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A Combined Adaptive Neural Network and Nonlinear Model Predictive Control for Multirate Networked Industrial Process Control Tong Wang, Huijun Gao, Fellow, IEEE, and Jianbin Qiu, Senior Member, IEEE

Abstract— This paper investigates the multirate networked industrial process control problem in double-layer architecture. First, the output tracking problem for sampled-data nonlinear plant at device layer with sampling period Td is investigated using adaptive neural network (NN) control, and it is shown that the outputs of subsystems at device layer can track the decomposed setpoints. Then, the outputs and inputs of the device layer subsystems are sampled with sampling period Tu at operation layer to form the index prediction, which is used to predict the overall performance index at lower frequency. Radial basis function NN is utilized as the prediction function due to its approximation ability. Then, considering the dynamics of the overall closed-loop system, nonlinear model predictive control method is proposed to guarantee the system stability and compensate the network-induced delays and packet dropouts. Finally, a continuous stirred tank reactor system is given in the simulation part to demonstrate the effectiveness of the proposed method. Index Terms— Adaptive neural networks (NNs), industrial processes, multirate sampling, networked control systems (NCSs), nonlinear model predictive control (NMPC).

I. I NTRODUCTION URING the past decades, industrial operation control has attracted a great deal of attention from both academic and industrial communities. The traditional industrial process control is based on a single-layer architecture where the given operation index is decomposed manually. In addition, the traditional operational control scheme is open-loop control without feedback information. Thus, both open-loop nature and inaccurate manual decomposition result in the poor robustness of the corresponding plant. This type of architecture is also confronted with the objection that the nonlinear dynamics are too complex to be solved reliably,

D

Manuscript received August 10, 2014; revised December 19, 2014 and March 2, 2015; accepted March 6, 2015. Date of publication April 16, 2015; date of current version January 18, 2016. This work was supported in part by the National Natural Science Foundation of China under Grant 61333012 and Grant 61374031, and in part by the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, China. The authors are with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2411671

Fig. 1.

Setpoints compensation scheme for industrial process control.

efficiently, and safely; thus, the reliability issues arise [1]. Instead of the single-layer architecture, a double-layer strategy was first proposed in [2], which is also adopted in this paper. There are two different objectives to be achieved in double-layer architecture: 1) the objective at device layer is to achieve a stable operation state and guarantee that the outputs can track the decomposed setpoints, which should be implemented by the device layer controllers and should operate at higher frequency and 2) at operation layer, the objective is to maximize profits and the optimization model is optimized at lower frequency. However, at the first stage of double-layer architecture, the upper operation layer is an openloop optimization problem, and the setpoints decomposed from operation index r are still made manually according to the operators’ experience. Therefore, in order to dynamically change the decomposed setpoints, an overall closed-loop control scheme with double-layer architecture is proposed in this paper, as shown in Fig. 1. To the best of our knowledge, the problem that double-layer architecture with sampled-data nonlinear system existing at the device layer has not been solved. As shown in Fig. 1, the overall performance index r is first decomposed into a set of desired local control output references as setpoints, then the device layer can be formulated as a tracking control problem where the decomposed setpoints

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WANG et al.: COMBINED ADAPTIVE NN AND NMPC FOR MULTIRATE NETWORKED INDUSTRIAL PROCESS CONTROL

can be seen as the corresponding reference signal. Several literatures have done the work under the above scheme, for example, Chai et al. [3] investigated the state feedback control problem at device layer, and a model predictive control (MPC) [4] compensator is designed at operation layer to dynamically change the decomposed setpoints. Liu et al. [5] extended this problem to output feedback sampled-data situation at device layer. However, these papers have only considered that the local industrial processes at device layer are linear systems, and the index prediction module adopted linear functions to predict the performance index. In practical industrial processes, the controlled plants are more or less nonlinear, and numerous control methods have been proposed for such nonlinear systems. Adaptive control has received considerable attention for controlling strict-feedback nonlinear systems, and many significant results have been achieved [6], [7]. However, they cannot solve the problem when there are unknown terms existing in the nonlinear system. Thus, intelligence methods like neural network (NN) [8]–[16] and fuzzy control method [17]–[23] were proposed to solve this problem. In this paper, an adaptive sampled-data NN control scheme is proposed to solve the tracking problem for nonlinear systems at device layer, where the NNs are used to approximate the unknown nonlinear functions. According to Fig. 1, the outputs of local plants are sampled at operation layer to form the index prediction, which is then used to evaluate the given overall performance index. Since performance index r is closely relevant to the outputs of device layer, the price of raw, and final products known as boundary conditions, which may be nonlinear, it is obvious that a nonlinear index prediction model can lead to better performance compared with a linear one in [3] and [5]. Despite the various applications of NN [24], [25], we use only the approximation ability of NN in this paper. Thus, radial basis function (RBF) NNs are utilized in this paper. An RBF has the power of the universal function approximation and its training is faster than that of a multilayer NNs. This faster learning speed comes from the fact that RBF has just two layers of weights and each layer can be determined sequentially. In addition, nonlinear MPC (NMPC) is proposed at operation layer to optimize the overall performance index and compensate the packet dropouts and transmission delays among the device layer and operational layer channel. As industrial automation systems develop due to their economic efficiency and flexibility in modularization, components of industrial processes, such as actuators, sensors, controllers, and other components, are often connected over industrial Ethernet, and communicated by exchanging packet-based messages. Despite the advantages that industrial Ethernet has brought, the utilization of Ethernet also results in several network-induced problems, hence giving rise to the so-called networked control systems [26]–[30]. Because of the extensive data exchange over the Ethernet, there is a strong possibility that random package dropout happens [31]. Thus, it is necessary to incorporate such effect into the overall consideration. In this paper, both network-induced delays and packet dropouts are considered at upper operation layer. As stated

