Modeling and Control of Wiener Systems Using Multiple Models and

Article pubs.acs.org/IECR

Modeling and Control of Wiener Systems Using Multiple Models and Neural Networks: Application to a Simulated pH Process Bi Zhang† and Zhizhong Mao*,†,‡ †

College of Information Science & Engineering, Northeastern University, Wenhua Road, Heping District, Shenyang, Liaoning, People’s Republic of China ‡ State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, Liaoning, People’s Republic of China ABSTRACT: This paper describes an efficient multiple models control scheme based on a bank of neural Wiener models and its application. Each Wiener model consists of a linear dynamic part in connection with a nonlinear static part which is approximated by a radial basis function (RBF) neural network. The control algorithm is developed from a Clarke performance index and can copy with unstable zero-dynamics problems. An extended system that is associated with the criterion function is defined for system parameters identification and control law design. After establishing multiple neural Wiener models for several distinct operation regions, a robust switching mechanism is introduced for the selection of the best controller. The overall scheme is applied to a simulated pH neutralization process. The set-point tracking, disturbance rejection, and robustness performance of this scheme are compared to several alternatives.

1. INTRODUCTION Modeling, identification, and control of the so-called blockoriented models have become an interesting research topic in recent years. Within these models, one simple but effective structure is the Wiener system, which is composed of a linear subsystem followed by a nonlinear function.1 This cascade system is demonstrated to be able to describe the nonlinear dynamic properties of many industrial processes, such as pH neutralization, polymerization reactor, distillation column, fuel cell system, and so on.2 The control problems of Wiener-type nonlinear processes are gradually attracting the attention of researchers. Plenty of interesting results have been achieved based on the output nonlinearity inversion concept, for instance, the modified PID control,3 the modified feedforward and feedback control,4 the robust internal model control,5 the nonlinear model predictive control (MPC),6−8 and the adaptive control.9−11 Recently, some advanced MPC algorithms12,13 based on the online trajectory linearization are also developed without an inverse of the nonlinear block. Despite the merits of these controllers, some key issues remain to be addressed: (i) we need to obtain as precise models as possible and (ii) we expect that controllers can operate in distinct operation regions. Accomplishing these aims in an intelligent control fashion is our objective. Intelligent control has long been considered as a promising scheme to meet the production requirements and improve the performance of nonlinear processes.14 In this field, multiple © 2016 American Chemical Society

models and neural networks are two powerful tools which are favored by researchers. The multiple models approach enables an integrated controller to effectively operate in multiple distinct operating environments by identifying which condition is currently in existence and servicing it appropriately.15 Neural networks gradually become an attractive way for modeling and controlling complex systems due to the capability of approximating arbitrary map defined on a compact set to any desired accuracy.16,17 The past decade has witnessed considerable intelligent control strategies and their applications.18−22 However, until now, the multiple models and neural networks based intelligent control concepts have seldom been systematically extended to Wiener nonlinear systems. A key challenge is that the analysis of the closed-loop properties is significantly different from the previous framework. To this end, this work proposes a novel intelligent control scheme for Wiener models. The parametrized model is based on the neural networks approximation of the inverse output nonlinearity. For a single model structure, the identification and control algorithms are based on a Clarke performance index,23 which can copy with unstable zero-dynamics problems. Unlike similar neural Wiener models based control methods,8,11,24 Received: Revised: Accepted: Published: 10147

June 8, 2016 September 1, 2016 September 1, 2016 September 1, 2016 DOI: 10.1021/acs.iecr.6b02214 Ind. Eng. Chem. Res. 2016, 55, 10147−10159

Article

Industrial & Engineering Chemistry Research the control law derives from a straightforward equation and requires no inverse network training. Then based on the online integrations, the multiple Wiener models control scheme is set up to account for distinct operation regions. A hysteresis function is also incorporated into the switching mechanism to increase the system robustness and avoid the extreme switch chattering phenomenon. Another objective of this paper is to address the pH control problem by the above intelligent controller. The control of pH plays an important role in several industrial processes, such as wastewater treatments, polymerization reactions, ore flotation, fatty acid production and biochemical processes.25 It is a consensus that pH processes show extremely strong nonlinear behaviors and time-varying characteristics with respect to the variation of the feed components or total ion concentrations.26 Thus, the traditional linear PI or PID controller can hardly ensure satisfactory performance in a pH control problem. More advanced controllers applicable to pH processes have become a focus of researchers. An excellent review has discussed possible approaches to control the pH processes.27 Among the methods, the idea of using a bank of separate submodels has been proven promising,28,29 but these publications mostly describe pH processes with linear submodels. Alternatively, in our approach the pH nonlinear behavior is captured by separating the range into different regions and obtaining a neural Wiener model for each region. After establishing the corresponding nonlinear controllers, the robust switching mechanism is introduced for the selection of the best one. The article is structured in the following way. The neural networks based Wiener model representation is derived in section 2, the modeling and control scheme for a single Wiener model is introduced in section 3, the multiple neural Wiener models based control scheme is established in section 4, simulation studies of a pH process are performed in section 5, and a concise summary is given in section 6.

