# A SVR Learning Based Sensor Placement Approach for Nonlinear Spatially Distributed Systems

Authors: Xian-xia Zhang; Zhi-qiang Fu; Wei-lu Shan; Bing Wang; Tao Zou
Journal: Applied Computational Intelligence and Soft Computing
 (2016)

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## Abstract

Many industrial processes are inherently distributed in space and time and are called spatially distributed dynamical systems (SDDSs). Sensor placement affects capturing the spatial distribution and then becomes crucial issue to model or control an SDDS. In this study, a new data-driven based sensor placement method is developed. SVR algorithm is innovatively used to extract the characteristics of spatial distribution from a spatiotemporal data set. The support vectors learned by SVR represent the crucial spatial data structure in the spatiotemporal data set, which can be employed to determine optimal sensor location and sensor number. A systematic sensor placement design scheme in three steps (data collection, SVR learning, and sensor locating) is developed for an easy implementation. Finally, effectiveness of the proposed sensor placement scheme is validated on two spatiotemporal 3D fuzzy controlled spatially distributed systems.


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## Body

## 1. Introduction

Many industrial processes are inherently distributed in space and time, such as fluid flow process, spray deposition process, heat exchange process, and snap curing process. These systems are usually called spatially distributed dynamical systems (SDDSs) or distributed parameter systems (DPSs) [1]. Traditionally, the spatially distributed nature is ignored, and then an SDDS is simplified into a lumped parameter system (LPS). However, its performance will deteriorate if the system dynamics significantly vary with space [2]. To satisfy tighter product quality requirements, the spatial nature should be considered in the modeling and control. Consequently, the spatiotemporal modeling and control for SDDSs have become the highlight in the modern modeling and control theory.On the research of SDDS, since sensor placement affects capturing the spatial distribution, it becomes one of key issues to influence the performance of spatiotemporal modeling or control. Sensor placement is a difficult problem, particularly in nonlinear and multivariable case [3].In the past several decades, sensor placement has been studied and can be classified into two categories, that is, model-based method and data-driven based method. For the first category, the sensor placement is dependent on an accurate mathematical model. Some methods aim at achieving better parameter estimation with different criteria, for example, scalar criteria of the covariance matrix [4] and scalar measure of performance defined on the Fisher information matrix [5]. Some methods intend to improve state estimation or state observer with possible criteria, such as observability measures [6], convergence properties of observer [7], and cost function related to Gramian observability matrix [8]. Some methods integrate optimal actuator and sensor placement with nonlinear output feedback control [9–11] and improve control performance, where the penalty on the response of the closed-loop system is used for cost function of actuator placement and the control action, and the estimation error in the closed-loop infinite-dimensional system is used for cost function of sensor placement.For the second category, only data information is used for sensor placement. Wouwer et al. [12] proposed the determinant of Fisher information matrix formed by sensitivity functions for parameter estimation and the Gram determinant formed by sensor responses for observer estimation. Zamprogna et al. [13] selected the most suitable secondary process variable as soft sensor inputs for batch distillation by exploiting the properties of principal component analysis (PCA) on the sensitivity matrix. Tongpadungroda et al. [14] identified the optimal position of an applied load by GA under the criterion related to PCA. Zhang et al. [15] proposed a sensor placement method for spatiotemporal 3D fuzzy control system based on spatial constrained fuzzy c-means algorithm. Compared with the model-based method, the data-driven method is very promising in practical engineering applications since no mathematical model is used.In this study, a new data-driven based sensor placement method is proposed. Support vector regression (SVR) algorithm is used to extract the main characteristics of spatial distribution from a spatiotemporal data set, which can be directly used for the sensor placement. The support vectors learned by SVR represent the crucial spatial data structure hidden in the spatiotemporal data set, which can be employed to determine the optimal sensor location and sensor number. A systematic sensor placement design scheme is developed for an easy implementation, which consists of data collection, SVR learning, and sensor locating. The effectiveness of the proposed sensor placement scheme is validated on two spatiotemporal 3D fuzzy controlled nonlinear spatially distributed systems.This paper is organized as follows. Preliminaries about nonlinear spatially distributed systems and SVR algorithm are described in Section2. In Section 3, the SVR learning based sensor placement method is presented in detail. Two illustrative examples are given in Section 4. Finally, Section 5 summarizes the conclusion.


## 2. Preliminaries

### 2.1. Nonlinear Spatially Distributed Dynamical Systems

Many industrial processes exhibit highly nonlinear behavior and strong spatial variations. They usually can be represented by nonlinear partial differential equations. Some processes have strong convection characteristics [16], some have strong diffusion phenomena [17], and some have both convection and diffusion characteristics [18]. These systems are usually installed with multiple spatially distributed actuators and multiple sensors and give rise to nonlinear control problems that involve the regulation of highly distributed control variables using these spatially distributed actuators and sensors.In the following subsections, two examples of SDDSs are presented and illustrated.


#### 2.1.1. A Catalytic Packed-Bed Reactor

The catalytic packed-bed reactor [17, 19] in Figure 1 is a typical spatiotemporal dynamic system. A reaction of the form C→D takes place on the catalyst. The reaction is endothermic and a jacket is used to heat the reactor. The dimensionless mathematical model that describes this nonlinear tubular chemical reactor is given as follows:(1)εp∂Tgz,t∂t=-∂Tgz,t∂z+αcTsz,t-Tgz,t-αgTgz,t-ut,∂Tsz,t∂t=∂2Tsz,t∂z2+B0exp⁡γTsz,t1+Tsz,t-βcTsz,t-Tgz,t-βpTsz,t-bzutsubject to the following boundary conditions:(2)z=0,Tgz,t=0,∂Tsz,t∂z=0,z=1,∂Tsz,t∂z=0,where Tg(z,t) and Ts(z,t) denote the dimensionless temperature of gas and catalyst, respectively, which are spatially dependent on z∈[0,1],  b(z)u(t) denotes the spatiotemporal heating source with the distribution b(z) and the manipulated input u(t), and u(t) denotes the dimensionless temperature of jacket. The values of the process parameters are given as follows:(3)εp=0.01,γ=21.14,βc=1.0,βp=15.62,B0=-0.003,αc=0.5,αg=0.5.Figure 1
Sketch of a catalytic packed-bed reactor.In this application, the control target is to control the catalyst temperatureTs(z,t) throughout the reactor in order to maintain a desired degree of reaction rate using the measurements of catalyst temperature from q sensing locations z′=z1z2⋯zq and manipulating one spatially distributed heating source (b(z)=1-cos⁡πz). The spatial reference profile is given as Tsdz=0.42-0.2cos⁡(πz),0≤z≤1, and then the entire spatial catalyst temperature should follow this reference.


