402 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
Nonlinear Model Predictive Control Based ona Self-Organizing Recurrent Neural NetworkHong-Gui Han, Senior Member, IEEE, Lu Zhang, Ying Hou, and Jun-Fei Qiao, Member, IEEE
Abstract— A nonlinear model predictive control (NMPC)scheme is developed in this paper based on a self-organizingrecurrent radial basis function (SR-RBF) neural network,whose structure and parameters are adjusted concurrently inthe training process. The proposed SR-RBF neural network isrepresented in a general nonlinear form for predicting the futuredynamic behaviors of nonlinear systems. To improve the modelingaccuracy, a spiking-based growing and pruning algorithm and anadaptive learning algorithm are developed to tune the structureand parameters of the SR-RBF neural network, respectively.Meanwhile, for the control problem, an improved gradientmethod is utilized for the solution of the optimization problemin NMPC. The stability of the resulting control system is provedbased on the Lyapunov stability theory. Finally, the proposedSR-RBF neural network-based NMPC (SR-RBF-NMPC) isused to control the dissolved oxygen (DO) concentration ina wastewater treatment process (WWTP). Comparisons withother existing methods demonstrate that the SR-RBF-NMPCcan achieve a considerably better model fitting for WWTP anda better control performance for DO concentration.
Index Terms— Dissolved oxygen (DO) concentration, nonlinearmodel predictive control (NMPC), recurrent radial basis func-tion (SR-RBF) neural networks, self-organizing, wastewatertreatment process (WWTP).
I. INTRODUCTION
MODEL predictive control (MPC), a powerful modelbased control technique, has proved its continuing
success in industrial applications, particularly in the pres-ence of constraints and varying operating conditions [1]–[3].MPC has been well developed in the last decades, both in theindustry and in academia. Recent brief survey and review ofMPC technologies and methods can be found in [4] and [5].
Manuscript received August 11, 2014; accepted July 30, 2015. Date ofpublication August 27, 2015; date of current version January 18, 2016.This work was supported in part by the Beijing Nova Program underGrant Z131104000413007, in part by the China Post-Doctoral Science Foun-dation through the Hong Kong Scholar Program under Grant 2014M550017and Grant XJ2013018, in part by the Beijing Science and TechnologyProject under Grant Z141100001414005 and Grant Z141101004414058, inpart by the National Science Foundation of China under Grant 61203099,Grant 61225016, and Grant 61533002, in part by the Ph.D. Program Founda-tion from Ministry of Chinese Education under Grant 20121103120020, andin part by the Beijing Municipal Education Commission Foundation underGrant km201410005001 and Grant KZ201410005002.
H.-G. Han is with the Beijing Key Laboratory of Computational Intelligenceand Intelligent System, College of Electronic and Control Engineering,Beijing University of Technology, Beijing 100124, China, and also with theDepartment of Mechanical and Biomedical Engineering, City University ofHong Kong, Hong Kong (e-mail: [email protected]).
L. Zhang, Y. Hou, and J.-F. Qiao are with the Beijing Key Laboratory ofComputational Intelligence and Intelligent System, College of Electronic andControl Engineering, Beijing University of Technology, Beijing 100124, China(e-mail: [email protected]; [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2015.2465174
In MPC, at each instant of sampling time, the control lawsare computed by solving a constrained finite horizon optimalcontrol problem in which the current state is set as an initialstate (see [6], [7]). This requires a reasonably accurate modelof the plant dynamics that captures the essential dynamicsof the process, to predict multistep-ahead dynamic behaviors.However, many reported applications of MPC are based onlinear models and only the systems’ behaviors over a givenhorizon can be predicted. In this case, MPC often results inpoor control performance for highly nonlinear processes due tothe considerable modeling error caused by the linear predictedmodel [8], [9]. Moreover, in many practical applications,e.g., wastewater treatment process (WWTP), a linear modelbased on physical principles is either not available or toocomplicated to be used for control [10]. In this scenario,nonlinear system modeling methods are desirable in nonlinearMPC (NMPC). Therefore, an efficient and effective modelingmethod for nonlinear systems is critical to the success ofNMPC [11].
To identify complex nonlinear systems for NMPC, manyresearchers have studied various modeling methods andsearched for suitable mathematical methodologies to improveaccuracy [12], [13]. However, unlike linear system modelingmethods, there is no uniform way to identify generalnonlinear dynamic systems. Recently, based on a biologicalprototype of the human brain, the neural networks haveattracted considerable attention for modeling uncertain,nonlinear, and complex systems, owing to their learning andadaptation capabilities [14], [15]. In addition, compared withother techniques, the neural networks have massive parallelprocessing capabilities, which can be exploited in practicalcontroller implementation. In general, the structures of neuralnetworks can be classified as feedforward and recurrent. Mostof the works on NMPC use feedforward neural networkswith backpropagation or its variations for modeling nonlinearsystems. For example, Nikdel et al. [16] present a newneural network model predictive controller, developed by afeedforward neural network model, to control a single degreeof freedomrotary manipulator. In [17], a neural network modelpredictive controller, still realized by using the feedforwardneural network model, is utilized for predictive control of thepower system to improve its transient stability. However, thefeedforward neural networks used in [16] and [17] are trainedby the backpropagation algorithm, which has a slow conver-gence speed and may be trapped into local minima in trainingprocess. To improve modeling accuracy, Yan and Wang [18]introduce a robust MPC based on a feedforward neuralnetwork but with the extreme learning machine to model the
HAN et al.: NMPC BASED ON A SELF-ORGANIZING RECURRENT NEURAL NETWORK 403
linearization residue. The results show that this robust MPCcould improve computational efficiency and shed a light forreal-time implementation. Then, a multilayer feedforwardneural network-based MPC scheme is developed for amultivariable nonlinear steel pickling process in [19]. In thismultivariable MPC, the feedforward neural network is adjustedby the Levenberg–Marquardt algorithm to predict the statevariables. Recently, some other kinds of feedforward neuralnetworks have been introduced for NMPC. In [20], an adaptivestate observer with radial basis function (RBF) neural networkis developed to estimate the unknown state of a class of time-varying delay nonlinear systems. Peng et al. [21] propose amulti-input and multioutput (MIMO) RBF neural network tomodel nonlinear systems. These feedforward neural networksused in [16]–[21] and some others [22]–[24] lead to signifi-cantly improved modeling performance than linear models andthus are able to achieve better control performance of nonlinearsystems. However, the main drawback of feedforward neuralnetworks is that they are essentially static input-to-outputmaps and their capability for representing nonlinear systems,especially complex and/or time-varying, is limited [25], [26].Besides, they can only provide predictions for a predeterminedfinite number of steps, in most cases, it is only one step.