The nonlinear static function is defined as y(t ) = f [v(t )]

(2)

In this work, the nonlinearity f [·] is assumed to be continuous and invertible. Such an assumption is reasonable for many industrial processes, for instance, the titration curve of a pH process is usually a monotone function. To parametrize the controlled plant, the inverse output nonlinearity f−1[·] is approximated by a radial basis function (RBF) network. This artificial neural network is able to approximate complex nonlinear functions adequately. More interestingly, it is of the following pseudolinear form30,31 p

v(t ) = f −1 [y(t )] = c1y(t ) +

∑ ciΦi(|y(t ) − ϖi|)

(3)

i=2

where p is the number of neurons in the hidden layer; c1,c2,...,cp are the weights of the RBF network; ϖ2,ϖ3,...,ϖp, are the centers of the basis functions; Φi(|y(t) − ϖi|), i = 2,3,...,p, are Gaussian functions, denoted as follows: Φi(|y(t ) − ϖi|) =

⎛ (y(t ) − ϖ )2 ⎞ 1 i ⎟, exp⎜ − 2 2π λ 2λ ⎝ ⎠

λ>0

The variable λ is a user-designed parameter that specifies the form of the function. Thus, from eq 3, the unmeasurable signal v(t) can be regarded as a nonlinear function of the measurable output y(t), which are weighted sums of the bias c1y(t) and the hidden unit p outputs ∑i = 2 ci Φi(|y(t ) − ϖi|). For ease of derivations, some definitions are made as follows: g1(y(t)) = y(t), g2(y(t)) = Φ2(|y(t) − ϖ2|), g3(y(t)) = Φ3(|y(t) − ϖ3|), ..., gp(y(t)) = Φp(|y(t) − ϖp|). Combing eqs 1 to 3 yields the following relationship p

A(z −1) ∑ ci gi(y(t )) = B(z −1) u(t − 1)

2. WIENER MODEL DESCRIPTION BASED ON NEURAL NETWORKS Figure 1 depicts a class of Wiener nonlinear systems, with the linear block G(z−1) in connection with the nonlinear block f [·].

(4)

i=1

where the system orders (na,nb,p) are known a priori beforehand, and the system parameters (a1,a2,...,ana, b1,b2,...,bnb, c1,c2,...,cp) may exhibit distinct values during the whole range. It is obvious that the system eq 4 could be represented as follow p

A(z −1) y(t ) = −A̅ (z −1) ∑ cigi(y(t )) + B̅ (z −1) u(t − 1)

Figure 1. Discrete-time Wiener-type plant.

i=2

(5)

In this cascade system, the input signal u(t) and the output signal y(t) can be measured, whereas the internal signal v(t) is assumed to be unavailable. The linear dynamic subsystem is expressed as A(z −1)v(t ) = B(z −1)u(t − 1) −1

−1

(1)

where A(z ) and B(z ) are polynomials in the unit time delay operator z−1 (e.g., z−1u(t) = u(t − 1)) which are defined as follows A(z −1) = 1 + a1z −1 + a 2z −2 + ... + anaz −na B(z ) = b1 + b2z G(z −1) =

−1

+ ... + bnbz

−1

−1

where A̅ (z ) = 1/c1A(z ) and B̅ (z ) = 1/c1B(z ). Remark 1. The benefits of the Wiener model (5) lie in two aspects: (i) The linear transfer function (1) requires no information about the process states, for instance, the concentrations of some species. (ii) The RBF network (3) can precisely approximate severe nonlinearities and thus enhance the applicability greatly. Meanwhile thanks to the feature of being linearly described by (g1(·),g2(·),...,gp(·)), an explicit control law is obtainable and will be derived in the next section.