#### 2.1.2. A Rapid Thermal Chemical Vapor Deposition System

Consider a rapid thermal chemical vapor deposition (RTCVD) reactor with three-zone heating banks (Adomaitis 1995; Theodoropoulou et al., 1998). The sketch of the RTCVD system is depicted in Figure2, where the lamp bank A mainly heats the total area of the wafer, the lamp bank B mainly heats the wafer edge, and the lamp bank C is used to coarsely adjust the wafer temperature average. When a wafer is being heated, it is rotated for azimuthal temperature uniformity. 10% silane feed in inert gas is inputted into the reactor from the top, and the silane is decomposed into silicon and hydrogen. A 0.5 μm film of polysilicon is expected to deposit on the wafer when deposition temperature appears at near 800 K or higher. The temperature is controlled by operating the power to lamp banks A, B, and C. A dimensionless wafer thermal dynamics equation (Adomaitis 1995) over the wafer domain is given as follows:(4)∂Tr′∂t′=κ01r′∂Tr′∂r′+∂2Tr′∂r′2+σ01-Tr′4+ωrqar′ua+ωrqbr′ub+ωrqcr′ucsubject to the boundary condition(5)∂Tr′∂r′=σed1-Tr′4+qedubwhenr′=1∂Tr′∂r′=0whenr′=0,where Tr′=Tr/Tamb denotes dimensionless wafer temperature, Tr denotes the actual wafer temperature, and Tamb=300 K denotes ambient temperature; r′=r/Rw denotes the dimensionless radial position of the wafer with r denoting the actual radial position of the wafer and Rw denoting the wafer radius (Rw=7.6cm); t′=t/τ denotes dimensionless time, t denotes the actual time, and τ=2.9s denotes residence time of upper chamber; qa(r′), qb(r′), and qc(r′) denote the radiant energy flux on the radial position r′ from the lamp banks A, B, and C to the wafer, respectively (the distributions are shown in Figure 3); ua, ub, and uc denote the percentage of the lamp power for the three-zone lamp banks. The values of the process parameters are given as follows:(6)κ0=0.0021,σ0=0.0012,σed=0.0037,qed=4.022,ωr=0.0256.Figure 2
Structure of a RTCVD system.Figure 3
Radiation flux distribution of three-zone heating lamp banks.From (4) and (5), we can find that the wafer temperature is a variable varying with time and space. The concerned control problem is to control the wafer temperature throughout the wafer radius to arrive at the set temperature 1000 K quickly and uniformly using the measurements of wafer temperature from q sensing locations z′=z1z2⋯zq and manipulating power percentage (ua, ub, and uc) to three zones of lamp banks.


### 2.2. SVR Algorithm

Support vector regression (SVR) is a learning technique that originated from theoretical foundations of the statistical learning theory [20]. Over the past decade, SVR has become one of the most popular regression techniques and has been employed in a variety of applications. It performs a new inductive principle (structural risk minimization) for learning from finite training data sets and chooses a model of the right complexity from a large number of candidate models (learning machines) to describe training data pairs.Suppose we have a training setD={[xi,yi]∈Rs×R,i=1,…,q} consisting of q pairs (x1,y1),(x2,y2),…,(xq,yq), where the inputs are s-dimensional vectors and the labels are continuous values. SVR builds a function as follows:(7)fx,w:Rs⟶Rx⟼w,ψx℘+b,where b is a bias term and ψ(·):Rs→℘ is an application mapping the space of factors into a feature space ℘.The regression problem can be formulated as a convex optimization problem as follows:(8)minw,b=12w2+C∑i=1lfxi,w-yiε,where C is a design parameter chosen by the user, which determines the trade-off between the complexity of f(x,w) and the approximate error, and [·]ε is the ε-insensitive loss function which is defined as follows:(9)fx,w-yε=0iffx,w-y≤εfx,w-y-εotherwise.In practice,℘,w∈℘ and ψ(x)∈℘ are deduced from a kernel K(·,·):Rs×Rs→R involved in the computation of a scalar product:(10)w=∑iwiKxi,·,ψx=Kxi,·,w,ψx℘=∑iwiKxi,x.The above optimization problem can be solved in a dual space. By introducing the Lagrange multipliersαi (αi∗), the primal optimization problem can be formulated in its dual form as follows:(11)maxαi,αi∗-12∑i=1q∑j=1qαi∗-αiαj∗-αjKxi,xj-ε∑i=1qαi∗+αi+∑i=1qαi∗-αiyisubjectto∑j=1qαi∗=∑i=1qαi0≤αi∗≤C,0≤αi≤C,i=1,…,q.Solving the dual quadratic programming problem, we can find an optimal weight vectorw and an optimal bias b of the regression hypersurface given in (12)w=∑i=1qαi∗-αiψxib=1q∑i=1qyi-w·ψxi.Then, the best regression hypersurface is given by(13)fx,w=∑i=1qαi∗-αiKxi,x+b=∑i∈SVαi∗-αiKxi,x+b.The training pattern xi with nonzero (αi∗-αi) is called support vector (SV).


## 2.1. Nonlinear Spatially Distributed Dynamical Systems

Many industrial processes exhibit highly nonlinear behavior and strong spatial variations. They usually can be represented by nonlinear partial differential equations. Some processes have strong convection characteristics [16], some have strong diffusion phenomena [17], and some have both convection and diffusion characteristics [18]. These systems are usually installed with multiple spatially distributed actuators and multiple sensors and give rise to nonlinear control problems that involve the regulation of highly distributed control variables using these spatially distributed actuators and sensors.In the following subsections, two examples of SDDSs are presented and illustrated.


