Consistency and Asymptotic Property of a Weighted Least Squares Method for Networked Control System s☆

Cong Zhang,Hao Ye

Tsinghua National Laboratory for Information Science and Technology,Depart mentof Automation,Tsinghua University,Beijing 100084,China

Consistency and Asymptotic Property of a Weighted Least Squares Method for Networked Control System s☆

Cong Zhang,Hao Ye*

Tsinghua National Laboratory for Information Science and Technology,Depart mentof Automation,Tsinghua University,Beijing 100084,China

A R T I c L E IN F o

Article history:

Received 27 December 2013

Received in revised form 26 February 2014 Accepted 5March 2014

Available on line 25 June 2014

In this paper,we study the problems related to parameter estimation of a single-input and single-output networked control system,which contains possible network-induced delays and packet dropout in both of sensor-to-controller path and controller-to-actuator path.A weighted least squares(W LS)method is designed to estimate the parameters of plant,which could overcome the data uncertainty problem caused by delays and dropout.This WLS method is proved to be consistent and has a good asymptotic property.Simulation examples are given to validate the results.

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

1.Introduction

In a networked control system(NCS),there may exist random network-induced delays and packet dropout in the sensor-to controller(S-C)path and the controller-to-actuator(C-A)path due to the communication access constraints[1,2].Network-induced delays and packet dropout could cause data uncertainty problems and bring difficulties for system identification,control,or fault detection.

Problems related to NCS identification have drawn more and more attentions in recent years.Particularly,Fei et al.[3]utilized a discard packet strategy,a cubic spline interpolation,and buffers to the actuator to overcome the data uncertainty problem in an NCS and proposed a recursive estimation method.Wang et al.[4]formulated an NCS in a continuous-time system with non-uniformly non-synchronized sampled data,and proposed a modified version of the simplified refined instrumental variable method to identify the parameter offline.Liu and Wang[5]extended the results[4]to the case with colored noise.Shi et al.[6]gave a recursive parameter estimator for closed-loop system with randomly missing output data.Shi and Fang[7]proposed a recursive method for open-loop system with randomly missing measurements of plant’s input and output.

In our previous work[8],we have proved that the dataset of a single-input and single-output(SISO)NCS with delays and dropout in both of the S-C and C-A paths is informative under very weak conditions, but we did not design an identification method.The result[8]is useful to design a consistent parameter estimator for the NCS in this paper, since the setup of NCSs is the same and the in form ative dataset isa necessary condition for consistent parameter estimation.

The results in Refs.[3-7]cannot be applied to the NCS in this paper due to the following four reasons:(1)the M sequence used as the controlsignal[3]cannot be generated by feedback controllers;(2)with the remote computers used in closed loop to generate control signals[4,5], and those computers’outputs are required to be independent of their inputs,which could not be achieved by feedback controllers;(3)an adaptive controller instead of a linear time-invariant(LTI)one was required[6];and(4)the problem in Ref.[7]was for open-loop setup.

In this paper,we consider system identification of a SISONCS with a common setup,which includes an LTI plant,an LTI controller,and network transmissions in both the S-C and C-A paths that contain random delays and dropout.Motivated by Isaksson’s work[11],in which a modification idea for the standard least squares(LS)method was proposed to estimate the parameter of an open-loop system with randomly missing output data,we design a weighted least squares(WLS)method.This WLS method could overcome the data uncertainty problems caused by random delays and dropout.Based on our previous result[8],we prove that this WLS method is consistent,i.e.its parameter estimate converges to the“true”value[9],and has a good asymptotic property in the sense that the product of estimation error and square root of data length converges to a Gaussian distribution,which means that the estimation error decays with the reciprocal of the square root of data length[10].

2.NCS Setup and Problem Statement

Notation 1.Throughout the paper,“q”,“E”,and“Pr”rep resent“forward shift operator,i.e.q χk=χk+1”,“a symbol introduced by E χk= Ref.[10],where{χk}is a sequence of quasi-stationarysignal”,and“probability”,respectively.

The SISONCS considered is shown in Fig.1,where the reference input is zero,and the network-induced delays and packet dropout may occur randomly in both of the S-C and C-A paths.We assume that the actuator and the sensor are clock d riven with a fixed sampling interval.