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in [32] and [33], different control layers lie in different time scales in double-layer architecture, which means that the operation layer sampling period is much larger than that of the device layer and leads to multirate sampled-data problems. To solve this problem, lifting method adopted in [34] is proposed to describe the device layer subsystems at the operation sampling instant. Furthermore, considering the effect of network-induced packet dropout phenomenon, a stochastic variable satisfying Bernoulli random binary distribution is employed to model the packet dropout phenomenon at operation layer. Notations: (a, b) for vectors a and b denotes [a T , b T ]T. The superscript T stands for matrix and vector transpose. The notation P > 0(≥ 0) means that matrix P is positive (semi) definite. I denotes an identity matrix with appropriate dimensions. Furthermore, ∗ denotes the symmetric terms in a block matrix and diag{· · · } denotes a block-diagonal matrix. |x| refers to the usual Euclidean vector norm and represents the Euclidean norm of corresponding matrix. II. L OCAL C ONTROLLER D ESIGN In modern practical industrial processes, the intelligent control methods and MPC methods have already been used in many factories. These methods have been successfully applied into industrial processes like roasting process of shaft furnace in the mineral processing plant. Compared with traditional PID control method, the advanced control methods increase the metal recovery ratio and concentrate grade in the mineral processing plant. Another example to apply the intelligent control methods is the multicapacity interacting liquid level, which consists of a series of three water tanks and three pneumatic valves. Thus, the investigation of new advanced method and its industrial application is significant. In this section, we will investigate the local controller design problem. For brevity, we first consider the case that a single subsystem exists at the device layer and the results obtained can be extended to the multiple subsystems case in a similar way. The following nonlinear plant is considered:  x˙ = f (x) + g(x)u (1) y = h(x) where u and y are input and output of the system, respectively. x = [x 1 , . . . , x n ]T denotes the corresponding state vector. f (·), g(·), and h(·) are smooth vector fields. The control task at device layer is to design an adaptive NN-based controller to guarantee the output y(t) to track the decomposed setpoint yc . A. Coordinate Transformation Let L f h denote the Lie derivative of the function h(x) with respect to the vector field f (x) Lfh =

∂h(x) f (x). ∂x

(2)

High-order Lie derivatives can be defined recursively L kf h = L f (L k−1 f h), k > 1. Before further analysis, we have Definition 1 and Assumption 1.

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Definition 1: System (1) is said to have a relative degree ρ at (x 0 , u 0 ) if there exists a positive integer 1 ≤ ρ < ∞ such that   ρ   ∂ Lfh ∂ L if h = 0, (i = 0, 1, . . . , ρ − 1), = 0 (3) ∂u ∂u for all (x, u) ∈ B(x 0 , u 0 ), a ball centered at (x 0 , u 0 ). Let x ∈ R n and u ∈ R be compact subsets containing x 0 and u 0 , respectively. System (1) is said to have a strong relative degree ρ in a compact set U = x × u if it has a relative degree ρ at every point (x 0 , u 0 ) ∈ U . Assumption 1: System (1) possesses a strong relative degree ρ = n, ∀(x, u) ∈ U . Under Assumption 1, (1) can be transformed under the mapping (x) = [h(x), φ2 (x), . . . , φn (x)] with φ j (x) = j −1 L f h, j = 2, 3, . . . , n, which has a nonsingular Jacobian matrix for all x ∈ x . By setting z = (x), (1) can be transformed into the following canonical form: ⎧ ⎨z˙ i = z i+1 , i = 1, . . . , n − 1 z˙ n = a(z) + b(z)u (4) ⎩ y = z1 −1 with a(z) = L nf h(x), b(z) = L g L n−1 f h(x), and x =  (z). Assumptions 1 and 2 are considered for (4) and the desired reference trajectory yc . Assumption 2: The function b : R n → R is positive ∀z ∈ R n and there exists an unknown constant b and a known ¯ ∀z ∈ R n . constant b¯ such that 0 < b < b(z) < b, Assumption 3: The nonlinear functions a(·) and b(·) are continuous ∀z ∈ R n . Since the decomposed signal for device layer is yc , the reference vector can be defined as yd = [yc , . . . , yc(n−1) ]T and the error vector e = [e1 , . . . , en ]T with ei = z i − yc(i−1) , where (i) yc denotes the i th derivative of yc . Using backstepping, one has the following state transformation:

χ1 = z 1 − yc , χi = z i − αi−1 , i = 2, . . . , n

(5)

where αi are virtual control laws and χi are corresponding virtual errors. The virtual control laws can be designed as α0 = yc , αi = yc(i) +

i 

(−1)i+1 ci,k χk , 2 ≤ i ≤ n − 1

(6)

k=1

(7)

with ci,k = 0 if min{i · k, i + 1 − k} ≤ 0 and δik is the Kronecker delta function defined as  1, if i = k (8) δik = 0, if i = k. Then, by selecting c1 < c2 < · · · < cn−1 , the dynamics of the first n − 1 error variables are given by χ˙ i = −ci χi + χi+1 , 1 ≤ i ≤ n − 1

χ˙ n = z˙ n − yc(n) + (−1)n+1

n−1 

cn−1,k (−ck χk + χk+1 )

k=1

= a(z) + b(z)u − v(t, z)

(10)

yc(n)

− [0 C T ]e is a known signal that where v(t, z) =

can be used in the control design, with C T = [C¯ T cn,n−1 ]=





[cn,1 cn,2 · · · cn,n−1 ] and satisfies ci,k = 0 for k > i and

= 1 for k = i ci,k

cn,k = (−1)n+1

n−1 

cn−1, j c j,k , 1 ≤ k ≤ n − 1.