### 2.1.1. A Catalytic Packed-Bed Reactor

The catalytic packed-bed reactor [17, 19] in Figure 1 is a typical spatiotemporal dynamic system. A reaction of the form C→D takes place on the catalyst. The reaction is endothermic and a jacket is used to heat the reactor. The dimensionless mathematical model that describes this nonlinear tubular chemical reactor is given as follows:(1)εp∂Tgz,t∂t=-∂Tgz,t∂z+αcTsz,t-Tgz,t-αgTgz,t-ut,∂Tsz,t∂t=∂2Tsz,t∂z2+B0exp⁡γTsz,t1+Tsz,t-βcTsz,t-Tgz,t-βpTsz,t-bzutsubject to the following boundary conditions:(2)z=0,Tgz,t=0,∂Tsz,t∂z=0,z=1,∂Tsz,t∂z=0,where Tg(z,t) and Ts(z,t) denote the dimensionless temperature of gas and catalyst, respectively, which are spatially dependent on z∈[0,1],  b(z)u(t) denotes the spatiotemporal heating source with the distribution b(z) and the manipulated input u(t), and u(t) denotes the dimensionless temperature of jacket. The values of the process parameters are given as follows:(3)εp=0.01,γ=21.14,βc=1.0,βp=15.62,B0=-0.003,αc=0.5,αg=0.5.Figure 1
Sketch of a catalytic packed-bed reactor.In this application, the control target is to control the catalyst temperatureTs(z,t) throughout the reactor in order to maintain a desired degree of reaction rate using the measurements of catalyst temperature from q sensing locations z′=z1z2⋯zq and manipulating one spatially distributed heating source (b(z)=1-cos⁡πz). The spatial reference profile is given as Tsdz=0.42-0.2cos⁡(πz),0≤z≤1, and then the entire spatial catalyst temperature should follow this reference.


### 2.1.2. A Rapid Thermal Chemical Vapor Deposition System

Consider a rapid thermal chemical vapor deposition (RTCVD) reactor with three-zone heating banks (Adomaitis 1995; Theodoropoulou et al., 1998). The sketch of the RTCVD system is depicted in Figure2, where the lamp bank A mainly heats the total area of the wafer, the lamp bank B mainly heats the wafer edge, and the lamp bank C is used to coarsely adjust the wafer temperature average. When a wafer is being heated, it is rotated for azimuthal temperature uniformity. 10% silane feed in inert gas is inputted into the reactor from the top, and the silane is decomposed into silicon and hydrogen. A 0.5 μm film of polysilicon is expected to deposit on the wafer when deposition temperature appears at near 800 K or higher. The temperature is controlled by operating the power to lamp banks A, B, and C. A dimensionless wafer thermal dynamics equation (Adomaitis 1995) over the wafer domain is given as follows:(4)∂Tr′∂t′=κ01r′∂Tr′∂r′+∂2Tr′∂r′2+σ01-Tr′4+ωrqar′ua+ωrqbr′ub+ωrqcr′ucsubject to the boundary condition(5)∂Tr′∂r′=σed1-Tr′4+qedubwhenr′=1∂Tr′∂r′=0whenr′=0,where Tr′=Tr/Tamb denotes dimensionless wafer temperature, Tr denotes the actual wafer temperature, and Tamb=300 K denotes ambient temperature; r′=r/Rw denotes the dimensionless radial position of the wafer with r denoting the actual radial position of the wafer and Rw denoting the wafer radius (Rw=7.6cm); t′=t/τ denotes dimensionless time, t denotes the actual time, and τ=2.9s denotes residence time of upper chamber; qa(r′), qb(r′), and qc(r′) denote the radiant energy flux on the radial position r′ from the lamp banks A, B, and C to the wafer, respectively (the distributions are shown in Figure 3); ua, ub, and uc denote the percentage of the lamp power for the three-zone lamp banks. The values of the process parameters are given as follows:(6)κ0=0.0021,σ0=0.0012,σed=0.0037,qed=4.022,ωr=0.0256.Figure 2
Structure of a RTCVD system.Figure 3
Radiation flux distribution of three-zone heating lamp banks.From (4) and (5), we can find that the wafer temperature is a variable varying with time and space. The concerned control problem is to control the wafer temperature throughout the wafer radius to arrive at the set temperature 1000 K quickly and uniformly using the measurements of wafer temperature from q sensing locations z′=z1z2⋯zq and manipulating power percentage (ua, ub, and uc) to three zones of lamp banks.


## 2.1.1. A Catalytic Packed-Bed Reactor

The catalytic packed-bed reactor [17, 19] in Figure 1 is a typical spatiotemporal dynamic system. A reaction of the form C→D takes place on the catalyst. The reaction is endothermic and a jacket is used to heat the reactor. The dimensionless mathematical model that describes this nonlinear tubular chemical reactor is given as follows:(1)εp∂Tgz,t∂t=-∂Tgz,t∂z+αcTsz,t-Tgz,t-αgTgz,t-ut,∂Tsz,t∂t=∂2Tsz,t∂z2+B0exp⁡γTsz,t1+Tsz,t-βcTsz,t-Tgz,t-βpTsz,t-bzutsubject to the following boundary conditions:(2)z=0,Tgz,t=0,∂Tsz,t∂z=0,z=1,∂Tsz,t∂z=0,where Tg(z,t) and Ts(z,t) denote the dimensionless temperature of gas and catalyst, respectively, which are spatially dependent on z∈[0,1],  b(z)u(t) denotes the spatiotemporal heating source with the distribution b(z) and the manipulated input u(t), and u(t) denotes the dimensionless temperature of jacket. The values of the process parameters are given as follows:(3)εp=0.01,γ=21.14,βc=1.0,βp=15.62,B0=-0.003,αc=0.5,αg=0.5.Figure 1
Sketch of a catalytic packed-bed reactor.In this application, the control target is to control the catalyst temperatureTs(z,t) throughout the reactor in order to maintain a desired degree of reaction rate using the measurements of catalyst temperature from q sensing locations z′=z1z2⋯zq and manipulating one spatially distributed heating source (b(z)=1-cos⁡πz). The spatial reference profile is given as Tsdz=0.42-0.2cos⁡(πz),0≤z≤1, and then the entire spatial catalyst temperature should follow this reference.


