Identification of LPV Models with Non-uniformly Spaced Operating Points by Using Asymmetric Gaussian Weights☆

Jie You,Qinm in Yang,Jiangang Lu*,Youxian Sun

State Key Laboratory of Industrial Control Technology,Department of Control Science and Engineering,Zhejiang University,Hangzhou 310027,China

Identification of LPV Models with Non-uniformly Spaced Operating Points by Using Asymmetric Gaussian Weights☆

Jie You,Qinm in Yang,Jiangang Lu*,Youxian Sun

State Key Laboratory of Industrial Control Technology,Department of Control Science and Engineering,Zhejiang University,Hangzhou 310027,China

A R T I c L E IN F o

Article history:

Received 5 June 2013

Received in revised form 4 October 2013 Accepted 12 November 2013

Available on line 19 June 2014

Identification

Multi-model linear parameter varying system

Asymmetric Gaussian weight

Continuous stirred tank reactor

In this paper,asymmetric Gaussian weighting functions are introduced for the identification of linear parameter varying systems by utilizing an input-output multi-model structure.It is not required to select operating points with uniform spacing and more flexibility is achieved.To verify the effectiveness of the proposed approach,several weighting functions,including linear,Gaussian and asymmetric Gaussian weighting functions,are evaluated and compared.It is demonstrated through simulations with a continuous stirred tank reactor model that the proposed approach provides more satisfactory approximation.

?2014 Chemical Industry and Engineering Society of China,and Chemical Industry Press.All rights reserved.

1.Introduction

Modern industrial processes are operated over a wide operating range and often display strong static and dynamic nonlinearities. The traditional linear models can no longer meet the requirement for model-based control.Accordingly,finding a sound and low cost nonlinear identification approach to approximate nonlinear processes over a broad operating regime is crucial and indispensable[1].

In the identification of nonlinear systems,several black box modeling approaches characterized by the usage of theoretically sound nonlinear functions such as nonlinear AR(MA)X[2],artificial neural network models[3],and blocked-oriented models such as Hammerstein and Wiener models[4]have been studied.Since these model shave complex structure and need dif fi cu lt computation,their applications to industrial processes are limited.

Recently,linear parameter varying(LPV)model identification has attracted great attention from academia and industry[5].The terminology of LPV was first introduced by Shamma and Athans[6]in the study of gain scheduling control.The study on LPV systems has been extended to the theory of linear systems[1].Much work has been on the identification of LPV systems[7,8].LPV approaches are also applied to aerospace systems including high performance aircraft,missiles and turbofan engines[9].

Most available references on input-output LPV are based on parameter interpolation,assuming that the scheduling parameter varies continuously[10].However,nonlinear functions are complex in the denominator of transfer function,which may cause numerical problems during model identification[10].Besides,the input excitation signal for this representation causes too much upset,which may be costly or even unrealistic in practice[11].To circumvent these difficulties,a multimodel LPV model is proposed by interpolating local linear models [10].With local linear models and model interpolation philosophy,the identification method is relatively simple and the stability of this LPV models is guaranteed[12].

Essentially,p roper weighting functions are required to combine local linear models in to a global LPV model in the multi-model LPV structure.The available options are linear weight function[10]and Gaussian weight function[12].The linear weight functions can be used conveniently,but it is not sufficient to capture the full dynamic behavior of a nonlinear process.Owing to relative small number of para meters and superior performance,Gaussian weighting functions have been widely adopted in the multi-model structures[13]and fuzzy sets[14].A drawback of Gaussian weighting functions is that the operating points for local linear models should have an equal distance,which limits their feasibility and causes large inconvenience in applications.

To overcome the disadvantage of Gaussian weights,asymmetric Gaussian weighting function is introduced for the identification of multi-model LPV structure in this paper.The locations of operating points can be selected freely.Linear,Gaussian and asymmetric Gaussian functions are used and compared in simulation study of a continuous stirred tank reactor(CSTR)system to demonstrate the accuracy andeffectiveness of the multi-model LPV model with asymmetric Gaussian weighting.

2.Description of LPV Model

For a multi-input single-output LPV system,let m inputs be{u1(t),…,um(t)}at time t and the output be y(t).One type of the input-output LPV system can be described as[1]

where

is a LPV transfer function from ui(t)to y(t)that is stable,Bi(q,w)and A(q,w)are polynomials of q?1,which denotes unit delay operator,diis the delay from the i th input to the output,v(t)is a stationary stochastic process with zero mean and bounded variance,n is the order of the model,and w(t)is the scheduling variable,which is measurable or can be calculated from other measurable process variables.In this paper,we assume that w(t)∈[wmin,wmax],where wminand wmaxare the low and high limits of w(t),respectively.

Polynomial method is a commonly used parameterization method to rep resent the LPV model,with parameters( w)and aj(w)rep laced by polynomial functions of w(t).

where nαand nβare the orders of polynomial functions.

Eqs.(1)-(3)form u late the common approach of current LPV methods,called para meter-interpolation input-output LPV model[1]. However,it is not easy to identify parameter-interpolation LPV structure for its complex structure and fails to obtain acceptable performance in a case study[12].

3.Multi-Model LPV Model Identification

Motivated by identification practice,Zhu and Xu[10]proposed a simpler LPV model structure,which is called multi-model LPV model by Huang et al.[12].Its basic principle is to identify several local linear models at fixed operating points,and then achieve the global model by interpolation via certain weighting functions.It has been verified [1,11,12]that the multi-model LPV is a good approximation of real processes along its operating-trajectory and the stability of this LPV model can be guaranteed easily[12].

