Orthogonal projection based subspace identification against colored noise

Jie HOU,Tao LIU,Fengwei CHEN

School of Control Science and Engineering,Dalian University of Technology,Dalian Liaoning 116024,China

Orthogonal projection based subspace identification against colored noise

Jie HOU,Tao LIU?,Fengwei CHEN

School of Control Science and Engineering,Dalian University of Technology,Dalian Liaoning 116024,China

In this paper,a bias-eliminated subspace identification method is proposed for industrial applications subject to colored noise.Based on double orthogonal projections,an identification algorithm is developed to eliminate the influence of colored noise for consistent estimation of the extended observability matrix of the plant state-space model.A shift-invariant approach is then given to retrieve the system matrices from the estimated extended observability matrix.The persistent excitation condition for consistent estimation of the extended observability matrix is analyzed.Moreover,a numerical algorithm is given to compute the estimation error of the estimated extended observability matrix.Two illustrative examples are given to demonstrate the effectiveness and merit of the proposed method.

Subspace identification,colored noise,orthogonal projection,extended observability matrix,consistent estimation

1 Introduction

Owing to the convenience of using a state space model to describe a multivariable system,increasing attentions[1,2]have been devoted to state space model identification.The state space identification methods(SIMs)have been increasingly explored in the past two decades owing to the robust properties and relatively low computational complexities[1,3,4].A few subspace identification methods have been widely recog recognized for engineering applications with white noise,e.g.,the canonical variate analysis(CVA)approach[5],the multiple-input-multiple-output error state space model identification(MOESP)method[6],the numerical subspace state space identification(N4SID)algorithm[7],and the instrumental variable method(IVM)[8].It was pointed out [9] that the aforementioned SIMs differ from each other by using different weighting matrices to construct the instrumental variables (IVs) for consistent estimation of the extended observability matrix of the plantstate-space model.The asymptotic properties of these identification algorithms were analyzed in[10,11].

Since there are industrial systems likely subject to colored noise,e.g.,harmonic signals are usually involved with industrial electric circuits and mechanical systems,identification of these systems with colored noise have therefore received increasing attentions[12–14]in the recent years.Although the existing SIMs can guarantee consistent estimation in the presence of white noise,biased estimation may be obtained when these SIMs are applied to these systems due to the autocorrelation between sampled output data arising from colored noise.A feasible approach to eliminate the estimation bias is the use of the IV technique.By taking the past input sequence as the IV to eliminate the influence of noise to the system output,an extended SIM named PIMOESP was proposed in[15]to guarantee consistent estimation.By projecting the observed data onto the past input sequence,an orthogonal subspace identification method named ORT-CN was developed in[16]to eliminate the influence of colored noise.However,this method requires the input excitation to be a zero-mean uncorrelated stationary sequence to ensure identification accuracy.

In this paper,a subspace identification method based on double orthogonal projections is proposed to realize consistent estimation of the extended observability matrix in the presence of colored noise,by projecting the observed data onto the orthogonal complement of the future input sequence to eliminate the influence from the future input,and then projecting the data onto the past input sequence to eliminate the noise effect.Compared to the existing SIMs, e.g., PI-MOESP and ORT-CN, an important merit of the proposed method is that there is no limit on the input correlation as long as the persistent excitation condition is satisfied. Consistency analysis of the proposed algorithm is given with a proof.Moreover,an explicit formula of the estimation error of the extended observability matrix is derived,which can be easily used to evaluate the estimation errors of the system matrices.Two illustrative examples are given to demonstrate the effectiveness of the proposed method.The paper is organized as follows.The identification problem is introduced in Section 2.Section 3 gives a brief review of the IV-4SID algorithm and then presents the proposed method.Furthermore,the asymptotic properties of the proposed method are analyzed in Section 4.Two illustrative examples are given in Section 5.Finally,some conclusions are drawn in Section 6.

2 Problem description

Consider the following linear discrete-time invariant state-space model:

wherex(t)∈Rnx,u(t)∈Rnu,andy(t)∈Rnydenote the system state,input,and output vectors,respectively.The process noisew(t)∈Rnuand measurement noisev(t)∈Rnuare assumed to be colored noise with unknown variance.The system matrices are denoted by(A,B,C,D)with appropriate dimensions.The following assumptions are considered in the paper.

A1)The system is asymptotically stable,i.e.,all the eigenvalues ofAlie inside the unit circle.

A2)The pair(A,C)is observable and the pair(A,B)is reachable.

A3)The noisesw(t),v(t)and system input are statistically independent of each other,i.e.,

whereE(·)is the expectation operator,and where

denotes the autocorrelation matrix.