−1

−1

−1

3. WIENER MODEL CONTROL SCHEME 3.1. Optimal Control Law for the Wiener System. Consider the Clarke criterion function:23

−nb + 1

B(z −1) A(z −1)

J[u(t )] = [P(z −1) y(t + 1) − Ry*(t + 1) + Q (z −1) u(t )]2 (6) 10148

DOI: 10.1021/acs.iecr.6b02214 Ind. Eng. Chem. Res. 2016, 55, 10147−10159

Article

Industrial & Engineering Chemistry Research ϒ(t ) = P(z −1) y(t ) + Q (z −1) u(t − 1)

where y*(t) is a known and bounded reference, the weighting polynomials P(z−1) and Q(z−1) are of the following forms P(z−1) = p0 + p1z−1 + ... + pnpz−np and Q(z−1) = q0 + q1z−1 + ... + qnqz−nq, and R is a scalar. Obviously, by setting J[u(t)] = 0, the one-step-ahead control law for known Wiener systems should be derived from Q(z−1) u(t) = Ry*(t + 1) − P(z−1) y(t + 1). By using the similar approach as that introduced by Clarke et al.,23 the control law that ensures the criterion function (6) to be minimized is

By combing eqs 5, 7, and 10, we have ϒ(t ) = S(z −1)y(t − 1) + p0 A(z −1)y(t ) + Q (z −1)u(t − 1) p −1

i=2 −1

= s1y(t − 1) + s2y(t − 2) + ... + snsy(t − ns) + t1u(t − 1) + t 2u(t − 2) + ... + tntu(t − nt )

i=2

− p0 c 2/c1·g2(y(t )) − p0 c3/c1·g3(y(t )) − ...

p

+ p0 A (z −1) ∑ cigi(y(t )) − S(z −1)y(t )

− p0 cp/c1·gp(y(t )) − ... − p0 anac 2/c1·g2(y(t − na))

(7)

i=2

where S(z−1) = s1 + s2z−1 + ... + snsz−ns + 1 and T(z−1) = t1 + t2z−1 + ... + tntz−nt + 1. Both of the parameters (s1,s2,...,sns, t1,t2,...,tnt) and the orders (ns,nt) are determined by P(z−1) = p0A(z−1) + z−1S(z−1) and T(z−1) = p0B̅ (z−1) + Q(z−1). Also the polynomial A̅ (z−1) is divided into A̅ (z−1) = 1/c1 + z−1A̿ (z−1); the term gi(y*(t + 1)) represents gi(y*(t + 1)) = Φi(|y*(t + 1) − ϖ|), i = 2,3,...,p. Remark 2. Most interestingly, the optimal control law (7) copies with unstable zero-dynamics problem and requires no inverse network training. By substituting the desired relationship Ry*(t) = P(z−1) y(t) + Q(z−1) u(t − 1) into the Wiener system (5), we have the following closed-loop equations −1

− p0 anac3/c1·g3(y(t − na)) − ... − p0 anacp/c1·gp(y(t − na))

ϒ(t ) = φ T(t )θ

(12)

in which the elements θ and φ(t) represent

−1

θ = [t1 , t 2 , ..., t nβ̅ , p0 c 2/c1 , p0 c3/c1 , ..., p0 cp/c1 ,

−1

−1

= −A̅ (z ) Q (z ) ∑ cigi(y(t )) + B̅ (z ) Ry*(t ) i=2

−s1 , p0 a1c 2/c1 , ..., p0 a1cp/c1 , ...p0 a nα̅ cp/c1]T

(8)

= [θ1 , θ2 , ..., θnβ̅ , θnβ̅ + 1 , θnβ̅ + 2 , ..., θnβ̅ + p − 1 , θnβ̅ + p ,

[A(z −1) Q (z −1) + B̅ (z −1) P(z −1)]u(t − 1)

θnβ̅ + p + 1 , ...θnβ̅ + 2p − 1 , ...θnβ̅ + (nα̅ + 1)p − 1]T

p −1

−1

= A̅ (z ) P(z ) ∑ cigi(y(t )) + A(z ) Ry*(t ) i=2

(9)

P(z −1) B̅ (z −1) + Q (z −1) A(z −1) ≠ 0,

(13)

φ(t ) = [u(t − 1), u(t − 2), ..., u(t − nβ̅ ),

From the closed-loop eqs 8 and 9, in order to ensure the closed-loop system stability, the weighting polynomials P(z−1) and Q(z−1) should be chosen such that Condition 1.