(11)

j =1

As shown in Fig. 1, a continuous-time nonlinear plant is sampled via analog-to-digital and digital-to-analog converters. An internal clock synchronizes the operation of the system while sampling instants tk are typically equidistant, i.e., tk = kTd , k = 0, 1, 2, . . . , where Td > 0 is the sampling period for device layer. The stability of the first n − 1 error system can be obtained, and we only need to consider the nth error system. We use T instead of Td in this section for brevity, and the discrete-time closed-loop system is described by χn (k + 1) = χn (k) + Ik (12) (k+1)T where Ik = kT [a(z(s)) + b(z(s))u(k) − v(s, z(s))]ds = (k+1)T [B1 (z(s)) + b(z(s))u(k)]ds and Ik1 + Ik2 , with Ik1 = kT (k+1)T (n) (k+1)T Ik2 = − kT [yc (s) + [0 C T ]yd (s)]ds = kT B2 (s)ds. Developing B1 (z(s)) and b(z(s)) in a Taylor series expansion, we can obtain the following:

d B1

(s − kT ) B1 (z(s)) = B1 (z(kT )) + dt t =kT

1 d 2 B1

+ (s − kT )2 + · · · (13) 2! dt 2 t =kT

db

(s − kT ) b(z(s)) = b(z(kT )) + dt t =kT

1 d 2 b

+ (s − kT )2 + · · · (14) 2! dt 2 t =kT with

n−1  ∂a ∂a d B1 T

¯ = z j +1 + [0 0 C ]z + cn,n−1 + dt ∂z j ∂z n j =1

where ci,k , 1 ≤ k ≤ i, 1 ≤ i ≤ n − 1 are defined as ci,k = −δik ck + ck ci−1,k − ci−1,k−1

and the nth error variable holds that

(9)

× (a(z) + b(z)u) n−1  db ∂b = z j +1 . dt ∂z j

(15)

j =1

Then, we can obtain the following: Ik1 = T (B1 (z(kT )) + b(z(kT ))u(k))



T 2 d 2 B1

d 2 b

+ O(T 3 ). + + u(k) 2! dt 2 t =kT dt 2 t =kT (16) For a control law of the form u(k) = u 0 (k) + T u 1 (k)

(17)

WANG et al.: COMBINED ADAPTIVE NN AND NMPC FOR MULTIRATE NETWORKED INDUSTRIAL PROCESS CONTROL

the integral Ik1 takes the form

C. Stability Analysis

Ik1 = T (B1 (z(kT )) + b(z(kT ))u 0 (k)) 1 + T 2 b(z(kT ))u 1 (k) + G(z, u 0 ) + O(T 3 ) 2

(18)

For the closed-loop system (12) with controller and NN update law (23), the overall Lyapunov functional is chosen as V (k) = Vn (k) + (1/2)||Wˆ (k) − W ∗ ||2 −1 , with

where   ∂a ∂a

a(z) + 0 0 C¯ T z j +1 + cn,n−1 + G(z, u 0 ) = ∂z j ∂z n j =1 ⎡ ⎤ n−1  ∂b ∂a

b(z) + + z j +1⎦. × z + u 0 ⎣ cn,n−1 ∂z n ∂z j n−1 

j =1

Similarly, we take the Taylor series expansion of B2 (s) and the integral Ik2 takes the form

1 d B2

Ik2 = B2 (kT )T + + O(T 3 ). (19) 2 dt t =kT Combining Ik1 and Ik2 , we have

+ O(T 3 ).

(20)

B. Controller Design In the continuous-time control case with Lyapunov functional V = (χn4 /4b(z)) + (1/2)||Wˆ (k) − W ∗ ||2 −1 , it can be easily proved that the following controller and NN weight update laws: u(t) = −cn χn (t) − Wˆ T (t)(z(t), χn (t), v(t)) ˙ Wˆ (t) = (t)χn3 (t) − σ Wˆ (t)

(21)

with > 0, σ > 0 ensures the semiglobal uniform ultimate boundedness of all the error variables and the NN approximation error, where NN is used to approximate the following nonlinear function: n−1 χn  ∂b a(z) − v z j +1 − 2 U (s) = b(z) 4b (z) ∂z j j =1

=W

(s) + (s)

(22)

where s = [z T , χn , v]. The approximation error is assumed to be bounded by (s) < 0 . The ideal weight W ∗ is the optimal approximation of U (s). However, one cannot directly prove similar relations in continuous case for sampled-data nonlinear systems. Then, according to the sampling behavior at local device layer, the sampled-data controller (17) and the NN update law are designed as follows: u 0 (k) = −cn χn (k) − Wˆ T (k)(s(k)), u 1 (k) = 0   (23) Wˆ (k + 1) = Wˆ (k) + T (s(k))χn3 (k) − σ Wˆ (k) where W (0) = 0.