2.1.Closed-loop model

The LTI plant and the LTI controller are described by

where yk∈R and uk∈R are the plant’s out put and input at time instant k,∈R and∈R are the controller’s input and out put at k,and ek∈R｛｝is a sequence of independent and identically distributed(i.i.d.)random signals with zero mean values,variances λ2,and bounded moments of order 4+δ with some δ＞0[10],respectively.

The structure of the plant in E q.(1)is assumed to be auto regressive exogenousof generality.

The parameter to be estimated is

The candidate parameter space used for estimation is denoted as DM

Remark 1.According to Refs.[6]and[7],we assume that the polynomial orders of the ARX plant,naand nb,are known,since they can be determined by using the statistical F-test[12-14]or the Akaike in formation criterion[15].

Fig.1.SISONCS.

2.2.Network transmission

The maximal steps of possible delays in the S-C and C-A paths are assumed to be τsteps,respectively.Then ykandwith delays longer than τsteps will be d iscarded when they fi nally arrives.

Remark 2.According to Refs.[1]and[2],it is common to assume that the delay sand dropout in network transmission satisfy Bernoulli distri-not available to the controller (or the actuator)at time instant k,multiple update mechanisms can be used by the controller(or the actuator)to

If“latest packet in the buffer”,or“previous step value”update mechanisms[1,2].In this paper,for the sake of generalization,we do not make any assumption on the update mechanisms adopted by the controller and the actuator.

2.3.Recovery of dataset

Due to the influences of delays and dropout in the S-C and C-A paths, the plant’s output may be received disorderly or lost on the controller side,and it is uncertain which packet sen t from the controller is used by the plant.These data uncertainty problems bring difficulties for parameter estimation.

Fortunately,some techniques about NCS have been provided for data recovery on the controller side,such as the sequence numbering technique[4]and the smart sensor technology[6,16-19].By using the sequence numbers of the packets received by the controller,the disorderinstants at most;using the smart sensor technology(i.e.the actuator feedbacks the sequence number of plant’s input to controller by sending it to the sensor and further adding it to the packet transmitted by the sensor),we could verify the packet used by the plant.

We make following assumption to recover the plant’s input and output data on the controller side.

Assumption 1.The sequence numbering technique[4]and the smart sensor technology[6,16-19]are used for data recovery on the controller side.Except for the influence of dropout on the S-C path,all the other data uncertain ties caused by unreliable transmission can be recovered.side for parameter estimation at time instant L

Com pared with the plant’s input and output dataset,{y1,u1,…,yL,is obviously incomplete lacking of dropped plant’s output data.

2.4.Formulation

We also have following assumption on the NCS.

Assumption 2.

(1)Delays and dropout occur independently of{ek};

(2)The proportional term of the controller,denoted as f0,is nonzero, i.e.f0≠0;

(3)The NCS is stable.

Next,we will propose a WLS method that uses the datasetto estimate the parameter of NCS’s plant and then analyze its consistency and asymptotic property.

3.AW LS Method for Parameter Estimation of NCS

3.1.A parameter estimation idea for open-loop systems with randomly missing output data

The parameter estimate of a standard LS method is given by[10]

Isaksson[11]proposed to modify the standard LS method for parameter estimation of an open-loop system with randomly missing output data.With the idea,the parameter estimate is given by Eq.(4),but both of the summations in it are only calculated over those time instant k that none of ykand yk?na，…，yk?1in φkis missing.

3.2.AWLS method for parameter estimation of NCS

The joint dynamics consisting of the LTI controller in E q.(1),the delays in the S-C path,and the delay sand dropout in the C-A path can be regarded as a linear time-varying(LTV)controller,denoted asThen the NCS defined in Section 2 is equivalent to a closed-loop system consisting of an LTI plant,an LTV controller,and randomly missing output.Thus,an be regarded as the dataset of this closedloop system,and we can use a strategy similar to the one in Ref.[11] to estimate the plant’s parameter.In the following,a WLS method using the datasetand the parameter space DMis designed to estimate the parameter of NCS’s plant,referred as WLS1 for convenience.