−g2(y(t )), −g3(y(t )), ..., −gp(y(t )), −y(t − 1), −g2(y(t − 1)), ..., −gp(y(t − 1)), ..., −gp(y(t − nα̅ ))]T

for|z| ≥ 1

(14)

In eqs 13 and 14, the constants nα̅ and nβ̅ are sure to satisfy nα̅ = na and nβ̅ = nb. Then the predicted output can be expressed as ϒ̂ (t + 1) ≜ T φ (t + 1)θ̂(t), where θ̂(t) is an estimate of θ at the current instant t

Meanwhile, to achieve the zero output tracking error between the output y(t) and the reference y*(t), these weighting terms should also satisfy that Condition 2. P(1) = R ,

−1

From eq 10 and 11, the orders of P(z ) and Q(z ) should be chosen as np ≤ na and nq ≤ nb to guarantee ns = na and nt = nb. Otherwise, if np > na and nq > nb, then the system (11) will induce additional computational burden in the sense ns > na and n t > nb. Further, by introducing the parameter vector θ and the information vector φ(t), the system 11 is redefined as follows

p

−1

(11) −1

[A(z ) Q (z ) + B̅ (z ) P(z )]y(t ) −1

−1

+ (p0 B̅ (z ) + Q (z ))u(t − 1)

T (z −1)u(t ) = Ry*(t + 1) + p0 /c1 ∑ cigi(y*(t + 1))

−1

−1

= S(z )y(t − 1) − p0 A̅ (z ) ∑ cigi(y(t ))

p

−1

(10)

Q (1) = 0

θ(̂ t ) = [θ1̂ , θ2̂ , ...θn̂ β̅ , θn̂ β̅ + 1 , θn̂ β̅ + 2 , ...θn̂ β̅ + p − 1 , θn̂ β̅ + p ,

Remark 3. The weighting polynomials P(z−1), Q(z−1), and R are quite easy to choose in practical applications. To meet Condition 1, P(z−1) and Q(z−1) are often chosen as stable polynomials. On the other hand, the key to meet Condition 2 is to guarantee Q(1) = 0. This property is generally prespecified by setting Q(z−1) as Q(z−1) = (1 − z−1)·Q′(z−1) for a polynomial Q′(z−1). 3.2. System Parameters Identification. Let us first define an extended system ϒ(t), which is associated with eq 6.

T

θ n̂ β̅ + p + 1 , ...θn̂ β̅ + 2p − 1 , ...θn̂ β̅ + (nα̅ + 1)p − 1]

(15)

The estimate θ̂(t) can be offline identified by the following weighted recursive least-squares algorithm32 a(t ) M(t − 1) φ(t ) e(t ) θ(̂ t ) = θ(̂ t − 1) + 1 + φ T(t ) M(t − 1) φ(t ) 10149

(16)


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Industrial & Engineering Chemistry Research M(t ) = M(t − 1) −

a(t ) M(t − 1) φ(t ) φT (t )M(t − 1) 1 + φ T(t ) M(t − 1) φ(t )

4.1. Integrations of Multiple Wiener Models. Several prediction models are established based on different operating conditions. This work employs L neural networks-based Wiener models M1,M2,...,ML, which are denoted as

(17)

e(t ) = P(z ) y(t ) + Q (z ) u(t − 1) − φ (t ) θ(̂ t − 1) −1

−1

T

i i M i : ϒ̂ (t + 1) = φ T(t + 1)θ ̂ (t ), i = 1, 2, ..., L (21) where ϒ̂ i(t + 1) is the ith prediction output; θ̂i(t) is the estimation of θ for the ith Wiener model and is defined by eq 22. Obviously, it has a similar form as eq 15 and can be offline identified by eqs 16−18:

(18)

where e(t) denotes the model error and a(t) is a user-chosen weighting coefficient. 3.3. Control Law Design. On the basis of the estimate θ̂(t), we can obtain the following equation Ry*(t + 1) = ϒ̂(t + 1) = φ T(t + 1) θ(̂ t )

i

i

1 {Ry*(t + 1) − θ1̂

i

nβ

∑ θk̂ ·u(t + 1 − k) k=2

+

u i (t ) =

∑ (θn̂ ̅ + r− 1·gr (y*(t + 1))) β

r=2 nα

i

i

1 i {Ry*(t + 1) − θ̂ 1

p

p

∑ ∑ (θn̂ ̅ + lp + r− 1·gr (y(t + 1 − l)))} β

l=1 r=1

i

i

i

i

T

Equating ϒ̂ i(t + 1) with Ry*(t + 1) yields that

p

+

i

θ n̂ β̅ + p + 1 , ...θn̂ β̅ + 2p − 1 , ...θn̂ β̅ + (nα̅ + 1)p − 1]

From eq19, the explicit control law u(t) is then calculated by u(t ) =

i

θ ̂ (t ) = [θ1̂ , θ2̂ , ...θn̂ β̅ , θn̂ β̅ + 1 , θn̂ β̅ + 2 , ...θn̂ β̅ + p − 1 , θn̂ β̅ + p ,

(19)

+ (20)

+

i

∑ θk̂ ·u(t + 1 − k) k=2

i

∑ (θn̂ ̅ + r− 1·gr (y*(t + 1))) β

r=2 nα

where g2(y*(t + 1)) = Φ2(|y*(t + 1) − ϖ2|), g3(y*(t + 1)) = Φ3(|y*(t + 1) − ϖ3|), ...gp(y*(t + 1)) = Φp(|y*(t + 1) − ϖp|). Remark 4. The above control law (20) is of a straightforward form and is easy to obtain. Note that compared with similar neural network Wiener models-based control methods,8,11,24 which computes the control signal u(t) via training another inverse network, the explicit control law (20) also reduces the computational burden and avoids additional modeling errors.

nβ

(22)

p

i

∑ ∑ (θn̂ ̅ + lp + r− 1·gr (y(t + 1 − l)))} β

l=1 r=1

(23)

where g2(y*(t + 1)) = Φ2(|y*(t + 1)−ϖ2|), g3(y*(t + 1)) = Φ3(|y*(t + 1)−ϖ3|), ...gp(y*(t + 1)) = Φp(|y*(t + 1)−ϖp|); ui(t) is the control signal obtained from the i − th Wiener model. 4.2. The Robust Switching Mechanism. Only one control channel is “open” for the current instant t, and the control law u(t) = u(t)i with the model Mi exhibiting the most appropriate performance. Similar to the previous technique,33,34 we define a switching criterion as below

4. MULTIPLE WIENER MODELS CONTROL SCHEME For many complex processes, the multiple models scheme has gradually been considered as an efficient tool to further improve the performance.33 The intelligent control scheme based on multiple models and RBF networks is plotted in Figure 2.

J i (t ) = || ei(t )||,

i = 1, 2, ..., L

(24)

̂i

with e (t) = ϒ(t) − ϒ (t). i

Figure 2. Control scheme based on multiple Wiener models and neural networks. 10150


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Industrial & Engineering Chemistry Research

Figure 3. Schematic diagram of the pH process.

Table 1. Nominal Operating Conditions u3 = 16.6 mL/s u1 = 15.55 mL/s Wa1 = −3.05 × 10−3 mol Wa3 = 3 × 10−3 mol Wb1 = 5 × 10−5 mol Wb3 = 0 mol pK1 = 6.35 y ̅ = 7.0

u2 = 0.55 mL/s V = 2900 mL Wa2 = −3 × 10−2 mol Wa = −4.32 × 10−4 mol Wb2 = 3 × 10−2 mol Wb = 5.28 × 10−4 mol pK2 = 10.25

Figure 4. Titration curve (input−output map) for the pH process.

Meanwhile, to guarantee the robustness of this switching law and reduce the extreme switch chattering phenomenon, a hysteresis function is incorporated into the selection of control law.35 Two variables are introduced: (i) SWITCH(t) = i denotes the selected model number i at the instant t; (ii) J ̅ denotes the hysteresis constant. Then u(t) = u(t)i with the model number i being determined by the following: if

Figure 5. Excitations for the offline identification procedure.

4.3. Overall Control Scheme. The proposed multiple models and neural networks Wiener model-based control scheme (MM-NN-W-C) is composed of two procedures: the offline identification part and the online implementation part. In sum, this method can be summarized as the following steps: (i) offline identification Step 1. Measure input−output data {u(t),y(t)} and obtain φ(t) from eq 14. Step 2. Identify θ̂1(t),θ̂2(t),...,θ̂L(t) for M1,M2,...,ML from eq 16−18.