Recurrent neural networks, on the other hand, are capable ofproviding long-range predictions even in the presence of mea-surements noise due to their structures. Therefore, recurrentneural networks are better suited to model nonlinear systemsfor NMPC. Pan and Wang [27] use an echo state network toidentify unknown nonlinear dynamical systems for NMPC.The results show that the echo state network-based NMPCcan reach the global convergence [27]. Al Seyab and Cao [28]develop a continuous time recurrent neural network as aninternal model for NMPC. This continuous time recurrentneural network, represented in a general nonlinear state spaceform, obtains good performance under different operatingconditions [28]. In [29], two recurrent neural networkswith different time-scales are proposed for nonlinear systemmodeling, and the corresponding online learning algorithmsare designed based on Lyapunov functions and singularlyperturbed techniques. Then, with the identified models, twoindirect predictive controllers are developed for nonlinearsystems containing both slow and fast dynamic processes.Furthermore, a NMPC, based on dynamic recurrent neuralnetworks, is used for the industrial baker’s yeast dryingprocess in [30]. Two prediction models have been realizedusing the recurrent neural network and external recurrentneural network, respectively. The case studies show thatthe performances of the two prediction models are moresuccessful than those of the feedforward neural networkmodels and the nonlinear partial differential equation-basedmodels. In addition, other applications of recurrent neuralnetworks for NMPC have also been investigated in [31]–[33].It can be seen from all these works that recurrent neuralnetworks are useful for modeling nonlinear dynamic systems,and are thus feasible for NMPC techniques. However, howto achieve better modeling accuracy of nonlinear dynamicsystems is still a challenge [34], [35]. When recurrent neuralnetworks are used to predict system behaviors, the prediction
accuracy may be degraded when structures and/or parametersof nonlinear systems change. In most existing works, thestructure of recurrent neural networks and thus the numberof hidden neurons are fixed after they are initially determinedfor modeling nonlinear systems. With the fixed structures,NMPC might run into trouble [36]. Ideally, when a recurrentneural network is used to model nonlinear systems, thenumber of hidden neurons should be self-organized online toget the sufficient modeling accuracy [37]–[39].
Motivated by the above review and analysis, a NMPCstrategy, based on a self-organizing recurrent RBF (SR-RBF)neural network, is developed for nonlinear systems modelingin this paper. The advantages of the SR-RBF neuralnetwork-based NMPC (SR-RBF-NMPC) can be summarizedas follows. First, a novel spiking-based growing and pruningalgorithm is proposed to optimize the structure of the SR-RBFneural network. The neurons in the hidden layer of SR-RBFneural network are self-organized through adaptive strategies.That is, the number of hidden neurons can be increased orpruned depending on the particular condition of the concernedsystem as time goes on. It is expected that better modelingaccuracy can be achieved by utilizing the SR-RBF neuralnetwork. Second, based on an improved gradient method,the solution of the optimization control problem is obtained.This gradient method relatively allows to consider the con-straints imposed on process variables. Third, the proposedSR-RBF-NMPC has been designed based on the Lyapunovstability theory such that the closed loop control system isasymptotically stable.
The remainder of this paper is as follows. Section IIgives the problem formulation and preliminaries for the MPCstrategy and the recurrent RBF neural networks. In Section III,the basic idea of the spiking-based growing and pruning algo-rithm and the details of SR-RBF-NMPC are presented. Then,in Section IV, the stability of the proposed SR-RBF-NMPC isdiscussed. The simulation results are presented to demonstratethe effectiveness of SR-RBF-NMPC in Section V, and the finalconclusions are given in Section VI.
II. PROBLEM FORMULATION AND PRELIMINARIES
A. Model Predictive Control
Consider an MPC formulation based on the followingconstrained finite-horizon optimization [40]:
J (t) =Hp∑
i=1
[r(t + i) − y(t + i)]T W yi [r(t + i) − y(t + i)]
+Hu∑
j=1
�u(t + j − 1)T W uj �u(t + j − 1) (1)
subject to
|�u(t)| ≤ �umax
umin ≤ u(t) ≤ umax
ymin ≤ y(t) ≤ ymax (2)
where r , y, u, and y represent the reference, measuredoutput, control input, and predicted output, respectively.
404 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
Fig. 1. MISO recurrent RBF neural network.
�u(t) and �u(t + 1), . . . ,�u(t + Hu − 1) are the setof present and future incremental control moves, y(t + 1),y(t + 2), . . . , y(t + Hp) are the future outputs of the process,Hp is the prediction horizon, Hu is the control predictionhorizon (Hu < Hp), W y
i and W ui are the weighting parameters,
and J is the cost function. According to the minimizationof a performance criterion, MPC is an iterative optimizationstrategy repeatedly at each time step t over a finite predictionhorizon. The solution to such an optimization control problemdepends on current state and leads to an optimal controlsequence.
In this paper, the nonlinear dynamic system is supposed tobe represented as a nonlinear autoregressive exogenous modelof the following form:
y(t) = g(y(t − 1), . . . , y(t − ny),
u(t − 1), . . . , u(t − nu − td )) + d(t) (3)
where u(t) and y(t) denote the process control input and mea-sured output, respectively, d(t) is the noise, the function g(·)is assumed to be unknown, ny and nu are the maximum lagsin the outputs and inputs, respectively, and td is the time delay.
B. Recurrent Radial Basis Function Neural Network
Fig. 1 shows the basic structure of a recurrent RBF neuralnetwork, which consists of one input layer, one output layer,and one hidden layer [51], [52]. In the feedforward phase, theneurons in the input layer receive the information from outside.The neurons in the hidden layer simultaneously receive andprocess the data from the combination of input layer andoutput layer. The output neurons respond to input informationvia a linear combination of nonlinear hidden basis functions.In the feedback phase, the prior outputs of the neural networkare returned back to the hidden layer.
