Huabo LIU,Tong ZHOU

1.Department of Automation,Tsinghua University,Beijing 100084,China;

2.College of Automation Engineering,Qingdao University,Qingdao Shandong 266071,China;

3.Tsinghua National Laboratory for Information Science and Technology(TNList),Tsinghua University,Beijing 100084,China

Robust state estimation for uncertain linear systems with deterministic input signals

Huabo LIU1,2†,Tong ZHOU1,3

1.Department of Automation,Tsinghua University,Beijing 100084,China;

2.College of Automation Engineering,Qingdao University,Qingdao Shandong 266071,China;

3.Tsinghua National Laboratory for Information Science and Technology(TNList),Tsinghua University,Beijing 100084,China

In this paper,we investigate state estimations of a dynamical system in which not only process and measurement noise,but alsoparameteruncertaintiesanddeterministicinputsignalsareinvolved.Thesensitivitypenalizationbasedrobuststateestimation is extended to uncertain linear systems with deterministic input signals and parametric uncertainties which may nonlinearly affect a state-space plant model.The form of the derived robust estimator is similar to that of the well-known Kalman filter with a comparable computational complexity.Under a few weak assumptions,it is proved that though the derived state estimator is biased,the bound of estimation errors is finite and the covariance matrix of estimation errors is bounded.Numerical simulations show that the obtained robust filter has relatively nice estimation performances.

Robust estimation;Deterministic input;Regularized least-squares

DOI10.1007/s11768-014-4072-4

1 Introduction

State estimation plays an important role in signal processing and control system design.It is known that the Kalman filter is the optimal estimator under the criterion of mean-squares and widely applied in numerous fields such as target tracking,global positioning systems,hydrological modelling,atmospheric observations,timeseries analyses in systems biology and econometrics,automated drug delivery,and so on[1–3].As modelling errors are generally unavoidable,robust state estimators such as H2/H∞filtering,set-valued estimation,and guaranteed-cost designs,which do not vary appreciably when actual plant parameters deviate from their nominal ones in a reasonable way,have been developed,see[1,4–8]and the references therein.Particularlyworth mentioning is that a regularized least-squares(RLS)based framework is suggested in[1]for robust filter designs,whose attractive characteristic is that the filter shares the same form of the well known Kalman filterwithcorrectedparameters.However,inthisframework the plant parameters are required to depend linearly on an uncertainty block,which may be a restrictive condition.Theotherpossiblelimitationisthattherobust estimator needs to optimize a cost function at every estimation step,whose unique minimum has no analytic expression.

There is another paradigm in robust filter designs which is based on sensitivity penalization of estimation errors to parameter variations.In[9],it is employed for single-input single-output systems in the frequency domain with transfer function representation and spectral factorization.In[2,10],it is adopted for multiinput multi-output time varying dynamical systems under state-space framework and the plant parameters are affected by modelling errors in a relatively arbitrary way.BasedontherelationshipbetweenKalmanfilterandregularized least-squares,as well as sensitivity penalization on estimation errors to parameter variations,an analytic expression of the robust state estimator has been derived[2,10].The estimator can be recursively implementedandhasacomparablecomputationalcomplexity with the widely applied Kalman filter.

The works aforementioned normally assume that the system is driven only by noise processes without deterministic input signals.It is true that the existence of a known deterministic input does not affect the estimation errors if the signal process does not involve parameter uncertainties,on the contrary it is not valid because the superposition principle is no longer established for the existence of parameter uncertainties[11].On this occasion,it is of significance to take robust state estimation with deterministic input signals into account and analyze the asymptotic properties of the derived estimator.The H∞filtering approach[7]was extended to provide a guaranteed H∞bound for estimation errors in the presence of both parameter uncertainties and known input signals for continuous time varying uncertain systems[12]and discrete time varying uncertain systems[11],respectively.In[13]a robust H∞state estimator is investigated for a class of uncertain discrete time piecewise affine systems with partitioned state space based on which the filter implementation may not be synchronized with state trajectory transitions.

