)

2

ζ 5η52

(66) 2 2 Using (61), (64),(65) and (66), (63) can be repressed as 3 3 5 ζ η j k z2 j V
32 ≤ −∑ 2 i1 i1 2 − ∑ ki∗2 zi22 − ∑ 2 i =1 kci − zi1 i =1 j =1

where L5 = L4 +

ζ 5η52 2

+

* 2 5

(ε ) 2

2

+ L5 (67)

and k32 is chosen to satisfy k32∗ = k32 −

1 . 2

Theorem 1: For nonlinear system (4), under Assumption 1, by constructing α i1 , i = 1, 2,3 and ui , i = 1, 2,3 , and

choosing the adaptation laws ηˆi , i = 1, … ,5 . Then, the adaptive neural control scheme can guarantee that all the signals in the closed-loop system are bounded, the tracking errors can converge to a small neighbourhood of zero, and the output constraints are not violated. Proof: According to the definition of Lyapunov function candidate Vij , i = 1, 2,3, j = 1, 2 , we obtain 2

V32 =

kc 1 3 1 log 2 i 2 ∑ kci − zi1 2 i =1 g i1 +

1 3 1 2 1 5 T −1 η j Γ j η j ∑ zi 2 + 2 ∑ 2 i =1 g i 2 j =1

(68)

Using Lemma 1, (68) can be expressed as follows V
≤ −κV + µ 32

where κ =

min

i =1,2,3, j =1,,5

(69)

32

{2k g i1

i1

}

, 2k gi 2 , ζ j λmin ( Γ j ) and µ = L5 . * i2

Define φ = µ / κ . Then, (69) can be represented as

V32 ( t ) ≤ φ + (V32 ( 0 ) − φ ) exp ( −κ t ) From the above inequality, it can be seen that V32 is bounded. This implies that all the signals in V32 are bounded, i.e.,

log

kc2i kc2i − zi21

, zi 2 and η j are bounded. Then, ηˆ j , j = 1,  ,5 are also bounded. According to the definition of α11 , α11 is

a function of x11 . Thus, α11 is bounded. From x12 = z12 − α11 , it has that x12 is bounded. u1 is a function of x11 , x12 and

ηˆ1 . Then, u1 must be bounded. Similarly, we can prove that u2 and u3 are also bounded. The proof is completed. 4. Simulation

In order to verify the effectiveness of the developed approach, a simulation study is described in this section. Consider the system (4) with the initial parameters as x11 ( 0 ) = −0.5 , x12 ( 0 ) = 0 , x21 ( 0 ) = 0.3 , x22 ( 0 ) = −2 , x31 ( 0 ) = −0.5 ,

x32 ( 0 ) = 0.5 . The outputs are constrained in x11 ≤ 0.6 , x21 ≤ 0.3 and x31 ≤ 0.5 . The design parameters are chosen as k11 = 5,

k12 = 3,

k21 = 2,

k22 = 1,

k31 = 6,

k32 = 10

,

ζ 1 = 3,

ζ 2 = 5,

ζ 3 = 10,

ζ 4 = 6,

ζ 5 = 8, Γ i = diag ( 2,, 2 ) , i = 1,,5 . The initial values of the adaptive parameters are chosen as ηˆ1 ( 0 ) = 0.2, ηˆ2 ( 0 ) = 0.1, ηˆ3 ( 0 ) = 0.3, ηˆ4 ( 0 ) = 0.1 and ηˆ5 ( 0 ) = 0.05 . The values for the parameters and steady-state values of the system variables are given in Table 1. Table. 1. Parameters of the CSTR system system parameters 10

α = 7.08 × 10 h

−1

ρ = 800.9189kg / m3

ρ j = 997.9450kg / m3

Ea = 3.1644 × 107 J / mol

℃

λ = −3.1644 × 107 J / mol 7

U = 1.3625 × 10 J / hm

℃ = 1860.3 J / kg℃

cρ = 1395.3 J / kg

R = 1679.2 J / mol

℃

2

cj

q = 2.8317 m3 h

C Ad 0 = 18.3728 mol m3

q j1 = 1.4130 m3 h

C Ad1 = 12.3061 mol m3

q j 2 = 1.4130 m3 h

C Ad 2 = 10.4178 mol m3

qR = 1.4158 m3 h

T jd10 = 629.2

T jd20 = 608.2

d 0

T

T2d

T jd2

℃ = 703.7℃ = 737.5℃ = 727.6℃

d 1

T

T jd1

℃ = 750℃ = 740.8℃

V j = 0.1090m3

V = 1.3592m3

A = 23.2m2

0.2

0.1

The system outputs

0

−0.1

−0.2

−0.3

−0.4

−0.5

0

1

2

3

4

5

Time(sec)

Fig. 2. The system output y1 (-), y2 (--) and y3 (-.).

