# An Improved Generalized Predictive Control in a Robust Dynamic Partial Least Square Framework

Authors: Jin Xin; Chi Qinghua; Liu Kangling; Liang Jun
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

To tackle the sensitivity to outliers in system identification, a new robust dynamic partial least squares (PLS) model based on an outliers detection method is proposed in this paper. An improved radial basis function network (RBFN) is adopted to construct the predictive model from inputs and outputs dataset, and a hidden Markov model (HMM) is applied to detect the outliers. After outliers are removed away, a more robust dynamic PLS model is obtained. In addition, an improved generalized predictive control (GPC) with the tuning weights under dynamic PLS framework is proposed to deal with the interaction which is caused by the model mismatch. The results of two simulations demonstrate the effectiveness of proposed method.


---

## Body

## 1. Introduction

In the past decades, PLS, a multivariable regression method, has been applied to many areas, such as quality prediction, process monitoring, and chemometrics [1]. PLS is not only able to extract principle component from both input and output datasets, but also able to determine the direction on which input and output data have the largest covariance [2]. Considering the advantage of dimension reduction and automatic decoupling, many researchers applied PLS to the modeling and control of dynamic systems. Kaspar and Ray [3, 4] proposed a dynamic PLS framework by utilizing the PLS loading matrices to construct precompensators and postcompensators. Chen et al. [5] proposed another dynamic PLS framework with ARX model. Laurí et al. [6] proposed a PLS-based model predictive control (MPC) relevant identification method.There are many uncertain factors in industrial fields which make the clean data cannot be acquired directly from actual systems. It is well known that the standard algorithms for PLS regression (NIPALS and SIMPLS) [7] are very sensitive to outliers in the dataset. Outliers can distort the curve of the data. It may lead to undesirable results if standard PLS is directly applied in practical applications. To overcome this problem, several robust PLS methods are proposed in recent years. One of them is obtained by robustifying the sample covariance matrix between the input and output datasets based on SIMPLS algorithm [8, 9]. Another is PLS calibration with outliers detection, which selects a subset of samples dataset randomly and gets the initial residual set and then detects and discards outliers in the subset [10]. These methods are not suitable for the dynamic PLS control framework because the samples are a time sequence series and cannot be selected randomly. Moreover, these methods only focus on the regression; they do not fit for dynamic modeling.There is one way to get a robust dynamic PLS model. In contaminated data, several data preprocessing methods could be applied to detect outliers and eliminate the influence of them. Various methods for outliers detection have been proposed in the literature, such as statistics-based algorithms, machine learning-based algorithms, and neural network-based algorithms. In these methods, outliers are the point far away from the majority of the data; however this situation is not always right [11]. Outliers can be close to the major curve of data as well. In order to detect general outliers in the dataset which is used for modeling, the relationship between inputs and outputs should be known. Liu et al. [11, 12] proposed an autoregression structure RBFN to predict the outputs of the system and detected outliers with this predictive model combined with HMM. The limitation is that the output of the RBFN is composed of the inputs and outputs of the process, but the prediction of system input is useless. Because outliers are usually contained in the output dataset, outliers in input dataset are commonly calculated by the controllers. Therefore, it is not useful to increase the dimension of the network weights to estimate the value of the inputs. In addition, Liu et al. [11] analyzed the performance of the algorithm with single-input single-output (SISO) system at low rate of outliers. There is no analysis with more complex situations such as multiple-input multiple-output (MIMO) systems and high rate of outliers.On the other hand, many control schemes are applied under the dynamic PLS framework. Kaspar and Ray proposed a PID control scheme under this framework [3], and Chen et al. [5] designed multiloop adaptive PID controllers based on an modified decoupling PLS framework. Hu et al. [13, 14] proposed a multiloop internal model controller in the dynamic PLS framework and achieved better performance for disturbance rejection. Lü and Liang [15] proposed a multiloop constrained MPC scheme. Ideally, with the PLS decoupled structure, there should be no interaction between PLS components [3]. However, there are always some interactions observed due to the plan/model mismatch which is caused by contaminated data and filtering color noises incompletely.Aiming at these limitations, a robust dynamic PLS modeling and improved GPC control scheme is proposed in this paper. A new RBFN structure is applied to predict the outputs of the system. With the difference between the prediction and the real data, a simple HMM algorithm is used to detect the outliers, and then they are replaced by the mean of clean data nearby. After outliers detection, dynamic PLS modeling would be more robust. Based on this model, a dynamic tuning weight sequence is introduced into the cost functions of improved GPC under the PLS latent space to reduce the interaction of different outputs which is caused by mismatch between the PLS model and real process.The rest of paper is organized as follows: in Section2, an improved RBFN training method and an HMM based outliers detection algorithm are presented. In Section 3, dynamic PLS modeling method is introduced and an improved GPC scheme is proposed. In Section 4, two examples are used to demonstrate the effectiveness and better control performance of the proposed method. In the last section, the conclusions are summarized.


## 2. Outliers Detection for Modeling Data

### 2.1. Improved Radial Basis Function Network

As a feedforward network, the RBFN neural network has three layers: input layer, hidden layer, and output layer. And, in many practical applications, Gaussian RBF is used in the hidden layer. It can approximate the linear and nonlinear functions with arbitrary precision in theory. When fitting the dynamic systems, several past input and output samples should be used as the input of the RBFN. To introduce the new structure of RBFN, 3 assumptions are firstly given as follows:(1) the system has m inputs and n outputs; (2) only the outputs of the system contain outliers; and (3) each input signal and each output signals have the same lags, respectively. With assumption (1), arbitrary input and output dimension system could be involved. Outliers in outputs are usually caused by some unmeasured reasons such as unknown sensor failures. Hence it is what we focus on (with assumption (2)). While it is easy to detect outliers in inputs, for the inputs can be calculated with set points and detected values by controllers. In order to simplify the analysis, assumption (3) is employed. Signal with different lags can be treated as a special case of that signal with the same lags. Past samples used as the inputs of the network have the form(1)O=xt-1,xt-2,…,xt-nx,yt-1,yt-2,…,yt-nyT,where x is an m-dimension vector and y is an n-dimension vector. [xt-1,xt-2,…,xt-nx] denotes the input values from t-1 to t-nx, [yt-1,yt-2,…,y(t-ny)] denotes the output values from t-1 to t-ny. nx and ny denote the lag of input and output, respectively.The structure of the modified RBFN is shown in Figure1. nx past input values and ny past output values enter the RBFN. The output of RBFN y^(t) denotes the predictive vector, which is calculated by summing the product of the hidden layer outputs φ and the output layer weights ω. The superscripts φ and ω denote which input or output it belongs to, and the subscript denotes which hidden node it belongs to. φiCi is computed by the hidden node center vector CiCi with the Gaussian function, where i=1,…,nx+ny and ic=1,…,Ci. Then the output of the RBFN is (2)y^t=∑i=1nx+ny∑iC=1CiφiCi·ωiCi.When O(i) is inputted into the RBFN, there are Ci hidden nodes for it to cluster. The outputs of hidden layers with Gaussian function RBFN for the inputs are(3)φiCi=exp⁡Oi-CiCi2-2σi2,wherei=1,…,nx+ny,iC=1,…,Ci.It is noted that CiCi is an m-dimension vector for i=1,…,nx or an n-dimension vector for i=nx+1,…,nx+ny because RBFN input O(i) represents both x and y vectors. The input of the conventional RBFN should be a (m·nx+n·ny)-dimensional vector [O(1),O(2),…,O(nx+ny)], and the center vector of it has the same dimension with it, which is denoted by CoCoi=[Co1,Co2,…,Conx+ny]. Coi should be chosen from [C1i,…,CCii]; therefore for each Coi there are Ci possible. In other words, conventional RBFN needs ∏i=1nx+nyCi hidden nodes to be equal to the improved RBFN with ∑i=1nx+nyCi hidden nodes. It is obvious that ∑i=1nx+nyCi is less than ∏i=1nx+nyCi; therefore, the improved RBFN has less hidden nodes.Figure 1
The structure of proposed RBFN.