In this paper we generalize the robust state estimator sensitivity penalization based[2,10]to cope with the cases where deterministic input signals are considered.An analytic expression has been derived for the robust estimator,which can be recursively implemented and hasasimilarformandacomparablecomputationalcomplexity with the Kalman filter.In[2,14]it is proved that under some assumptions,as well as conditions like detectability and stabilizability,the robust state estimator sensitivity penalization based is asymptotically unbiased whentherearenodeterministicinputsignalsinprocess.Our main contribution lies in the fact that the estimation errors are proved to be bounded and have a bounded covariance matrix though the robust state filter is biased owing to the existence of modelling errors when there exist deterministic input signals.Some numerical simulationsshowthatthisrobustestimatorhasrelativelynice estimation performances and can be widely applied.

The rest of this paper is organized as follows.In Section 2,a state-space plant model is given and the robust state estimator sensitivity penalization based is derived.Some important properties such as convergence and boundednessarediscussedinSection3.Numericalsimulation results are reported in Section 4.Finally,Section 5 concludes this paper.Two appendices are included to give a derivation of the recursive estimation procedure and a proof of the theoretical result.

Notation Given a column vector x and a positivedefinite matrix W,‖x‖2and ‖x‖2Ware defined to denote the Euclidean norm and its weighted version,namely,xTx and xTWx,respectively.

2 Plant dynamics description and robust state estimator design

Consider the following uncertain linear system,

where x is the state,w is the process noise,u is a deterministic input signal,y is the measurement,and v is the measurement noise.x0,wiand viare uncorrelated random vectors with E(x0)=0,E(wi)=0,E(vi)=0 and E((col(x0,wi,vi))(＊)T)=in whichknown positive definite matrices and δijrepresents the Kronecker delta function.Moreover,εidenotes parametric modelling errors at the ith sampled instant which is composed of L real valued scalar uncertainties,k=1,...,L.It is assumed that the L uncertainties are independent of each other and all the entries of matrices Ai(εi),B1i(εi),B2i(εi)and Ci(εi)are differentiable functions of εi.

Compared system(1)with the one in[2,10],the known deterministic input is considered in this paper.It is obvious that system(1)collapses to the one in[2,10]when there is no deterministic input,therefore,system(1)can be regarded as a generalization of the one in[2,10].

From[1,3],we know that the Kalman filter admits a deterministic interpretation as the solution to a regularized least-squares problem,as follows:

wherexˆi|lstands for the optimal estimator of xibased on measurementsand Pi|lthe corresponding estimation errors covariance matrix.The cost function of the regularized least-squares problem is the regularized squares residual norm.The interpretation means that given an initial estimateone seeks to meliorate it by incorporating the additional information provided by the new measurement yi+1and deterministic input

We improve the cost function of the regularized least-squares problem considering the estimation performances appreciable deterioration because of model uncertainties which are generally unavoidable.For notational simplicity,define matrices respectively as follows:

Then the cost function can be rewritten as+‖Hi(0,0)αi− βi(0and an analytic solution to this regularized least-squares problem can be obtained.In our improvement,denoteCi+1(εi+1)[Ai(εi)B2i(εi)]αias ei(εi,εi+1)which is generally called innovation process,the new cost function of the RLS at every instant is suggested to be(3)to reduce the sensitivity of estimation performances to modelling errors.

From the cost function we can conclude that the deviations of the innovation process from yi+1−reflect contributionsofmodellingerrorstopredictionerrorsbasedon yi+1,and the design parameter γitakes account in both theimportanceofnominalestimationperformancesand that of estimation performance degradation due to modelling errors.Generally,the design parameter γihas an empirical value[2,10]and can be adjusted according to the relative magnitude of modelling errors in practical application.The bigger the amplitude of modelling errors is,the smaller the parameter is.This means that the estimation performance degradation owing to modelling errors plays a more important role.It is consistent with physical intuitions.When there are no modelling errors and γi=1,the state estimator through minimizing the cost function(3)collapses to the standard Kalman filter.

Define matrices Si,T1iand T2irespectively as follows:

where

Then,we can obtain

According to the above analysis we can provide the following recursive procedure to compute the estimate of the plant state when there exist a deterministic input signal and parameter uncertainties.The derivative details are provided in Appendix A.Denote λi=(1−γi)/γi.

1)Initialization.Designate P0|0andˆx0|0as

respectively,in which

2)Parameter modification.Define matricesˆT2i,ˆAi(0),ˆB1i(0),ˆB2i(0),ˆPi|iandˆQirespectively as follows:

3)State estimate updating.Calculateˆxi+1|i+1and Pi+1|i+1respectively as

Based on the above explanation,the form of the estimation procedure is consistent with the time and measurement update form of the robust estimator derived in[1]and has a similar form with the one in[2,10]increasedbysometermsrelativetothedeterministicinput ui.When ui=0,the derived state estimation procedure collapses to the robust filter in[2,10],which means that it is a generalization of the one in[2,10].