In this multi-input and single-output (MISO) recurrentRBF neural network, the output can be described by
y(t) = f (w, θ) =m∑
j=1
w j (t) × θ j (t), j = 1, . . . , m (4)
where w = [w1, w2, . . . , wm ]T is the connectionweights between the hidden neurons and output neuron,θ = [θ1, θ2, . . . , θm ]T is the output vector of the hidden layer,m is the number of hidden neurons, the function f (·) is theoutput expression of recurrent RBF neural network, and θ j is
Fig. 2. Approximation of the membrane potential in the biological neuralnetwork.
the output value of the j th hidden neuron
θ j (t) = e−‖h j (t)−c j (t)‖/2σ 2j (t) (5)
where c j denotes the center vector of the j th hidden neuron,c j = [c j1, c j2, . . . , c jn+1]T , ‖h j − c j‖ is the Euclideandistance between h j and c j , and σ j is the radius or widthof the j th hidden neuron, h j is input vector of the j th hiddenneuron, described as
h j (t) = [x1(t), x2(t), . . . , xn(t), v j (t) × y(t − 1)]T (6)
where v j denotes the connection weight from output layer tothe j th hidden neuron, and v = [v1, v2, . . . , vm ]T.
Remark 1: It is clear that this recurrent RBF neural networkhas an internal feedback loop between hidden layer and outputlayer. It is expected that the internal feedback loop is able toenhance its capability of modeling.
III. NMPC BASED ON SR-RBF NEURAL NETWORK
A. Spiking-Based Growing and Pruning Mechanism
In the biophysical spike-processing neural circuits,correlations between neuronal spike trains affect neuralnetwork dynamics and population coding [41]. The spikingand plasticity rules are that the exchanged information canbe maximized, while the number of costly spikes betweensuccessive layers of neurons should be minimized. Based onthis observation, the spiking strength (ss) of hidden neurons isused to tune the structure of this recurrent RBF neural network.
Before introducing the spiking-based growing and pruningmechanism, the leaky integrated and fire (LIF) model ofbiological neurons is described by
τmd L(t)
dt= −L(t) + G I (t) (7)
where L(t) is the membrane potential of biological neurons,τm denotes the time span, G is the membrane resistance,and I (t) is an injected current at time t . Based on the resultsin [42], the membrane potential behavior of biological neuronsis shown in Fig. 2.
In general, the LIF model is popular in the computationalneuroscience community, because it can be simulatedefficiently and, perhaps more important for exploringinformation transmission. In the LIF model, the input and
HAN et al.: NMPC BASED ON A SELF-ORGANIZING RECURRENT NEURAL NETWORK 405
Fig. 3. Spiking strength of hidden neuron varies with the hidden outputvalues.
output signals of spiking neurons are represented by thetiming of spikes. The firing mechanism of neurons has acrucial influence on the computation power of the biophysicalneural network. Traditional rate-coded neural networksrepresent an analog variable through the firing rate of thebiological neuron. That is, the output of the biological neuralnetwork is a representation of the firing rate of the biologicalneuron. In order to increase the computational power ofthe network, experimental evidence suggests that the exacttime of the biological neural network can be used to encodeinformation [42]. The membrane potential of each biologicalneuron depends on the stimulus strength, which keepsincreasing until it reaches a spiking threshold. The neuron isthen in the state of action potential. Later, the neural membranepotential will be reset to a silent state, and the correspondingneurons will be in the refractory period and no spiking duty.
Following the characteristics of spiking neurons, in thispaper, the definition of ss is given as:
ss j (t) = −0.6 ln
(2
sin(eln(θ j (t))+8.16)− 3.8
)+ 0.7 (8)
where ss j is the spiking strength of the j th hidden neuron andθ j is the output value of the j th hidden neuron. The detailsof ss j on the outputs of the j th hidden neuron are illustratedin Fig. 3.
The curve of ss, which represents the exciting level ofhidden neurons, with the output values of hidden neuronsas the horizontal axis, can approximate the LIF model.If the value of ss j is larger than the firing threshold ss0, thej th hidden neuron is an active one and will be split. In contrast,if the value of ss j is lesser than the resting potential value,the j th hidden neuron is inactive and will be removed.
Combining ss and the characteristics of recurrentRBF neural networks, the spiking-based growing andpruning algorithm will be presented as follows.
1) Growing Phase: Based on the aforementioned discus-sion, if the spiking strength of the j th hidden neuron satisfies
ss j (t) ≥ ss0 (9)
where ss0 ∈ (0, 0.1) is the growing threshold of the j th hiddenneuron ( j = 1, 2, . . . , m), the j th hidden neuron will be
divided and new neurons will be added to the hidden layer. Thenew neurons are designed by the following initial parameters:
⎧⎪⎨
⎪⎩
c j−i (t) = αi c j (t) + βi h j (t)
σ j−i (t) = αiσ j (t), v j−i(t) = αi v j (t)
w j−i (t) = w j (t)θ j (t)(Nnewθ j−i(t))−1
(10)
where c j and σ j represent the center and radius of presplitj th neuron, and c j−i and σ j−i stand for the center and radiusof the new added i th neuron (i = 1, 2, . . . , Nnew), respectively,with αi ∈ [0.95, 1.05] and βi ∈ [0, 0.1]; v j−i and v j are thefeedback weights of the new i th hidden neuron and the presplitj th neuron, respectively; w j−i and w j represent the connec-tion weights of the new i th hidden neuron and the presplitj th neuron respectively, θ j−i denotes the output of new addedi th neuron, and Nnew is the number of new added neurons.
2) Pruning Phase: In this phase, the redundant neurons areremoved, if
ss j (t) ≤ ssr (11)
where ssr ∈ (0, Ed) is the preset pruning threshold and Ed isthe preset constant. If the j th hidden neuron will be pruned,then the connection weights of the j ′th hidden neuron, denotedas the least Euclidean distance from the j th neuron, will beupdated as
w′j ′(t) = w j ′(t) + w j (t)θ j (t)(θ j ′(t))−1 (12)
where w′j ′ and w j ′ stand for the connection weights between
the j ′th hidden neuron and the output layer before and afterthe j th hidden neuron is cut off, θ j ′ and θ j are the outputsof the j ′th and the j th hidden neurons before the j th hiddenneuron is cut off. The center, radius, and feedback weight ofthe j ′th hidden neuron remain unchanged after the j th hiddenneuron is deleted.