## 2.1.2. A Rapid Thermal Chemical Vapor Deposition System

Consider a rapid thermal chemical vapor deposition (RTCVD) reactor with three-zone heating banks (Adomaitis 1995; Theodoropoulou et al., 1998). The sketch of the RTCVD system is depicted in Figure2, where the lamp bank A mainly heats the total area of the wafer, the lamp bank B mainly heats the wafer edge, and the lamp bank C is used to coarsely adjust the wafer temperature average. When a wafer is being heated, it is rotated for azimuthal temperature uniformity. 10% silane feed in inert gas is inputted into the reactor from the top, and the silane is decomposed into silicon and hydrogen. A 0.5 μm film of polysilicon is expected to deposit on the wafer when deposition temperature appears at near 800 K or higher. The temperature is controlled by operating the power to lamp banks A, B, and C. A dimensionless wafer thermal dynamics equation (Adomaitis 1995) over the wafer domain is given as follows:(4)∂Tr′∂t′=κ01r′∂Tr′∂r′+∂2Tr′∂r′2+σ01-Tr′4+ωrqar′ua+ωrqbr′ub+ωrqcr′ucsubject to the boundary condition(5)∂Tr′∂r′=σed1-Tr′4+qedubwhenr′=1∂Tr′∂r′=0whenr′=0,where Tr′=Tr/Tamb denotes dimensionless wafer temperature, Tr denotes the actual wafer temperature, and Tamb=300 K denotes ambient temperature; r′=r/Rw denotes the dimensionless radial position of the wafer with r denoting the actual radial position of the wafer and Rw denoting the wafer radius (Rw=7.6cm); t′=t/τ denotes dimensionless time, t denotes the actual time, and τ=2.9s denotes residence time of upper chamber; qa(r′), qb(r′), and qc(r′) denote the radiant energy flux on the radial position r′ from the lamp banks A, B, and C to the wafer, respectively (the distributions are shown in Figure 3); ua, ub, and uc denote the percentage of the lamp power for the three-zone lamp banks. The values of the process parameters are given as follows:(6)κ0=0.0021,σ0=0.0012,σed=0.0037,qed=4.022,ωr=0.0256.Figure 2
Structure of a RTCVD system.Figure 3
Radiation flux distribution of three-zone heating lamp banks.From (4) and (5), we can find that the wafer temperature is a variable varying with time and space. The concerned control problem is to control the wafer temperature throughout the wafer radius to arrive at the set temperature 1000 K quickly and uniformly using the measurements of wafer temperature from q sensing locations z′=z1z2⋯zq and manipulating power percentage (ua, ub, and uc) to three zones of lamp banks.


## 2.2. SVR Algorithm

Support vector regression (SVR) is a learning technique that originated from theoretical foundations of the statistical learning theory [20]. Over the past decade, SVR has become one of the most popular regression techniques and has been employed in a variety of applications. It performs a new inductive principle (structural risk minimization) for learning from finite training data sets and chooses a model of the right complexity from a large number of candidate models (learning machines) to describe training data pairs.Suppose we have a training setD={[xi,yi]∈Rs×R,i=1,…,q} consisting of q pairs (x1,y1),(x2,y2),…,(xq,yq), where the inputs are s-dimensional vectors and the labels are continuous values. SVR builds a function as follows:(7)fx,w:Rs⟶Rx⟼w,ψx℘+b,where b is a bias term and ψ(·):Rs→℘ is an application mapping the space of factors into a feature space ℘.The regression problem can be formulated as a convex optimization problem as follows:(8)minw,b=12w2+C∑i=1lfxi,w-yiε,where C is a design parameter chosen by the user, which determines the trade-off between the complexity of f(x,w) and the approximate error, and [·]ε is the ε-insensitive loss function which is defined as follows:(9)fx,w-yε=0iffx,w-y≤εfx,w-y-εotherwise.In practice,℘,w∈℘ and ψ(x)∈℘ are deduced from a kernel K(·,·):Rs×Rs→R involved in the computation of a scalar product:(10)w=∑iwiKxi,·,ψx=Kxi,·,w,ψx℘=∑iwiKxi,x.The above optimization problem can be solved in a dual space. By introducing the Lagrange multipliersαi (αi∗), the primal optimization problem can be formulated in its dual form as follows:(11)maxαi,αi∗-12∑i=1q∑j=1qαi∗-αiαj∗-αjKxi,xj-ε∑i=1qαi∗+αi+∑i=1qαi∗-αiyisubjectto∑j=1qαi∗=∑i=1qαi0≤αi∗≤C,0≤αi≤C,i=1,…,q.Solving the dual quadratic programming problem, we can find an optimal weight vectorw and an optimal bias b of the regression hypersurface given in (12)w=∑i=1qαi∗-αiψxib=1q∑i=1qyi-w·ψxi.Then, the best regression hypersurface is given by(13)fx,w=∑i=1qαi∗-αiKxi,x+b=∑i∈SVαi∗-αiKxi,x+b.The training pattern xi with nonzero (αi∗-αi) is called support vector (SV).


## 3. SVR Learning Based Sensor Placement Scheme

### 3.1. Design Methodology

Consider a nonlinear SDDS in Figure4, u(t)∈RL_ is the temporal input, and y(z,t)∈R is the spatiotemporal output, where t is the time variable, z∈Z- is the spatial variable, and Z- is the spatial domain. Here suppose the system is controlled by the L_ actuators with implemental temporal signal ut=u1tu2t⋯uL_t and certain spatial distribution. The output is measured at the p spatial locations z1,z2,…,zp (p≥1); therefore, let Z-=[z1,z2,…,zp] and let y(Z-,t)=[y(z1,t),y(z2,t),…,y(zp,t)] be spatial sensing output. Let S=S1S2⋯Sp be a collected data set, where Sj=[Xj,zj], Xj=[xj1,…,xjN] is the collected measurement data from the jth (1≤j≤p) sensing location in the time interval [t0,t1], and xjs=y(zj,s) is the output measurement at the sth sampling period, 1≤s≤N. The sensor placement problem is to learn optimal sensor locations from the spatiotemporal data set S.Figure 4
Methodology of SVR learning based sensor placement.The design methodology can be depicted in Figure4. S is taken as the training set. ε-SVR is used to find support vectors and produce a function so that all training patterns X have a maximum deviation ε from the target values zi and have a maximum margin. The support vectors, as crucial data points that represent principle spatial structures, are employed to determine the optimal sensor location and sensor number.


