State Estimation Moving Window Gradient Iterative Algorithm for Bilinear Systems Using the Continuous Mixed p-norm Technique

1  Introduction

Although many physical dynamic behaviors are characterized as linear systems in the neighborhood of a single operating point, when they exhibit strong nonlinearities or must be described over the entire operating range, linear models may not yield appropriate results [1–3]. Owing to that, the study of system identification and parameter estimation for nonlinear systems has drawn considerable attention of academic researchers [4–6]. Various models are exploited to describe actual nonlinear systems with relatively simple structures [7–11] such as Hammerstein systems, Bilinear systems, and Wiener systems.

The parameter estimation of the system models is important for control system analysis and design. The parameters of the models can be estimated by using some identification methods [12–15] such as the hierarchical algorithms [16–19]. In this paper, we confine our discussion to bilinear systems, which are simplicity in model structures and capable of approaching arbitrary nonlinear systems with much higher accuracy than traditional linear approximations theoretically [20]. Bruni et al. designed a bilinear time-series model-based self-tuning controller for a multi-machine power system to enhance the region of stability of the system, and return the states to their stable equilibrium [21]. Yeo et al. developed the bilinear model predictive control algorithm for paper plant systems to control the grade change operations in paper production [22]. Wang et al. simulated the bridge nonlinear boundary as bilinear translational, and applied the nonlinear least squares optimization algorithm to identify the nonlinear translational and rotational boundary parameters of the bridge [23].

Many researches have been studied in the parameter estimation of bilinear state space systems. A stochastic gradient algorithm and a gradient-based iterative algorithm have been proposed for the parameter identification of bilinear systems by using the auxiliary model. The gradient-based iterative algorithm uses fixed batch data to update the parameters, so that the parameter estimation accuracy can be greatly improved compared with the stochastic gradient algorithm. Li et al. combined the maximum likelihood theory with the data filtering technique for bilinear systems with colored noises. The state observer is vital in the field of state estimation of bilinear systems [24]. Tsai considered a H-infinity fuzzy observer for bilinear systems by means of a linear matrix inequality approach [25]. Phan et al. presented a full-order bilinear state observer for the bilinear system, which optimized the observer gain by interaction matrices [26]. Zhang et al. considered the state estimation problem of bilinear systems and proposed a state filtering method for the single-input single-output bilinear systems and multiple-input multiple-output bilinear systems by minimizing the covariance matrix of the state estimation errors [27]. Recently, some state and parameter estimation methods have been proposed for linear and bilinear state space systems in the presence of exogenous noises [28–30].

However, in industrial applications, the dynamical processes often work on various noise environments, and some works used nonlinear filtering technique such as median filtering and least mean p-norm to overcome the problems. Zayyani proposed a mixed-norm adaptive filter algorithm for systems identification based on minimization of the logarithmic continuous mixed p-norm [31]. Moreover, an improved mixed p-norm algorithm had been studied to combat non-Gaussian interference and its computational complexity as well as the mean convergence had also been analyzed [32,33].

Inspired by the above researches, we study the parameter estimation algorithms for bilinear state space models with the noise disturbances. Based on the iterative search and the state estimator, a bilinear state continues mixed p-norm gradient iteration (BSO-CMPN-GI) is proposed. The proposed algorithms control the proportions of the error norms and offer an extra degree of freedom within the adaptation. By taking into consideration each p-norm of errors for 1≤p≤2, the proposed algorithms combine the benefits of the variable error norms and thus are more robust against noise interference. Furthermore, the moving data window theory is introduced to improve the effectiveness of the proposed algorithms. The contributions of this paper are summarized as follows:

•   A state observer is designed to obtain the system states variables consisting of the product terms of state variables and control variables.

•   A bilinear state observer-based continues mixed p-norm gradient iteration (BSO-CMPN-GI) algorithm is presented for the bilinear state space system under the noise interference to estimate the unknown system parameters.

•   A bilinear state observer-based moving window CMPN-GI (BSO-MW-CMPN-GI) algorithm is derived to update collected data and thus maintain high data utilization.