### 2.2. Convergence of Improved RBFN

Above all, to analyze the convergence of improved RBFN, the relationship between improved RBFN and conventional RBFN is proposed in Lemma1.Lemma 1.
The improved RBFN is equivalent to the conventional RBFN.Proof.
Suppose that[O(1),O(2),…,O(nx+ny)] are independent of each other. The output of the hidden layer of the conventional RBFN is given by(4)φ-Coi=exp⁡O1-Co12+O2-Co22+⋯+Onx+ny-Conx+ny2-2σ2=exp⁡O1-Co12+O2-Co22+⋯+Onx+ny-Conx+ny2-2σ12+σ22+⋯+σnx+ny2=exp⁡O1-Co12-2σ12·exp⁡O2-Co22-2σ22·…·exp⁡Onx+ny-Conx+ny2-2σnx+ny2=φCo11·φCo22·…·φConx+nynx+ny,whereCoi=1,2,…,∏i=1nx+nyCi.
The output of conventional RBFN can be written as(5)y^t=∑iCo=1∏i=1nx+nyCiφ-iCo·ω-iCo=∑Co1=1C1⋯∑Conx+ny=1Cnx+nyφCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+ny,
whereω-Co1⋯Conx+ny is the weight of the output layer. For each φCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+ny there must be time-varying weights (ω-Co1⋯Conx+nyi,i=1,…,nx+ny) always found to make the following hold:(6)φCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+ny=φCo11·ω-Co1⋯Conx+ny1+φCo22·ω-Co1⋯Conx+ny2⋯+φConx+nynx+ny·ω-Co1⋯Conx+nynx+ny.According to (6), (5) can be rewritten as(7)y^t=∑Co1=1C1⋯∑Conx+ny=1Cnx+nyφCo11·ω-Co1⋯Conx+ny1+φCo22·ω-Co1⋯Conx+ny2⋯+φConx+nynx+ny·ω-Co1⋯Conx+nynx+ny.Equation (7) can be rewritten as(8)y^t=∑Co2=1C2⋯∑Conx+ny=1Cnx+nyφ11·ω-1,Co2⋯Conx+ny1+∑Co2=1C2⋯∑Conx+ny=1Cnx+nyφ21·ω-2,Co2⋯Conx+ny1+⋯+∑Co2=1C2⋯∑Conx+ny=1Cnx+nyφC11·ω-C1,Co2⋯Conx+ny1+∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyφ12·ω-Co1,1⋯Conx+ny2+∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyφ22·ω-Co1,2⋯Conx+ny2+∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyφC22·ω-Co1,C2⋯Conx+ny2+⋯+∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1φ1nx+ny·ω-Co1⋯Conx+ny-1,1nx+ny+∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1φ2nx+ny·ω-Co1⋯Conx+ny-1,2nx+ny+⋯+∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1φCnx+nynx+ny·ω-Co1⋯Conx+ny-1,Cnx+nynx+ny.Comparing (8) and (2), it is obvious that (6) always holds with Gaussian function RBFN:(9)∑Co2=1C2⋯∑Conx+ny=1Cnx+nyω-1,Co2⋯Conx+ny1=ω11,∑Co2=1C2⋯∑Conx+ny=1Cnx+nyω-2,Co2⋯Conx+ny1=ω21,⋮∑Co2=1C2⋯∑Conx+ny=1Cnx+nyω-C1,Co2⋯Conx+ny1=ωC11;∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyω-Co1,1Co3⋯Conx+ny2=ω12,∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyω-Co1,2Co3⋯Conx+ny2=ω22,⋮∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyω-Co1,C2Co3⋯Conx+ny2=ωC22;⋮∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1ω-Co1⋯Conx+ny-1,1nx+ny=ω1nx+ny,∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1ω-Co1⋯Conx+ny-1,2nx+ny=ω2nx+ny,∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1ω-Co1⋯Conx+ny-1,Cnx+nynx+ny=ωCnx+nynx+ny.
From (9), we can see the improved RBFN with time-variant output weights is completely equivalent to the conventional RBFN.To analyze the convergence of improved RBFN, we give two assumptions as follows:(1) each output must be bounded when t→∞. In other words, there must be a constant x, satisfying the relationship that xi(t)<x, i=1,…,m; (2) the first sample data of the time series is accurate and xi(1)≠0, i=1,…,m.From (6), we know that the output weights of improved RBFN are time-variant, which are decided by the output of hidden nodes. In order to analyze the convergence, we divided the output of improved RBFN into ∏i=1nx+nyCi parts corresponding to the ∏i=1nx+nyCi outputs of hidden nodes of conventional RBFN. We take only one output of the improved RBFN for analysis, and other parts of the RBFN have the same property. From (5) and (6), the ith hidden nodes for a output of the RBFN are denoted by f(t) for the left of (6) at time t, and the right are denoted by g(t):(10)ft=φCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+nyi=φCot·ωCo,gt=φCo11·ω-Co1⋯Conx+nyi,1+φCo22·ω-Co1⋯Conx+nyi,2⋯+φConx+nynx+ny·ω-Co1⋯Conx+nyi,nx+ny=φ-Cot·ω-CoTt,where ω-CoT(t)=[ω-Co1⋯Conx+nyi,1,ω-Co1⋯Conx+nyi,2,…,ω-Co1⋯Conx+nyi,nx+ny]T is the time-variant weight vector and φ-Co(t)=[φCo11,φCo22,…,φConx+nynx+ny] is the output of the hidden layer.From (10), we can get the output of the hidden layer of improved RBFN at time t+1:(11)ft+1=φCot+1·ωCo,gt+1=φ-Cot+1·ω-CoTt+1.From (6), (10), and (11), we know that f(t)=g(t) and f(t+1)=g(t+1), and then the output weights of the improved RBFN at time t+1 and t have the relation as follows:(12)ω-CoTt+1=φCot+1·ωCo·φ-Co-1t+1=φCot+1·ftφCot·φ-Co-1t+1=φCot+1·gtφCot·φ-Co-1t+1=φCot+1·φ-Cot·ω-CoTtφCot·φ-Co-1t+1.So the deviation of the improved RBFN at time t+1 is(13)σt+1=gt+1-ft+1=φ-Cot+1·ω-CoTt+1-ft+1=φ-Cot+1·φCot+1·φ-Cot·ω-CoTtφCot·φ-Co-1t+1-ft+1=φ-Cot+1·φCot+1·ωCo·φ-Cot·ω-CoTtφCot·ωCo·φ-Co-1t+1-ft+1=ft+1·gtft-ft+1=ft+1·gt-ftft=ft+1·σtft=ft+1·ft·σt-1ft-1·ft=⋯=ft+1·…·f2·σ1f1·…·ft=ft+1·σ1f1.Since f(t+1) is the hidden layer output of conventional RBFN and approximation property of conventional RBFN, there is relationship that f(t+1)=yi(t+1)+ε, where yi(t+1) denotes the true value of ith output. ε denotes an arbitrarily small value. σ(t+1) can be written as(14)σt+1=g1-f1·yit+1+εf1≤g1f1-1yit+1+ε,where yi is the true value of system output. According to assumption (2), we know that yi(1)≠0 and yi(1) is accurate. Thus yi(1) is used to replace g(1) and the equation is obtained as follows:(15)g1f1-1=yi1yi1+ε-1=-εyi1+ε.Because of the approximation property of the conventional RBFN, yi(1) is bounded. -ε/yi(1)+ε also can be arbitrarily small when t→∞. Thus the conclusion of the convergence of improved RBFN is proved.


### 2.3. Outliers Detection Using Hidden Markov Model

The real value ofy^(t) is denoted by y(t). From the approximation property of the RBFN, the prediction y^(t) should be close to its true value, y(t). For the jth output, it has the following relationship:(16)y^jt=yjt+ε^jt,where j=1,…,n and ε^j(t) is the estimation of the white noise which obeys the Gaussian distribution N(0,Uj). The probability density function of ε^j(t) is Gaussian-type membership function and given by [11, 12] (17)Pε^jt∣μj,Uj=exp⁡-12y^jt-yjtTUj-1y^jt-yjt,where μj=0. The higher the probability is, the closer ε^j(t) is equal to 0 (the closer y^j(t) is equal to yj(t)). In other words, if P(ε^j(t)∣μj,Uj) equals or is close to 0 at time t, yj(t) is considered to be a possible outlier.In order to make outliers detection accurately, HMM is introduced. HMM is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states. One of the problems in HMM is choosing a most likely state chain with the given observation chain and model. This problem can be solved by Viterbi algorithm [16]. In the observation chain, state transition probabilities are the only parameters. P(ε^j(t)∣μj,Uj) is fit for this character. Hence, an HMM is introduced to detect whether the data are outliers or not. It is noted that the dimension of the chains is the same and equal to the dimension of system output.An HMM for outliers detection is characterized by the following [17]:(1)
Outliers detection is done separately for each system output.(2)
P(ε^j(t)∣μj,Uj) represents the chain which has been observed. The hidden chain is the state of the observation.(3)
There are two states in the HMM. “1” denotes that the state is normal, and “0” means it is outlier.(4)
The transition probability of hidden chain is defined as follows:(18)a01=Na01Na01+a00,a11=Na11Na11+a10,a00=1-a01,a10=1-a11,where aisjs (is,js=0 or 1) denotes the transition probability from state is to js and N(aisjs) denotes the total transition times.To find the best detection result, the probability is defined as(19)ϕist=maxs1,s2,…,st-1⁡Ps1,s2,…,st=is,p1,p2,…,pt,where {p(1),p(2),…,p(t)} is a sequence of observation probabilities and calculated by (17); the element in sequence S={s(1),s(2),…,s(t)} denotes the state; and ϕis(t) is the best score (highest probability) for finding the best state sequence. Since the outliers are detected in a time sequence and the previous data before time t have already been detected, the conventional Viterbi algorithm can be simplified. Equation (19) is rewritten as(20)ϕist=max⁡Pst=is,pt.State transition probabilityaisjs is used to express all influences of historical observations probability and states, and (20) can be rewritten as(21)ϕ1t=ais1p1t;ϕ0t=ais0p0t.p
1
(
t
) obtained by (17) is the observation probability that state is “1” and p0(t)=1-p1(t). Then the state of the data at time t can be determined:Ifϕ1(t)<ϕ0(t), then s(t)=0, and x(t) is an outlier.Else thens(t)=1, x(t) is normal.Then the data which are detected as outliers are removed and replaced by the mean of clean data nearby.