3 Some properties of the estimator

In this section,some important asymptotic properties of the derived state estimator are investigated.Suppose εi,kis normalized in magnitude to be contractive and the set E is composed of these modelling errors.That is,E={ε||εi,k|≤ 1,k=1,...,L}.Moreover,we adopt two assumptions for the asymptotic behaviours analysis of the robust state estimator.

A1)Ai(0),B1i(0),B2i(0),Ci(0),Ri,Qi,Si,T1i,T2iand γiare time invariant.

A2)The uncertain linear system of(1)is exponentially stable in the sense of Lyapunov.Moreover,matrices Ai(0),B1i(0),B2i(0),Ci(0),Ri,Qiand Π0are bounded for i＞ 0 and εi∈ E.

Equation(5)can be rewritten as follows:

where

This expression is similar to equation(6)in[14]and the difference is that"the input"of(6)is yi+1instead of[ui;yi+1],thereforeTheorem1in[14]canbegeneralized directly to system(5)as follows when the convergence of system(1)is considered.

Theorem 1 Assume that condition A1)is satisfied,is detectableis stabi-Pi|i−1converges exponentially to a unique positive semidefinite matrix P,while Apiconverges to a constant stable matrix Ap.Here,

We know that the convergence of Pi|i−1is equivalent to that of Pi|ifrom the relation between Pi|i−1and Pi|i.Therefore,the derived robust state estimator converges to a time-invariant stable system when the conditions of Theorem 1 are satisfied.

We then consider the boundedness and biasness of estimation errors for this robust state filter.For simple denotations,define¯xi,ˆ¯xi|iand˜¯xi|irespectively as¯xi=[I+Ωi(0)]xi,ˆ¯xi|i=[I+Ωi(0)]ˆxi|iand ˜¯xi|i=¯xi−

ˆ¯xi|i,

where Ωi(εi)=It is obvious thatThen from equation(5)we can directly prove that

From equation(6),we have with the whiteness of wiand vi,as well as the assumption of unrelated wiand vi,that

we can conclude that

where N1is a finite positive constant.It means that the summation of series in the last term of equation(7)is finite.The fact that Ai(εi)and B1i(εi)are bounded leads to thatare bounded,which implies that there exists a finite positive constant N2≥0 suchTherefore,the estimation errors of the robust filter are bounded.When there are no deterministic input signals,that is,ui=0,we can obtainwhich means that the robust state estimator sensitivity penalization based is asymptotically unbiased.It is consistent with the result derived in[2,14].

Based on the relations in(6)and the stability of matrix Ai(εi),as well as the aforementioned derivation,we achieve a condition for the boundedness of estimation errors of the robust filter as follows.Its proof is deferred to Appendix B.

4 Numerical simulations

In this section,we compare the performances of the derived state estimator with those of the Kalman filter based on actual parameters and nominal parameters by some examples.In these simulations,it is assumed that modelling errors are time-invariant,and every uncertainty parameter is contractive,that is,it belongs to the interval[−1,1].Furthermore,1000 time-domain inputoutput data pairs are generated for plant state estimation,in which all the initial states are set to zero,while disturbances wiand viare produced according to normal distributions.The deterministic input signal uiis fixed or produced according to normal distributions.

Thisexampleisimprovedfromtheonein[1]and[10],in which it is assumed that,

In the first set of simulations the modelling error ε is fixed to be−0.8508 and the input signal uiis also fixed,ui=[1.0;0.1].Fig.1 shows the variations of estimation error variances with respect to time samples and the filter design parameter γ.When the design parameter γ takes the empirical value which is approximately 0.8,the difference between the performances of the Kalman filter with actual parameter values and the robust state estimator derived in this paper is only 1dB and nearly 10dB performance improvement is obtained compared with the Kalman filter based on nominal parameter values.The same conclusion can be drawn from Fig.2.

Fig.2 shows that at the sampled instants i=500 and i=1000,if γ takes any value between 0.0000 and 1.0000,the performance of the derived robust filter is better than that of the Kalman filter based on nominal parameter values and the optimal γ is approximately 0.8300.