J i (t ) ≤ J ̅ and SWITCH(t − 1) = i

then SWITCH(t ) = i else if

J i (t ) = min[J1(t ), J 2 (t ), ..., J L (t )]

then SWITCH(t ) = i

(25) 10151


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Figure 6. Convergence properties of the parameter estimations.

(ii) online implementation Step 1. Online calculate J1(t),J2(t),...,JL(t) from eq 24. Step 2. Select the best model number i from eq 25. Step 3. Solve the control law u(t) = u(t)i from eq 23. Step 4. Apply u(t) to the plant and return to Step 1 by t = t + 1.

5. APPLICATION TO A PH NEUTRALIZATION PROCESS PH Neutralization Process.36 The schematic diagram of this process is depicted in Figure 3. The constant volume (V) stirred tank consists of three streams: the acid (HNO3), the base (NaOH), and the buffer (NaHCO3). The pH value is the effluent solution. In this study, the process output pH (y)̅ is controlled by manipulating the base flow rate (u1), while the 10152


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Figure 7. Set-point tracking results.

buffer flow rate (u2), and the acid flow rate (u3) are considered as system parameters.

This work borrows the first principle-based simulation model from Henson and Seborg.36 This nonlinear 10153


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Industrial & Engineering Chemistry Research 2

dynamic model can be expressed in the following state-space form x1̇ =

M 2 : θ ̂ (t ) = [0.2879, − 0.2613, − 0.2848, − 0.2952,

u u1 u ·(Wa1 − x1) + 2 ·(Wa2 − x1) + 3 ·(Wa3 − x1) V V V

−0.2895, 0.1134, 0.8537, −0.8259, −1.4348,

(26)

x 2̇ =

−1.4073, −1.4467, 0.7252, 2.9777, 0.3669, − 1.8153, −1.8245, −1.7518, 1.9592, 3.1494]T

u u1 u ·(Wb1 − x 2) + 2 ·(Wb2 − x 2) + 3 ·(Wb3 − x 2) V V V

3 M3: θ ̂ (t )

(27)

g (x , y ̅ ) = x1 + 10 y ̅ − 14 − 10−y ̅ 1 + 2 × 10 y ̅ − pK 2 + x 2· =0 1 + 10 pK1− y ̅ + 10 y ̅ − pK 2 T

T

x = [x1 , x 2] = [Wa , Wb]

= [0.3219, − 0.2895, 8.3905, − 0.6687, − 2.5420, −2.5385, −2.5216, −1.0685, − 3.8501, 2.4305, 0.4516, 0.4833, 0.4481, 0.5267, − 4.0399,

(28)

1.1674, 0.9722, 0.9696, 0.9645]T (29)

Later, we will show that M1,M2, and M3 are more suitable when the pH is near 7, 10, and 3, respectively. For the online implementation, the switching mechanism is designed as eq 24−25 with a hysteresis constant J ̅ = 0.011. After selecting the best Wiener model number i(1,2,3), the corresponding controller u(t) = u(t)i is calculated from eq 23. For comparisons, three other control methods are applied to this process. (i) The multiple models-based linear control (MM-L-C):29 This method handles the titration curve by separating the range into different regions, obtaining a linear model for each region, and designing the corresponding linear controller. In sum, this highly nonlinear behavior is intended to be captured by a piecewise-linear representation.27 (ii) The neural networks Wiener model based control (NN−W-C):24 In this scheme, the main idea is to train inverse neural networks to eliminate the static nonlinear block. More precisely, an offline identification procedure is first carried out. The linear and nonlinear blocks can be identified separately by a two-step technique. Then based on the identification results, the control of the Wiener model relies on a linear controller incorporated with an inverse network. (iii) The polynomial Wiener model based adaptive control (PL-W-AC):9 This method approximates the static nonlinear block by a polynomial-basis function and may lead to unsatisfactory modeling error. Note that this method is presented in an adaptive scheme, which will account for possible parametric uncertainties to some extent. Set-Point Tracking. Set-point responses of these four alternatives are shown in Figure 7 with process output, process