## 2.1. Improved Radial Basis Function Network

As a feedforward network, the RBFN neural network has three layers: input layer, hidden layer, and output layer. And, in many practical applications, Gaussian RBF is used in the hidden layer. It can approximate the linear and nonlinear functions with arbitrary precision in theory. When fitting the dynamic systems, several past input and output samples should be used as the input of the RBFN. To introduce the new structure of RBFN, 3 assumptions are firstly given as follows:(1) the system has m inputs and n outputs; (2) only the outputs of the system contain outliers; and (3) each input signal and each output signals have the same lags, respectively. With assumption (1), arbitrary input and output dimension system could be involved. Outliers in outputs are usually caused by some unmeasured reasons such as unknown sensor failures. Hence it is what we focus on (with assumption (2)). While it is easy to detect outliers in inputs, for the inputs can be calculated with set points and detected values by controllers. In order to simplify the analysis, assumption (3) is employed. Signal with different lags can be treated as a special case of that signal with the same lags. Past samples used as the inputs of the network have the form(1)O=xt-1,xt-2,…,xt-nx,yt-1,yt-2,…,yt-nyT,where x is an m-dimension vector and y is an n-dimension vector. [xt-1,xt-2,…,xt-nx] denotes the input values from t-1 to t-nx, [yt-1,yt-2,…,y(t-ny)] denotes the output values from t-1 to t-ny. nx and ny denote the lag of input and output, respectively.The structure of the modified RBFN is shown in Figure1. nx past input values and ny past output values enter the RBFN. The output of RBFN y^(t) denotes the predictive vector, which is calculated by summing the product of the hidden layer outputs φ and the output layer weights ω. The superscripts φ and ω denote which input or output it belongs to, and the subscript denotes which hidden node it belongs to. φiCi is computed by the hidden node center vector CiCi with the Gaussian function, where i=1,…,nx+ny and ic=1,…,Ci. Then the output of the RBFN is (2)y^t=∑i=1nx+ny∑iC=1CiφiCi·ωiCi.When O(i) is inputted into the RBFN, there are Ci hidden nodes for it to cluster. The outputs of hidden layers with Gaussian function RBFN for the inputs are(3)φiCi=exp⁡Oi-CiCi2-2σi2,wherei=1,…,nx+ny,iC=1,…,Ci.It is noted that CiCi is an m-dimension vector for i=1,…,nx or an n-dimension vector for i=nx+1,…,nx+ny because RBFN input O(i) represents both x and y vectors. The input of the conventional RBFN should be a (m·nx+n·ny)-dimensional vector [O(1),O(2),…,O(nx+ny)], and the center vector of it has the same dimension with it, which is denoted by CoCoi=[Co1,Co2,…,Conx+ny]. Coi should be chosen from [C1i,…,CCii]; therefore for each Coi there are Ci possible. In other words, conventional RBFN needs ∏i=1nx+nyCi hidden nodes to be equal to the improved RBFN with ∑i=1nx+nyCi hidden nodes. It is obvious that ∑i=1nx+nyCi is less than ∏i=1nx+nyCi; therefore, the improved RBFN has less hidden nodes.Figure 1
The structure of proposed RBFN.


## 2.2. Convergence of Improved RBFN

Above all, to analyze the convergence of improved RBFN, the relationship between improved RBFN and conventional RBFN is proposed in Lemma1.Lemma 1.
The improved RBFN is equivalent to the conventional RBFN.Proof.
Suppose that[O(1),O(2),…,O(nx+ny)] are independent of each other. The output of the hidden layer of the conventional RBFN is given by(4)φ-Coi=exp⁡O1-Co12+O2-Co22+⋯+Onx+ny-Conx+ny2-2σ2=exp⁡O1-Co12+O2-Co22+⋯+Onx+ny-Conx+ny2-2σ12+σ22+⋯+σnx+ny2=exp⁡O1-Co12-2σ12·exp⁡O2-Co22-2σ22·…·exp⁡Onx+ny-Conx+ny2-2σnx+ny2=φCo11·φCo22·…·φConx+nynx+ny,whereCoi=1,2,…,∏i=1nx+nyCi.
The output of conventional RBFN can be written as(5)y^t=∑iCo=1∏i=1nx+nyCiφ-iCo·ω-iCo=∑Co1=1C1⋯∑Conx+ny=1Cnx+nyφCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+ny,
whereω-Co1⋯Conx+ny is the weight of the output layer. For each φCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+ny there must be time-varying weights (ω-Co1⋯Conx+nyi,i=1,…,nx+ny) always found to make the following hold:(6)φCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+ny=φCo11·ω-Co1⋯Conx+ny1+φCo22·ω-Co1⋯Conx+ny2⋯+φConx+nynx+ny·ω-Co1⋯Conx+nynx+ny.According to (6), (5) can be rewritten as(7)y^t=∑Co1=1C1⋯∑Conx+ny=1Cnx+nyφCo11·ω-Co1⋯Conx+ny1+φCo22·ω-Co1⋯Conx+ny2⋯+φConx+nynx+ny·ω-Co1⋯Conx+nynx+ny.Equation (7) can be rewritten as(8)y^t=∑Co2=1C2⋯∑Conx+ny=1Cnx+nyφ11·ω-1,Co2⋯Conx+ny1+∑Co2=1C2⋯∑Conx+ny=1Cnx+nyφ21·ω-2,Co2⋯Conx+ny1+⋯+∑Co2=1C2⋯∑Conx+ny=1Cnx+nyφC11·ω-C1,Co2⋯Conx+ny1+∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyφ12·ω-Co1,1⋯Conx+ny2+∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyφ22·ω-Co1,2⋯Conx+ny2+∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyφC22·ω-Co1,C2⋯Conx+ny2+⋯+∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1φ1nx+ny·ω-Co1⋯Conx+ny-1,1nx+ny+∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1φ2nx+ny·ω-Co1⋯Conx+ny-1,2nx+ny+⋯+∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1φCnx+nynx+ny·ω-Co1⋯Conx+ny-1,Cnx+nynx+ny.Comparing (8) and (2), it is obvious that (6) always holds with Gaussian function RBFN:(9)∑Co2=1C2⋯∑Conx+ny=1Cnx+nyω-1,Co2⋯Conx+ny1=ω11,∑Co2=1C2⋯∑Conx+ny=1Cnx+nyω-2,Co2⋯Conx+ny1=ω21,⋮∑Co2=1C2⋯∑Conx+ny=1Cnx+nyω-C1,Co2⋯Conx+ny1=ωC11;∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyω-Co1,1Co3⋯Conx+ny2=ω12,∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyω-Co1,2Co3⋯Conx+ny2=ω22,⋮∑Co1=1C1∑Co3=1C3⋯∑Conx+ny=1Cnx+nyω-Co1,C2Co3⋯Conx+ny2=ωC22;⋮∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1ω-Co1⋯Conx+ny-1,1nx+ny=ω1nx+ny,∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1ω-Co1⋯Conx+ny-1,2nx+ny=ω2nx+ny,∑Co1=1C1⋯∑Conx+ny-1=1Cnx+ny-1ω-Co1⋯Conx+ny-1,Cnx+nynx+ny=ωCnx+nynx+ny.
From (9), we can see the improved RBFN with time-variant output weights is completely equivalent to the conventional RBFN.To analyze the convergence of improved RBFN, we give two assumptions as follows:(1) each output must be bounded when t→∞. In other words, there must be a constant x, satisfying the relationship that xi(t)<x, i=1,…,m; (2) the first sample data of the time series is accurate and xi(1)≠0, i=1,…,m.From (6), we know that the output weights of improved RBFN are time-variant, which are decided by the output of hidden nodes. In order to analyze the convergence, we divided the output of improved RBFN into ∏i=1nx+nyCi parts corresponding to the ∏i=1nx+nyCi outputs of hidden nodes of conventional RBFN. We take only one output of the improved RBFN for analysis, and other parts of the RBFN have the same property. From (5) and (6), the ith hidden nodes for a output of the RBFN are denoted by f(t) for the left of (6) at time t, and the right are denoted by g(t):(10)ft=φCo11·φCo22·…·φConx+nynx+ny·ω-Co1⋯Conx+nyi=φCot·ωCo,gt=φCo11·ω-Co1⋯Conx+nyi,1+φCo22·ω-Co1⋯Conx+nyi,2⋯+φConx+nynx+ny·ω-Co1⋯Conx+nyi,nx+ny=φ-Cot·ω-CoTt,where ω-CoT(t)=[ω-Co1⋯Conx+nyi,1,ω-Co1⋯Conx+nyi,2,…,ω-Co1⋯Conx+nyi,nx+ny]T is the time-variant weight vector and φ-Co(t)=[φCo11,φCo22,…,φConx+nynx+ny] is the output of the hidden layer.From (10), we can get the output of the hidden layer of improved RBFN at time t+1:(11)ft+1=φCot+1·ωCo,gt+1=φ-Cot+1·ω-CoTt+1.From (6), (10), and (11), we know that f(t)=g(t) and f(t+1)=g(t+1), and then the output weights of the improved RBFN at time t+1 and t have the relation as follows:(12)ω-CoTt+1=φCot+1·ωCo·φ-Co-1t+1=φCot+1·ftφCot·φ-Co-1t+1=φCot+1·gtφCot·φ-Co-1t+1=φCot+1·φ-Cot·ω-CoTtφCot·φ-Co-1t+1.So the deviation of the improved RBFN at time t+1 is(13)σt+1=gt+1-ft+1=φ-Cot+1·ω-CoTt+1-ft+1=φ-Cot+1·φCot+1·φ-Cot·ω-CoTtφCot·φ-Co-1t+1-ft+1=φ-Cot+1·φCot+1·ωCo·φ-Cot·ω-CoTtφCot·ωCo·φ-Co-1t+1-ft+1=ft+1·gtft-ft+1=ft+1·gt-ftft=ft+1·σtft=ft+1·ft·σt-1ft-1·ft=⋯=ft+1·…·f2·σ1f1·…·ft=ft+1·σ1f1.Since f(t+1) is the hidden layer output of conventional RBFN and approximation property of conventional RBFN, there is relationship that f(t+1)=yi(t+1)+ε, where yi(t+1) denotes the true value of ith output. ε denotes an arbitrarily small value. σ(t+1) can be written as(14)σt+1=g1-f1·yit+1+εf1≤g1f1-1yit+1+ε,where yi is the true value of system output. According to assumption (2), we know that yi(1)≠0 and yi(1) is accurate. Thus yi(1) is used to replace g(1) and the equation is obtained as follows:(15)g1f1-1=yi1yi1+ε-1=-εyi1+ε.Because of the approximation property of the conventional RBFN, yi(1) is bounded. -ε/yi(1)+ε also can be arbitrarily small when t→∞. Thus the conclusion of the convergence of improved RBFN is proved.