### 3.2. Systematic Design Scheme

The systematic sensor placement design scheme is composed of three steps shown in Figure5, including data collection, SVR learning, and sensor locating. Firstly, a set of data is collected as the fundamental data for sensor locating. To acquire sufficiently rich information from the system, the system should be excited by adding persistently exciting perturbation signals. Secondly, SVR is used to extract crucial data points from the collected data set. Thirdly, the sensor location is determined in terms of the extracted crucial data.Figure 5
Systematic design for sensor placement.(  1) Data Collection. The idea data should contain sufficiently rich information of dynamic characteristics. To acquire the sufficiently rich data, one possible method is to add persistently exciting perturbation signals to the input u(t). In this study, we use white noise and the pseudorandom multilevel signals (PRMSs) with maximal length [21] as the input perturbation signals for two application examples, respectively. When input perturbation signals are sequentially added to the input u(t), the system output is collected. The collected data and its corresponding spatial location consist of fundamental data set S for the subsequent support vector learning.(  2) SVR Learning. Before the learning, Gaussian kernel function with its spread σ is selected and two learning parameters C and ε are required to be set. We use K-Fold Cross-Validation method to acquire C, ε, and σ. Root mean-squared error (RMSE) is employed as quantitative performance criteria for test, which is defined as follows:(14)RMSE=∑k=1pzk∗-zk2p,where zk∗ denotes the predicted spatial point after learning, zk denotes the practical spatial point, and p denotes the number of sensors.(  3) Sensor Locating. After the second step, we can obtain some data points that represent the spatial structure implied in the data set. In this step, we can easily determine the sensor locations, namely, the spatial locations in these support vectors. The number of support vectors is equal to the number of sensors.


## 3.1. Design Methodology

Consider a nonlinear SDDS in Figure4, u(t)∈RL_ is the temporal input, and y(z,t)∈R is the spatiotemporal output, where t is the time variable, z∈Z- is the spatial variable, and Z- is the spatial domain. Here suppose the system is controlled by the L_ actuators with implemental temporal signal ut=u1tu2t⋯uL_t and certain spatial distribution. The output is measured at the p spatial locations z1,z2,…,zp (p≥1); therefore, let Z-=[z1,z2,…,zp] and let y(Z-,t)=[y(z1,t),y(z2,t),…,y(zp,t)] be spatial sensing output. Let S=S1S2⋯Sp be a collected data set, where Sj=[Xj,zj], Xj=[xj1,…,xjN] is the collected measurement data from the jth (1≤j≤p) sensing location in the time interval [t0,t1], and xjs=y(zj,s) is the output measurement at the sth sampling period, 1≤s≤N. The sensor placement problem is to learn optimal sensor locations from the spatiotemporal data set S.Figure 4
Methodology of SVR learning based sensor placement.The design methodology can be depicted in Figure4. S is taken as the training set. ε-SVR is used to find support vectors and produce a function so that all training patterns X have a maximum deviation ε from the target values zi and have a maximum margin. The support vectors, as crucial data points that represent principle spatial structures, are employed to determine the optimal sensor location and sensor number.


## 3.2. Systematic Design Scheme

The systematic sensor placement design scheme is composed of three steps shown in Figure5, including data collection, SVR learning, and sensor locating. Firstly, a set of data is collected as the fundamental data for sensor locating. To acquire sufficiently rich information from the system, the system should be excited by adding persistently exciting perturbation signals. Secondly, SVR is used to extract crucial data points from the collected data set. Thirdly, the sensor location is determined in terms of the extracted crucial data.Figure 5
Systematic design for sensor placement.(  1) Data Collection. The idea data should contain sufficiently rich information of dynamic characteristics. To acquire the sufficiently rich data, one possible method is to add persistently exciting perturbation signals to the input u(t). In this study, we use white noise and the pseudorandom multilevel signals (PRMSs) with maximal length [21] as the input perturbation signals for two application examples, respectively. When input perturbation signals are sequentially added to the input u(t), the system output is collected. The collected data and its corresponding spatial location consist of fundamental data set S for the subsequent support vector learning.(  2) SVR Learning. Before the learning, Gaussian kernel function with its spread σ is selected and two learning parameters C and ε are required to be set. We use K-Fold Cross-Validation method to acquire C, ε, and σ. Root mean-squared error (RMSE) is employed as quantitative performance criteria for test, which is defined as follows:(14)RMSE=∑k=1pzk∗-zk2p,where zk∗ denotes the predicted spatial point after learning, zk denotes the practical spatial point, and p denotes the number of sensors.(  3) Sensor Locating. After the second step, we can obtain some data points that represent the spatial structure implied in the data set. In this step, we can easily determine the sensor locations, namely, the spatial locations in these support vectors. The number of support vectors is equal to the number of sensors.


## 4. Case Studies

In this section, we take the two practical applications described in Section2 as examples. The mathematical model (1) and (2) in the catalytic packed-bed reactor and the mathematical model (4) and (5) in the RTCVD system are only for the process simulation for evaluation of the sensor placement scheme. The method of lines [22] is used to simulate the two models.