In Fig.3,the input signal uiis fixed to be[1.0;0.1]and the modelling error εis produced randomly and independently in each simulation according to a normal distribution with truncations.The mean and the standard variance of the normal distribution are set respectively to 0.0000 and 1.0000.In case that a generated εhas a magnitude greater than 1,it will be got rid of and reproduced until an εwith magnitude not greater than 1 is obtained.From Fig.4,we can see that there exists a large interval of γ which leads to a robust estimator with better performance than the Kalman filter based on nominal parameters at the sampled instants i=500 and i=1000.

Fig.1 Estimation error variance with fixed γs.

Fig.2 Estimation error variance at fixed instants.

Fig.3 Estimation error variance with fixed γs.

Fig.4 Estimation error variance at fixed instants.

The deterministic input signal uiin Fig.5 is produced randomly and independently according to a normal distribution with truncations and the modelling error εis produced randomly and independently in each simulation according to a normal distribution with truncations.The mean and the standard variance of the normal distribution εare set respectively to 0.0000 and 1.0000.

Fig.5 Estimation error variance with fixed γs.

Fig.6 Estimation error variance at fixed instants.

From Figs.1–6,it is obvious that the robust state estimatorbasedonthesensitivitypenalizationofestimation errors to modelling uncertainties can bring about significant robustness improvements in plant state estimator designs.The optimal design parameter γ may lead to the robust estimator with performances close to those of the Kalman filter based on actual plant parameter values.Moreover,there are quite a lot of selections for the parameter γ for that the performances of the robust state estimator are continuous functions of the design parameter.These properties are attractive in actual filter designs and next we aim to find the optimal filter design parameter.

5 Conclusions

This paper investigates a robust state estimator based on modelling errors sensitivity penalization for uncertain linear systems subject to deterministic input signals and norm-bounded parametric uncertainties.The derived state estimator is biased owing to the existence of modelling errors in the input matrix,but the covariance matrix of estimation errors is proved to be bounded.The simulation examples show that this approach significantly improved the estimator’s robustness to model uncertainties compared with the designs only based on nominal systems.

It also remains challenging to give an estimate for the interval of desirable penalizing factor γ,as well as an estimate for the size of tolerable modelling errors.

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Appendix A Derivation of the estimation procedure

To estimate the initial state x0,the cost function J(α0)is set as follows,in which e0(ε0)=y0− C0(ε0)x0.

We obtain the following estimate of the initial state

instituting(a1)to(4),and multiplyingthe left sides of(4),we can obtain that

Therefore,

Moreover,

Therefore,

Appendix B Proof of Theorem 2

Based on this relation,a direct application of mathematical inductions shows that

Then,from the boundedness of B1i(εi),B2i(εi),Ci(εi),Qiand Ri,as well as the boundedness of estimation errors for the derived estimator,we have that N3is a finite positive number and,

That is,the covariance matrix of estimation errors is always upper bounded.

28 May 2014;revised 28 October 2014;accepted 28 October 2014

†Corresponding author.

E-mail:liu-hb10@mails.tsinghua.edu.cn,liuhuabo1979@qdu.edu.cn.

This work was supported in part by the 973 Program(Nos.2009CB320602,2012CB316504),in part by the National Natural Science Foundation of China(Nos.61174122,61021063,60721003,60625305),and in part by the Specialized Research Fund for the Doctoral Program of Higher Education,China(No.20110002110045).

©2014 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag Berlin Heidelberg

Huabo LIU received his B.Sc.and M.Sc.degrees in Engineering from Chongqing University,China,in 2001 and 2005,respectively.He is a Ph.D.candidate at the Department of Automation,Tsinghua University,and also a lecture at the college of Automation Engineering,Qingdao University,China.His research interest covers robust performance analysis and robust state estimation of multidimensional distributed systems.E-mail:liuhb10@mails.tsinghua.edu.cn,liuhuabo1979@qdu.edu.cn.

Tong ZHOU is currently a professor in the Department of Automation,Tsinghua University.He received his B.Sc.and M.Sc.degrees from University of Electronic Science and Technology of China in 1984 and 1989,respectively,and his Ph.D.degree from Osaka University,Osaka,Japan in 1994.His research interest covers robust control,system identification,signal processing,hybrid systems,communication systems,and their applications to real-world problems.E-mail:tzhou@mail.tsinghua.edu.cn.