The outlines of this paper are organized as follows. Section 2 introduces some definitions and proposes the identification model of the bilinear state space system and introduces a bilinear state observer for the state estimation. Sections 3 and 4 derive a BSO-CMPN-GI and a MW-CMPN-GI algorithms based on the bilinear state estimator, respectively. An example to illustrate the effectiveness of the proposed algorithms in Section 5. Finally, Section 6 gives some concluding remarks.

2  System Description and Identification Model

Consider the following single-input single-output bilinear state space modelblue:

xt+1=Axt+Bxtut+gut+wt,(1)

yt=cxt+vt,(2)

where ut∈Rn is the system input variables, yt∈Rn is the measurement output variables, xt is the state vector, wt is an uncorrelated process noise with zero mean, and vt is an uncorrelated measurement noise with zero mean. A∈Rn×n, B∈Rn×n, g∈Rn and c∈R1×n are system parameter matrix and parameter vector. Consider the observer canonical form of bilinear systems, the system parameters can be expressed as

A:=[−a110⋯0−a2010⋮⋮⋱⋮−an−10⋯01−an0⋯00]∈Rn×n,B:=[b1b2⋮bn−1bn]∈Rn×n,g:=[g1g2⋮gn−1gn]∈Rn,c:=[1,0,⋯,0,0]∈R1×n,(3)

where bj:=[bj1,bj2,…,bjn]∈R1×n,j=1,2,…,n.

Assumption 1: The dimension n of the system state vector is known, ut=0, xt=0, yt=0, wt=0 and vt=0 for t≤0.

Assumption 2: The bilinear system in Eqs. (1) and (2) is observable and controllable.

Eq. (1) can be written as

{x1,t+1=−a1x1,t+x2,t+b1xtut+g1ut+w1,t,x2,t+1=−a2x1,t+x3,t+b2xtut+g2ut+w2,t,⋮xn−1,t+1=−an−1x1,t+xn,t+bn−1xtut+gn−1ut+wn−1,t,xn,t+1=−anx1,t+bnxt+gnut++wn,t.(4)

Adding both sides of the above equations has

x1,t+1=−a1x1,t−a2x1,t−1−⋯−anxt−n+1+b1xtut+b2xt−1ut−1+⋯+bnxt−n+1ut−n+1+g1ut+g2ut−1+⋯+gnut−n+1+w1£s¬t+w2,t−1+⋯+wn,t−n+1=−∑j=1najx1,t−j+1+∑j=1nbjxt−j+1ut−j+1+∑j=1ngjut−j+1+∑j=1nwj,t−j+1.(5)

Substituting Eq. (5) into Eq. (2) obtains the regression form of the bilinear state model

yt=−∑j=1najx1,t−j+∑j=1nbjxt−jut−j+∑j=1ngjut−j+∑j=1nwj,t−j+vt.(6)

Define the information vector ψt and parameter vector ϑ as

ψt:=[dt,kt,ht]T∈Rn2+2n,(7)

dt:=[−x1,t−1,−x1,t−2,…,−x1,t−n]T∈Rn,(8)

kt:=[xt−1Tut−1,xt−2Tut−2,…,xt−nTut−n]T∈Rn2,(9)

ht:=[ut−1,ut−2,…,ut−n]T∈Rn,(10)

ϑ:=[a,B,g]T∈Rn2+2n,(11)

a:=[a1,a2,…,an]T∈Rn,(12)

B:=[b1,b2,…,bn]T∈Rn2,(13)

g:=[g1,g2,…,gn]T∈Rn.(14)

Redefine the noise term as

et:=∑j=1nwj,t−j+vt.(15)

Eq. (6) can be rewritten as

yt=ψtTϑ+et.(16)

The proposed parameter estimation algorithms in this paper are based on this identification model in (16). Many identification methods are derived based on the identification models of the systems [34–43] and these methods can be used to estimate the parameters of other linear systems and nonlinear systems [44–50] and can be applied to other fields such as chemical process control systems. The purpose of this paper is to obtain the estimates of the unknown parameters in the bilinear state space model by means of the measurement data {ut,yt}. However, the state space model contains not only unknown parameters, but also unknown state variables. That means the information vector ψt in Eq. (16) contains the unknown state vector xt−i. Given this, we adopt the state observer for estimating the unknown states variables [51]. According to the bilinear state space model, the state observer is designed as

x^t+1=A^x^t+B^x^tut+g^ut,(17)

where

A^=[−a^110⋯0−a^2010⋮⋮⋱⋮−a^n−10⋯01−a^n0⋯00]∈Rn×n,B^=[b^1b^2⋮b^n−1b^n]∈Rn×n,g^=[g^1g^2⋮g^n−1g^n]∈Rn.