Equation 28 implicitly connects the process output with the process states. Under the nominal operating condition, the values of the parameters in eq 26−28 are presented in Table 1. Note that all the parameters remain unchanged except that u3 varies for the acid flow rate disturbances situation. It is a consensus that the control of the pH processes has always been a challenge due to the strong nonlinear behaviors and time-varying characteristics with respect to different values of the manipulated variables. The titration curve (input−output map) for this process is given in Figure 4. It is seen that this process exhibits distinct nonlinear behaviors during a wide operation range. Despite this severe nonlinearity, it has been verified that a Wiener model structure is able to represent the nonlinear property of this process.37,38 Control Schemes. Four representative controllers are applied to this pH process. The proposed scheme is used to model and control the pH process which is sampled every 0.25 min. The base flow rate u1 is constrained to [0 mL/s, 31 mL/s]. For the Wiener model, the input and output variables are respectively normalized by u = (u1 − 15.5)/15.5 and y = (y ̅ − 7)/7. The model orders are first chosen as na = nb = 2, p = 6. That is, a RBF network with 5 neurons in the hidden layer is used. The centers are ϖ2 = −0.4, ϖ3 = −0.2, ϖ4 = 0, ϖ5 = 0.2, ϖ6 = 0.4. Φi(·) is chosen as Gaussian functions. The variable λ = 5. The weighting polynomials are chosen as P(z−1) = 1 − 0.4z−1, Q(z−1) = 0.25(1 − z−1), R = 0.6. For the offline identification procedure, the weighting coefficient is a(t) = 1. The excitations come from process input−output data around some different operation regions. The first, second, and third groups of process input− output data are shown in Figure 5, respectively. On the basis of these data, three groups of parameter estimations are conducted offline to build up Wiener models M1,M2,M3. To show the convergence properties, the parameter estimations θ̂1(t) for M1, θ̂2(t) for M2, and θ̂3(t) for M3 are plotted in Figure 6, respectively. Finally, three models M1,M2,M3 can be obtained as follows: 1

M1: θ ̂ (t ) = [0.3605, − 0.3052, 3.7924, 3.7984, 3.2060, − 3.3352, −7.3923, −1.0342, 1.3335, 1.3721, − 2.2476, −1.9291, 1.3327, 0.5360, 2.2080, Figure 8. Non-Gaussian measurement noise added at the process output pH.

2.2149, −2.8327, −3.7296, 2.2072]T 10154


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Figure 9. Disturbance rejection performance.

input, and switching sequences. The corresponding statistical results can also be found in the figure with the root-mean-square errors (RMSE) and the mean absolute differences (MAD). It is seen that all these methods guarantee the system stability and achieve convergent tracking performance. Compared with

the other controllers, the NN-W-C method requires a bit longer settling time and leads to more oscillation. Its statistical property is the poorest. The obtained results indicate that one single neural networks-based Wiener model can hardly precisely describe such a highly nonlinear process for the 10155


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Industrial & Engineering Chemistry Research whole range and the control performance may deteriorate sometimes. The PL-W-AC method achieves a satisfactory output performance for the set-point pH = 4 but leads to a quite sluggish output tracking for the set-point pH = 10. The results show that the response speed is slow for large set-point changes and the operation region of this method is narrow. One reason is that the adaptation costs a lot of time. The other reason is that a polynomial Wiener model is subject to various modeling errors, which confines the applicability. Thus, the reduction of modeling errors by a bank of neural network Wiener models is necessary and meaningful. On the other hand, both of the two multiple models based controllers (MM-L-C and MM-NN-W-C) possess good setpoint tracking performance, especially for the faster response speed and the larger operation region. It can be seen that the multiple models scheme is suitable for distinct operation regions. Moreover, the MM-NN-W-C method outperforms the MM-L-C method. It is clear that a bank of neural network Wiener models is more promising than a bank of simple linear models, due to good approximation accuracy and possibility of incorporating prior process knowledge. Meanwhile, the switching sequence for the MM-NN-W-C method is shown in Figure 7c with 1,2,3 representing the corresponding controllers for M1,M2,M3. This result verifies the effectiveness of the switching mechanism since no extreme switch chattering phenomenon exists. Non-Gaussian Measurement Noise. Stochastic disturbances are added to the process output pH value, which denotes that there exists measurement noise. Figure 8 shows the non-Gaussian disturbance signal. The reference of pH process is fixed at pH = 7. Disturbance rejection performance of these four methods is shown in Figure 9. The statistical results about RMSE and MAD are also given in this figure. From the obtained results, it is seen that the PL-W-AC method is sensitive to measurement noise especially at 5, 15, and 25 minutes. Its statistical property is the poorest. This is because the adaptive controller continuously identifies system parameters based on process input and output signals. The online recursive estimator is greatly affected by stochastic noises, especially when non-Gaussian disturbances exist. Thus, the applicability of the PL-W-AC method has always been a big concern. The other three methods (MM-L-C, NN-W-C and MM-NN-W-C) can effectively reject stochastic disturbances. However, the differences are also visible: the MM-L-C method rejects the noises a bit slower and the regulation is not that timely especially during 5th to 25th minutes. On the other hand, the disturbance rejection performances of the NN-W-C and MM-NN-W-C methods are both promising. These two neural networks Wiener-based controllers exhibit similar statistical properties. Moreover, in this study, the hysteresis constant for the MM-NN-W-C method has been increased to J ̅ = 0.045. As shown in Figure 9c, this modification can ensure the robustness of the switching mechanism. Acid Flow Rate Variations and Gaussian Measurement Noise. In this study, a more complex situation is considered. Assume there exist disturbances in both the acid flow rate u3 (its variations are shown in Figure 10, which can cause parametric uncertainties) and process output pH (it is corrupted by Gaussian white noise with a variance of 0.022).