Remark 2: The spiking strength ss can recognize the activeand inactive neurons. Then the growing and pruning algorithmcan find the suitable structure of the recurrent RBF neuralnetwork. These observations led to a new way of designingneural network based on temporal encoding of individualhidden neurons.
Remark 3: The proposed growing and pruning algorithm hastwo major advantages: 1) the definition ofss, as shown in (8),can mimic the dynamics of hidden neurons to choose the rightactive or inactive neurons. The proposed method may alsolead to the structure design of other kinds of recurrent neuralnetworks and 2) adjustments to the structure are performedwithout partitioning the input space. This feature is useful fornonlinear systems modeling.
B. SR-RBF Neural Network
In the proposed SR-RBF neural network, there are fourtypes of parameters: 1) the output weights w; 2) the feedbackconnection weights v; 3) the centers c = [c1, c2, . . . , cm ]T ;and 4) the radii σ = [σ1, σ2, . . . , σm ]T of hidden neurons.Therefore, there are four adaptive learning algorithms,
406 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
where η1, η2, η3, and η4 are the positive learning rates, ande(t) is the current modeling error of the neural network attime t with e(t) = y(t) − y(t). The partial derivative of thenetwork output with respect to the centers of hidden neuronsis given as
and the partial derivative of the network output with respectto the output weights is
ϕw(t) = [ϕw1(t), ϕw2(t), . . . , ϕwm (t)]T
ϕw j (t) = θ j (t)−ϕTc j
(t)c j (t)−ϕσ j (t)σ j (t)−ϕv j (t)v j (t) (19)
where j = 1, 2, . . . , m.
TABLE I
DETAILS OF SR-RBF NEURAL NETWORK TRAINING PROCESS
In the proposed SR-RBF neural network, the spiking-basedgrowing and pruning algorithm is used to tune the networkstructure. Meanwhile, the parameter adaptation strategy isapplied to tune the parameters. The main procedures oftraining process of the proposed SR-RBF neural network aresummarized in Table I.
Remark 4: One may notice that the training of the SR-RBFneural network contains two parts: 1) structure organizing and2) parameters adjusting. It is expected that with the suitablestructure and the updated parameters w, c, σ , and v, themodeling performance can be improved.
C. SR-RBF-NMPC
The objective of MPC is to determine a control signal u(t)via an online optimization to make the output of thesystem y(t) follow a reference r(t) as closely as possible.Therefore, another critical issue of the SR-RBF-NMPCscheme would be the feasibility of the online optimization.The method for improving optimization performance is alsoconsidered in this paper. The MPC formulations (1) and (2)can be rewritten in the following compact form:J(t) = ρ1[r(t) − y(t)]T [r(t) − y(t)] + ρ2�u(t)T �u(t) (20)
subject to
|�u(t)| ≤ �umax
umin ≤ u(t) ≤ umax
ymin ≤ y(t) ≤ ymax
r(t + Hp + i) − y(t + Hp + i) = 0, i ≥ 1 (21)
HAN et al.: NMPC BASED ON A SELF-ORGANIZING RECURRENT NEURAL NETWORK 407
ρ1 and ρ2 are the control weighting factors.To reduce the number of iterations required to reach
the optimum, the following gradient method is used forSR-RBF-NMPC [43]:
u(t + 1) = u(t) + �u(t) = u(t) + ξ
(− ∂ J(t)
∂u(t)
)(23)
where
�u(t) = (1 + ξρ2)−1ξρ1
((∂ y(t)
∂u(t)
)T
[r(t) − y(t)])
(24)
where ξ > 0 is the learning rate for controlling inputsequences, ∂ y(t)/∂u(t) is a Jacobian matrix, which can bederived from the SR-RBF neural network model based on thecontrol input sequences in (25), as shown at the bottom ofthis page.
The basic idea of this gradient method is that the currentcontrol inputs are chosen to minimize the cost function J overseveral steps for tracking the set-points. Then, the optimalcontrol input sequences can be calculated. The first elementsof the updated control input sequences are applied as thecontrol signals into the nonlinear system [43]. The proposedSR-RBF-NMPC scheme is summarized in Table II.
Remark 5: Using the cost function (20) and neural predictorbased on (4), the optimization problem u(t) = arg minu(t) J(t)can be solved at each sample time based on a sequence offuture controls u(t), where the first element is taken to controlthe process. Based on the results in [53], a popular approachfor considering constraints is to transform the original problemto its alternative unconstrained form using a penalty cost.Then, the improved gradient method can be used to minimizethe compact cost (20) and (21).
IV. STABILITY ANALYSIS
In this section, we give the convergence analysis of theSR-RBF neural network model in the cases with and withoutstructural changes. Moreover, the stability of the proposedSR-RBF-NMPC is also discussed.
A. Convergence Analysis of SR-RBF Neural Network
In order to discuss the convergence convenient, let us definethe discrete Lyapunov function candidate as
V (t) = 1
2e2(t) (26)
where e(t) is the current modeling error of the neural networkat time t with e(t) = y(t) − y(t). Then, the change of theLyapunov function V is obtained by
�V (t) = V (t + 1) − V (t) = 1
2(e2(t + 1) − e2(t)) (27)
and the modeling error difference can be represented by
�e(t) = e(t + 1) − e(t). (28)
In order to make it easier to analyze the convergence, themodeling error difference due to the learning can berepresented by
e(t + 1) = e(t) + �e(t) = e(t) +[
∂e(t)
∂W(t)
]T
�W(t) + O(t)
(29)
where O(t) indicates the higher order terms of the remainderof the Taylor series expansion, and
W(t) = [c(t), σ (t), v(t), w(t)]T
∂e(t)
∂W(t)=
[∂e(t)
∂c(t),
∂e(t)
∂σ (t),∂e(t)
∂v(t),
∂e(t)
∂w(t)
]T
. (30)
∂ y(t)
∂u(t)=
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
∂ y(t + 1)
∂u(t)0 0 · · · 0
∂ y(t + 2)
∂u(t)
∂ y(t + 2)
∂u(t + 1)0 · · · 0
......
.... . .
...∂ y(t + Hu)
∂u(t)
∂ y(t + Hu)
∂u(t + 1)
∂ y(t + Hu)
∂u(t + 2)· · · ∂ y(t + Hu)
∂u(t + Hu − 1)...
......
......