## 2.3. Outliers Detection Using Hidden Markov Model

The real value ofy^(t) is denoted by y(t). From the approximation property of the RBFN, the prediction y^(t) should be close to its true value, y(t). For the jth output, it has the following relationship:(16)y^jt=yjt+ε^jt,where j=1,…,n and ε^j(t) is the estimation of the white noise which obeys the Gaussian distribution N(0,Uj). The probability density function of ε^j(t) is Gaussian-type membership function and given by [11, 12] (17)Pε^jt∣μj,Uj=exp⁡-12y^jt-yjtTUj-1y^jt-yjt,where μj=0. The higher the probability is, the closer ε^j(t) is equal to 0 (the closer y^j(t) is equal to yj(t)). In other words, if P(ε^j(t)∣μj,Uj) equals or is close to 0 at time t, yj(t) is considered to be a possible outlier.In order to make outliers detection accurately, HMM is introduced. HMM is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states. One of the problems in HMM is choosing a most likely state chain with the given observation chain and model. This problem can be solved by Viterbi algorithm [16]. In the observation chain, state transition probabilities are the only parameters. P(ε^j(t)∣μj,Uj) is fit for this character. Hence, an HMM is introduced to detect whether the data are outliers or not. It is noted that the dimension of the chains is the same and equal to the dimension of system output.An HMM for outliers detection is characterized by the following [17]:(1)
Outliers detection is done separately for each system output.(2)
P(ε^j(t)∣μj,Uj) represents the chain which has been observed. The hidden chain is the state of the observation.(3)
There are two states in the HMM. “1” denotes that the state is normal, and “0” means it is outlier.(4)
The transition probability of hidden chain is defined as follows:(18)a01=Na01Na01+a00,a11=Na11Na11+a10,a00=1-a01,a10=1-a11,where aisjs (is,js=0 or 1) denotes the transition probability from state is to js and N(aisjs) denotes the total transition times.To find the best detection result, the probability is defined as(19)ϕist=maxs1,s2,…,st-1⁡Ps1,s2,…,st=is,p1,p2,…,pt,where {p(1),p(2),…,p(t)} is a sequence of observation probabilities and calculated by (17); the element in sequence S={s(1),s(2),…,s(t)} denotes the state; and ϕis(t) is the best score (highest probability) for finding the best state sequence. Since the outliers are detected in a time sequence and the previous data before time t have already been detected, the conventional Viterbi algorithm can be simplified. Equation (19) is rewritten as(20)ϕist=max⁡Pst=is,pt.State transition probabilityaisjs is used to express all influences of historical observations probability and states, and (20) can be rewritten as(21)ϕ1t=ais1p1t;ϕ0t=ais0p0t.p
1
(
t
) obtained by (17) is the observation probability that state is “1” and p0(t)=1-p1(t). Then the state of the data at time t can be determined:Ifϕ1(t)<ϕ0(t), then s(t)=0, and x(t) is an outlier.Else thens(t)=1, x(t) is normal.Then the data which are detected as outliers are removed and replaced by the mean of clean data nearby.


## 3. Dynamic PLS Modeling and Controller Design

### 3.1. Dynamic PLS Modeling

The conventional PLS consists of an outer model which extracts the latent variables and an inner model to estimate the relation between the input and output scores. Considering anl×m dimension input dataset X and an l×n dimension output dataset Y,l,m, and n denote the number of samplings, input dimension, and output dimension, respectively. It is also assumed that the dataset has been processed by the outliers detection algorithm and scaled to zero mean and unit variance. As for outer model, it is built by decomposing matrices X and Y as follows: (22)X-E∗=TPT=∑r=1RtrprT,Y-F∗=UQT=∑r=1RurqrT,where R is the number of latent variables; pr and qr are the rth loading vectors of loading matrices P and Q, respectively; tr and ur are the rth score vectors of matrices T and U, respectively; and E∗ and F∗ are residual matrices of X and Y, respectively.In inner model, the score matricesP and Q are related by a diagonal matrix B which is obtained by least squares (LS) method:(23)ur=brtr,U=TB,Y-F∗=TBQT=∑r=1RbrtrqrT,where br is the rth element of inner relation matrices B.It is noted that PLS has the effect of dimensionality reduction.R<max⁡(m,n) is met in most cases. And nonsquare and ill-conditioned system control problem can be dealt with due to the fact that PLS decomposes the multivariate regression problem into a series of univariate regression problems [2].The major drawback of PLS is that it is not suitable for dynamic systems. Researchers have proposed many different dynamic PLS models by employing structures like time series term or dynamic filters into PLS structure [4, 18]. In this paper, integrating autoregressive exogenous (ARX) model is applied into inner model of PLS [19] to represent the dynamic character of process. It can be expressed as follows:(24)ur=HrtrU=HT,where HT=diag⁡H1t1,H2t2,…,HRtR. Hr· denotes the ARX model of the rth subsystem in the latent variable space. With (24), (23) is transformed to(25)Y-F∗=HTQT=∑r=1RHrtrqrT.An LS algorithm is used to identify the parameters of ARX model in the latent space.


### 3.2. Dynamic PLS Framework for Controller Design

The controller design in latent variable space proposed by Kaspar and Ray [4] is shown in Figure 2. Wx and Wy are scaling matrices of X and Y; Gc, Gp, and Gd are the transfer function matrices of the controller in latent variable space, the controlled plant, and disturbances, respectively. P and Q are the loading matrices of the input data and output data, respectively. Q+ is the appropriate inverse of Q. Since the MIMO system is decomposed into multiple SISO subsystems in latent variable space, the original MIMO controller problem is decomposed into multiple independent SISO control problems. A variety of conventional control schemes could be applied in this control framework [4]. In this paper, GPC would be employed.Figure 2
PLS control framework proposed by Kaspar and Ray.