Remark 1: Differently from the identification of bilinear-in-parameter systems which involve the product terms of the parameter vectors and the information matrices, this paper studies the identification problem of bilinear state-space systems with the product terms of state variables and control variables, which makes it difficult for the parameter and state estimation.

3  Bilinear State Observer Based Continuous Mixed p-norm Gradient Iterative Identification Algorithm

Set the data length be L, based on the input and output data {ut,yt,1⩽t⩽L}, define the stacked input vector YL and the stacked information matrix ΨL as

YL:=[yLyL−1⋮yt⋮y1]∈RL,ΨL:=[ψLTψL−1T⋮ψtT⋮ψ1T]∈RL×(n2+2n).

To suppress the effect of noise interference, define the continuous mixed p-norm cost function

J1(ϑ):=∫12λ(p)|VL|pdp,(18)

where λ(p) is the probability density-like weighting function which is constrained by ∫12λt(p)dp=1, and VL:=YL−ΨLϑ.

Using the negative gradient search method to minimize the cost function J1(ϑ) with respect to ϑ gives

grad[J1(ϑ)]=∂J1(ϑ)∂ϑ=[∂J1(ϑ)∂a∂J1(ϑ)∂B∂J1(ϑ)∂g](19)

where

∂J1(ϑ)∂a=∫12pλ(p)|VL|p−1sgn(VL)∂(VL)∂adp=−∫12pλ(p)|VL|p−1sgn(VL)dp[dL,dL−1,…,dt,…,d1]T,∂J1(ϑ)∂B=−∫12pλ(p)|VL|p−1sgn(VL)dp[kL,kL−1,…,kt,…,k1]T,∂J1(ϑ)∂g=−∫12pλ(p)|VL|p−1sgn(VL)dp[hL,hL−1,…,ht,…,h1]T.

Therefore, Eq. (19) can be expressed as

grad[J1(ϑ)]=−∫12pλ(p)|VL|p−1sgn(VL)ΨLdp=−ΓLsgn(VL)ΨL,

where

ΓL=∫12pλ(p)|VL|p−1dp.(20)

Let s=1,2,3,… be the iteration variables, ϑ^s be the parameter estimation vector at iteration s. Using the negative gradient search and minimizing J1(ϑ) yield

ϑ^s=ϑ^s−1−μsgrad[J1(ϑ^s−1)]=ϑ^s−1+μsΓLsgn(VL)ΨL,(21)

where μs is the iteration step size [52–57], which satisfies

0<μs⩽2λmax[ΨLTΨL].(22)

Eqs. (21) and (22) cannot figure out the parameter estimation vector ϑ^s, because the information matrix ΨL contains the unknown states xt−j. The solution is to replace the unknown states xt−j in the information matrix with their estimates obtained by the state observer at the previous iteration. Define the estimated information matrix as

Ψ^Ls:=[(ψ^Ls)T,(ψ^L−1s)T,…,(ψ^ts)T,…,(ψ^1s)T]T∈RL×(n2+2n),ψ^ts=[−x^1,t−1s−1,…,−x^1,t−ns−1,(x^t−1s−1)Tut−1,…,(x^t−ns−1)Tut−n,ut−1,…,ut−n]T∈Rn2+2n,

where x^t−js−1 is given by

x^t−js−1=A^s−1x^t−j−1s−1+B^s−1x^t−j−1s−1ut−j−1+g^s−1ut−j−1,j=1,2,…,L.