∂ y(t + Hp)
∂u(t)
∂ y(t + Hp)
∂u(t + 1)
∂ y(t + Hp)
∂u(t + 2)· · · ∂ y(t + Hp)
∂u(t + Hu − 1)
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
Hp×Hu
(25)
408 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
Assumption 1: The higher order terms of the remainder ofthe Taylor series expansion O(t) is assumed to be bounded
−[
1 − PT (t)ηP(t)
2
]e(t) − A(t) < O(t)
< −[
1 − PT (t)ηP(t)
2
]e(t) + A(t) (31)
where
η =
⎡
⎢⎢⎣
η1 0 0 00 η2 0 00 0 η3 00 0 0 η4
⎤
⎥⎥⎦, P(t) = ∂ y(t)
∂W(t)
A(t) =√
[1−PT (t)ηP(t)/2]2e2(t)+PT (t)ηP(t)e2(t). (32)
Theorem 1: Consider the SR-RBF neural network describedin Section III. If Assumption 1 is valid, the number of hiddenneurons is fixed, and the learning rates of parameters arechosen to satisfy
0 < ηi < 1/4υi,max (33)
where i = 1, 2, . . . , 4, and
υmax = [υ1,max, υ2,max, υ3,max, υ4,max]T
=[
maxt
∥∥∥∥∂ y(t)
∂c(t)
∥∥∥∥2
, maxt
∥∥∥∥∂ y(t)
∂σ (t)
∥∥∥∥2
,
maxt
∥∥∥∥∂ y(t)
∂v(t)
∥∥∥∥2
, maxt
∥∥∥∥∂ y(t)
∂w(t)
∥∥∥∥2]T
(34)
then the convergence of the SR-RBF neural network can beguaranteed and e(t) → 0 as t → ∞.
Proof: From (28), one has
V (t) > 0. (35)
Using (27) and (28), the change of the Lyapunov function is
�V (t) = �e(t)(
e(t) + 1
2�e(t)
). (36)
Combining the weight adaptation strategy as (13), (36) canbe expressed as
�V (t) =[−PT (t)ηP(t)e(t) + O(t)
]
×[
e(t) − 1
2PT (t)ηP(t)e(t) + 1
2O(t)
]
= −PT (t)ηP(t)e(t)
[e(t) − 1
2PT (t)ηP(t)e(t)
]
+ 1
2O2(t) + O(t)[e(t) − PT (t)ηP(t)e(t)]
= −e2(t)PT (t)ηP(t)
[1 − 1
2PT (t)ηP(t)
]
+ 1
2O2(t) + O(t)e(t)
[1 − 1
2PT (t)ηP(t)
]. (37)
If the learning rates of parameters satisfy (30), one has
0 < PT (t)ηP(t) < 1 (38)
and
�V (t) < −1
2e2(t)PT (t)ηP(t)
+ 1
2O2(t) + O(t)e(t)
[1 − 1
2PT (t)ηP(t)
]. (39)
If Assumption 1 is satisfied, it can be concluded
�V (t) < −1
2e2(t)PT (t)ηP(t)
+ 1
2O(t)
{O(t)+2e(t)
[1 − 1
2PT (t)ηP(t)
]}
< −1
2e2(t)PT (t)ηP(t)
+ 1
2
{A(t) −
[1 − PT (t)ηP(t)
2
]e(t)
}
×{
A(t) +[
1 − PT (t)ηP(t)
2
]e(t)
}
= −1
2e2(t)PT (t)ηP(t) + 1
2A2(t)
− 1
2
[1 − PT (t)ηP(t)
2
]2
e2(t)
= −1
2e2(t)PT (t)ηP(t)− 1
2
[1− PT (t)ηP(t)
2
]2
e2(t)
+ 1
2
[1− PT (t)ηP(t)
2
]2
e2(t)+ 1
2PT (t)ηP(t)e2(t)
= 0. (40)
Hence, by the Lyapunov like lemma [14]
limt→∞ e(t) = 0. (41)
By now we have proved the convergence result of theSR-RBF neural network with a fixed structure. �
Theorem 2: Consider the SR-RBF neural networkdescribed in Section III. If the number of hidden neurons isself-organized in the learning process and the learning ratesof parameters are chosen to satisfy
0 < η1 <1
2m|wmax|20 < η2 <
|σmin|22m|wmax|2
0 < η3 <1
2m2|wmax|4 , 0 < η4 <1
2m(42)
where m is the number of hidden neurons, then theconvergence of the SR-RBF neural network can be guaranteed,and e(t) → 0 as t → ∞.
Proof: For the modeling error e(t), if the hidden neuronsare added or deleted, based on (10) and (12), there exists
e(t) − e′(t) = 0 (43)
where e(t) and e′(t) are the modeling errors before and afterthe hidden neurons are added or deleted.
To prove Theorem 2, the following conclusions are used:| f (z)| = |e−z | < 1, |z f (z)| = |ze−z| < 1 (44)
where z > 0.
HAN et al.: NMPC BASED ON A SELF-ORGANIZING RECURRENT NEURAL NETWORK 409
Then, combining the weight adaptation strategy asformula (14), it leads to
From Theorem 1, if the learning rates of parameterssatisfy (42), we find �V (t) < 0, and limt→∞ e(t) = 0. Then,Theorem 2 can be established. �
Remark 6: Theorems 1 and 2 show the convergence of theproposed SR-RBF neural network. Moreover, one of the keyadvantages of the approach is that the modeling error canasymptotically converge to zero.
B. Stability Analysis for SR-RBF-NMPC
The stability analysis of SR-RBF-NMPC is discussedas Theorem 3.
Theorem 3: Consider the constrained finite-horizon optimalcontrol problem represented by (20) and (21). If Theorems 1and 2 hold, and the control law is designed as (24), then theasymptotic stability of the proposed SR-RBF-NMPC can beguaranteed.
Proof: The proof consists of two parts: 1) if u(0) isfeasible for r(0), then u(t) are feasible at all time stepst ≥ 0 (recursive feasibility) and 2) if Theorems 1 and 2hold, then the SR-RBF-NMPC closed loop control system isasymptotically stable.
1) By taking into account [43, Th. 4], it is easy to see thatif u(t − 1) satisfies (20) and (21) for r(t − 1), then u(t)satisfies (20) and (21) for r(t). As u(0) is feasible att = 0, it can be proved recursively that u(t) is feasibleat all time steps t ≥ 0.