### 3.3. Improved GPC Controller Design in Dynamic PLS Framework

GPC is one of the model predictive control methods. It has been successfully implemented in many industry and academia, showing good performance and a certain degree of robustness [20]. The GPC controller in dynamic PLS is proposed by Chi et al. [2]. This method has some special advantages, such as simple controller design procedure and low computing time. The predictive model used in GPC is called controlled autoregressive integrated moving averaged (CARIMA) model:(26)Az-1Yt=Bz-1z-dXt-1+Cz-1ξtΔ,where A(z-1)=I+A1z-1+⋯+Anaz-na, B(z-1)=I+B1z-1+⋯+Bnbz-nb, C(z-1)=C0+C1z-1+⋯+Cncz-nc, and Δ=1-z-1.As demonstrated by Laurí et al. [21], a filtered version ARX is equivalent to the CARIMA model for both SISO case and MIMO case. Hence, the ARX models which are obtained in the latent variable space (Section 3.1) can be used for GPC. Replacing the controller Gc in Figure 2, the GPC scheme in dynamic PLS framework is illustrated in Figure 3. The setpoints in the original space are transferred into latent variable space; then the output scores are subtracted to be the inputs of the GPC controllers. The outputs of the GPC controllers are transferred back to the original space as the input of the real process.Figure 3
GPC in dynamic PLS framework.The cost function for therth GPC controller is(27)Jr=∑j=1N1rδrjusetrt+j-u^rt+j22+∑j=1N2rλrjΔtrt+j-122,where u^r(t+j) is the predictive output of j step ahead. N1rand N2r are the control horizon and predictive horizon, respectively. To make the discussion simple, let N1=minr⁡N1r, N2=minr⁡N2r in the following discussion. δr(j) and λr(j) are weighting sequences. δr(j)(j=1,…,N1) is usually a constant weighting sequence that considers future behavior of controllers. It is chosen based on the system features and experience of the engineers while, in this paper, it is dynamically tuned based on the residual of setpoints and output of the systems. In order to denote time-varying weighting sequence, it is rewritten as δr(t+j). Then, (27) is rewritten as(28)Jr=∑j=1N1δrt+j·usetrt+j-u^rt+j22+∑j=1N2λrj·Δtrt+j-122.δr(t+j) is calculated by the following equation:(29)δrt+j=δr0+∑p=1Rθrpusetpt+j-u^pt+j22,where δr0 is a threshold and θrp is a weight coefficient. And θrp=0, when r=p.δ
r affects the weight on the residual of setpoints and prediction outputs. In the case of incompletely decoupling caused by model mismatch, this method can restrain the interaction in the presence of part coupling. The principle of interaction rejection with δr(t+j) is described as follows. When the residual of pth output and setpoint increases, the weight δr(t+j) is adjusted with (29). The adjustment speed of outputs except pth output is accelerated because weight δrt+j(r≠p) is increased according to (29). Then adjustment time and overshoot value will be reduced. In this way, the pth output interaction on other output would be reduced.Take a two-input two-output latent variable model as example. When the 1st output deviation betweenuset1(t+j) and u^1(t+j) is produced, the weight δ2(t+j) of 2nd output is increased. Conversely, when the 1st output deviation is reduced to zero, δ2(t+j) is reduced to the threshold δ20. δ2(t+j) is adjusted with uset1(t+j)-u^1(t+j)22 as shown in Figure 4.Figure 4
δ
2
(
t
+
j
) is adjusted with the 1st output deviation.According to the prediction scheme of GPC,δrt+j in (29) consists of u^p(t+j) of other outputs. The cost function (28) is a multivariate minimization problem. These variables influence each other, which reflects variables interactions due to the plan/model mismatch. There are R cost functions for the R GPC controllers in latent space; these series of minimized problem min⁡(Jr)  (r=1,…,R) are equal to the minimized problem of min⁡(J) as follows:(30)J1+J2+⋯+JR=J.And the total cost function J is(31)J=∑r=1R∑j=1N1δrt+j·usetrt+j-u^rt+j22+∑r=1R∑j=1N2λrj·Δtrt+j-122.Equation (31) is rewritten in the form of matrix as(32)J=Uset-U^TδUset-U^+ΔTTλΔT,where δ=δ10⋱0δR, δr=δr(t+1)0⋱0δr(t+N1), r=1,…,R.Letzpj=[0,…,0︸(p-1)N1+j-1,1,0,…,0︸(R-p)N1-j+N1], θrpj=θrpzpjT, θrj=[θr1j,…,θrRj], and zrj=[z1jT,…,zRjT]T; (29) is rewritten as(33)δrt+j=δr0+∑p=1RθrpUset-U^TzpkTzpkUset-U^=δr0+Uset-U^TθrjzrjUset-U^.Let δr0=δr00⋱0δr0, θr=θr10⋱0θrN1, U-setr-U^-r=Uset-U^0⋱0(Uset-U^), and zr=zr10⋱0zrN1; then the rth output weight is δr=δr0+(U-setr-U^-r)Tθrzr(U-setr-U^-r).Letδ0=diag⁡[δ10,…,δR0], θ=diag⁡θ1,…,θR, Z=diag⁡z1,…,zR, and U-set-U^-=diag⁡[U-set1-U^-1,…,U-setR-U^-R]; weight matrix has the form(34)δ=δ0+Y^--W-TθZY^--W-.Replace the weight factor in (32) with (34); the cost function of improved MIMO GPC in latent variable space is written as follows:(35)J=Uset-U^Tδ0+Y^--W-TθZY^--W-Uset-U^+ΔTTλΔT.Equation (35) is a nonlinear optimal problem; interior point method [22] is a typical method to solve this problem.From the discussion above, the improved GPC has 4 types of parameters (predictive horizonN1, control horizon N2, weight threshold δr0, and weight coefficient θrp) which would influence the control result. They should be chosen properly. For each step of the control period, the following procedures are done for control incremental update.(1)
Get the predictionU^ of the outputs.(2)
Calculate (35) by interior point method.(3)
Send the first element of everyΔT to the score vector T as tr(t+1)=tr(t)+Δtr(t), where r=1,…,R.


## 3.1. Dynamic PLS Modeling

The conventional PLS consists of an outer model which extracts the latent variables and an inner model to estimate the relation between the input and output scores. Considering anl×m dimension input dataset X and an l×n dimension output dataset Y,l,m, and n denote the number of samplings, input dimension, and output dimension, respectively. It is also assumed that the dataset has been processed by the outliers detection algorithm and scaled to zero mean and unit variance. As for outer model, it is built by decomposing matrices X and Y as follows: (22)X-E∗=TPT=∑r=1RtrprT,Y-F∗=UQT=∑r=1RurqrT,where R is the number of latent variables; pr and qr are the rth loading vectors of loading matrices P and Q, respectively; tr and ur are the rth score vectors of matrices T and U, respectively; and E∗ and F∗ are residual matrices of X and Y, respectively.In inner model, the score matricesP and Q are related by a diagonal matrix B which is obtained by least squares (LS) method:(23)ur=brtr,U=TB,Y-F∗=TBQT=∑r=1RbrtrqrT,where br is the rth element of inner relation matrices B.It is noted that PLS has the effect of dimensionality reduction.R<max⁡(m,n) is met in most cases. And nonsquare and ill-conditioned system control problem can be dealt with due to the fact that PLS decomposes the multivariate regression problem into a series of univariate regression problems [2].The major drawback of PLS is that it is not suitable for dynamic systems. Researchers have proposed many different dynamic PLS models by employing structures like time series term or dynamic filters into PLS structure [4, 18]. In this paper, integrating autoregressive exogenous (ARX) model is applied into inner model of PLS [19] to represent the dynamic character of process. It can be expressed as follows:(24)ur=HrtrU=HT,where HT=diag⁡H1t1,H2t2,…,HRtR. Hr· denotes the ARX model of the rth subsystem in the latent variable space. With (24), (23) is transformed to(25)Y-F∗=HTQT=∑r=1RHrtrqrT.An LS algorithm is used to identify the parameters of ARX model in the latent space.


## 3.2. Dynamic PLS Framework for Controller Design

The controller design in latent variable space proposed by Kaspar and Ray [4] is shown in Figure 2. Wx and Wy are scaling matrices of X and Y; Gc, Gp, and Gd are the transfer function matrices of the controller in latent variable space, the controlled plant, and disturbances, respectively. P and Q are the loading matrices of the input data and output data, respectively. Q+ is the appropriate inverse of Q. Since the MIMO system is decomposed into multiple SISO subsystems in latent variable space, the original MIMO controller problem is decomposed into multiple independent SISO control problems. A variety of conventional control schemes could be applied in this control framework [4]. In this paper, GPC would be employed.Figure 2
PLS control framework proposed by Kaspar and Ray.