Figure 10. Variations in the acid flow rate u3.

In this study, the reference is fixed at pH = 7, and the regulation results under both uncertainties are shown in Figure 11. The statistical results about RMSE and MAD are also given in this figure. As expected, the PL-W-AC method can effectively account for the parametric uncertainties since this controller is derived from a self-tuning control scheme32 and it can update the parameters online according to the variations of u3. Its steady state tracking performances are better than all the other nonadaptive controllers since zero tracking error property is likely to be ensured. However, the prices it paid are stronger excitations and more regulations. This phenomenon is consistent with the dual control problem.39 Although there is no update in controller parameters, the MM-NN-W-C method copies with parametric uncertainties to some extent, thanks to the multiple models approach. Generally speaking, it is seen that the MM-NN-W-C method causes milder overshoot and oscillation than the PL-W-AC method, and possesses a better statistical results. Meanwhile, it is worth noting that the possible steady state tracking errors can be further reduced by offline design of more Wiener models. On the other hand, due to the multiple models approach, the MM-L-C method can also handle parametric uncertainties to some extent but its performance is not as good as the PL-W-AC and MM-NN-W-C methods. Obviously, the NN-W-C method exhibits the poorest performance since it is based on a nonadaptive Wiener model. It does not address time-varying process behavior caused by acid flow rate changes. Summary. Some observations from these results are given as follows: (i) The NN-W-C method is applicable in some narrow regions, but sensitive to parametric uncertainties caused by acid flow rate variations. (ii) The PL-W-AC method rejects parametric uncertainties effectively, but achieves unsatisfactory tracking performance for a wide operation region and deteriorates greatly under non-Gaussian disturbances. (iii) The MM-L-C method exhibits quite plain performances for set-point tracking, disturbance rejection, and parametric uncertainties, which deserves an improvement by the combination of using neural Wiener models and regional handling of pH processes. (iv) In general, the MM-NN-W-C method has wider applicability than the other three alternatives. It can operate well for distinct regions, reject non-Gaussian disturbances, and account for parametric uncertainties to some extent. The intelligent approach adds only a few additional costs than a single model approach as it requires some further steps in the model selection, but the overall computational complexities are still quite low and acceptable. 10156


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Figure 11. Regulation results under both u3 and pH uncertainties.

6. CONCLUSION This work addresses the modeling and control problem of Wiener-type nonlinear processes. The salient features of the

proposed scheme can be summarized as follows: (i) the neural networks reduce approximation error and improve control effects; (ii) the explicit control law is straightforward in form 10157


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and requires no inverse network training; (iii) the method copies with possible unstable zero-dynamics problems in industrial processes; (iv) this method operates in distinct regions and has reliable closed-loop properties. Furthermore, this controller is also applied to a simulated pH neutralization process and compared with three other representative control methods. The numerical results have confirmed the remarkable advantages of the proposed intelligent controller. Hopefully, this interesting modeling and control scheme may also find wider applicability in other Wiener-type industrial processes.40−43

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the National Natural Science Foundation of China (61473072). REFERENCES

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