The objective of this paper is to propose a new SIM method to estimate the system matrices based on the measured input and output data.

Denote bypandfthe past and future horizons,respectively.For convenience,we assumep=f(p＞nx).Denote the stacked future and past output vectors byyp(t)=[y(t?p)T···y(t? 2)Ty(t? 1)T]Tandyf(t)=[y(t)T···y(t+f? 2)Ty(t+f? 1)T]T,respectively.Similar definitions are given forwp(t),wf(t),vp(t),vf(t),up(t)anduf(t).By iterating the state-space model in(1),we have

where Γ =[CT···(CAf?1)T]Tis the extended observability matrix.L1=[Af?1B···ABB],L2=[Af?1···A I]are the extended controllability matrices.The lower triangular Toeplitz matrices are,respectively,

Suppose that there areN+p+f?1 sampled data and introduce the output block Hankel matricesYp=[yp(t) ···yp(N)]andYf=[yf(t) ···yf(N)].Similar definitions are given forWp,Wf,Vp,Vf,UpandUf.DenoteXp=[x(t?p) ···x(t?p+N? 1)]andXf=[x(t) ···x(t+N? 1)].It follows from(4)and(5)that

3 Orthogonal projection based identification method

First,a brief review of the well recognized IV-4SID method is presented to explain why most of existing SIMs are biased in the presence of colored noise.Then the proposed identification method is given accordingly.

3.1 Brief review of the IV-4SID algorithm

The key idea of IV-4SID is to estimate the range space of Γ.For this purpose,it eliminates both effects of the future input and future noise fromYf.The first step is to annihilate the input term in(7)by projecting the data onto the orthogonal complement ofUf,i.e.,

Then,the following IV is used to annihilate the noise term in(8),

It follows that

The noise part in (10) can be asymptotically expressed as

If the input is a persistent excitation of orderf,which means thatRufis positive definite[10],we have

Substituting(2)into(4)yields

Then,substituting(13)into(12)yields

3.2 Proposed method

The noise effect can be removed from(10)to obtain consistent estimation by projectingYfΠ⊥Ufonto the column space ofUp,i.e.,

where ΠUp=UTp(UpUTp)?1Updenotes an orthogonal projection matrix ofUp.

Correspondingly,the noise part in(15)can be asymptotically expressed as

Performing an SVD for the left-hand side of(17),we obtain

where is the firstnxeigenvalues of(18).

The range space of extended observability matrix Γ is therefore obtained as

With the estimated?Γ,the estimations of?Aand?Ccan be extracted as

The last step is to estimateBandD.By postmultiplying (Uf)?and pre-multiplying ?Γ⊥to both sides of(7)and using?Γ⊥Γ=0,we have

For abbreviation,denote

The estimation ofBandDcan be extracted by

Hence,the proposed double orthogonal projections based subspace identification method,named as 2ORTSIM,can be summarized as follows:

1)Eliminate the influence of the future input and colored noise to the future output by using(15).

2)Calculate the SVD of the projection matrix in(18).

4 Asymptotic properties

The asymptotic properties including consistency and asymptotic error for estimating the extended observability matrix are studied below.

4.1 Consistent estimation

The following theorem is given for consistent estimation of the extended observability matrix by using the proposed method.

ProofIt can be seen from(17)and(18)that a consistent estimate of?Γ can be obtained if

It can be derived from(3)that

Ifpis sufficiently large and the input is a persistent excitation of order max(p,f),it follows

Therefore,we have

According to the assumptions of A1)and A2),we are sure thatL1is a full row rank matrix.Hence,the rank condition in(26)is equivalent to

Note that

which is equivalent to

In fact,it holds that

If the input is a persistent excitation of orderp+f,the condition in(30)can surely be satisfied.This completes the proof. □

4.2 Estimation error

The true estimation of Γ can be computed from an SVD as following,

Obviously,it can be simply taken as

It follows from(38)–(40)that

After the asymptotic error of the estimated extended observability matrix is computed,the asymptotic errors of the system matrices can be computed using the numerical methods given in the references[20,21].Then the asymptotic error of the plant transfer function matrix can be easily computed[22].

5 Illustration

Two examples are used to demonstrate the effectiveness and merit of the proposed method.One is a benchmark example studied in the reference[23],and the other is an injection molding process in the reference[24].

Example 1Consider a benchmark example studied in[23],

wherew(t)andv(t)are colored noises,which are independently generated by(1 ? 0.75q?1? 2.5q?2)/(1 ?1.5q?1+0.8q?2)e1(t),wheree1(t)is a white noise with variance of 0.05.For illustration,the input excitation is taken asu(t)=((1 ? 0.8q?1+0.6q?2)e2(t))wheree2(t)is a white noise with variance of 5.