### 4.1. The Catalytic Packed-Bed Reactor

The pseudorandom quinary signal (PRQS) with maximum length of 624 is used as the perturbation signal. The parameters of PRQS are chosen as follows: the lowest and uppermost values of input are 0.1474 and 0.0632, respectively, the number of the levels is 5, the sampling time is 0.1, the length of the period is 624, and the minimum switching time (clock period) is 0.1. The perturbation signal generated by PRQS is depicted in Figure6.Figure 6
Perturbation signal generated by PRQS.The space domain of the system is discretized uniformly into 81 points; that is,Z-=[z1,z2,…,z81], the sampling period is set as 0.1, and the time interval of simulation is [0,62.3]. We collect the catalytic temperature measurements from the system over the space domain and constitute the data set S=S1S2⋯S81, where Sj=[Xj,zj], Xj=[xj1,…,xj624] (1≤j≤p), xjs=y(zj,s) is the catalytic temperature measurement at the sth sampling period from the jth sensing location, 1≤s≤624.Then, the SVR is used to learn support vectors from the data setS. As for model selection problem of the SVR, we employ 5-Fold Cross-Validation method to select proper C from [22,215] with step-size set as 10, ε from [2-10,22] with step-size set as 0.01, and σ from {σ0/100,σ0/95,…,σ0/n+5,σ0/n,σ0/n-5,…,σ0/10,σ0/5,σ0}, where σ0 is set as 10% of the length of the input interval of y(Z-,t) denoted by(15)σ0=max1≤j≤pσj=max1≤j≤pymaxzj-yminzj10,where ymax(zj) and ymin(zj) are the maximum and the minimum bound value of temperature measurements from the jth spatial input variable, respectively. In this application, σ0=0.03969.After 5-Fold Cross-Validation, we can find a group of best parameters; that is,C=4096, ε=0.174, and σ=σ0/19=0.00209. The learned support vectors are S=S6,S23,S39,S44,S47,S62,S66,S81. Therefore, the optimal sensor locations are z6,z23,z39,z44,z47,z62,z66,z81; that is, {0.0625,0.275,0.475,0.5375,0.575,0.7625,0.8125,1}. At the same time, we can find the optimal sensor number is eight.The effectiveness of the proposed sensor locating method is validated on the 3D fuzzy controlled catalytic packed-bed reactor. The design of 3D FLC is given in AppendicesA and B. We carried out two different sensor placement schemes on the 3D fuzzy controlled catalytic packed-bed reactor. One is that two to nine sensors are evenly distributed in the space domain, respectively. The other is the proposed sensor locating scheme. The control performance comparison of the two schemes is given in Table 1, where SSE, IAE, and ITAE [23] stand for steady-state error, integral of the absolute error, and integral of time multiplied by absolute error for spatiotemporal dynamic systems, respectively. Graph comparison is given in Figure 7, where eight sensors in (a) are placed in the proposed optimal locations, while eight sensors in (b) are evenly placed. In terms of Table 1 and Figure 7, we can find that the proposed sensor placement scheme has improved the control performance.Table 1
Performance index comparisons (SVR learning based sensing locations versus evenly distributed locations).
Performance index Sensor number withevenly distributed locations SVR learning based sensor number (sensor location) 2 3 4 5 6 7 8 9 8{0.0625,0.275,0.475,0.5375,0.575,0.7625,0.8125,1} SSE (×10−2) 3.332 1.73 1.74 1.74 1.74 1.74 1.75 1.74 1.70 IAE (×10−1) 3.382 2.355 2.362 2.366 2.36 2.364 2.365 2.362 2.341 ITAE (×10−1) 10.751 5.715 5.749 5.761 5.754 5.753 5.771 5.754 5.615Figure 7
Control performance comparisons with different sensor placement schemes: (a1)–(a3) using proposed sensor locations (eight optimal locations); (b1)–(b3) using eight evenly distributed sensor locations. (∗1) denotes catalyst temperature evolution profile, (∗2) denotes manipulated input signal, and (∗3) denotes catalyst temperature under steady state, where ∗ may be (a) or (b).


### 4.2. The RTCVD System

To acquire sufficiently rich information from the system, white noise signals, whose values are less than 10% steady state, are added as perturbation signals to the manipulated variablesua, ub, and uc. Therefore, the manipulated variables with perturbation signals are given as follows:(16)ua_excitet=0.2028+0.1∗0.2028∗rand⁡∗sgn⁡rand⁡-0.5,ub_excitet=0.1008+0.1∗0.1008∗rand⁡∗sgn⁡rand⁡-0.5,uc_excitet=0.2245+0.1∗0.2245∗rand⁡∗sgn⁡rand⁡-0.5.In this application, the sample interval is 0.5 s, and the simulation duration is 100 s. The evolution profiles of manipulated variables are shown in Figure 8, and the measured wafer temperature y(z,t) is shown in Figure 9.Figure 8
Evolution profiles of manipulated variablesua, ub, and uc with perturbed signals.Figure 9
The measured wafer temperature with perturbed signals.The space domain of the system is discretized uniformly into 61 points; that is,Z-=[z1,z2,…,z61]. We collect the wafer temperature measurements from the system over the wafer radius and constitute the data set S=S1S2⋯S61, where Sj=[Xj,zj], Xj=[xj1,…,xj200] (1≤j≤p), and xjs=y(zj,s) is the wafer temperature measurement at the sth sampling period from the jth sensing location, 1≤s≤200.Then, the SVR is used to learn support vectors from the data setS. As for model selection problem of the SVR, we employ 5-Fold Cross-Validation method to select proper C from [22,215] with step-size set as 10, ε from [2-10,22] with step-size set as 0.01, and σ from {σ0/130,σ0/125,…,σ0/n+5,σ0/n,σ0/n-5,…,σ0/10,σ0/5,σ0}, where σ0 is the initial set as in (15). In this application, σ0=1.2816.Using 5-Fold Cross-Validation, we can find a group of best parameters; that is,C=7000, ε=0.146, and σ=σ0/50=0.00256. The learned support vectors are S=S5,S13,S27,S39,S49,S61. Therefore, the optimal sensor locations are z5,z13,z27,z39,z49,z61; that is, {0.507,1.52,3.29,4.81,6.08,7.60}. At the same time, we can find the optimal sensor number is six.The effectiveness of the proposed sensor placement method is validated on the space decomposition and coordination based 3D fuzzy controlled RTCVD system. The design of space decomposition and coordination based 3D FLC is given in AppendixC. We carried out two different sensor placement schemes on the 3D fuzzy controlled RTCVD system. One is that three to eleven sensors are evenly distributed in the space domain, respectively. The other is the proposed sensor placement scheme. The control performance comparison of the two schemes is given in Table 2. Graph comparison is given in Figure 10, where six sensors in (a) are placed in the proposed optimal locations, while nine sensors in (b) are evenly placed. In the case of six sensors located at the proposed optimal locations, that is, {0.507,1.52,3.29,4.81,6.08,7.60}, the RTCVD system is decomposed into three subsystems with one distributed control source, that is, ua and sensing locations {0.507,1.52,3.29} for subsystem 1, ub and sensing locations {3.29,4.81,6.08,7.60} for subsystem 2, and uc and sensing locations {4.81,6.08,7.60} for subsystem 3. In the case of nine sensors located at evenly placed locations, that is, {1/9,2/9,3/9,4/9,5/9,6/9,7/9,8/9,1}, the decomposed three subsystems are given as follows: ua and sensing locations {1/9,2/9,3/9,4/9} for subsystem 1, ub and sensing locations {3/9,4/9,5/9,6/9,7/9} for subsystem 2, and uc and sensing locations {5/9,6/9,7/9,8/9,1} for subsystem 3. In terms of Table 2 and Figure 10, we can find that the proposed sensor locating scheme has improved the control performance in the RTCVD system.Table 2
Performance index comparisons (SVR learning based sensing locations versus evenly distributed locations).
Performance index Sensor number with evenly distributed locations SVR learning based sensor number (sensor location) 3 4 5 6 7 8 9 10 6{0.507,1.52,3.29,4.81,6.08,7.60} Max nonuniformity (K) 3.48 84.58 4.46 1.013 6.687 3.905 2.795 5.993 0.8620 SSE 8.352 56.34 5.49 4.37 18.42 7.77 6.038 10.16 4.2195 IAE (×104) 3.327 3.64 3.285 3.268 3.313 3.276 3.271 3.285 3.268 ITAE (×105) 1.764 4.00 1.51 1.473 1.737 1.505 1.485 1.593 1.469Figure 10
Control performance comparisons with different sensor placement schemes: (a1)–(a3) using proposed sensor locations (six optimal locations); (b1)–(b3) using nine evenly distributed sensor locations. (∗1) denotes wafer temperature evolution profile, (∗2) denotes three input signals, and (∗3) denotes wafer temperature under steady state, where ∗ may be (a) or (b).