The simulation figures 2-5 are obtained by Matlab program. Fig. 2 shows the output trajectories of the systems and it can be seen that the system output can be very good to converge to zero. At the same time, it can be seen that the constraints of the output variables are not violated. The control inputs are illustrated in Figs. 3 and the trajectories of the adaptation laws are given in Figs. 4 and 5. It is verified that the control inputs and the adaptation laws are bounded.

20 10 0

The control laws

−10 −20 −30 −40 −50 −60 −70

0

1

2

3

4

5

Time(sec)

Fig. 3. The controllers u1 (-), u2 (--) and u3 (-.). 0.9 0.8

The adaptive laws

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

Time(sec)

Fig. 4. The controllers η1 (-), η2 (--) and η3 (-.). 0.35

0.3

The adaptive laws

0.25

0.2

0.15

0.1

0.05

0

0

1

2

3

4

5

Time(sec)

Fig. 5. The controllers η4 (-) and η5 (--).

5. Conclusion

In this paper, we have proposed an adaptive neural network control algorithm for a class of a two continous stirred tank reactors with unknown functions. The neural networks are used to approximate the unknown funcitons. The output constraints are considered in the reactors. The reactor modelling have nonlinearity property and the unknown funciton to be appeared in each subsystem. The Lyapunov stability analysis is proved that all the signals are bounded. The system output is proved to converges to a small neighborhood of zero. The effectiveness has been demonstrated through a simulation example.

Acknowledgements

This work was supported in part by the Foundation of Educational Department of Liaoning Province L2013243, the Natural Science Foundation of China under Grant 61473139, and Program for Liaoning Excellent Talents in University under Grant LR2014016. References References [1]

C.L.P. Chen, Y.H. Pao, An integration of neural network and rule-based systems for design and planning of mechanical assemblies, IEEE Transactions on Systems, Man and Cybernetics, 23 (5) (1993) 1359–1371.

[2]

C.L.P. Chen, A rapid supervised learning neural network for function interpolation and approximation, IEEE Transactions on Neural Networks, 7 (5) (1996) 1220–1230.

[3]

M. Fei, H. Zhang, D.P. Cao, H.Y. Chen, System modeling and pressure control of a clutch actuator for heavy-duty automatic transmission systems, IEEE Transactions on Vehicular Technology, accepted and press, DOI: 10.1109/TVT.2005. 2404857. 2015.

[4]

D. Zhang, L. Yu, Exponential state estimation for Markovian jumping neural networks with time-varying discrete and distributed delays, Neural Networks, 35 (11) (2012) 103-111.

[5]

H. Zhang, W. Junmin, Vehicular lateral dynamics control through AFS/DYC and robust gain-scheduling approach, IEEE Transactions on Vehicular Technology, DOI: 10.1109/TVT.2015. 2391184.2015.

[6]

H. Zhang, X.T. Liu, H. R. Karimi, Robust H∞ sliding mode control with pole placement for a fluid power electrohydraulic actuator (EHA) system, International Journal of Advanced Manufacturing Technology, 73 (5-8) (2014) 1095-1104.

[7]

H. Zhang, J. Wang, State estimation of discrete-time takagi–sugeno fuzzy systems in a network environment, IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2014.2354431, 2014.

[8]

C.L.P. Chen, S.R. LeClair, Y.H. Pao, An incremental adaptive implementation of functional-link processing for function approximation, timeseries prediction, and system identification, Neurocomputing, 18 (1-3) (1998) 11–31.

[9]

H. Zhang, J. Wang, Combined feedback–feedforward tracking control for networked control systems with probabilistic delays, Journal of the Franklin Institute, 351 (6) (2014) 3477–3489.