Vn (k) =

χn4 (k) . 4b(z(k))

(24)

Then, according to (20), χn4 (k + 1) and 4b(z(k + 1)) can be written as χn4 (k + 1) = [χn (k) + Ik ]4 = χn4 (k) + 4χn3 (k)Ik + 6χn2 (k)Ik2 + 4χn (k)Ik3 + Ik4 = χn4 (k) + 4T χn3 (k) (B1 (z(kT )) + B2 (z(kT )) 2 T + b(z(kT ))u 0 (k)) + 2T 2 z n3 (k) 2

d B2

× 2b(z)u 1(k) + + G(z, u 0 ) + 6T 2 χn2 (k) dt

t =kT

Ik = T (B1 (z(kT )) + B2 (z(kT )) + b(z(kT ))u 0 (k))

2 d B2

T 2b(z)u 1 (k) + + G(z, u 0 ) + 2 dt t =kT

∗T

419

× (B1 (z(kT )) + B2 (z(kT )) + b(z(kT ))u 0 (k))2 + O(T 3 ) = R0 (k) + R1 (k)T + R2 (k)T 2 + O(T 3 )

(25)

and 4b(z(k + 1))

 n−1  ∂b z i+1 T = 4b(z(k)) + 4 ∂z i i=1 n−2  ∂b ∂b z i+2 + +2 (a(z(k)) + b(z(k))u 0) ∂z i ∂z n−1 i=1 ⎤ n−1 n−1   ∂ 2b + z i+1 z j +1 ⎦ T 2 + O(T 3 ) ∂z i z j 

i=1 j =1

= S0 (k) + S1 (k)T + S2 (k)T 2 + O(T 3 )

(26)

respectively. Then, considering (25) and (26), (24) can be rewritten as Vn (k + 1) R0 + R1 T + R2 T 2 + O(T 3 ) = S0 + S1 T + S2 T 2 + O(T 3 )   R0 R1 R0 S1 = + − 2 T S0 S0 S0   R0 S12 R2 R1 S1 R0 S2 + − 2 + − 2 T 2 + O(T 3 ). S0 S0 S03 S0

(27)

Then, applying (23) into the difference of the overall Lyapunov functional, one can obtain V (k +1)−V (k) ≤ 0 |χn3 | − cn χn4 − σ W˜ T Wˆ + O(T ). (28) T Applying Young’s inequality with appropriate parameters, one has 1 27 4 (29) 0 |χn3 | ≤ cn χn4 + 2 32cn3 0

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and λmin ( ) ˜ T −1 ˜ ||Wˆ ||2 ||W ∗ ||2 W˜ T Wˆ ≥ W W+ − . 2 2 2 The above three inequalities yield

(30)

V (k + 1) − V (k) ≤ −μV (k) + λ + Q(T, z, Wˆ ) T

(31)

where λ and μ are two constants satisfying λ = (27/32cn3 ) 04 + (σ/2)||W ∗ ||2 and μ = min{2cn b, σ λmin ( )}, respectively. Q(T, z, Wˆ ) is the function with first-order of T . Before proceeding further, we introduce the following definition. Definition 2: A Lyapunov function V has a discrete-time rate with Mth-order perturbed linearly decreasing upper bound if there exist two constants θ1 , θ2 > 0 and an order T M function Q(T, t, x) = O(T M ), M ≥ 1 such that V (k + 1) − V (k) ≤ −θ1 V (k) + θ2 + Q(T, t, x). (32) T From the above definition, no conclusion can be made directly with respect to the boundedness of the states even for the discrete-time system. Assume that the control law u(k) is smooth and bounded, then, motivated by the results of [10] and [35], for the sampled-data closed-loop system (12), if there exists some Lyapunov function, which has a discretetime rate with Mth-order perturbed linearly decreasing upper bound and a sufficiently small sampling period, the closedloop sampled-data system is semiglobal uniformly ultimately bounded. Remark 1: As shown from the above stability analysis, a Lyapunov function that has a discrete-time rate with first-order perturbed linearly decreasing upper bound. By redesigning the adaptive controller and NN update law, one can increase the order of the perturbation in the bound of the discrete-time rate. III. O PERATIONAL L AYER D ESIGN In this section, the operation layer compensation and performance tracking problems will be investigated. As shown in Fig. 1, the outputs and inputs of local plants are sampled and transmitted through operation layer Ethernet to form the index prediction with sampling period Tu with Tu = aTd . Then, by combining RBF NN, one can design the index prediction as m  ωi ϕi (ξ(n)) (33) r¯ = i=1

where ϕi (ξ(n)) = exp(− ξ(n) − c¯i 2 /2bi2 ), in which ξ(n) = (u(n), y(n)) is the input signal of index prediction module. u(n) and y(n) can be seen as the overall consumption and benefit of industrial processes, respectively. ωi are the weights representing the corresponding consumption coefficient and benefit coefficient, respectively. Then, the prediction error can be obtained as r = r − r¯ . The dynamics of the operation layer system is given as follows: yc (n + 1) = yc (n) + F(n)r (n) where F(n) is the gain matrix to be designed.

(34)

Considering (23) and using a lifting method, the following dynamics can be obtained with sampling period Tu : ⎧ a−1  ⎪ a ⎪ ⎪ (1− σ T )i T

