The Bias Compensation Based Parameter and State Estimation for Observability Canonical State-Space Models with Colored Noise

Abstract: This paper develops a bias compensation-based parameter and state estimation algorithm for the observability canonical state-space system corrupted by colored noise. The state-space system is transformed into a linear regressive model by eliminating the state variables. Based on the determination of the noise variance and noise model, a bias correction term is added into the least squares estimate, and the system parameters and states are computed interactively. The proposed algorithm can generate the unbiased parameter estimate. Two illustrative examples are given to show the effectiveness of the proposed algorithm.


Introduction
Mathematical models play a great role in adaptive control, online prediction and signal modeling [1][2][3][4].The system models of physical plants can be constructed from first principle, but this method cannot work well when the physical mechanism of the true plants is not very clear [5,6].Thus, data-driven-based black-box or gray-box state-space modeling can be used to approximate the law of motion [7][8][9].The advantages of state-space modeling lie in that the model maps the relationship from inputs to outputs, and the states reflect the internal dynamic behavior of the plants.The estimation of state-space models may include both the unknown system parameters and the unmeasurable states [10,11].It is well known that the Kalman filtering method captures the optimal state estimator based on the measurement data [12][13][14].However, the model parameters of the system are generally assumed to be known in advance, and the statistical characteristics of measurement noise and process noise have to be chosen, which are hard to satisfy in practice [15][16][17].
On the identification of state-space models, the classical identification methods are the prediction-error methods, which require adequate knowledge of the system structures and parameters [18,19], and the subspace identification methods, which ignore the consistency of system parameter estimates [20][21][22].Since the system states and parameters are involved in the state-space model, the simultaneous estimation of them is a feasible choice [23,24].In this aspect, Pavelková and Kárný investigated the state and parameter estimation problem of the state-space model with bounded noise, providing a maximum estimator based on Bayesian theory [25].By transforming the single-input multiple-output Hammerstein state-space model into the multivariable one, Ma et al. used the Kalman smoother to estimate the system states and the expectation maximization algorithm to compute parameter estimates [26].Li et al. presented a maximum likelihood-based identification algorithm for the Wiener nonlinear system with a linear state-space subsystem [27].Wang and Liu applied the recursive least squares algorithm to non-uniformly sampled state-space systems with the states being available and combined the singular value decomposition with the hierarchical identification principle for the identification of the system with the states being unavailable [28].These studies were mainly focused on the state-space system with white noise.
In industrial processes, the observation data are often corrupted by colored noises [29][30][31].As we have known, the parameter estimates generated by the recursive least squares algorithm for the system with colored noise are biased.To obtain the unbiased estimates, the bias compensation method was proposed [32].The basic idea is to compensate the biased least squares estimate by adding a correction term [33][34][35].Zhang investigated the parameter estimation of the multiple-input single-output linear system with colored noise and introduced a stable prefilter to preprocess the input data for the purpose of obtaining the unbiased estimate [36].Zheng extended the bias compensation method to the errors-in-variables output-error system, where the input was contaminated by white noise and the output errors were modeled by colored noise [37].On the identification of state-space models with colored noise, Wang et al. applied the Kalman filter algorithm to estimate the system states and developed a filtering-based recursive least squares algorithm for observer canonical state-space systems with colored noise [38].On the basis of the work in [38], this paper discusses the state and parameter identification of the state-space model with colored noise and presents a bias compensation-based identification algorithm for jointly estimating the system parameters and states based on the bias compensation.The main contributions of the paper are as follows.

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By using the bias compensation, this paper derives the identification model and achieves the unbiased parameter estimation for observability canonical state-space models with colored noise.

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By employing the interactive identification, this paper explores the relationship between the noise parameters and variance and the bias correction term and realizes the simultaneous estimation of the system parameters, noise parameters and system states.
The rest of this paper is organized as follows.Section 2 demonstrates the problem formulation about the observability canonical state-space system and derives the identification model.Section 3 develops a bias compensation-based parameter and state estimation algorithm.Section 4 provides two illustrative examples to show that the proposed algorithm is effective.Finally, Section 5 offers some concluding remarks.