## 3.3. Improved GPC Controller Design in Dynamic PLS Framework

GPC is one of the model predictive control methods. It has been successfully implemented in many industry and academia, showing good performance and a certain degree of robustness [20]. The GPC controller in dynamic PLS is proposed by Chi et al. [2]. This method has some special advantages, such as simple controller design procedure and low computing time. The predictive model used in GPC is called controlled autoregressive integrated moving averaged (CARIMA) model:(26)Az-1Yt=Bz-1z-dXt-1+Cz-1ξtΔ,where A(z-1)=I+A1z-1+⋯+Anaz-na, B(z-1)=I+B1z-1+⋯+Bnbz-nb, C(z-1)=C0+C1z-1+⋯+Cncz-nc, and Δ=1-z-1.As demonstrated by Laurí et al. [21], a filtered version ARX is equivalent to the CARIMA model for both SISO case and MIMO case. Hence, the ARX models which are obtained in the latent variable space (Section 3.1) can be used for GPC. Replacing the controller Gc in Figure 2, the GPC scheme in dynamic PLS framework is illustrated in Figure 3. The setpoints in the original space are transferred into latent variable space; then the output scores are subtracted to be the inputs of the GPC controllers. The outputs of the GPC controllers are transferred back to the original space as the input of the real process.Figure 3
GPC in dynamic PLS framework.The cost function for therth GPC controller is(27)Jr=∑j=1N1rδrjusetrt+j-u^rt+j22+∑j=1N2rλrjΔtrt+j-122,where u^r(t+j) is the predictive output of j step ahead. N1rand N2r are the control horizon and predictive horizon, respectively. To make the discussion simple, let N1=minr⁡N1r, N2=minr⁡N2r in the following discussion. δr(j) and λr(j) are weighting sequences. δr(j)(j=1,…,N1) is usually a constant weighting sequence that considers future behavior of controllers. It is chosen based on the system features and experience of the engineers while, in this paper, it is dynamically tuned based on the residual of setpoints and output of the systems. In order to denote time-varying weighting sequence, it is rewritten as δr(t+j). Then, (27) is rewritten as(28)Jr=∑j=1N1δrt+j·usetrt+j-u^rt+j22+∑j=1N2λrj·Δtrt+j-122.δr(t+j) is calculated by the following equation:(29)δrt+j=δr0+∑p=1Rθrpusetpt+j-u^pt+j22,where δr0 is a threshold and θrp is a weight coefficient. And θrp=0, when r=p.δ
r affects the weight on the residual of setpoints and prediction outputs. In the case of incompletely decoupling caused by model mismatch, this method can restrain the interaction in the presence of part coupling. The principle of interaction rejection with δr(t+j) is described as follows. When the residual of pth output and setpoint increases, the weight δr(t+j) is adjusted with (29). The adjustment speed of outputs except pth output is accelerated because weight δrt+j(r≠p) is increased according to (29). Then adjustment time and overshoot value will be reduced. In this way, the pth output interaction on other output would be reduced.Take a two-input two-output latent variable model as example. When the 1st output deviation betweenuset1(t+j) and u^1(t+j) is produced, the weight δ2(t+j) of 2nd output is increased. Conversely, when the 1st output deviation is reduced to zero, δ2(t+j) is reduced to the threshold δ20. δ2(t+j) is adjusted with uset1(t+j)-u^1(t+j)22 as shown in Figure 4.Figure 4
δ
2
(
t
+
j
) is adjusted with the 1st output deviation.According to the prediction scheme of GPC,δrt+j in (29) consists of u^p(t+j) of other outputs. The cost function (28) is a multivariate minimization problem. These variables influence each other, which reflects variables interactions due to the plan/model mismatch. There are R cost functions for the R GPC controllers in latent space; these series of minimized problem min⁡(Jr)  (r=1,…,R) are equal to the minimized problem of min⁡(J) as follows:(30)J1+J2+⋯+JR=J.And the total cost function J is(31)J=∑r=1R∑j=1N1δrt+j·usetrt+j-u^rt+j22+∑r=1R∑j=1N2λrj·Δtrt+j-122.Equation (31) is rewritten in the form of matrix as(32)J=Uset-U^TδUset-U^+ΔTTλΔT,where δ=δ10⋱0δR, δr=δr(t+1)0⋱0δr(t+N1), r=1,…,R.Letzpj=[0,…,0︸(p-1)N1+j-1,1,0,…,0︸(R-p)N1-j+N1], θrpj=θrpzpjT, θrj=[θr1j,…,θrRj], and zrj=[z1jT,…,zRjT]T; (29) is rewritten as(33)δrt+j=δr0+∑p=1RθrpUset-U^TzpkTzpkUset-U^=δr0+Uset-U^TθrjzrjUset-U^.Let δr0=δr00⋱0δr0, θr=θr10⋱0θrN1, U-setr-U^-r=Uset-U^0⋱0(Uset-U^), and zr=zr10⋱0zrN1; then the rth output weight is δr=δr0+(U-setr-U^-r)Tθrzr(U-setr-U^-r).Letδ0=diag⁡[δ10,…,δR0], θ=diag⁡θ1,…,θR, Z=diag⁡z1,…,zR, and U-set-U^-=diag⁡[U-set1-U^-1,…,U-setR-U^-R]; weight matrix has the form(34)δ=δ0+Y^--W-TθZY^--W-.Replace the weight factor in (32) with (34); the cost function of improved MIMO GPC in latent variable space is written as follows:(35)J=Uset-U^Tδ0+Y^--W-TθZY^--W-Uset-U^+ΔTTλΔT.Equation (35) is a nonlinear optimal problem; interior point method [22] is a typical method to solve this problem.From the discussion above, the improved GPC has 4 types of parameters (predictive horizonN1, control horizon N2, weight threshold δr0, and weight coefficient θrp) which would influence the control result. They should be chosen properly. For each step of the control period, the following procedures are done for control incremental update.(1)
Get the predictionU^ of the outputs.(2)
Calculate (35) by interior point method.(3)
Send the first element of everyΔT to the score vector T as tr(t+1)=tr(t)+Δtr(t), where r=1,…,R.


## 4. Examples

In this section, two examples are presented to demonstrate the advantages of proposed modeling and control strategy.


### 4.1. Example  1

The first example is an industrial-scale polymerization reactor control problem, which is proposed by Chien et al. [23]. The basic control strategy is similar to the terpolymerization reactor example by Ogunnaike et al. [24]. The advanced product quality control loops are used to track the setpoints of the reactor condition control loops. The process model for the reactor setpoints control loops is (36)
Y
s
=
22.89
e
-
0.2
s
4.572
s
+
1
-
11.64
e
-
0.4
s
1.807
s
+
1
4.689
e
-
0.2
s
2.174
s
+
1
5.80
e
-
0.4
s
1.801
s
+
1
X
s
+
-
4.243
e
-
0.4
s
3.445
s
+
1
-
0.601
e
-
0.4
s
1.982
s
+
1
d
s
.Controlled variables are the two measurements representing the reactor condition; and manipulated variables are the setpoints of two reactor feed flow loops; load disturbance is the purge flow of the reactor.In order to build up the dynamic PLS model, two random step signals with magnitude-0.50.5 are applied to excite the system. And random signals with -0.050.05 and -0.10.1 are added to both disturbance inputs separately in order to simulate the disturbance influence. The generated datasets are plotted in Figure 5.Figure 5
Modeling dataset for Wood and Berry process.In order to demonstrate the performance of outliers detection method, a number of outliers are added to the outputs of the modeling dataset and several performance criteria are involved to evaluate the performance.In this example, the number of latent variables is 2. And an outlier is defined as 3 times bigger than the true value to simulate the failure that some samples are measured abnormally. Outliers are randomly added to the outputs at different rate. The data with more than 50% outliers are considered invalid because the useful information it carried is not sufficient. Since the numbers of outliers are added randomly, the results of different simulations may be different. For example, the result with 100 discontinuous outliers is better than the continuous ones. A discontinuous outlier can be detected and imputed easily based on the adjacent true points.For each rate of outliers, 3 performance criteria are calculated to evaluate the performance of outliers detection method. LetH, α, β, and χ denote the number of samples, outliers that are detected, error detection (normal samples detected to be outliers), and missing detection (outliers that are not detected), respectively. The criteria are described as follows:(1)
Accurate rate is the ratio of actual outliers among (α-β) and actual outliers (α-β+χ): (37)accurate  rate=α-βα-β+χ.(2)
Error detection means the true value is considered as an outlier. Error rate is the ratio of the number of error detections (β) and the number of actual true samples (H-α+β-χ): (38)error  rate=βH-α+β-χ.(3)
Missing detection means outliers are considered as true samples. Missing rate is the ratio of the number of missing detections (χ) and the number of actual outliers (α-β+χ):(39)missing  rate=χα-β+χ.The simulation result is shown in Table1. It is clear that the less outliers contained, the better accurate rate is obtained. It is noted that the results in Table 1 could have a bit different because of the randomness of outliers.Table 1
The detection result of different rates of outliers.
Percentage of outliers
Output number
Accurate rate
Error rate
Missing rate
1%
Output 1
100%
2.68%
0%
Output 2
90%
1.24%
10%
5%
Output 1
68.75%
6.22%
31.25%
Output 2
77.08%
4.08%
22.92%
10%
Output 1
36.96%
5.51%
63.04%
Output 2
16.44%
1.01%
83.56%
25%
Output 1
11.32%
2.60%
88.68%
Output 2
14.05
2.77%
85.95%
50%
Output 1
7.09%
2.50%
92.91%
Output 2
12.76%
3.02%
87.24%The detected results of 10% outliers are shown in Figures6(b) and 6(d). As observed in Figure 6 and Table 1, most outliers have been detected out correctly when the rate of outliers is small. That is because when outliers’ rate increases, the probability that samples nearby outliers are also outliers is bigger. That is to say, the useful information nearby would be missing; the RBFN method would not get enough information to predict the next output correctly. The samples of y1 from 600 to 750 are all detected as true values because the values of these samples are very small, and the outliers are submerged in the noises.Figure 6
Detected results for 10% outliers.
(a)
The data ofy1 with outliers
(b)
The detection result ofy1
(c)
The data ofy2 with outliers
(d)
The detection result ofy2The data with 10% outliers would be used for further dynamic PLS modeling. The outliers that have been detected as outliers would be replaced with the corresponding samples next to them. In order to evaluate the robustness of dynamic PLS modeling with outliers detection, a criterion of comparison is applied [25]. The angle between loading vectors of the improved method with outliers detection and conventional method is chosen and is defined as angr=angr(qr,qr′), where angr(qr,qr′) is the function that calculates the angle between the loading vectors qr and qr′. qr denotes the loading vector which is derived from the data without outliers. qr′ denotes the loading vectors of modeling method with outliers detection or conventional modeling method.The simulation result is shown in Table2. From Table 2, it is clear that the improved method has smaller angle than the conventional method. A model validation test of the two methods is done and shown in Figure 7. The other 500 samples without outliers are used to perform this test. The improved dynamic PLS without outliers detection method and conventional method are denoted as ODPLS and DPLS for short. The integral of squared error (ISE) is applied to evaluate the deviation between real data and modeling result which is described as(40)ISEi=∫0∞y^τi-yiτ2dτ,where yi(τ) denotes the real data, y^(τ)i denotes the modeling result data, and i=1,…,n denotes the output number.Table 2
The comparison of the two methods.
Conventional method
Improved method
Ang1
13.8
2.46
Ang2
9.31
2.75Figure 7
The validation for the improved dynamic PLS modeling method and conventional method.In Figure7, the two ISEs of ODPLS are 1.23×103 and 0.09×103, while ISEs of DPLS are 1.38×104 and 0.12×104. It is clear that the dynamic PLS modeling with outliers detection is more robust than conventional method.In addition, the SISO GPC (SGPC) controllers and improved MIMO GPC (MGPC) controller in latent variable space are implemented with the two modeling methods. The results are shown in Figures8 and 9. The predictive horizon N1 and control horizon N2 have the same value in the 4 simulations (N1=2, N2=1). It is noted that the SGPC and MGPC with DPLS model would be unstable when N1 and N2 are bigger because of the mismatch between the DPLS model and the real process which is caused by outliers. It is observed that SGPC and MGPC with ODPLS obtain better control performance than that with DPLS in terms of rise time and ISE. That is because ODPLS partly removes outliers away and make the modeling more accurate than DPLS. With the same model (DPLS or ODPLS), it can also be observed that MGPC obtains better result than SGPC in terms of rise time and ISE. In summary, it can be concluded that MGPC with ODPLS has the least interaction between the two outputs among the 4 simulations.Figure 8
The simulation result of SGPC with ODPLS and DPLS.Figure 9
The simulation result of MGPC with ODPLS and DPLS.