⎨Wˆ ((n + 1)Tu ) = (1− σ T ) Wˆ (nTu ) + i=0   × (s(nTu + i Td ))χn3 (nTu + i Td ) ⎪ ⎪ ⎪ ⎩ u(nTu ) = −cn χn (nTu ) − Wˆ T (nTu )(s(nTu )). (35) The dynamics of device layer plant is ⎧ ⎨z 1 ((n + 1)Tu ) = [1, 0, . . . , 0]χ((n + 1)Tu ) + yc ((n + 1)Tu ) (36) ⎩ χ((n + 1)Tu ) = Aχ(nTu ) + h(z(nTu ), u(nTu )) T ¯ ¯ where h(z, u) = ( 0 u e At dt)B(a(z) + b(z)u − v), A = e ATu T with B = [0, 0, . . . , 1] and ⎡ ⎤ −c1 1 ··· 0 ⎢ ∗ −c2 ··· 0⎥ ⎢ ⎥ ⎢ . . . .. ⎥. .. .. A¯ = ⎢ .. .⎥ ⎢ ⎥ ⎣ ∗ ∗ −cn−1 1 ⎦ ∗ ∗ 0 0 By introducing a new vector as ζ = [yc , χ]T , the dynamics of the overall closed-loop system can be written as F(n)r (n) 1 0 ζ(n + 1) = ζ(n) + 0 −c2 −Wˆ (n)(s(n)) = R(ζ(n), F(n)). (37) Remark 2: According to the definition of r (n), r(n) ¯ is the function of input signals u(n) and y(n), while u(n) and y(n) = z 1 (n) are the functions of variables χ(n) and yc (n). Thus, as shown in (37), r (n) can be viewed as the function of variable ζ(n). Denote X and U as the compact set of state vector ζ and input vector F, respectively. For any time n ≥ 0, ζ (n, ζ (0), F) is denoted as the solution of (37) with initial time t = 0, initial value ζ(0), and input sequence F, obtained by iterating (37) for 0, 1, . . . , n. A. Compensation of Delays According to Fig. 1, both operation layer to device layer channel and device layer to operation layer channel are connected by industrial Ethernet. Let τdo (t) ∈ [0, τ¯do ] represent the up-link delay, while τod (t) ∈ [0, τ¯od ] denotes the downlink delay. For sake of simplicity, the delays will be referred as τdo and τod , respectively. At the time instant i Tu , which can be simply written as ti , the signals are sampled and transmitted from device layer to operation layer and are available at controller F at time instant ti + τdo . In addition, τdo can be compensated by means of suitable prediction. In addition, for the nonlinear system (37), a sampled-data NMPC, which is based on the repeated solution of an open-loop control problem, is proposed in this paper. Based on the states at time instant ti , the controller predicts the system behavior in the future over a prediction horizon T p , such that a certain objective cost functional is minimized. The procedure is repeated at every sampling time instant i Tu .

WANG et al.: COMBINED ADAPTIVE NN AND NMPC FOR MULTIRATE NETWORKED INDUSTRIAL PROCESS CONTROL

At time ti , the state ζ (ti ) is measured, and this can be mathematically formulated as the following optimal control problem (OCP): ¯ min J (ζ¯ (n), F(n)) ˙ ¯ )), ζ¯ (ti ) = ζ (ti ), ζ¯ (ti + T p ) ∈ E ¯ ¯ s.t. ζ = R(ζ (τ ), F(τ ¯ ) ∈ U, ζ¯ (τ ) ∈ X, τ ∈ [ti , ti + T p ] (38) F(τ ¯ denotes the controller internal variables and where () E is the terminal region satisfying E ⊂ X, and a typical t +T ¯ (τ ) + ¯ T (τ )R1 r cost functional is given by J = tii p (r t +T ¯ ))dτ + E(ζ¯ (ti + T p )) = i p Q(ζ¯ (τ ), F(τ ¯ ))d F¯ T (τ )R2 F(τ ti

τ + E(ζ¯ (ti + T p )), where R1 and R2 are positive definite matrices and E(ζ¯ (ti + T p )) is the terminal cost functional. For t ∈ [ti , ti + T p ], the solution of the OCP is an optimal control signal F ∗ (t; ζ (ti )), which is then implemented for t ∈ (ti + τdo + τ¯od , ti+1 + τdo + τ¯od ]. Under certain rather mild conditions, stability in the nominal case (no delays and no packet dropouts) can be guaranteed [36], [37]. The idea is to consider the available model of the plant and use it to compensate the measurement delay by predicting ζˆ (ti + τdo ) from the last available measurement. The timestamp can be used to calculate the delay just by comparing with the inner clock of prediction module. In this way, the OCP can be solved by the predicted state. Then, the measurement delay can be compensated in this way. Then, for the additional delay τod , the smart controllers are introduced. They are able to store the control information and apply it at the proper moment. Then, it is possible to solve the delay by considering the delay bound τ¯od . In fact, since τ¯od is assumed to be known, the state prediction ζˆ (ti +τdo + τ¯od ) can be calculated and used to solve the OCP. The timestamp of the corresponding control packet is set equal to ti + τdo + τ¯od , since it cannot be used before that time. It is the responsibility of the controller to apply the input when the timestamp matches the controller’s inner clock. Although performance might slightly worsen, it is possible for the controller to reconstruct exactly the applied input. Thus, we will adopt the algorithm and compensation strategy proposed in [38], and the overall procedure can be summarized as Algorithm 1 in the Appendix. Using the mentioned algorithm, even though the up-link and down-link delays are present, the stability can be proved. Theorem 1: Consider the nonlinear system (37) and compensation strategy described in the Appendix, the system is stable in the sense of asymptotic convergence, if the nominal controller, i.e., the controller subject to no delays, stabilizes the system. Proof: The proof is composed by two parts, feasibility and convergence, and it can be reconducted to the nominal case. Feasibility: Consider any time ti such that the OCP has a solution. The corresponding optimal input {[F ∗ (τ ; ζ(ti ))]} for ζ(ti ) is implemented for τ ∈ (ti + τdo + τ¯od , ti+1 + τdo + τ¯od ], by sorting the corresponding piece of trajectory in the actuator’s buffer. Since it is assumed that there is no model mismatch, the prediction state ζˆ (ti+1 ) at ti+1 must be equal