Fig.1 Magnitude plot of the identified transfer function matrix for Example1.

Fig.2 Plot of the standard deviation of model error for Example1.

It is seen that both the proposed 2ORT-SIM and PIMOESP give consistent estimations,while the proposed method gives an improved accuracy compared to PIMOESP.Note that the N4SID and ORT-SIM give biased estimation.Note that the ORT-SIM can only be used to obtain consistent estimation when the input excitation is a zero-mean uncorrelated stationary sequence.

Furthermore,to assess the accuracy of proposed method for estimating the asymptotic error of the extended observability matrix,the estimated errors of(δA,δC)are computed directly from δΓ through a linear operation as[25],

Tables 1 shows the mean values along with the Std of(δA,δC)by using the proposed asymptotic error estimation method.The true values obtained by using the true plant model to compute the estimation errors are also listed in Table 1,which demonstrates the effectiveness of the proposed estimation of asymptotic error.

Example 2Consider an injection molding process studied in the reference[24],

Using the same input excitation as Example1,one thousand MC tests are carried out for model identification.The above four methods are used again for comparison.The averaged TFM magnitude plots are shown in Fig. 3. It is seen that both the proposed 2ORT-SIM and PI-MOESP give consistent estimations,while the proposed method gives an improved accuracy compared to PI-MOESP.In contrast,the N4SID and ORT-SIM give biased estimation.Fig.4 shows the Stds of model errors by using the proposed 2ORT-SIM and PI-MOESP.The mean values along with the Stds of(δA,δC)are listed in Table 2,well demonstrating the effectiveness of the proposed estimation of asymptotic error.

Fig.3 Magnitude plot of the identified transfer function matrix for Example2.

Fig.4 Plot of the standard deviation of model error for Example2.

Table 1 Estimation error of δA and δC by using the proposed 2ORT-SIM for Example 1.

Table 2 Estimation error of δA and δC by using the proposed 2ORT-SIM for Example 2.

6 Conclusions

A bias-eliminated subspace identification method has been proposed for industrial applications subject to colored noise,to overcome the deficiency of existing SIMs that could not provide consistent estimation.An identification algorithm based on double orthogonal projections is developed by using the past input sequence rather than the output sequence to eliminate the influence of colored noise,such that consistent estimation of the extended observability matrix can be obtained.The persistent excitation condition for consistent estimation of the extended observability matrix is analyzed with a strict proof.Moreover,a numerical algorithm is given to compute the asymptotic error of the estimated extended observability matrix,which can be easily applied to compute the estimation errors of the system matrices.The applications to two illustrative examples have well demonstrated the effectiveness and good accuracy of the proposed identification method.

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4 Janurary 2016;revised 22 July 2016;accepted 25 July 2016

DOI10.1007/s11768-017-6003-7

?Corresponding author.

E-mail:liurouter@ieee.org.Tel.:+86-411-84706465.

This work was supported by the National Thousand Talents Program of China, the National Natural Science Foundation of China (Nos.61473054,61633006),and the Fundamental Research Funds for the Central Universities of China(No.DUT15ZD108).

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Jie HOUreceived the B.Eng.degree in Automation from Beifang University of Nationalities,Yinchuan,China,in 2010,the M.Sc.degree in Control Science and Engineering from Chongqing University,Chongqing,China,in 2013.He is currently a Ph.D.candidate in the School of Control Science and Engineering,Dalian University of Technology.His research interest covers system identification.E-mail:jiehou.phd@hotmail.com.

Tao LIUreceived his Ph.D.degree in Control Science and Engineering from Shanghai Jiaotong University,Shanghai,China,in 2006.He is a professor in the Institute of Advanced Control Technology at Dalian University of Technology.His research interests include chemical and industrial process identification& modeling,robust process control,iterative learning control, batch process optimization.He is a member of the Technical Committee on Chemical Process Control of IFAC,Technical Committee on System Identification and Adaptive Control of the IEEE Control System Society,and Chinese Process Control Committee.E-mail:liurouter@ieee.org.

Fengwei CHENreceived the B.Eng.and M.Eng.degrees from Wuhan University,Wuhan,China,in 2009 and 2011,respectively,and the Ph.D.degree from Université de Lorraine,Nancy,France,in 2014.He is currently working with Dalian University of Technology,Dalian,China.His research interests include system identification and signal processing.E-mail:fengwei.chen@dlut.edu.cn.