Replace the information matrix ΨL in Eqs. (21) and (22) with its estimated value Ψ^L, the estimate V^Ls at iteration s is YL−Ψ^Lsϑ^s−1. Replacing VLs in (20) with its estimates gives

Γ^Ls=∫12pλ(p)|V^Ls|p−1dp.(23)

To obtain a closed form formula for Γ^Ls, a uniform weighting function λ(p)=1 is assumed. Then, the bilinear state observe-based continuous mixed p-norm gradient iterative (BSO-CMPN-GI) identification algorithm (24)–(30) for the bilinear system is summarized in the following:

ϑ^s=ϑ^s−1+μsΓ^Lssgn(V^Ls)Ψ^Ls,(24)

0<μs⩽2λmax[(Ψ^Ls)TΨ^Ls],(25)

Γ^Ls=(2|V^Ls|−1)ln⁡(|V^Ls|)−|V^Ls|+1ln2(|V^Ls|),(26)

V^Ls=YL−Ψ^Lsϑ^s−1,(27)

Ψ^Ls=[(ψ^Ls)T,(ψ^L−1s)T,…,(ψ^ts)T,…,(ψ^1s)T]T,(28)

ψ^ts=[−x^1,t−1s−1,…,−x^1,t−ns−1,(x^t−1s−1)Tut−1,…,(x^t−ns−1)Tut−n,ut−1,…,ut−n]T,(29)

x^t−js=A^sx^t−j−1s+B^sx^t−j−1sut−j−1+g^sut−j−1.(30)

The flowchart of the BSO-CMPN-GI algorithm in Eqs. (24)–(30) is shown in Fig. 1.

Figure 1: The flowchart of the BSO-CMPN-GI algorithm

Remark 2: The parameter p in BSO-CMPN-GI is adapted by continuous p-norm without resorting to a priori knowledge of the noise.

Remark 3: The BSO-CMPN-GI algorithm makes full use of the measurement data in each iteration of the calculation process, but it requires a batch of data to be collected in advance, and thus is implemented offline. Therefore, an on-line identification algorithm derived from the BSO-CMPN-GI algorithm will be introduced by exploiting the past and current measurement data to estimate the unknown parameters in Section 4.

4  Bilinear State Observe Moving Window Continuous Mixed p-norm Gradient Iterative Algorithm

In this section, we introduce the moving window method to derive an on-line identification algorithm and enhance the performance of the BSO-CMPN-GI algorithm. The length of the moving window is set as a fixed value m. Define the stacked output vector Yt,m and the stacked information matrix Ψt,m as

Yt,m:=[ytyt−1⋮yt−m+1]∈Rh,Ψt,m:=[ψtTψt−1T⋮ψt−m+1T]∈Rh×(n2+2n).

Consider the measurements from t−m+1 to t and define the cost function

J2(ϑ):=∫12λt(p)|Vt,m|pdp,(31)

where

Vt,m=Yt,m−Ψt,mϑ(32)

=[etet−1⋮et−m+1]=[yt−dtTa−ktTB−htTgyt−1−dt−1Ta−kt−1TB−ht−1Tg⋮yt−m+1−dt−m+1Ta−kt−m+1TB−ht−m+1Tg],(33)

Taking the gradient of J2(ϑ) gives

grad[J2(ϑ)]=∂J2(ϑ)∂ϑ=[∂J2(ϑ)∂a∂J2(ϑ)∂B∂J2(ϑ)∂g],(34)

where

∂J2(ϑ)∂a=∫12pλt(p)|Vt,m|p−1sgn(Vt,m)∂(Vt,m)∂adp=−∫12pλt(p)|Vt,m|p−1sgn(Vt,m)dp[dt,dt−1,dt−m+1]T,∂J2(ϑ)∂B=−∫12pλt(p)|Vt,m|p−1sgn(Vt,m)dp[kt,kt−1,…,kt−m+1]T,∂J2(ϑ)∂g=−∫12pλt(p)|Vt,m|p−1sgn(Vt,m)dp[ht,ht−1,…,ht−m+1]T.