### 4.3. Discussions

According to the experiment results in the two examples, we can find that the proposed sensor placement scheme outperforms all evenly distributed sensor placement scheme. The results validate the effectiveness of the proposed sensor placement method in terms of the spatiotemporal 3D fuzzy control system.


## 4.1. The Catalytic Packed-Bed Reactor

The pseudorandom quinary signal (PRQS) with maximum length of 624 is used as the perturbation signal. The parameters of PRQS are chosen as follows: the lowest and uppermost values of input are 0.1474 and 0.0632, respectively, the number of the levels is 5, the sampling time is 0.1, the length of the period is 624, and the minimum switching time (clock period) is 0.1. The perturbation signal generated by PRQS is depicted in Figure6.Figure 6
Perturbation signal generated by PRQS.The space domain of the system is discretized uniformly into 81 points; that is,Z-=[z1,z2,…,z81], the sampling period is set as 0.1, and the time interval of simulation is [0,62.3]. We collect the catalytic temperature measurements from the system over the space domain and constitute the data set S=S1S2⋯S81, where Sj=[Xj,zj], Xj=[xj1,…,xj624] (1≤j≤p), xjs=y(zj,s) is the catalytic temperature measurement at the sth sampling period from the jth sensing location, 1≤s≤624.Then, the SVR is used to learn support vectors from the data setS. As for model selection problem of the SVR, we employ 5-Fold Cross-Validation method to select proper C from [22,215] with step-size set as 10, ε from [2-10,22] with step-size set as 0.01, and σ from {σ0/100,σ0/95,…,σ0/n+5,σ0/n,σ0/n-5,…,σ0/10,σ0/5,σ0}, where σ0 is set as 10% of the length of the input interval of y(Z-,t) denoted by(15)σ0=max1≤j≤pσj=max1≤j≤pymaxzj-yminzj10,where ymax(zj) and ymin(zj) are the maximum and the minimum bound value of temperature measurements from the jth spatial input variable, respectively. In this application, σ0=0.03969.After 5-Fold Cross-Validation, we can find a group of best parameters; that is,C=4096, ε=0.174, and σ=σ0/19=0.00209. The learned support vectors are S=S6,S23,S39,S44,S47,S62,S66,S81. Therefore, the optimal sensor locations are z6,z23,z39,z44,z47,z62,z66,z81; that is, {0.0625,0.275,0.475,0.5375,0.575,0.7625,0.8125,1}. At the same time, we can find the optimal sensor number is eight.The effectiveness of the proposed sensor locating method is validated on the 3D fuzzy controlled catalytic packed-bed reactor. The design of 3D FLC is given in AppendicesA and B. We carried out two different sensor placement schemes on the 3D fuzzy controlled catalytic packed-bed reactor. One is that two to nine sensors are evenly distributed in the space domain, respectively. The other is the proposed sensor locating scheme. The control performance comparison of the two schemes is given in Table 1, where SSE, IAE, and ITAE [23] stand for steady-state error, integral of the absolute error, and integral of time multiplied by absolute error for spatiotemporal dynamic systems, respectively. Graph comparison is given in Figure 7, where eight sensors in (a) are placed in the proposed optimal locations, while eight sensors in (b) are evenly placed. In terms of Table 1 and Figure 7, we can find that the proposed sensor placement scheme has improved the control performance.Table 1
Performance index comparisons (SVR learning based sensing locations versus evenly distributed locations).
Performance index Sensor number withevenly distributed locations SVR learning based sensor number (sensor location) 2 3 4 5 6 7 8 9 8{0.0625,0.275,0.475,0.5375,0.575,0.7625,0.8125,1} SSE (×10−2) 3.332 1.73 1.74 1.74 1.74 1.74 1.75 1.74 1.70 IAE (×10−1) 3.382 2.355 2.362 2.366 2.36 2.364 2.365 2.362 2.341 ITAE (×10−1) 10.751 5.715 5.749 5.761 5.754 5.753 5.771 5.754 5.615Figure 7
Control performance comparisons with different sensor placement schemes: (a1)–(a3) using proposed sensor locations (eight optimal locations); (b1)–(b3) using eight evenly distributed sensor locations. (∗1) denotes catalyst temperature evolution profile, (∗2) denotes manipulated input signal, and (∗3) denotes catalyst temperature under steady state, where ∗ may be (a) or (b).