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to the actual state ζ(ti+1 ). Therefore, the remaining piece of optimal input {[F ∗ (τ ; ζ(ti ))]}, τ ∈ [ti+1 , ti + T p ], where T p is the prediction horizon, satisfies the state and input constraints. By considering the input  ∗ ¯ ) = F (τ ; ζ(ti )), τ ∈ [ti+1 , ti + T p ] (39) F(τ FE (τ ), τ ∈ [ti + T p , ti+1 + T p ] ¯ satisfying the constraints, the state ζ(ti+1 + T p ; ζti+1 , F(·)) is reached. Thus, feasibility for ti implies feasibility at ti+1 . Convergence: By denoting the optimal cost function at ti as the value function V (ζ(ti )) = J ∗ (F ∗ (·), ζ(ti )), it can be shown that the value function is strictly decreasing. Therefore, the states converge to the origin. In fact, the value function at ti is given by  ti +T p V (ζ(ti )) = Q(ζˆ (τ ; ζ(ti )), F ∗ (·; ζ(ti )), F ∗ (τ ; ζ (ti )))dτ ti

¯ + E(ζˆ (ti + T p ; ζ(ti ), F(·))).

(40)

Then, consider the cost function resulting from the application of F¯ starting from ζ(ti+1 )  ti+1 +T p ¯ ¯ ¯ ))dτ J (ζ(ti+1 ), F(·)) = Q(ζˆ (τ ; ζ(ti+1 )), F(·), F(τ ti+1

¯ + E(ζˆ (ti+1 + T p ; ζ(ti+1 ), F(·))).

(41)

It can be reformulated as ¯ J (ζ(ti+1 ), F(n))



= V (ζ(ti )) −

ti+1

Q(ζˆ (τ ; ζ(ti )), F ∗ (·; ζ (ti )))

ti

−E(ζˆ (ti + T p ; ζ(ti ), F ∗ (·; ζ(ti ))))  ti+1 +T p ¯ ¯ ))dτ + Q(ζˆ (τ ; ζ(ti+1 )), F(·), F(τ ti +T p

¯ + E(ζˆ (ti+1 + T p ; ζ(ti+1 ), F(·))).

(42)

By integrating the inequality (∂ E/∂ζ ) f (ζ(τ ), FE (τ )) + Q(ζ(τ ), FE (τ )) ≤ 0, where, as previously mentioned, FE (τ ) is such that the set E is an invariant region, the last three terms of the former cost function can be upper bounded by zero. Thus ¯ V (ζ(ti )) − J (ζ(ti+1 ), F(·))  ti+1 ≤− Q(ζˆ (τ ; ζ(ti )), F ∗ (·; ζ(ti )))dτ

(43)

ti

and this is equal to V (ζ(ti )) − V (ζ(ti+1 ))  ti+1 ≤− Q(ζˆ (τ ; ζ(ti )), F ∗ (·; ζ(ti )))dτ

(44)

ti

which is strictly decreasing for (ζ, F) = (0, 0). B. Compensation of Packet Dropouts In Section III-A, the transmission delays between device layer and operation layer are compensated using the proposed compensation strategy. However, in the presence of packet

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dropouts between device layer and operation layer, the compensation strategy for delays is too conservative. In this paper, both communication channels between device layer and operation layer are assumed subject to random packet dropouts, which can be modeled as Bernoullian variables S1i ∼ B(1 − ps1), S2i ∼ B(1 − ps2 ) with ps1, ps2 ∈ [0, 1]. When S1i = 0, i.e., a down-link packet is dropped, the model (37) can be still used using the last available setpoint as internal variable. Instead, if S2i = 0, i.e., no feedback information, the controller F does not have any new information to use and the controller cannot be sure about what is applied to the system. The idea is to keep compensating the delays as described in the compensation strategy for delays, and at the same time use prediction consistent feedbacks, i.e., ∀ti ∈ [ti , ti + T p ), F ∗ (τ ; ζ (·)) is calculated and dispatched. When a packet is lost, the smart controller can continue to apply the remainder of the old input, i.e., if S2i = 0, F ∗ (τ ) = F ∗ (τ ; ζ (·)) for τ ∈ [ti−1 , ti+1 ). However, to guarantee the stability even if packet dropout happens, a more elaborated version of NMPC is required, such that the input trajectories are able to keep consistency between consecutive state predictions. The following definitions are useful to delineate the prediction consistency. Definition 3: The nonempty set  ⊂ R n is a control invariant set for the discrete system x k+1 = f˜(x k , u k ) if and only if there exists a feedback control law u k = g(x k ) such that  is a positive invariant set for the closed-loop system x k+1 = f˜(x k , u k ), and u k ∈ U, ∀x k ∈ . Definition 4: The set Si (, T ) is the i -step reachable set contained in  for the system x k+1 = f˜(x k , u k ) if and only if T is a control invariant subset of  and Si (, T ) contains all the states in  for which there exists an admissible control sequence of length i which will drive the states of the system in i or less steps, while keeping the evolution of the states inside . Definition 5: Given the sampling time instants ti , and the terminal set T ⊂ X ⊂ Rn , the feedback F(·) is called prediction consistent if for every sampling time instants ti , F(·; ζ (ti )) ∈ U , and, given two input trajectories Fk (·; ζ (tk )), Fh (·; ζ (th )) obtained at the sampling time instants tk < th , the predicted states of (37), ζˆ (t j ; Fk (·; x(tk ))), ζˆ (t j ; Fh (·; x(th ))) at time instants t j > th , obtained by applying the former inputs, belong to the same reachable set Si (X, T ) of the corresponding sampled-data system. Theorem 2: System (37) is stable if: 1) the feedback prediction is consistent; 2) the controller F stabilizes the nominal system, i.e., the system without delays; 3) compensation and smart controllers are used to counteract delays; and 4) the number of consecutive packet dropouts is less than the prediction horizon. Proof: Since the prediction consistent feedback is ensured by definition that ∀ti , ζˆ (ti ; Fi∗ (·)) ∈ Si (X), from Theorem 1, the feasibility and convergence of the packet dropout compensation scheme are guaranteed under the assumption that the number of consecutive packet dropouts is less than the prediction horizon. Thus, the stability of nonlinear system (37) subject to both delays and data packet dropouts can be obtained.