2) Consider the cost function at time t as
J(t) = ρ1εT (t)ε(t) + ρ2�u(t)T �u(t)
= ρ1
Hp∑
i=1
ε2(t + i) + ρ2
Hu∑
i=1
�u2(t + i − 1) (50)
where ε(t) = r(t) − y(t) is the tracking error at time t .
Let us assume that u(t) = [u(t), u(t+1), …, u(t+Hu−1)]T
is the optimal control at time t found by the optimiza-tion procedure. Now, let us introduce the suboptimal controlus(t + 1) postulated at time t + 1
The control sequence us(t + 1) is formed based on thecontrol derived at time t . Therefore, for the suboptimal controlus(t + 1), the cost function can be defined as
Js(t + 1) = ρ1
Hp+1∑
i=2
ε2(t + i) + ρ2
Hu∑
i=2
�u2(t + i − 1).
(52)
Then, the difference of cost function J s(t +1) and J (t) canbe calculated as
Furthermore, if u(t + 1) is the optimal solution of theoptimization problem time (t + 1) based on the improvedgradient method (24), J (t + 1) ≤ Js(t + 1) as us(t + 1) isthe suboptimal one. Then, one has
J (t + 1) − J(t) ≤ Js(t + 1) − J (t) ≤ 0. (55)
Hence, by taking (55), Theorem 3 can be proved. �
410 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
Fig. 4. General overview of the BSM1 plant.
Remark 7: As the proposed SR-RBF-NMPC requires tosolve a constrained nonlinear optimization problem, the prob-lems of feasibility and stability are both nontrivial. Theorem 3shows both the feasibility and the stability of SR-RBF-NMPC.Moreover, the gradient method allows to consider constraintsimposed on process variables, the cost functions (20) and (21)can be optimized at each sample time.
V. SIMULATION STUDIES
In this section, the proposed SR-RBF-NMPC strategy isused to control the dissolved oxygen (DO) concentration ina WWTP. In order to show the evaluation and comparisonwith other control strategies, the SR-RBF-NMPC is appliedto benchmark simulation model 1 (BSM1) [45] to showwhether it is able to improve the control performance. All thesimulations were programmed with MATLAB, version 2010,and were run on a PC with a clock speed of 2.6 GHz and4 GB RAM, in a Microsoft Windows 8.0 environment. Theexperiment also shows the effect of set-point changes and loaddisturbances on the control systems.
A. Wastewater Treatment Process
WWTP is a dynamic, nonlinear, and complex system inwhich large disturbances take place. This is partly due tothe variability of the influent, the complexity of physicaland biochemical phenomena, and the large range of timeconstants inherent in the activated sludge process. The centralissue to be resolved in WWTP control is the translation ofimplicit operating objectives into feedback-controlled variablesof individual control loops. To evaluate the control strategiesof WWTP, a benchmark simulation system, BSM1, shownin Fig. 4, has been proposed within the framework of COSTActions 682 and 624 [45]. Based on the most common WWTP,BSM1 is composed of two anoxic zones, three aerobic zonesand a secondary settler. The system for each bioreactor zoneis based on the International Association on Water QualityActivated Sludge Model 1 [45]. Moreover, BSM1 defines someperformance criteria to characterize the global effluent quality,the energy consumption, and the respect of norms. The centralissue to be resolved in WWTP control is the translation ofimplicit operating objectives into sets of feedback-controlledvariables of individual control loops.
The aim of this section is to evaluate the proposedSR-RBF-NMPC method for controlling DO concentrationin BSM1. The control system, based on SR-RBF-NMPC,
TABLE III
LIMITS ON THE EFFLUENTS
is connected to a process computer, which is in charge ofthe key process parameters.
B. System Conditions
The system conditions of WWTP are described as follows.Samples: To get an objective view of the applied control
strategy performance in different situations, the simulatedinfluent data are available in two-week files derived fromreal operating data. These files are generated to simulate twodifferent weather situations (dry weather and rain weather).
Bounds: The limits on the effluents, such as ammo-nium (NH) concentration, total nitrogen (Ntot) concentration,suspended solid (SS) concentration, biological oxygendemand over a 5-day period (BOD5), and chemical oxygendemand (COD), are shown in Table III.
Inputs: The two manipulated variables are the internalrecycle flow rate Qa and the oxygen transfer coefficients inthe fifth aerated reactor KLa5. The mass transfer coefficientcorresponds to the efficiency of aeration in a given aeratedtank.
Outputs: The control variable is the DO concentration in thelast units of the bioreactor. Five effluent variables—the NH,the Ntot, the SS, the BOD5, and the COD concentrations—areused as the constrained conditions of the control system.
Control Loop: The control loop tunes the DO concentrationin the fifth tank (SO5) by manipulating the oxygen transfercoefficient of the same tank (KLa5) and the internal recycleflow rate (Qa).
Disturbances: To derive the robustness of the proposedcontrol strategy, the measurable disturbances have been con-sidered: the influent flow rate Qo and the influent ammoniumconcentration NHo in the time interval [0, 72] hour. Theamplitude of the disturbances occurs after t = 72 h (twiceof Qo and three times of NHo).
C. SR-RBF-NMPC Results
1) Modeling Results: To verify the performance ofthe described strategy for controlling WWTP, TheSR-RBF-NMPC method is proposed for controlling the SO5.For BSM1, a MISO SR-RBF neural network has beendeveloped for predicting the SO5 (as shown in Fig. 4).Prior to on-line control, the parameters to be determinedfor the MISO model are the model input and output orders(ny = 2, and nu = 2), the learning rates η1 = η2 = 0.01,η3 = η4 = 0.02, ξ = 0.2, the delay time td = 4, thepreset growing and pruning thresholds ss0 = 0.05 andssr = 0.001. Since the SR-RBF neural network contains the
HAN et al.: NMPC BASED ON A SELF-ORGANIZING RECURRENT NEURAL NETWORK 411
TABLE IV
COMPARISON OF PERFORMANCE OF DIFFERENT MODELS (ALL RESULTS
WERE AVERAGED BASED ON 20 INDEPENDENT RUNS)
recurrent mechanism, the number of inputs is 4, thus x(t) =[KLa5(t − 5), KLa5(t − 6), Qa(t − 5), Qa(t − 6)]. Initially,there are five initial hidden neurons with initial σ = 2.5. Theeffectiveness of SR-RBF neural network model is evaluatedin terms of online modeling and multistep-ahead prediction.