According to Ref.[10],the prediction error with parameter θi∈DMand datasetis defined as

Motivated by the strategy in Ref.[11],we calculate the parameterestimate only with those prediction errorswithout any of yk?na，…，ykbeing dropped.We introduce a weighted factor αk,which is determined by the dropout state of…，ykjoin tly,

Rem ark 3.The differences between WLS1 method and conventional WLS method[10]are as follows:

(1)Conventional WLS method uses plant’s input and output datasetwhile WLS1 uses the incomplete dataset reconstructed on the controller side,

(2)The weighted factor of WLS1method,αk,is a stochastic process determined by the join t state of the dropouts in the S-C path from time instant k?nato k,while that of conventional WLS method is a deterministic time sequence,which can be set by user before the identification experiment[10].

Thus,in spite of the existing results of consistency and asymptotic property of conventional W LS method in Ref.[10]and other literature, we still need to reconsider these properties for WLS1 method.

Remark 4.The differences between WLS1 method and the modification idea of standard LS method[11]are as follows:

(1)Isaksson[11]considered an open-loop identification problem,

while our problem belongs to closed-loop identification;

(2)Isaksson[11]only gave an idea for modification of the standard LS method,while we give an actual W LS1 method and prove its consistency and asymptotic property.

4.Consistency and Asymptotic Property of WLS1Method

4.1.An equivalent form of WLS 1method

In order to analyze the consistency and asymptotic property of LS1 method in Ljung’s identification framework[10],we propose an equivalent form of W LS1 method.According to Ref.[10],the prediction error with the parameter θi∈DMand the datasetis

where the regression vector φkis defined in Eq.(5).According to Eq s.(3)Eqs.(9)and(10),we have the equivalent form of W LS1method

We can use Eqs.(11)-(13)to analyze the consistency and asymptotic property of W LS1method in Ljung’s identification framework.4.2.Consistency of WLS1 method

Because the closed-loop models of the NCS are LTV,and{ek}is the only excitation signal,ukand ykin Fig.1 can be written as the two convolutions of LTV filters and{ek},respectively

where the time-varying coefficients∈R are determined by the models of plant and controller in Eq.(1)and transmission states in both paths from time instant k?j to k.

4.3.Asymptotic property of estimation error of W LS1method

Theorem 2.IfW LS1method is consistent,the distribution of the product of its estimation error and the square root of data lengthis asymptotically normal with mean values 0 and covariance matrix P0,i.e.

where AsN(0,P0)denotes a asymptotic Gaussian distribution[10],and

Proof.Please refer to Appendix A.2.

Rem ark 7.According to Theorem 2,we know that the estimation error of W LS1method,converges toθ0very fast.We can conclude that W LS1 method has a good asymptotic property.

Accord ing to Eqs.(9.18)and(9.19)in Ref.[10],the approximation of the asymptotic covariance matrix(i.e.P0)can be calculated from experiment data,i.e.

5.Simulation

5.1.Closed-loop model

We assume that the orders of the ARX plant in Eq.(1)are na=2 and nb=2,{ek}is an independent and identically distributed Gaussian noise sequence with zero mean values and variance 0.01,and the parameter to be identified are specified as[1.4,0.45,1,0.7]T. The LTI controller in Eq.(1)is selected as

5.2.Network transmission

We assume that both the update mechanisms adopted by the controller and the actuator are the“latest packet in the buffer”[1].Both the maximum steps of delays in the S-C and C-A paths are assumed to be three,i.e.==3.Accord ing to Notation 2 and Rem ark 2, the probability distributions of the Bernoulli processes∈{?1,0,1, 2,3}and∈｛?1，0，1，2，3｝are specified as=[0.02,0.8, 0.08,0.06,0.04]and=[0.01,0.82,0.09,0.05,0.03],?k,respectively.

5.3.Estimates of parameter and noise variance

Fig.3.Noise variance estimates and relative errors.(dotted line:λ2;solid line:

Fig.4.Parameter estimates by conventional WLS method.

Fig.5.Noise variance estimates by conventional WLS method.

The estimates of plant’s parameter and noise’s variance are presented here.In order to show the consistency and the asymptotic property of WLS1 method clearly,we use itsstep delayed recursive form for parameter estimation and we calculate the estimate of noise’s variance according to Eq.(19).

The simulation is run in Matlab environment,and the data lengthis selected as 1000,i.e.L=1000.The initial conditions for estimation of parameter and covariance matrix of error are selected as^θ0=[0.1，0.1，0.1，0.1]Tand P0=103?I4×4,where I4×4is a 4×4 identity matrix.