Remark 3: According to [3] and [5], the analysis for a single subsystem case can be easily extended to multisubsystem case in a similar way. IV. S IMULATION S TUDIES Consider a continuous stirred tank reactor (CSTR) system model in which the first-order irreversible exothermic reaction occurs. The mass balance and energy balance equations in a well-stirred region uniform in composition C and temperature T are governed by [39] dC = λ(C0 − C) − Vol R dt dT = λ(T0 − T ) + (H )Vol R − U (T − Tw ) (45) Vol C p dt respectively, where Vol is the constant reaction volume, λ is the flow rate in and out, C0 is the feed concentration of the component of interest, R is the reaction rate per unit volume, C p is the heat capacity per unit volume of the flowing materials as well as the contents of the reactor, T0 is the feed temperature, H is the molar heat of reaction, taken to be positive for exothermic systems, U is the overall heat transfer coefficient, and Tw is coolant or heating fluid temperature. Denoting the first-order kinetics for reaction rate as R = k0 Cexp(−Q/T ), where k0 is the reaction velocity constant, Q is the ratio of an Arrhenius activation energy to the gas constant. Using variable transformation as follows: Vol

x1 =

λt C0 − C T0 − T Tw − T0 , s= . , x2 = , u= C0 Q Vol Q (46)

Then, the dynamic model can be rewritten as d x1 1 = −x 1 + Da (1 − x 1 )exp − ds x2 + ρ d x2 −1 = −(1 + κ)x 2 + H (1 − x 1 )exp + κu ds x2 + ρ

(47)

where Da = (k0 Vol )/λ, H = ((H )C0/QC p )Da , ρ = T0 /Q, and κ = U/λC p . Using the feedback linearization method, the following dynamic model can be obtained: z˙ 1 = z 2 z˙ 2 = a(z) + b(z)u y = z1

(48)

where a(z) = L f h(x) = −(1 + κ)x 2 + H (1 − x 1 )exp(−1/ x 2 + ρ), b(z) = L g L f h(x) = κ(−(1 + κ) + (1/(x 2 + ρ)2 ) H (1 − x 1 )exp(−1/x 2 + ρ)) which is difficult to obtain, and thus can be assumed as unknown. The parameters for the CSTR system is designed as Da = 0.072, H = 0.576, ρ = 20, κ = 0.3. Assume that there are two CSTR subsystems existing at device layer and the parameter variables in the controller design process can be given as: 1) virtual control law gains are c11 = c21 = 2.8; 2) control law gains are c12 = c22 = 3.35; 3) Lyapunov function gain matrix for both subsystems are of the same value = 0.1 ∗ I ; 4) σ -modification values are

WANG et al.: COMBINED ADAPTIVE NN AND NMPC FOR MULTIRATE NETWORKED INDUSTRIAL PROCESS CONTROL

Fig. 2. Tracking of operation index with networked multirate setpoints compensation.

Fig. 3.

Tracking performance of subsystems at device layer.

σ1 = σ2 = 0.3; 5) device layer sampling period is Td = 1 min; 6) operation layer sampling period is Tu = 40 min; and 7) the total step for operation layer is 100 steps. The operation index is set as 5, 8, 6, and 10 for every 25 operation steps, respectively. The initial condition for the two subsystems is set as [0, 0.3] and [0, 0.5], respectively. The simulation results are shown as follows. Fig. 2 shows the overall index prediction performance. As shown in Fig. 2, the index prediction module can track the overall index even if it changed every 25 steps. Fig. 3 shows the tracking performance for two subsystems at device layer. The outputs of the two subsystems can track the decomposed setpoints and did not jump at the instant of setpoints value changing. In addition, Fig. 4 gives the information of control signal for two subsystems, and Fig. 5 shows the data dropouts series, which happened during data transmission between the channel of device layer and operation layer.

Fig. 4.

Control signal for device layer subsystems.

Fig. 5.

Data dropout series.

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Remark 4: Compared with the existing results [3], [5], the control method proposed in this paper solved the problem of nonlinear system at device layer instead of the linear one. In addition, the simulation results in this paper further demonstrate the proposed method, and lead to better performance utilizing RBF NNs index prediction module. V. C ONCLUSION In this paper, a combined adaptive NN control and NMPC approach is proposed for a class of multirate networked nonlinear systems with double-layer architecture. At the device layer with sampling period Td , an adaptive NN control scheme was proposed to guarantee the stability and tracking performance of the local nonlinear industrial plant. Combining the overall performance index and index prediction unit, which is formed using RBF NNs, an NMPC method is proposed to optimize the performance index under the effects of network-induced delays and packet dropouts. In the simulation part, a CSTR system