Therefore, Eq. (4) can be expressed as

grad[J2(ϑ)]=−∫12pλt(p)|Vt,m|p−1sgn(Vt,m)Ψt,mdp=−Γt,msgn(Vt,m)Ψt,m,

where

Γt,m=∫12p|Vt,m|p−1dp.(35)

Let s=1,2,3,… be iteration variables. Similar to the derivation of the BSO-CMPN-GI algorithm, using the negative gradient search method and minimizing J2(ϑ) get

ϑ^ts=ϑ^ts−1−μtsgrad[J2(ϑ^ts−1)]=ϑ^ts−1+μtsΓt,msgn(Vt,m)Ψt,m,(36)

where

0<μts⩽2λmax[Ψt,mTΨt,m].(37)

As pointed earlier in Section 3, the unknown states xt−j in the information matrix Ψt,m are replaced with their estimates. The estimated information matrix Ψ^t,ms is redefined as

Ψ^t,ms:=[(ψ^ts)T,(ψ^t−1s)T,…,(ψ^t−m+1s)T]∈RL×(n2+2n)ψ^ts=[−x^1,t−1s−1,…,−x^1,t−ns−1,(x^t−1s−1)Tut−1,…,(x^t−ns−1)Tut−n,ut−1,…,ut−n]T∈R(n2+2n).

where xt−js−1 is estimated by the state observer:

x^t−js−1=A^ts−1x^t−j−1s−1+B^ts−1x^t−j−1s−1ut−j−1+g^ts−1ut−j−1.(38)

Based on the above derivation, a bilinear state observe-based moving window continuous mixed p-norm gradient iterative (BSO-MW-CMPN-GI) algorithm proposed for the bilinear state space system is summarized as follows:

ϑ^ts=ϑ^ts−1+μtsΓt,mssgn(Vt,ms)Ψ^t,ms,(39)

0<μts⩽2λmax[(Ψ^t,ms)TΨ^t,ms],(40)

Γ^t,ms=(2|V^t,ms|−1)ln⁡(|V^t,ms|)−|V^t,ms|+1ln2(|V^t,ms|),(41)

V^t,ms=Yt,m−Ψ^t,msϑ^ts−1,(42)

Ψ^t,ms:=[(ψ^ts)T,(ψ^t−1s)T,…,(ψ^t−m+1s)T],(43)

ψ^ts=[−x^1,t−1s−1,…,−x^1,t−ns−1,(x^t−1s−1)Tut−1,…,(x^t−ns−1)Tut−n,ut−1,…,ut−n]T,(44)

x^t−js=A^tsx^t−j−1s+B^tsx^t−j−1sut−j−1+g^tsut−j−1.(45)

The flowchart of the BSO-MW-CMPN-GI algorithm in Eqs. (39)–(45) is shown in Fig. 2.

Figure 2: The flowchart of the BSO-MW-CMPN-GI algorithm for computing ϑ^ts

Remark 4: The parameter estimate given by the BSO-MW-CMPN-GI algorithm depends only on the iterative counter s, but also time t. As the sampling time t increases, the BSO-MW-CMPN-GI algorithm can utilize a batch of data to calculate the parameter estimate simultaneously.

5  Simulation

Case 1: About parameter estimation

Consider a second-order bilinear state space system:

xt+1=[−0.801−0.450]xt+[0.040.150.250.18]xtut+[1.271.16]ut+wt,yt=[1,0]xt+vt.

The parameter vector to be estimated is

ϑ=[0.80,0.45,0.04,0.15,0.25,0.18,1.27,1.16]T

In simulation, the input {ut} is taken as an uncorrelated uniform disturbance random signal sequence with zero mean and unit variance, {wt} is taken as an uncorrelated process white noise vector sequence with zero mean and variance Q=[σw2,0;0,σw2], σw2=0.022, {vt} is taken as a white noise vector sequence with zero mean and variance σv2=1.002 and σv2=1.502, respectively. Apply the BSO-CMPN-GI algorithm in Eqs. (24)–(30) and the BSO-MW-CMPN-GI algorithm in Eqs. (39)–(45) with the data length L=500 to estimate the states xt and parameters ϑ of this bilinear system, respectively. For comparison with the different algorithm, we introduce the bilinear state observe-based gradient iterative (BSO-GI) algorithm. In the BSO-MW-CMPN-GI algorithm, s=15 represents the iteration variable, and m=60 represents the recursive variable. The parameter estimates and errors δ:=‖ϑ^t−ϑ‖/‖ϑ‖ vs. s of the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm with different noise variances are shown in Tables 1 and 2. The parameter estimation errors δ vs. s of the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm with different noise variances are shown in Figs. 3–6, respectively. The estimated output (EO) and the actual output (AO) of the four algorithms are shown in Figs. 7 and 8. From Tables 1 and 2 and Figs. 3–8, the following conclusions can be drawn:

•   The parameter estimation errors given by the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm become smaller as the iteration s increases. It thus to say the proposed algorithms are effective for bilinear systems.

•   The state estimates are close to their true values with t increasing.

•   Under the same data length, a lower noise variance leads to higher parameter estimation accuracy by the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm.

•   The BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm possess higher parameter estimation accuracy at the same noise variance compared with the BSO-GI algorithm.

•   When comparing the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm, the parameter estimation errors of the BSO-MW-CMPN-GI algorithm become smaller with m increasing, and approach to zero if m is large enough.

Figure 3: The estimation errors δ vs. t with σ2=1.002

Figure 4: The BSO-MW-CMPN-GI estimation errors δ vs. m and s (m=60 and s=15) with σ2=1.002

Figure 5: The estimation errors δ vs. t with σ2=1.502

Figure 6: The BSO-MW-CMPN-GI estimation errors δ vs. m and s (m=60 and s=15) with σ2=1.502

Figure 7: State x1,t and the estimated state x^1,t vs. t

Figure 8: State x2,t and the estimated state x^2,t vs. t

Case 2: About model validation

Applying the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm to construct the estimated model for the model validation, respectively. Take the data from t=1001 to t=2000 to calculate the root mean square errors (RMSEs) of the predicted output y^t. Using the BSO-CMPN-GI and BSO-MW-CMPN-GI estimates in Table 1 with the noise variance σv2=1.002. The actual output (AO) yt and the predicted output (PO) y^t are plotted in Fig. 9. The RMSEs of the BSO-GI, BSO-CMPN-GI and BSO-MW-CMPN-GI are

δ1=1Lt∑j=10012000[yj−y^BSO−GIj]2=1.0223,δ2=1Lt∑j=10012000[yj−y^BSO−CMPN−GIj]2=1.0187,δ3=1Lt∑j=10012000[yj−y^BSO−MW−CMPN−GIj]2=1.0127,

where y^BSO - GIj, y^BSO - CMPN - GIj and y^BSO - MW - CMPN - GIj represent the predicted output of the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm, respectively.

From Fig. 9, we can see that the predicted outputs of the BSO-CMPN-GI and the BSO-MW-CMPN-GI are very close to the true outputs, and the RMSEs of the two algorithms are very close to the noise standard deviation σv2=1.002. In other words, the estimated model can capture the dynamics of the system.

Figure 9: The system actual outputs and the predicted outputs vs. t with σv2=1.002

6  Conclusions

This paper studies the parameter identification problems of the nonlinear systems described by the bilinear state space models with the noise disturbances. A bilinear state observe-based continuous mixed p-norm gradient iterative algorithm and a bilinear state observe-based moving window continuous mixed p-norm gradient iterative algorithm are proposed to estimate the parameters of the bilinear system. The proposed optimal algorithms are robustness for stochastic white noise by means of the continuously mixed p-norms theory. In order to improve the performance of the proposed algorithms, the moving window identification theory is introduced into the proposed algorithm. Although the proposed algorithms are effective for identifying the bilinear system, it also has some limitations. For example, the disturbance noise is confined to the white noise. The proposed model parameter estimation methods in the paper can combine some mathematical strategies [58–60] and other estimation algorithms [61–65] to study the parameter identification problems of linear and nonlinear systems with different disturbances [66–68] and can be applied to other fields [69–74] such as engineering application systems.

Funding Statement: This research was funded by the National Natural Science Foundation of China (No. 61773182) and the 111 Project (B12018).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.