### 4.2. Example  2

The methanol-water distillation column model reported by Wood and Berry [26] is a typical 2-by-2 process with strong interaction. The reflux and the reboiler steam flow rates are the process inputs. The compositions of the top and bottom products are the process outputs. The discrete sample time Ts=1 min. The transfer function is given as [6](41)
G
p
s
=
12.8
e
-
s
16.7
s
+
1
-
18.9
e
-
3
s
21
s
+
1
6.6
e
-
7
s
10.9
s
+
1
-
19.4
e
-
3
s
14.4
s
+
1
.In order to build up the dynamic PLS model, two 1000 sampling random step signals with-0.50.5 are applied to excite the system and generate the outputs. And white noise in which the SNR (signal to noise ratio) is 10 DB is added to both outputs in order to simulate the actual outputs. The generated dataset is plotted in Figure 10.Figure 10
Modeling dataset for Wood and Berry process.In this example, the number of latent variables is 2. And a zero value or a small value is defined as an outlier to simulate the measuring equipment failure that some samples are detected abnormally. 10% and 25% rates of outliers are added. The effect of RBFN parameters on the outliers detection result is shown in Tables3 and 4. These parameters include the number of input and output time delays (nx,ny) and hidden node center (C), where the hidden node center of each RBFN input has the same number C. Comparing Tables 3 and 4, RBFN parameters have more impaction on the detection result at 25% rate of outliers detection than at 10%. That is because great rate of outliers makes outliers detection more difficult and results in the poor robustness of RBFN. It is obvious that RBFN with parameters (4, 4, 2) has better outliers detection result shown in Table 4. It is noted that the real detected result such as that in Table 4 or Table 3 cannot be obtained in the real process. Parameters may be selected based on the ratio of ISE and integral of output (ISE/IO) as shown in Table 5 which would be fitting the samples properly. Shown in Figure 11 are the detected results of 10% outliers with RBFN parameters (4, 4, 2) which has better fitting result and is used for the further control simulation.Table 3
Detection result of different RBFN parameters with 10% outliers.
RBFN parameters (nx, ny, C)
(3, 3, 2)
(4, 4, 2)
(5, 5, 2)
(3, 3, 3)
(3, 3, 4)
Output 1
Accurate rate
84.95%
82.61%
86.73%
84.69%
83.87%
Error rate
17.14%
14.86%
14.74%
16.09%
15.44%
Missing rate
15.05%
17.39%
13.27%
15.31%
16.13%
Output 2
Accurate rate
79.79%
75.76%
82.80%
88.30%
82.61%
Error rate
11.17%
10.44%
18.82%
16.93%
12.95%
Missing rate
20.21%
24.24%
17.20%
11.70%
17.39%Table 4
Detection result of different RBFN parameters with 25% outliers.
RBFN parameters (nx, ny, C)
(3, 3, 2)
(4, 4, 2)
(5, 5, 2)
(3, 3, 3)
(3, 3, 4)
Output 1
Accurate rate
74.07%
75.18%
75.57%
27.86%
15.56%
Error rate
16.21%
13.82%
3.41%
1.31%
3.08%
Missing rate
25.93%
24.82%
24.43%
72.14%
84.44%
Output 2
Accurate rate
69.50%
67.65%
66.67%
43.61%
13.33%
Error rate
12.27%
10.42%
12.62%
8.03%
7.89%
Missing rate
30.50%
32.35%
33.33%
56.39%
86.67%Table 5
ISE/IO of different RBFN parameters.
RBFN parameters (nx, ny, C)
10% outliers
25% outliers
Output 1
Output 2
Output 1
Output 2
(3, 3, 2)
1.60
1.26
3.29
2.74
(4, 4, 2)
1.29
1.24
3.23
2.74
(5, 5, 2)
1.45
1.17
3.24
2.74
(3, 3, 3)
1.51
1.42
3.24
2.75
(3, 3, 4)
1.67
1.29
3.26
2.77Figure 11
Detected results for 10% outliers with RBFN parameters (4, 4, 2).
(a)
The data ofy1 with outliers
(b)
The detection result ofy1
(c)
The data ofy2 with outliers
(d)
The detection result ofy2Figure12 shows the control result of MGPC with different parameters, where the solid line is the result of parameter, δ10=1, δ10=1, θ12=2, θ22=2(1,1,2,2), and dotted line is the result of parameter, δ10=1, δ10=2, θ12=5, θ22=5(1,2,5,5). It is clear to see the interaction between the 2 outputs at times 250 and 750, where y2 is disturbed by the setpoint changing of y1. From Figure 12, control result of dotted line has less overshot and shorter convergence time. It can conclude that the parameters δr0 and θrp in (31) have a great influence on the control result. With appropriate parameters in (31) a better control result can be obtained.Figure 12
The simulation result of MGPC with different parameters.In real process, there are some disturbance signals under the control framework. In order to get the disturbance rejection performance,y1 and y2 are separately set at 0.1 and 0, and a 50% change is added to y1. The result is shown in Figure 13. MGPC has less disturbance than SGPC for both y1 and y2. And the convergence time of MGPC is also shorter.Figure 13
The simulation result of disturbance rejection.