For k=1,…,1000,Fig.2 gives the results ofand its relative errors,converge to their true values quickly,which validates the conclusions in Theorems 1 and 2 that WLS1 method is consistent and has a good asymptotic property.

In order to show the superiority of WLS1 method,we also give the simulation results by using the conventional WLS method[10]. The forgetting factor of the conventional WLS method is selected as a constantλ=0.98[20],and initial conditions of its recursive form are the same as those in W LS1method,i.e.=0.1，0.1，0.1，0.1 [and P0=103?I4×4.Figs.4 and 5 give the parameter estimates and noise variance estimates by using the recursive form of the conventional W LS method,and both of these estimates do not converge to the true values and have large biases.Thus W LS1method is better than the conventional W LS method for estimating the parameter of the NCS’s plant.

Table 1Parameter and noise variance estimates of W LS1 method at step 1000

6.Conclusions

In this paper,we consider the parameter estimation problems of a SISONCS with random delays and dropout in both of the S-C and C-A paths.Firstly,we propose a data recovery method to overcome the uncertain ties of plant’s input and output data,which are caused by delays in the S-C path,and delays and dropout in the C-A path.However,the reconstructed dataset still lacks the dropped plant’s output data.Secondly,we design a W LS method to estimate the plant’s parameter, which could use the reconstructed dataset.Finally,the designed W LS method is proved to be consistent and have a good asymptotic property in the sense that the estimation error decays with the reciprocal of the square root of data length.

Some interesting,related problems still remain to be studied,e.g.extending the results here to multi variable case,or study new estimation method if the network techniques cannot be used to recover the plant’s data on the controller side.

Appendix

A.1.Proof of Theorem 1

Before giving the p roof,we define

The proof of Theorem 1 is through three steps.

θi∈DM

With Eqs.(2),(5),and(14),Eq.(11)becomes

Because the NCS is stable according to Assumption 2(3)and?θi∈DMwith‖θi‖2being bounded,from Eq.(A3),it is easy to verify that there exists a nonnegative sequence{μm},such that

Next,we use a derivation similar as Lemma 2B.1 and Theorem 2B.1 in Ref.[10]to prove the result of this step.Let L，t∈Z+and t≤L,then for θi∈DM,we denotes

According to Eq.(A2),we have

According to E qs.(8)and(A3),and Assumption 2(1),we knowWith Eq.(A4),Eq.(A5)turns to

In the following,we will prove θ1≡θ2from Eq.(A11). Let m=1.According to Lemma 1 in Ref.[8]and Eq.(A9),

According to Eq.(A12),we have

Substituting Eq.(A13)in to Eq.(A11)for m=1 and according to f0≠0 from Assumption 2(2)and conditionsin Eq.(15),we have

Fo r 2≤m≤na,it is assumed that=0

holds for j=1,…,m?1.According to Lemma 1 in Ref.

[8],and using a derivation similar to Eq s.(A12)-(A14),

Therefore,we can recursively

According to Eq s.(11),(A1),(A2),and(A3),we know that for any θi∈DM,

According to Eq.(A15),we know that θ0∈Dc.Then,?θj∈Dc, we have

Applying Step(2)to Eq.(A16),we have θj≡θ0,which implies that Dc={θ0}.Therefore,according to Step(1),we have“^θW1L→θ0，w.p.1，as L→∞”.

A.2.Proof of Theorem 2

According to Eq.(12),we have

Then,we prove Theorem 2 by the following two steps. (1)The asymptotic distribution of

According to Eqs.(5)and(14),we know that

According to Eq.(A21),and because the NCS is stale,we know that there exists a nonnegative sequence{ξm},such that

Define

where M≥na.It is assumed that

(2)The asymptotic distribution of

According to Eq.(13),we know that

Expanding this equation into Taylor series around θ0,we have

where ηLbelongs to a neighborhood of θ0with radius

Then we have

According to Eq.(A18),we know thatis continuous at?θi∈DMand according to Theorem 1,we know that“ηL→θ0, w.p.1,as L→∞”.Then,we have

Moreover,using a derivation similar to Section A1(1),we have

According to Eqs.(A33)and(A34),we have

where

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☆Supported by the National Natural Science Foundation of China(61290324).

*Corresponding author.

E-mailaddress:haoye@tsinghua.edu.cn(H.Ye).

Networked control system Network-induced delay Packet d ropou t

Weighted least squares Consistency

Asymptotic property