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Algorithm 1 Compensation Strategy Sensor: 1. Measure ζ (ti ), 2. Send the packet [ζ (ti )|ts], with ts = ti , 3. Go to 1. Operation Layer:control-input={[F(·)|ts0 ]}; buffer =[ζ(ti )|ts]old ; 1. A new packet [ζ(ti )|ts]new arrives. If tsnew ≤ tsold , then discard. Else buffer=[ζ (ti )|ts]new . 2. τdo = (t − ts). t s+τ 3. ζˆ (t + τ¯od ) = ζt s + t s do R(ζ (τ ), F(τ ))dτ , where F(τ ) ∈ control-input. 4. Solve the optimization control problem for ζˆ (t + τ¯od ) → F ∗ (τ ; ζ (ti )), for τ ∈ (ti + τdo + τ¯od ], with ti = ts. 5. Send {[F ∗ (τ ; ζ (ti ))|ts]}, with ts = (t + τ¯od ). 6. Insert {[F ∗ (τ ; ζ (ti ))|ts]} in control-input. 7. Go to 1. Device Layer: buffer={[F(·)|ts0 ], . . . , [F(·)|tsn ]}, for ts0 < · · · < tsn ; applied-input = [F(·)|ts0 ]. 1. A new packet [F(·)|ts]new arrives. Insert [F(·)|ts]new in buffer. Sort buffer by increasing ts 2. temp= first element of buffer. 3. If tstemp = t, applied-input=temp and remove first element from buffer. 4. Go to 1.

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[32] S. J. Qin and T. A. Badgwell, “A survey of industrial model predictive control technology,” Control Eng. Pract., vol. 11, no. 7, pp. 733–764, 2003. [33] J. Liu, D. Muñoz de la Peña, B. J. Ohran, P. D. Christofides, and J. F. Davis, “A two-tier control architecture for nonlinear process systems with continuous/asynchronous feedback,” Int. J. Control, vol. 83, no. 2, pp. 257–272, 2010. [34] M. Mizuochi, T. Tsuji, and K. Ohnishi, “Multirate sampling method for acceleration control system,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp. 1462–1471, Jun. 2007. [35] D. Neši´c, A. R. Teel, and E. D. Sontag, “Formulas relating K L stability estimates of discrete-time and sampled-data nonlinear systems,” Syst. Control Lett., vol. 38, no. 1, pp. 49–60, 1999. [36] L. Magni and R. Scattolini, “Stabilizing model predictive control of nonlinear continuous time systems,” Annu. Rev. Control, vol. 28, no. 1, pp. 1–11, 2004. [37] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000. [38] R. Findeisen and P. Varutti, “Stabilizing nonlinear predictive control over nondeterministic communication networks,” in Nonlinear Model Predictive Control. Berlin, Germany: Springer-Verlag, 2009, pp. 167–179. [39] F. Zheng, Q.-G. Wang, and T. H. Lee, “Output tracking control of MIMO fuzzy nonlinear systems using variable structure control approach,” IEEE Trans. Fuzzy Syst., vol. 10, no. 6, pp. 686–697, Dec. 2002.

Tong Wang received the B.S. degree in information and computing sciences from Shanxi University, Taiyuan, China, in 2010, and the M.S. degree in control theory and control engineering from the Liaoning University of Technology, Jinzhou, China, in 2013. He is currently pursuing the Ph.D. degree with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, China. His current research interests include fuzzy control, stochastic adaptive control, and networked control systems.

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Huijun Gao (SM’09–F’13) received the Ph.D. degree in control science and engineering from the Harbin Institute of Technology, Harbin, China, in 2005. He was a Research Associate with the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, from 2003 to 2004. He held a post-doctoral research position with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, from 2005 to 2007. He has been with the Harbin Institute of Technology, since 2004, where he is currently a Professor and the Director of the Research Institute of Intelligent Control and Systems. His current research interests include network-based control, time-delay systems, and their engineering applications. Prof. Gao is an Administrative Committee Member of the IEEE Industrial Electronics Society. He is the Co-Editor-in-Chief of the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS , and an Associate Editor of Automatica, the IEEE T RANSACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY, the IEEE T RANSACTIONS ON C YBERNETICS , and the IEEE/ASME T RANSACTIONS ON M ECHATRONICS .

Jianbin Qiu (M’11–SM’15) received the B.Eng. and Ph.D. degrees in mechanical and electrical engineering from the University of Science and Technology of China, Hefei, China, in 2004 and 2009, respectively, and the Ph.D. degree in mechatronics engineering from the City University of Hong Kong, Hong Kong, in 2009. He has been with the School of Astronautics, Harbin Institute of Technology, Harbin, China, since 2009, where is currently a Full Professor. He was a Senior Research Associate with the Department of Manufacturing Engineering and Engineering Management, The City University of Hong Kong, from 2010 to 2011. He is an Alexander von Humboldt Fellow with the Institute of Automatic Control and Complex Systems, University of Duisburg-Essen, Duisburg, Germany. His current research interests include intelligent and hybrid control systems, signal processing, and robotics. Dr. Qiu was a recipient of the Outstanding Doctoral Thesis Award from Anhui Province, China, in 2011. He was also a recipient the New Century Excellent Talents in University from the Ministry of Education, China, in 2012, and the Alexander von Humboldt Fellowship of Germany in 2013. He is also an Associate Editor of the IEEE T RANSACTIONS ON C YBERNETICS , the Journal of the Franklin Institute, the Journal of Intelligent and Fuzzy Systems, and Neurocomputing.

A Combined Adaptive Neural Network and Nonlinear Model Predictive Control for Multirate Networked Industrial Process Control.

This paper investigates the multirate networked industrial process control problem in double-layer architecture. First, the output tracking problem fo...
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