WANG Hongwei and CHEN Yuxiao

1.School of Electrical Engineering,Xinjiang University,Urumqi 830047,China;2.School of Control Science and Engineering,Dalian University of Technology,Dalian 110024,China

Abstract: The identification of nonlinear systems with multiple sampled rates is a difficult task. The motivation of our paper is to study the parameter estimation problem of Hammerstein systems with dead-zone characteristics by using the dual-rate sampled data. Firstly, the auxiliary model identification principle is used to estimate the unmeasurable variables,and the recursive estimation algorithm is proposed to identify the parameters of the static nonlinear model with the dead-zone function and the parameters of the dynamic linear system model. Then, the convergence of the proposed identification algorithm is analyzed by using the martingale convergence theorem. It is proved theoretically that the estimated parameters can converge to the real values under the condition of continuous excitation.Finally,the validity of the proposed algorithm is proved by the identification of the dual-rate sampled nonlinear systems.

Keywords:dual-rate sampled data,dead-zone nonlinearity,Hammerstein model,system identification,convergence analysis.

1.Introduction

Due to the hardware condition,economic costing or environment influence, the updating period of the input data and the sampled period of the output data are not always equal in the industrial production process [1–3]. If one system refreshes the input data at a fast rate and samples the output data at a slow rate, then that can be called the dual-rate sampled system [4–7]. Dual-rate sampled systems broadly exist in the chemical process,industrial process and biopharmac-eutical process.To identify dual-rate sampled systems, there have been some methods to deal with the systems such as the polynomial transformation technique,lifting technique,auxiliary model principle,and so on.Based on lifting technique,Ding et al.presented two algorithms:the recursive least squared algorithm when the system states are measurable, and the hierarchical identification algorithm when the system states are unmeasurable[8].Ding studied the extended recursive least squared algorithm with estimated residual for the dual-rate controlled auto regressive(CAR)system by using the polynomial transformation technique[9].Aiming at the problem from the existence of unmeasurable variables in the identification process, based on the auxiliary model principle,the output value of the auxiliary model is used to replace the unmeasurable variable value in[10].Further,Chen proposed the identification algorithm based on Kalman filtering to estimate the unmeasurable variables[11].

In actual industrial production,most systems are nonlinear in industrial processes. Thus, in recent years, more and more researchers have begun to study nonlinear systems.Nonlinear systems have none universal model to depict the dynamical process. In recent years, researchers have studied various nonlinear models, and Hammerstein is one of these models. For single-rate Hammerstein systems, many algorithms have been presented to identify methods such as the over-parameterization method [12],the hierarchical identification algorithm [13,14], the iterative method [15,16], the blind identification method[17,18] and so on. For single-rate Hammerstein systems with polynomial nonlinearity, Ding proposed the hierarchical multi-innovation stochastic gradient algorithm[19];considering particle variation in particle swarms,Xu studied improved particle swarm optimization [20]; Ding et al. presented an auxiliary model based least squared algorithm [21]. For Hammerstein systems with continuous nonlinearity, Chen et al. simplified the nonlinearity into polynomial and presented two algorithms to identify the parameters: the stochastic gradient algorithm and the particle swarm optimization algorithm [22]. Ding et al. proposed the hierarchical least squared identification algorithm for Hammerstein CAR systems with the key term separation technique[23].For extended Hammerstein systems,Fang et al.used the Preisach model to describe hysteresis nonlinearity and then the estimated parameters with the over-parameterized method [24]. However, these algorithms are the identification methods based on singlerate sampled data. Therefore, it is hard to apply these methods to dual-rate sampled Hammerstein systems directly. To identify dual-rate sampled Hammerstein systems, some researchers investigated the auxiliary model based recursive least squared algorithm[25,26].Considering the Hammerstein CAR moving average (CARMA)systems, Wang et al. studied two algorithms: the hierarchical least squared algorithm and the key term separation based least squared algorithm [27]; Wang et al. studied the maximum likelihood estimation method[28].For dualrate sampled Hammerstein systems with polynomial nonlinearity,Li presented three different methods,inluding the recursive least squared algorithm, the stochastic gradient algorithm,and the multi-innovation algorithm,to estimate unknown parameters by using the auxiliary model principle [29].The above algorithms are all proposed based on polynomial nonlinearity.Besides polynomial nonlinearity,however, there are many kinds of Hammerstein systems corresponding with different nonlinearities [30]. Among these systems, the identification of Hammerstein systems with dead-zone nonlinearity is more complicated. Due to dead-zone nonlinearity,the biggest hindrance to the identification process is the coupling of the information vector with unknown parameters which are hard to be identified directly.It brings more challenge when we identify singlerate systems with only the dual-rate sampled data.To sum up,the innovation in this paper mainly embodies in the following parts:

(i) Based on the auxiliary model principle, present one new algorithm to estimate parameters for Hammerstein systems with dead-zone nonlinearities.