[10] C. L. P. Chen and J. Z. Wan, A Rapid Learning and Dynamic Stepwise Updating Algorithm for Flat Neural Networks and its Application to Time-Series Prediction, IEEE Trans. on Systems, Man and Cybernetics, Part B, 29(1) (1999) 62-72. [11] D. Zhang, L. Yu, Q.G. Wang, C.J. Ong, Estimator design for discrete-time switched neural networks with asynchronous switching and timevarying delay, IEEE Transactions on Neural Networks and Learning Systems, 23 (5) (2012) 827 - 834. [12] C.L.P. Chen, C.Y. Zhang, Data-intensive applications, challenges, techniques and technologies: a Survey on big data, Information Sciences, 275 (2014) 314–347. [13] Z. G. Zeng, T. W. Huang, W. X. Zheng, Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function, IEEE Transactions on Neural Networks, 21 (8) (2010) 1371-1377. [14] Z.G. Zeng, W.X. Zheng, Multistability of neural networks with time-varying delays and concave-convex characteristics, IEEE Transactions on Neural Networks and Learning Systems, 23 (2) (2010) 293-305. [15] Z.G. Zeng, W.X. Zheng, Multistability of two kinds of recurrent neural networks with activation functions symmetrical about the origin on the phase plane, IEEE Transactions on Neural Networks and Learning Systems, 24 (11) (2013) 1749-1762. [16] H. Li, X. J. Jing, H.K. Lam, P. Shi, Fuzzy sampled-data control for uncertain vehicle suspension systems, IEEE Transactions on Cybernetics, 44 (7) (2014) 1111–1126. [17] L.X. Wang, Adaptive fuzzy systems and control: design and stability analysis, Prentice-Hall, Englewood Cliffs, NJ, 1994. [18] J. Park, I.W. Sandberg, Universal approximation using radial-basis-function network, Neural Computation, 3 (2) (1991) 246-257. [19] S.S. Ge, C. Wang, Direct adaptive NN control of a class of nonlinear systems, IEEE Transactions on Neural Networks, 13 (1) (2002) 214-221. [20] Y.J. Liu, S.C. Tong, D. Wang, T.S. Li, C.L.P. Chen, Adaptive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO systems, IEEE Transactions on Neural Networks, 22 (8) (2011) 1328–1334. [21] S.S. Ge, C. Wang, Adaptive NN control of uncertain nonlinear pure-feedback systems, Automatica, 38 (4) (2002) 671-682. [22] C.L.P. Chen, Y.J. Liu, G.X. Wen, Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems, IEEE Transactions on Cybernetics, 44 (5) (2014) 583-593. [23] S.S. Ge, C. Wang, Adaptive neural control of uncertain MIMO nonlinear systems, IEEE Transactions on Neural Networks, 15 (3) (2004) 674692.

[24] Y.J. Liu, S.C. Tong, W. Wang, Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems, Fuzzy Sets and Systems, 160 (19) (2009) 2727-2754. [25] Y.J. Liu, S.C. Tong, C.L.P. Chen, D.J. Li, Neural controller design-based adaptive control for nonlinear MIMO systems with unknown hysteresis inputs, IEEE Transactions on Cybernetics, DOI: 10.1109/TCYB.2015.2388582, 2015. [26] S.S. Ge, J. Zhang, T.H. Lee, Adaptive neural network control for a class of MIMO nonlinear systems with disturbances in discrete-time, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, , 34 (4) (2004) 1630-1645. [27] Y.J. Liu, C. Chen, G. Wen, S.C. Tong, Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems, IEEE Transactions on Neural Networks, 22 (7) (2011) 1162-1167. [28] Y.J. Liu, S. C. Tong, Adaptive neural network tracking control of uncertain nonlinear discrete-time systems with nonaffine dead-zone input, IEEE Transactions on Cybernetics, 45 (3) (2015) 497-505. [29] C.L.P. Chen, G.X. Wen, Y.J. Liu, F.Y. Wang, Adaptive consensus control for a class of nonlinear multiagent time-delay systems using neural networks, IEEE Transactions on Neural Networks and Learning Systems, 25 (6) (2014) 1217-1226. [30] Y.J. Liu, L. Tang, S.C. Tong, C.L.P. Chen, Adaptive NN controller design for a class of nonlinear MIMO discrete-time systems, IEEE Transactions on Neural Networks and Learning Systems, 26 (5) (2015) 1007-1018. [31] M. Chen, S.S. Ge, B.B. Ren, Adaptive tracking control of uncertain MIMO nonlinear systems with input constraints, Automatica, 47 (3) (2011) 452-465. [32] Y.J. Liu, L. Liu, S.C. Tong, Adaptive neural network tracking design for a class of uncertain nonlinear discrete-time systems with dead-zone, Science China Information Sciences, , 57 (3) (2014) 1-12. [33] M. Chen, S.S. Ge, B.B. Ren, Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities, IEEE Transactions on Neural Networks, 21 (5) (2010) 796-812. [34] Y.J. Liu, L. Tang, S.C. Tong, C.L.P. Chen, D.J. Li, Reinforcement learning design-based adaptive tracking control with less learning parameters for nonlinear discrete-time MIMO systems, IEEE Transactions on Neural Networks and Learning Systems, 26 (1) (2015) 165-176. [35] H. Li, J.Y. Yu, C. Hilton, H.H. Liu, Adaptive Sliding Mode Control for Nonlinear Active Suspension Systems using T-S Fuzzy Model, IEEE Transactions on Industrial Electronic, 60 (8) (2013) 3328–3338. [36] Y. J. Liu and S. C. Tong, Adaptive fuzzy identification and control for a class of nonlinear pure-feedback MIMO systems with unknown deadzones, IEEE Transactions on Fuzzy Systems, DOI: 10.1109/TFUZZ.2014.2360954, 2015. [37] Q. Zhou, P. Shi, S. Xu, H. Li, Adaptive output feedback control for nonlinear time-delay systems by fuzzy approximation approach, IEEE Transactions on Fuzzy Systems, 21 (2) (2012) 301-313. [38] S.C. Tong, C.Y. Li, Y.M. Li, Fuzzy adaptive observer backstepping control for MIMO nonlinear systems, Fuzzy Sets and Systems, 160 (19) (2009) 2755-2775. [39] S.C. Tong, Y.M. Li, Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone inputs, IEEE Transactions on Fuzzy Systems, 21 (1) (2013) 134-146. [40] S.C. Tong, X.L. He, H.G. Zhang, A combined backstepping and small–gain approach to robust adaptive fuzzy output feedback control, IEEE Transactions on Fuzzy Systems, 17 (5) (2009) 1059–1069. [41] S.C. Tong, B.Y. Huo, Y.M. Li, Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Transactions on Fuzzy Systems, 22 (1) (2014) 1–15 [42] S.C. Tong, H.X. Li, Fuzzy adaptive sliding-mode control for MIMO nonlinear systems, IEEE Transactions on Fuzzy Systems, 11 (3) (2003) 354–360. [43] Y. J. Liu, S. C. Tong, T. S. Li, Observer-based adaptive fuzzy tracking control for a class of uncertain nonlinear MIMO systems, Fuzzy Sets and Systems, 164 (1) (2011) 25-44. [44] S.C. Tong, C.L. Liu, Y.M. Li, Fuzzy-adaptive decentralized output feedback control for large-scale nonlinear systems with dynamical uncertainties, IEEE Transactions on Fuzzy Systems, 18 (5) (2010) 845–861. [45] S.C. Tong, Y. Li, Y.M. Li, Y. J. Liu, Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems, IEEE Transactions on Systems, Man and Cybernetics, Part B, 41 (6) (2011) 1693–1704. [46] S.C. Tong, T. Wang, Y.M. Li, B. Chen, A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control, IEEE Transactions on Fuzzy Systems, 21 (2) (2013) 314–327. [47] Y.J. Liu, S.C. Tong, C.L.P. Chen, Adaptive fuzzy control via observer design for uncertain nonlinear systems with unmodeled dynamics, IEEE Transactions on Fuzzy Systems, 21 (2) (2013) 275–288. [48] K. P. Tee, S. S. Ge, E. H. Tay, Barrier Lyapunov functions for the control of the output-constrained nonlinear systems, Automatica, 45 (4) (2009) 918-927. [49] Y.J. Liu, D.J. Li, S.C. Tong, Adaptive output feedback control for a class of nonlinear systems with full-state constraints, International Journal