## 4.2. The RTCVD System

To acquire sufficiently rich information from the system, white noise signals, whose values are less than 10% steady state, are added as perturbation signals to the manipulated variablesua, ub, and uc. Therefore, the manipulated variables with perturbation signals are given as follows:(16)ua_excitet=0.2028+0.1∗0.2028∗rand⁡∗sgn⁡rand⁡-0.5,ub_excitet=0.1008+0.1∗0.1008∗rand⁡∗sgn⁡rand⁡-0.5,uc_excitet=0.2245+0.1∗0.2245∗rand⁡∗sgn⁡rand⁡-0.5.In this application, the sample interval is 0.5 s, and the simulation duration is 100 s. The evolution profiles of manipulated variables are shown in Figure 8, and the measured wafer temperature y(z,t) is shown in Figure 9.Figure 8
Evolution profiles of manipulated variablesua, ub, and uc with perturbed signals.Figure 9
The measured wafer temperature with perturbed signals.The space domain of the system is discretized uniformly into 61 points; that is,Z-=[z1,z2,…,z61]. We collect the wafer temperature measurements from the system over the wafer radius and constitute the data set S=S1S2⋯S61, where Sj=[Xj,zj], Xj=[xj1,…,xj200] (1≤j≤p), and xjs=y(zj,s) is the wafer temperature measurement at the sth sampling period from the jth sensing location, 1≤s≤200.Then, the SVR is used to learn support vectors from the data setS. As for model selection problem of the SVR, we employ 5-Fold Cross-Validation method to select proper C from [22,215] with step-size set as 10, ε from [2-10,22] with step-size set as 0.01, and σ from {σ0/130,σ0/125,…,σ0/n+5,σ0/n,σ0/n-5,…,σ0/10,σ0/5,σ0}, where σ0 is the initial set as in (15). In this application, σ0=1.2816.Using 5-Fold Cross-Validation, we can find a group of best parameters; that is,C=7000, ε=0.146, and σ=σ0/50=0.00256. The learned support vectors are S=S5,S13,S27,S39,S49,S61. Therefore, the optimal sensor locations are z5,z13,z27,z39,z49,z61; that is, {0.507,1.52,3.29,4.81,6.08,7.60}. At the same time, we can find the optimal sensor number is six.The effectiveness of the proposed sensor placement method is validated on the space decomposition and coordination based 3D fuzzy controlled RTCVD system. The design of space decomposition and coordination based 3D FLC is given in AppendixC. We carried out two different sensor placement schemes on the 3D fuzzy controlled RTCVD system. One is that three to eleven sensors are evenly distributed in the space domain, respectively. The other is the proposed sensor placement scheme. The control performance comparison of the two schemes is given in Table 2. Graph comparison is given in Figure 10, where six sensors in (a) are placed in the proposed optimal locations, while nine sensors in (b) are evenly placed. In the case of six sensors located at the proposed optimal locations, that is, {0.507,1.52,3.29,4.81,6.08,7.60}, the RTCVD system is decomposed into three subsystems with one distributed control source, that is, ua and sensing locations {0.507,1.52,3.29} for subsystem 1, ub and sensing locations {3.29,4.81,6.08,7.60} for subsystem 2, and uc and sensing locations {4.81,6.08,7.60} for subsystem 3. In the case of nine sensors located at evenly placed locations, that is, {1/9,2/9,3/9,4/9,5/9,6/9,7/9,8/9,1}, the decomposed three subsystems are given as follows: ua and sensing locations {1/9,2/9,3/9,4/9} for subsystem 1, ub and sensing locations {3/9,4/9,5/9,6/9,7/9} for subsystem 2, and uc and sensing locations {5/9,6/9,7/9,8/9,1} for subsystem 3. In terms of Table 2 and Figure 10, we can find that the proposed sensor locating scheme has improved the control performance in the RTCVD system.Table 2
Performance index comparisons (SVR learning based sensing locations versus evenly distributed locations).
Performance index Sensor number with evenly distributed locations SVR learning based sensor number (sensor location) 3 4 5 6 7 8 9 10 6{0.507,1.52,3.29,4.81,6.08,7.60} Max nonuniformity (K) 3.48 84.58 4.46 1.013 6.687 3.905 2.795 5.993 0.8620 SSE 8.352 56.34 5.49 4.37 18.42 7.77 6.038 10.16 4.2195 IAE (×104) 3.327 3.64 3.285 3.268 3.313 3.276 3.271 3.285 3.268 ITAE (×105) 1.764 4.00 1.51 1.473 1.737 1.505 1.485 1.593 1.469Figure 10
Control performance comparisons with different sensor placement schemes: (a1)–(a3) using proposed sensor locations (six optimal locations); (b1)–(b3) using nine evenly distributed sensor locations. (∗1) denotes wafer temperature evolution profile, (∗2) denotes three input signals, and (∗3) denotes wafer temperature under steady state, where ∗ may be (a) or (b).


## 4.3. Discussions

According to the experiment results in the two examples, we can find that the proposed sensor placement scheme outperforms all evenly distributed sensor placement scheme. The results validate the effectiveness of the proposed sensor placement method in terms of the spatiotemporal 3D fuzzy control system.


## 5. Conclusion

The sensor placement for an unknown nonlinear SDDS is necessary for various applications (e.g., modeling and control design). In this paper, a data-based SVR learning based sensor placement method was proposed. The systematic design method is divided into three steps. Firstly, sufficiently rich data information is generated by adding persistently exciting perturbation signals and is collected as fundamental data for sensor placement. Secondly, SVR is used to extract the characteristics of spatial distribution that is expressed as support vectors. Thirdly, the learned support vectors are employed to determine optimal sensor location and sensor number. Simulations were presented to illustrate the effectiveness of this sensor placement method and its potential for a wide range of SDDSs.



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*Source: 5241279-2016-11-14.xml*