(ii)To identify the model of single-rate systems by using the dual-rate sampled data.

(iii) To give the convergence analysis of the proposed algorithm by using the martingale convergence theorem.

The rest content of this paper is organized as follows.Section 2 introduces Hammerstein systems with dead-zone nonlinearities.Section 3 presents an auxiliary model based recursive extended least squared algorithm for the dualrate sampled nonlinear systems.Section 4 presents convergence analysis of the proposed algorithm. Section 5 provides the results of simulating one actual Hammerstein system to demonstrate the effectiveness of the proposed method.Section 6 draws some conclusions and remarks.

2.Problem formulation

Consider dual-rate sampled Hammerstein systems with dead-zone nonlinearities as shown in Fig.1.It consists of the dynamic linear part and the static nonlinear part. In Fig.1,Hhmeans the zero-order holder whose holding period ish.Sqhis the sampler with sampling period asqh,u(k) andy(kq) are the discrete input data and the output data of systems, respectively.u(t)andy(t)are the corresponding continuous input data and output data, respectively.v(t) is uncorrelated white noise.andx(t) are the intermediate variables.

Fig.1 Dual-rate sampled Hammerstein system with dead-zone nonlinearities

According to Fig. 1, the Hammerstein system consists of a static nonlinear part and the dynamic linear part.The dynamic linear model of the controlled object can be modeled as

whereA(z) andB(z) are polynomials of the unit delay operatorz-1(z-1x(t)=x(t-1)),A(z)=a0+a1z-1+···+anaz-na,B(z)=b0+b1z-1+···+bnbz-nb.

The static nonlinear partf(·)is the function of the deadzone characteristic. The input/output relationship can be shown as Fig.2.

The functionf(·)can be described by

Fig.2 Dead-zone function of the static nonlinear part

wherem1andm2are the segment slopes of the nonlinear part;d1andd2are the dead-zone break-points,d1≥0,d2≤0.Equation(3)can also be written as

where

θf= [m1,m1d1,m2,m2d2,m1,m2]T, andhis the switching function,i.e.,

In this paper, the following transfer function is used to describe the dynamic linear part in Fig.1.

In order to have correct and sole parameters,we assumeb0=1.After that,substitute(5)into(2),and we have

Substituting(4)and(6)into(1),(1)can be rewritten as

where?(t) andθare the input information vector and the parameter vector, respectively,

The characteristic of the dual-rate sampled nonlinear systems given in Fig.1 is that the input signal is refreshed rapidly in the periodh, and the output signal is sampled slowly in the periodqh.The identification of the dual-rate sampled nonlinear system is to identify the static nonlinear module and the controlled objects by collecting sampled data{u(k),y(kq)}(k=1,2,...).Hence,replacingtwithkqin(7),(7)can be arranged as

where

Observing (4) and (8), it is obvious to find that there ave difficulties by using general identification methods. Firstly, the vector?fin (4) includes two elements:-(u(kq)-d1)h(u(kq))h(d1-u(kq)),-(u(kq)-d2)h(-u(kq))h(u(kq)-d2)with unknown parametersd1andd2,and therefore the auto-coupling parametersd1andd2cannot be directly identified.Secondly,there are mutual coupling between parameters in the hammerstein model with dead-zone non-linearities. Among the unknown parametersθf=[m1,m1d1,m2,m2d2,m1,m2]T,there are two elementsm1d1,m2d2existing in the form of product ofm1andd1,m2andd2, hence the identification results ofd1,m1andd2,m2are related.Finally,xanduare unmeasurable as intermediate variables.Therefore,our paper uses the output of the auxiliary model to be the estimates of intermediate variables,and then identify parameters by using the recursive least squared algorithm.