of Control, , 87 (2) (2014) 281-290. [50] B.B. Ren, S.S. Ge, K.P. Tee, T.H. Lee, Adaptive Neural Control for Output Feedback Nonlinear Systems Using a Barrier Lyapunov Function, IEEE Transactions on Neural Networks, 21(8) (2010) 1339-1344. [51] Z.J. Li, C.Q. Xu, Adaptive fuzzy logic control of dynamic balance and motion for wheeled inverted pendulums, Fuzzy Sets and Systems, 160 (12) (2009) 1787-1803. [52] Z.J. Li, C.Y. Su, Neural-adaptive control of single-master multiple slaves teleoperation for coordinated multiple mobile manipulators with timevarying communication delays and input uncertainty, IEEE Transactions on Neural Networks and Learning Systems, 24 (9) (2013) 1400-1413. [53] S.S. Ge, C.C. Hang, T. Zhang, Nonlinear adaptive control using neural networks and its application to CSTR systems, Journal of Process Control, 9 (4) (1999) 313-323. [54] S. Salehi, M. Shahrokhi, Adaptive fuzzy approach for H∞ temperature tracking control of continuous stirred tank reactors, Control Engineering Practice, 16 (9) (2008) 1101-1108. [55] D.J. Li. Adaptive neural network control for a class of continuous stirred tank reactor systems, Science China Information Sciences, , 57 (10) (2014) 1-8. [56] X. Liu, A. Jutan, S. Rohani, Almost disturbance decoupling of MIMO nonlinear systems and application to chemical processes, Automatica, 40 (3) (2004) 465-471. [57] D.J. Li, Neural network control for a class of continuous stirred tank reactor process with dead-zone input, Neurocomputing, 131 (5) (2014) 453-459. [58] D.J. Li, D.P. Li, Adaptive Controller Design-Based Neural Networks for Output Constraint Continuous Stirred Tank Reactor, Neurocomputing, 153 (4) (2015) 159-163.

Dong-Juan Li received the B.S. degree in Applied Chemistry from Shenyang University of Technology, Shenyang, China, in 2003 and the M. S. degree in Chemical Engineering and Technology from Dalian Polytechnic University, Dalian, China, in 2007. His research interests include the control of continuous stirred tank reactor and the chaotic control.