We choose l operating points:

The parameters to be estimated for each local model can be written as

Parameters to be estimated for all l local linear models can be denoted as

The values of all the parameters in ΘLcan be obtained with linear identifications using the data collected at each operating point test. Several linear identification methods can be used:prediction error method,subspace method and asymptotic method(ASYM)[15].In an operating point test,the scheduling variable is kept constant,while a norm al identification test is perform ed for local linear model identification using small test signals.

The multi-model global LPV model is obtained by interpolating local linear models,which can be exp ressed as

To combine all the local linear models in to a global multi-model LPV model in Eq.(8),p roper weighting functions are required for interpolation,which have a large effect on the accuracy of the global model.Some common weighting functions are available in literature,such as linear weight function[10]and Gaussian weight function[12].The structures of weighting functions will be specified later.

3.1.Multi-model LPV model with linear weights

The linear weight function is the simplest one that can be p reassigned.The weighting equals to the distance between current scheduling variable and operating points of the local linear models. With^y（ t｜w（ t））to be estimated where wk＜w(t)＜wk+1,(k=1,2,…, l?1),the weighted output y^（ t）is

Although linear weighting can be used conveniently in the multi model LPV model,it is not accurate enough to capture the full dynamic behavior of nonlinear process.

3.2.Multi-model LPV model with Gaussian weights

A p referable choice for deter mining model weights is Gaussian function,which can be written as

The choice of l is a trade-off of computing cost and model accuracy.

To identify the multi-model LPV model,local linear models at each fi xed operating point should be determined first.The transfer function of the k th local linear models can be expressed as

where

andσkrepresents the width coefficient of the k th local linear model.

In the realm of multi-model structures[13]and fuzzy sets[14], Gaussian weighting functions have been widely utilized,which haverelatively small number of parameters and natu rally better functions than linear functions.However,a disadvantage of Gaussian weighting functions is that the operating points for local linear models should have identical distance with respect to a scheduling variable,which is inconvenient in practice.

3.3.Multi-model LPV model with asymmetric Gaussian weights

To overcome the drawback of the Gaussian weights,asymmetric Gaussian functions are introduced here,which can be written as

where

σk,1andσk,2are the left and right variances,respectively,being the width coefficients of the scheduling variable.

The asymmetric Gaussian function in Eq.(13)is normalized in the range of zero to one,without negative values.Essentially,it maintains the constrained shape and smooth properties of the Gaussian function. Since asymmetric Gaussian function has different left and right variances,the operating points for local linear models may have different distances with respect to a scheduling variable.For traditional symmetric Gaussian functions,the range in w(t)should be divided with equal intervals.

Obviously,only two parameters need to be defined for each weighting function.The parameter vector for all l weighting functions can be written as

For a given number of operating points and a given operating range, the selection of operating points is based on trial-and-error method. From the empirical know ledge,if more operating points locate at the p lace where the systemdisp lays strong nonlinearity and parameter varying,more accuracy models can be achieved.Therefore,the flexibility of choosing uneven operation points for local linear models provides an effective approach to improve the accuracy of the LPV model.

4.Simulation Results and Discussion

CSTR is one of the common operations in chemical and petrochemical plant,which is used to validate the feasibility of the method.An irreversible,exothermic reaction A→B occurs in a constant-volume reactor cooled by a single coolant stream.The model of the process is as follows [11]

where CAis the concentration of component A,CA0is the initial concentration,t is the time,qcis the flow rate of coolant,V is the reaction volume,k0is the reaction rate constant,E is the activation energy,R is the universal gas constant,T is the temperature of material,T0and Tc0are the initial temperatures of material and coolant,respectively,ΔH is the heat of reaction,ρandρcare the densities of material and coolant,respectively,Cpand Cpcare the heat capacities of material and coolant,respectively,h is the heat transfer coefficient,and A is the heat transfer area.Their values are the same as those in literature[11].

The controlled variable is CA(t),mol·L?1,and the manipulated variable is qc(t),L·m in?1.For the LPV model design,coolant flow rate qc(t) is chosen as the scheduling variable,which is in the range of[97,109]. Without loss of generality,four typical operating points for asymmetric Gaussian weight method are pre-determined for qc(t),97,103,107 and 109 L·m in?1,while the counterparts for Gaussian weight method are 97,101,105 and 109 L·m in?1.

The identification test is performed for the coolant flow rate.The test signal is a generalized binary noise with average switch time of 40m in.output to represent measureable noise.The second order output-error model is identified using the data at each operating point.Their step responses are shown in Fig.1,which implies that the dynamic behavior of the process is distinct at different operating points.In this case,the parameter vector for all asymmetric Gaussian weighting functions is [0.5 0.5 0.5 1 1 2 2 2],while the counterpart for Gaussian weighting functions is[1 1 1 1].

Fig.2 shows the simulation for identified LPV model using linear, Gaussian and asymmetric Gaussian weights and com pares the simulated outputs with measured data.Table 1 com pares the output errors of LPV models with three weighting functions.The output error is defined as

One can see that the identified LPV model with asymmetric Gaussian weights is more accurate in cap tu ring the actual nonlinear process dynamics.

Fig.1.Step responses of local linear models at 4 different operating points.

Table 1Comparison for model output error

Fig.2.Comparison of the outputs of identified global LPV models using different weights with that of real process with noise.

5.Conclusions

In this paper,the asymmetric Gaussian weighting function is introduced in to nonlinear identification of the multi-model LPV structure. Choosing uneven operating points for local linear models over the scheduling variable,the accuracy of the multi-model LPV model can be improved.To verify the feasibility of the proposed approach,a simulation example for a CSTR is given.It demonstrates that the LPV model with asymmetric Gaussian weighting presents superior performance.

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☆Supported by the National Natural Science Foundation of China(21076179, 61104008),and National Basic Research Program of China(2012CB720500).

*Corresponding author.

E-mailaddress:jglu@iipc.zju.edu.cn(J.Lu).