The SR-RBF neural network is compared with a fixedstructure multilayer perceptron (MLP) model [46] and theself-organizing RBF neural network models—the minimalresources-allocating network (MRAN) [47], the growing andpruning algorithm for RBF (GGAP-RBF) [48], and the self-organizing RBF neural network (SORBF) [40]. The inputs ofthe neural networks are x(t) = [KLa5(t − 5), KLa5(t − 6),Qa(t − 5), and Qa(t − 6)], which are different from theinitial papers. The other parameters of the MLP, MRAN,GGAP-RBF, and SORBF are the same as in the initial papersto ensure a fair comparison. The prediction performance isassessed using the number of hidden neurons and the meansquared error (MSE) [40], and the results of these models aredisplayed in Table IV. Clearly, the predictions obtained usingthe proposed SR-RBF neural network model is more accuratethan those obtained from the other models. Moreover, thefinal structure of the proposed SR-RBF neural network is thesimplest among all the models. The number of input variablesis also lesser than that of SORBF in [40], and thus the storagespace of the real application can be saved. The comparisonsdemonstrate that the SR-RBF neural network, containing therecurrent connections, is more suitable for modeling WWTPthan other methods.
2) Control Results: Based on the predicting model, theSR-RBF-NMPC strategy has been carried out to evaluate itscontrol performance. In this approach, the SR-RBF neuralnetwork predicts the future process response over the specifiedhorizon, which is used in a numerical optimization routine.Then, the optimization based on the gradient method attemptsto minimize the cost functions (20) and (21) in searching for anoptimal control signal at each sample instance. At each controlinterval, the proposed SR-RBF-NMPC strategy attempts to
Fig. 5. Online tracking control performance of SO5 in case 1 (dry weather).
optimize future plant behavior by computing a sequence offuture manipulated variable adjustments. Since the dynamicsof nutrient-removal processes are on a time scale of hours, theprediction horizon is set to 1–1.5 h, which is 4–6 time steps(each time step was 15 min). Then, the prediction horizonis set to Hp = 5. In order to smooth the trajectory andreduce the aeration energy (AE) of the process, in our case,the control horizon is set as Hu = 4 at each sampling instant.AE consumption is described as [45]
AE = 24
7
∫ t=14
t=7
5∑
l=3
[0.0007 × KLal(t)
2(
Vl
1333
)
+ 0.3267 × KLal(t)
(Vl
1333
)]dt
(56)
where (KLa)l is the overall mass transfer coefficient in thelth unit and Vl is the volume of the lth unit. In this case,a rough estimate of average electricity price in the EU(0.1 C/kWh) was taken into account, and thus, all of theweights were multiplied by 0.1. Moreover, the controllerperformance is measured using the integral of absoluteerror (IAE) [45]
IAE(t) = 1
T
T∑
t=1
|r(t) − y(t)| (57)
where y(t) and r(t) are the output and desired output of thesystem, respectively, and; T is the total number of samples.
In this paper, two aspects of the evaluations are discussed:the set-point of SO5 is fixed on 2 mg/L (case 1) anddynamical in the process (case 2). The control resultsof the proposed SR-RBF-NMPC method and the formerSORBF-based predictive control (SORBF-MPC) method [40]for dry weather are displayed in Figs. 5 and 6 (case 1),including the process responses and the control variables usedto track the set-points with the above parameters setting.
The graphs in Fig. 5 show that the SR-RBF-NMPC strategycan track the set-point of SO5 in case 1. The errors in Fig. 6indicate the difference between the set-point and the controloutput remain within the range of ±0.05 mg/L (±2.5%).Figs. 5 and 6 demonstrate that the SR-RBF-NMPC method can
412 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
Fig. 6. Error values of SO5 in case 1 (dry weather).
Fig. 7. Online manipulated variables in case 1 (dry weather).
TABLE V
COMPARISON OF THE IAE AND AE OF DIFFERENT CONTROLLERS IN
CASE 1 (BOTH DRY WEATHER AND RAIN WEATHER)
track the SO5 when the set-point of SO5 remains unchanged.The control results of SO5 are smooth.
In addition, the two manipulated variables KLa5 andQa are shown in Fig. 7. The performance of SR-RBF-NMPC method is also compared with that of other existingcontrollers that are designed for the treatment process: theproportional-integral (PI) controller [49], MPC [50], andSORBF-MPC [40], The details of the comparisons in thedry weather and rain weather cases are shown in Table V.
Fig. 8. Online tracking control performance of SO5 in case 2 (rain weather).
Fig. 9. Error values of SO5 in case 2 (rain weather).
Fig. 10. Online manipulated variables in case 2 (rain weather).
The results demonstrate that the control performance ofSR-RBF-NMPC is excellent for DO concentration.
In case 2, in which the set-points of SO5 is dynamicalrather than fixed, the conditions are the same as in case 1.Figs. 8–10 show the corresponding control results of both theSR-RBF-NMPC method and the SORBF-MPC method. Thetracking results of the SO5 in case 2 are shown in Fig. 8.The errors indicating the difference between the set-pointsand the control output can also remain within the rangeof ±0.05 mg/L (±2.5%) in this case, which can be seenin Fig. 9. Meanwhile, the two manipulated variables areshown in Fig. 10. In addition, Table VI illustrates the
HAN et al.: NMPC BASED ON A SELF-ORGANIZING RECURRENT NEURAL NETWORK 413
TABLE VI
COMPARISON OF THE IAE AND AE OF DIFFERENT CONTROLLERS IN
CASE 2 (BOTH DRY WEATHER AND RAIN WEATHER)
control performances with respect to the different controlstrategies during the seven-day observation period. Theaverage IAE and AE values of the different control strategiesare specially considered to investigate the effectiveness of theproposed SR-RBF-NMPC control scheme.
D. Analysis of the Simulation Results
The results presented above show that the proposedSR-RBF-NMPC strategy can control the DO concentrationin WWTP even when the process is highly loaded withdisturbances. Based on the former results, the proposed controlstrategy obtains the following performances.
1) Good System Model Ability: Due to WWTP is a dynamicsystem that involves various physical and biological phenom-ena as well as large disturbances, it eliminates the difficultiesof finding appropriate methods for nonlinear system modeling.Moreover, the presence of nonlinearity, time-varying, anddelay characteristics in WWTP significantly complicates themissions of control system analysis and design. In this paper,an SR-RBF neural network model is developed for approxi-mating the online outputs of WWTP.