## 4.1. Example  1

The first example is an industrial-scale polymerization reactor control problem, which is proposed by Chien et al. [23]. The basic control strategy is similar to the terpolymerization reactor example by Ogunnaike et al. [24]. The advanced product quality control loops are used to track the setpoints of the reactor condition control loops. The process model for the reactor setpoints control loops is (36)
Y
s
=
22.89
e
-
0.2
s
4.572
s
+
1
-
11.64
e
-
0.4
s
1.807
s
+
1
4.689
e
-
0.2
s
2.174
s
+
1
5.80
e
-
0.4
s
1.801
s
+
1
X
s
+
-
4.243
e
-
0.4
s
3.445
s
+
1
-
0.601
e
-
0.4
s
1.982
s
+
1
d
s
.Controlled variables are the two measurements representing the reactor condition; and manipulated variables are the setpoints of two reactor feed flow loops; load disturbance is the purge flow of the reactor.In order to build up the dynamic PLS model, two random step signals with magnitude-0.50.5 are applied to excite the system. And random signals with -0.050.05 and -0.10.1 are added to both disturbance inputs separately in order to simulate the disturbance influence. The generated datasets are plotted in Figure 5.Figure 5
Modeling dataset for Wood and Berry process.In order to demonstrate the performance of outliers detection method, a number of outliers are added to the outputs of the modeling dataset and several performance criteria are involved to evaluate the performance.In this example, the number of latent variables is 2. And an outlier is defined as 3 times bigger than the true value to simulate the failure that some samples are measured abnormally. Outliers are randomly added to the outputs at different rate. The data with more than 50% outliers are considered invalid because the useful information it carried is not sufficient. Since the numbers of outliers are added randomly, the results of different simulations may be different. For example, the result with 100 discontinuous outliers is better than the continuous ones. A discontinuous outlier can be detected and imputed easily based on the adjacent true points.For each rate of outliers, 3 performance criteria are calculated to evaluate the performance of outliers detection method. LetH, α, β, and χ denote the number of samples, outliers that are detected, error detection (normal samples detected to be outliers), and missing detection (outliers that are not detected), respectively. The criteria are described as follows:(1)
Accurate rate is the ratio of actual outliers among (α-β) and actual outliers (α-β+χ): (37)accurate  rate=α-βα-β+χ.(2)
Error detection means the true value is considered as an outlier. Error rate is the ratio of the number of error detections (β) and the number of actual true samples (H-α+β-χ): (38)error  rate=βH-α+β-χ.(3)
Missing detection means outliers are considered as true samples. Missing rate is the ratio of the number of missing detections (χ) and the number of actual outliers (α-β+χ):(39)missing  rate=χα-β+χ.The simulation result is shown in Table1. It is clear that the less outliers contained, the better accurate rate is obtained. It is noted that the results in Table 1 could have a bit different because of the randomness of outliers.Table 1
The detection result of different rates of outliers.
Percentage of outliers
Output number
Accurate rate
Error rate
Missing rate
1%
Output 1
100%
2.68%
0%
Output 2
90%
1.24%
10%
5%
Output 1
68.75%
6.22%
31.25%
Output 2
77.08%
4.08%
22.92%
10%
Output 1
36.96%
5.51%
63.04%
Output 2
16.44%
1.01%
83.56%
25%
Output 1
11.32%
2.60%
88.68%
Output 2
14.05
2.77%
85.95%
50%
Output 1
7.09%
2.50%
92.91%
Output 2
12.76%
3.02%
87.24%The detected results of 10% outliers are shown in Figures6(b) and 6(d). As observed in Figure 6 and Table 1, most outliers have been detected out correctly when the rate of outliers is small. That is because when outliers’ rate increases, the probability that samples nearby outliers are also outliers is bigger. That is to say, the useful information nearby would be missing; the RBFN method would not get enough information to predict the next output correctly. The samples of y1 from 600 to 750 are all detected as true values because the values of these samples are very small, and the outliers are submerged in the noises.Figure 6
Detected results for 10% outliers.
(a)
The data ofy1 with outliers
(b)
The detection result ofy1
(c)
The data ofy2 with outliers
(d)
The detection result ofy2The data with 10% outliers would be used for further dynamic PLS modeling. The outliers that have been detected as outliers would be replaced with the corresponding samples next to them. In order to evaluate the robustness of dynamic PLS modeling with outliers detection, a criterion of comparison is applied [25]. The angle between loading vectors of the improved method with outliers detection and conventional method is chosen and is defined as angr=angr(qr,qr′), where angr(qr,qr′) is the function that calculates the angle between the loading vectors qr and qr′. qr denotes the loading vector which is derived from the data without outliers. qr′ denotes the loading vectors of modeling method with outliers detection or conventional modeling method.The simulation result is shown in Table2. From Table 2, it is clear that the improved method has smaller angle than the conventional method. A model validation test of the two methods is done and shown in Figure 7. The other 500 samples without outliers are used to perform this test. The improved dynamic PLS without outliers detection method and conventional method are denoted as ODPLS and DPLS for short. The integral of squared error (ISE) is applied to evaluate the deviation between real data and modeling result which is described as(40)ISEi=∫0∞y^τi-yiτ2dτ,where yi(τ) denotes the real data, y^(τ)i denotes the modeling result data, and i=1,…,n denotes the output number.Table 2
The comparison of the two methods.
Conventional method
Improved method
Ang1
13.8
2.46
Ang2
9.31
2.75Figure 7
The validation for the improved dynamic PLS modeling method and conventional method.In Figure7, the two ISEs of ODPLS are 1.23×103 and 0.09×103, while ISEs of DPLS are 1.38×104 and 0.12×104. It is clear that the dynamic PLS modeling with outliers detection is more robust than conventional method.In addition, the SISO GPC (SGPC) controllers and improved MIMO GPC (MGPC) controller in latent variable space are implemented with the two modeling methods. The results are shown in Figures8 and 9. The predictive horizon N1 and control horizon N2 have the same value in the 4 simulations (N1=2, N2=1). It is noted that the SGPC and MGPC with DPLS model would be unstable when N1 and N2 are bigger because of the mismatch between the DPLS model and the real process which is caused by outliers. It is observed that SGPC and MGPC with ODPLS obtain better control performance than that with DPLS in terms of rise time and ISE. That is because ODPLS partly removes outliers away and make the modeling more accurate than DPLS. With the same model (DPLS or ODPLS), it can also be observed that MGPC obtains better result than SGPC in terms of rise time and ISE. In summary, it can be concluded that MGPC with ODPLS has the least interaction between the two outputs among the 4 simulations.Figure 8
The simulation result of SGPC with ODPLS and DPLS.Figure 9
The simulation result of MGPC with ODPLS and DPLS.


## 4.2. Example  2

The methanol-water distillation column model reported by Wood and Berry [26] is a typical 2-by-2 process with strong interaction. The reflux and the reboiler steam flow rates are the process inputs. The compositions of the top and bottom products are the process outputs. The discrete sample time Ts=1 min. The transfer function is given as [6](41)
G
p
s
=
12.8
e
-
s
16.7
s
+
1
-
18.9
e
-
3
s
21
s
+
1
6.6
e
-
7
s
10.9
s
+
1
-
19.4
e
-
3
s
14.4
s
+
1
.In order to build up the dynamic PLS model, two 1000 sampling random step signals with-0.50.5 are applied to excite the system and generate the outputs. And white noise in which the SNR (signal to noise ratio) is 10 DB is added to both outputs in order to simulate the actual outputs. The generated dataset is plotted in Figure 10.Figure 10
Modeling dataset for Wood and Berry process.In this example, the number of latent variables is 2. And a zero value or a small value is defined as an outlier to simulate the measuring equipment failure that some samples are detected abnormally. 10% and 25% rates of outliers are added. The effect of RBFN parameters on the outliers detection result is shown in Tables3 and 4. These parameters include the number of input and output time delays (nx,ny) and hidden node center (C), where the hidden node center of each RBFN input has the same number C. Comparing Tables 3 and 4, RBFN parameters have more impaction on the detection result at 25% rate of outliers detection than at 10%. That is because great rate of outliers makes outliers detection more difficult and results in the poor robustness of RBFN. It is obvious that RBFN with parameters (4, 4, 2) has better outliers detection result shown in Table 4. It is noted that the real detected result such as that in Table 4 or Table 3 cannot be obtained in the real process. Parameters may be selected based on the ratio of ISE and integral of output (ISE/IO) as shown in Table 5 which would be fitting the samples properly. Shown in Figure 11 are the detected results of 10% outliers with RBFN parameters (4, 4, 2) which has better fitting result and is used for the further control simulation.Table 3
Detection result of different RBFN parameters with 10% outliers.
RBFN parameters (nx, ny, C)
(3, 3, 2)
(4, 4, 2)
(5, 5, 2)
(3, 3, 3)
(3, 3, 4)
Output 1
Accurate rate
84.95%
82.61%
86.73%
84.69%
83.87%
Error rate
17.14%
14.86%
14.74%
16.09%
15.44%
Missing rate
15.05%
17.39%
13.27%
15.31%
16.13%
Output 2
Accurate rate
79.79%
75.76%
82.80%
88.30%
82.61%
Error rate
11.17%
10.44%
18.82%
16.93%
12.95%
Missing rate
20.21%
24.24%
17.20%
11.70%
17.39%Table 4
Detection result of different RBFN parameters with 25% outliers.
RBFN parameters (nx, ny, C)
(3, 3, 2)
(4, 4, 2)
(5, 5, 2)
(3, 3, 3)
(3, 3, 4)
Output 1
Accurate rate
74.07%
75.18%
75.57%
27.86%
15.56%
Error rate
16.21%
13.82%
3.41%
1.31%
3.08%
Missing rate
25.93%
24.82%
24.43%
72.14%
84.44%
Output 2
Accurate rate
69.50%
67.65%
66.67%
43.61%
13.33%
Error rate
12.27%
10.42%
12.62%
8.03%
7.89%
Missing rate
30.50%
32.35%
33.33%
56.39%
86.67%Table 5
ISE/IO of different RBFN parameters.
RBFN parameters (nx, ny, C)
10% outliers
25% outliers
Output 1
Output 2
Output 1
Output 2
(3, 3, 2)
1.60
1.26
3.29
2.74
(4, 4, 2)
1.29
1.24
3.23
2.74
(5, 5, 2)
1.45
1.17
3.24
2.74
(3, 3, 3)
1.51
1.42
3.24
2.75
(3, 3, 4)
1.67
1.29
3.26
2.77Figure 11
Detected results for 10% outliers with RBFN parameters (4, 4, 2).
(a)
The data ofy1 with outliers
(b)
The detection result ofy1
(c)
The data ofy2 with outliers
(d)
The detection result ofy2Figure12 shows the control result of MGPC with different parameters, where the solid line is the result of parameter, δ10=1, δ10=1, θ12=2, θ22=2(1,1,2,2), and dotted line is the result of parameter, δ10=1, δ10=2, θ12=5, θ22=5(1,2,5,5). It is clear to see the interaction between the 2 outputs at times 250 and 750, where y2 is disturbed by the setpoint changing of y1. From Figure 12, control result of dotted line has less overshot and shorter convergence time. It can conclude that the parameters δr0 and θrp in (31) have a great influence on the control result. With appropriate parameters in (31) a better control result can be obtained.Figure 12
The simulation result of MGPC with different parameters.In real process, there are some disturbance signals under the control framework. In order to get the disturbance rejection performance,y1 and y2 are separately set at 0.1 and 0, and a 50% change is added to y1. The result is shown in Figure 13. MGPC has less disturbance than SGPC for both y1 and y2. And the convergence time of MGPC is also shorter.Figure 13
The simulation result of disturbance rejection.


## 5. Conclusions

Since dynamic PLS modeling is sensitive to outliers in the data, a new RBFN structure with HMM to detect outliers is proposed to deal with contaminated data. The improved RBFN considered past samples as the input of the network for the dynamic system prediction. It has less hidden nodes; the equivalent to the conventional RBFN and convergence is proved. With HMM, there is no need to preselect a threshold to detect outliers. With the processed data in this method, a more robust dynamic PLS model is obtained. Unfortunately, there is still mismatch between the model and the real process, caused by many reasons. The inner model cost function in dynamic PLS is treated as a special MIMO problem and a GPC based on the adaptive weights is used to tackle this problem. Two examples are used to demonstrate the performance of outlier detecting, accurate modeling, and control.



---

*Source: 923584-2015-10-22.xml*