3.Description of identification algorithm

To deal with the identification problem,define the following cost function as

The outputs of auxiliary models are described as

Thus, the recursive least squared estimation algorithm can be given as

According to the estimation values ofwe can get

The whole identification algorithm is summarized as

4.Convergence analysis

To understand convergence analysis of the proposed algorithm,some mathematical notations are given.‖X‖is the norm of matrixX,it is defined as

Inmeans the identity matrix of the ordernbyn;ξmax[X] andξmin[X] are maximum and minimum singular values of the matrixX,respectively.|X|= det[X]is the determinant of one matrix.f(k) =o(g(k)) means thatg(k)＞0 andmeans thatg(k)＞0 and there exists constantsε1andk0to satisfy|f(k)|≤ε1|g(k)|,k≥k0, whereε1＞0. Let

Suppose{v(kq),Fkq}is the martingale-difference sequence in the probability space[Ω,F,P], where{Fkq}is theσalgebra sequence generated from all measurable data till timekq.The noise sequence{v(kq)}satisfies the following conditions:

After assuming statistical conditions in (21) and (22)of the noise signal{v(kq)}, the following lemma can be given.

Lemma 1For the proposed algorithm in (10)–(20),the following inequalities hold:

ProofAccording to the definition ofP(kq)in(12),we can have

Take the determinant on both sides of the above equation, and use the determinant relationship|Im+DE|=|In+ED|,D ∈Rm×n,E ∈Rn×m,then

Therefore,

(i) Replacingkwithiin the above equation and then summing fromi= 1 toi=kon both sides of the above equation,we can get

(ii) Divide both sides of (24) by [ln|P-1(kq)|]c, and sum fromk=1 tok=∞on both sides of the above equation.Thus,

Theorem 1For the proposed algorithm in(10)–(20),suppose (A1) and (A2) in (21) and (22) hold. There exist constantsc1,c2,c3,c4and the integerk0such that the persistent excitation condition satisfies the following condition whenk≥k0,i.e.,

Then the parameter estimation errorconverges to zero.

ProofFirstly, define the following parameter estimation vector

Substituting(7)and(10)into the above equation,we can get

Define the following positive definite function

Substituting(7), (23)and(26)into the above equation,we have

Besides,the following inequality holds:

Using the above inequality,(28)can be written as

Using(23),we can get

Once persistent excitation condition(A3)holds, condition(A4)also holds, andc1≤c3,c2≥c4. Using condition(A3), the above equation is calculated iteratively and is written as

Thus,

Using the conclusion(1)in Lemma 1,we can get

Taking the condition expectation on both sides of (27),the following inequality holds:

Using

we can get

Taking the extremum of the above inequality,we get

5.Simulation

In this paper, we use an actual Hammerstein system with dead-zone nonlinearity as an simulation example. The function of dead-zone nonlinearity is described as follows:

The linear part model is shown as

where

v(k) is white noise with zero mean and different variances,σ2=0.12,σ2=0.152,σ2=0.252.

For the model of the simulation example, the real parameter vector is shown as follows:

In the process of the simulation, the input signalu(k)is a stochastic signal sequence which is independent, uncorrelated, evenly distributed with zero mean and unit variance. The noise-to-signal ratio values of the system areδns= 13.135 3%,δns= 32.403 8%,δns=54.768 8%, respectively. The parameter estimation error is.We use the proposed algorithm to identify simulated example.

The estimation results are shown in Tables 1–3,and the changing diagram of relative parameter estimation errors is shown as Fig. 3. Observing these tables and the figure,it can be obviously found that the estimation results are close to the true values.At the same data length,it is obviously shown by the three sets of identification results corresponding to the three systems noise variances that with the increasing noise variance the parameter estimation error becomes bigger.Furthermore,observing the identification results from each set respectively,it can be found that the parameter estimation errorsδdecrease when increasing the data lengtht.

Fig.3 Parameter estimation errors diagram with different variances

Table 1 Estimated parameters and errors by the proposed algorithm(σ2 =0.12,δns =13.135 3%)

Table 2 Estimated parameters and errors by the proposed algorithm(σ2 =0.152,δns =32.403 8%)

Table 3 Estimated parameters and errors by the proposed algorithm(σ2 =0.252,δns =54.768 8%)

6.Conclusions

Nonlinear systems have no certain, common models to depict until now. Nonlinearities do not just include deadzone characteristic. Their uncertainty and complexity increase the difficulty of research.In our paper,the auxiliary model idea and the recursive least squared algorithm are combined to deal with the difficulty.For algorithm convergence,the martingale convergence theorem is employed to prove convergence performance of the proposed method.It can only apply to dual-rate sampled Hammerstein system identification with the dead-zone characteristic.Therefore,it is worth for further research about the identification issues of the dual-rate sampled nonlinear system with different nonlinearities.