Table IV depicts the results of the modeling performance.The comparative results of various methods demonstrate thatthe SR-RBF neural network is more suitable for WWTPwith nonlinear and time delay (see Table IV). This proposedSR-RBF neural network can adjust the structure andparameters simultaneously by a spiking-based growing andpruning algorithm and an adaptive learning algorithm. Hence,the SR-RBF neural network could achieve a more satisfactoryaccuracy and better generalization capability.
2) Good Control Ability: the control ability evaluatesthe quality of responses to the controller that are usedfor WWTP. In order to show the complete results of theproposed SR-RBF-NMPC, the details of the comparisonsin both the dry weather and rain weather are given in theexperiments within fixed and time-varying set-points. Throughthe aforementioned two cases, it is evident that the proposedSR-RBF-NMPC strategy owns superior control performancefor DO concentration control. It reduces the tracking errors
(indicating the difference between the set-points and thecontrol output) for SO5 (see Figs. 5, 6, 8, and 9).
Moreover, according to the results, it is noted that theaverage IAE and AE values via the proposed SR-RBF-NMPCcan be controlled well and thereby save energy of theprocess. Please refer to Tables V and VI, which illustratethe average IAE and AE values according to the proposedSR-RBF-NMPC. In case 1, the IAE values are 0.046 mg/L(2.3%) and 0.063 mg/L (3.2%), and the AE values are667 C/kWh (−7.9%) and 670 C/kWh (−7.5%) in bothdry and rain weather cases. In case 2, the IAE values are0.066 mg/L (3.3%) and 0.072 mg/L (3.6%), and the AE valuesare 673 C/kWh (−7.0%) and 677 C/kWh (−6.5%) in both thecases. This is because the SR-RBF neural network can modelnonlinear dynamics, and a long control prediction horizonhas been used in the process. Obviously, the IAE and AEvalues mainly depend on the model accuracy and the controlprediction horizons, respectively. Thus, the SR-RBF neuralnetwork model can obtain better predicting values and controlprediction horizon. The average IAE and AE values confirmthat the SR-RBF-NMPC method behaves better than the othercontrol strategies. In addition, from Figs. 7 and 10, whichshow the results of the two manipulated variables in bothcases, it can be seen the details of the KLa5 and Qa . Thus,it is possible to keep acceptable levels of the average effluentqualities.
VI. CONCLUSION
In this paper, an SR-RBF-NMPC strategy, based on theSR-RBF neural network and gradient optimization method,is proposed, which is capable of modeling and controllingthe DO concentration in BSM1. Numerical simulations areprovided to demonstrate its superior performance under theconditions of different set-points, external disturbances andweathers changes. This SR-RBF-NMPC strategy is differentfrom traditional MPC approaches, since it actually addressesthe characteristics of nonlinear systems. The major advantageof SR-RBF-NMPC is that the SR-RBF neural network modelcan update the structure and parameters to keep the modelattuned with the current system dynamics. This methodenhances the capacity of the predicting model to adapt tononlinear systems. Moreover, a gradient optimization methodis developed to find the optimal control input sequences. Theadaptive strategies are designed based on Lyapunov stabilitytheory. The proposed SR-RBF-NMPC strategy is guaranteedto be asymptotic stable. Experimental results showed thatthe SR-RBF-NMPC strategy is competitive in terms ofperformance, compared with other existing methods. Thisstudy can be seen as an essential step before implementingthe methodology in a real plant, since the generation of datawas done as realistically as possible.
ACKNOWLEDGMENT
The authors would like to thank Prof. G. Feng for readingthe paper and providing valuable comments. The authorswould also like to thank the Editor-in-Chief of the IEEETRANSACTIONS ON NEURAL NETWORKS AND LEARNING
414 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 27, NO. 2, FEBRUARY 2016
SYSTEMS and the reviewers for their valuable comments andsuggestions, which helped improve this paper greatly.
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Hong-Gui Han (M’11–SM’15) received theB.E. degree in automatic from the Civil AviationUniversity of China, Tianjin, China, in 2005, and theM.E. and Ph.D. degrees from the Beijing Universityof Technology, Beijing, China, in 2007 and 2011,respectively.
He has been with the Beijing University ofTechnology since 2011, where he is currently aProfessor. His current research interests includeneural networks, fuzzy systems, intelligent systems,modeling and control in process systems, and civil
and environmental engineering.Prof. Han is a member of the IEEE Computational Intelligence Society.
He is a Reviewer of the IEEE TRANSACTIONS ON FUZZY SYSTEMS, theIEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS,the IEEE TRANSACTIONS ON CYBERNETICS, the IEEE TRANSACTIONS ON
CONTROL SYSTEMS TECHNOLOGY, Control Engineering Practice, and theJournal of Process Control.
Lu Zhang received the B.E. degree in automaticcontrol from Heze University, Heze, China, in 2014.She is currently pursuing the Ph.D. degree withthe College of Electronic and Control Engineering,Beijing University of Technology, Beijing, China.
Her current research interests include neuralnetworks, intelligent systems, modeling, and controlin process systems.
Ying Hou received the B.E. and M.E. degreesin control engineer from the Beijing Universityof Technology, Beijing, China, in 2004 and 2007,respectively, where she is currently pursuing thePh.D. degree with the College of Electronic andControl Engineering.
Her current research interests include neuralnetworks, self-adaptive/learning systems, modeling,control, and optimization in process systems.
Jun-Fei Qiao (M’11) received the B.E. andM.E. degrees in control engineer fromLiaoning Technical University, Fuxin, China, in1992 and 1995, respectively, and the Ph.D. degreefrom Northeast University, Shenyang, China,in 1998.
He was a Post-Doctoral Fellow with the School ofAutomatics, Tianjin University, Tianjin, China, from1998 to 2000. He joined the Beijing University ofTechnology, Beijing, China, where he is currentlya Professor. He is the Director of the Intelligence
Systems Laboratory. His current research interests include neural networks,intelligent systems, self-adaptive/learning systems, and process controlsystems.
Prof. Qiao is a member of the IEEE Computational Intelligence Society.He is a Reviewer for more than 20 international journals, such as the IEEETRANSACTIONS ON FUZZY SYSTEMS and the IEEE TRANSACTIONS ON