Problem Description and Identification Model
For narrative convenience, let us introduce some notation.The nomenclature is displayed in Table 1.Consider the following state-space system with colored noise, where x(k T ∈ R n is the state vector, u(k) ∈ R is the system input, y(k) ∈ R is the system output, e(k) ∈ R is a random noise and G ∈ R n×n , h ∈ R n and f ∈ R 1×n are the system parameter matrix and vectors, defined as: The external disturbance e(k) can be fitted by a moving average process, an autoregressive process or an autoregressive moving average process.Without loss of generality, we consider e(k) as a moving average noise process: where v(k) is the white noise with zero mean and variance δ.Assume that y(k) = 0, u(k) = 0 and v(k) = 0 for k 0. The objective is to estimate the parameters g i , h i and e i and the system state x(k) from the available input-output data {u(k), y(k) Note that the system in Equations ( 1) and ( 2) is an observability canonical form, and the observability matrix T is an identity matrix, i.e., From Equations ( 1) and (2), we have: Define the parameter matrix M and the information vectors φ y (k), φ u (k) and φ e (k) as: According to Equations ( 5)-( 8), we have: Equation ( 10) can be rewritten as: Define the parameter vectors ϑ, ϑ g and ϑ h and the information vectors φ s (k) and φ n (k) as: Inserting ( 11) into (9) yields: Define the intermediate variable: Replacing k + n in Equation ( 12) with k gives: Equation ( 14) or (15) is the system identification model of the state-space system in Equation ( 1), where the information vector φ s (k) is composed of the observed data.Define the parameter vector ϑ v and the information vector φ v (k) as: From Equation (3), we have: Thus, we obtain the noise identification model in Equation ( 16).The following derives the bias compensation-based identification algorithm based on the system model in Equation ( 15) and the noise model in Equation ( 16).

The Bias Compensation-Based Parameter and State Estimation Algorithm
The algorithm includes two parts: the parameter estimation algorithm and the state estimation algorithm.Two parts are implemented in an interactive way.

The Parameter Estimation Algorithm
According to the identification model in Equation (15), define the cost function: Using the least squares search and minimizing J(ϑ) give the least squares estimate θLS (k) of the parameter vector ϑ: Inserting ( 14) into (17) and using ( 13), we have: Obviously, the least squares estimate θLS (k) in Equation ( 18) is a biased estimate since e(k) is correlated noise.Equation ( 18) can be rewritten as: Dividing by k and taking limits on both sides give: Note that e(k) in Equation ( 19) is the moving average noise, and v(k) is white noise with zero mean and variance δ and is independent of the inputs.From Equation (3), we have: where R(i) is the autocorrelation function of the noise e(k) and R(i) = 0 when i > n e .Define the autocorrelation function vectors r and ζ and the autocorrelation function matrices R, Λ and Q as: In fact, Λ is a Toeplitz matrix consisting of n autocorrelation functions.Equation ( 19) can be rewritten as: or: It can be seen from Equation ( 25) that the bias of the least squares estimate θLS (k) can be eliminated by adding a compensation term ∆ϑ(k) := kP(k)(Rϑ − r).That is, the estimate θC (k) := θLS (k) + ∆ϑ(k) is an unbiased estimate of the true parameter vector ϑ.Thus, the unbiased estimate θC (k) can be computed by the following recursive expressions, Equation (26) shows that the unbiased estimate θC (k) is related to the estimates of R and r (i.e., the noise variance δ and the noise parameter vector ϑ v ).The following derives their estimates based on the interactive identification.
Let the least squares residual ε LS (i) := y(i) − φ T s (i) θLS (k).Using Equation ( 14) and the relation Noting that R is a symmetric matrix, and using Equations ( 20)-( 24), we have: Thus, we have: Let J(k) := ∑ k i=1 ε 2 LS (i) be the cost function at time k, Q(k) and ζ(k) be the estimates of Q and ζ at instant k, respectively.Then, the estimate of the noise variance δ can be computed by: Note that Equations ( 30)-( 34) involve the estimate θv (k) of the noise parameter vector ϑ v , which can be computed by the noise model in Equation (16).Let ê(k) and v(k) be the estimates of e(k) and v(k), respectively.Define the noise information vectors: From Equations ( 14) and ( 16), we have: Using the least squares principle, the estimate θv (k) of ϑ v in Equation ( 16) can be computed by: Thus, Equations ( 26)-( 40) form the bias compensation-based parameter estimation (BC-PE) algorithm for identifying the parameter vector ϑ.The BC-PE algorithm is an interactive estimation process: we can compute the estimates θv (k) and δ(k); with these obtained variable estimates, we derive the estimate of R and r and then update the unbiased estimate θC (k) using Equations ( 26)-(29).

The State Estimation Algorithm
When the system parameter estimation and the noise estimation ê(k) are obtained from the BC-PE algorithm, we can estimate the system state x(t) using Equation (11).
Post-multiplying Equation ( 4) by h and letting the elements of each row be equal, we have: Then, the parameter matrix M can be expressed as: Post-multiplying Equation ( 4) by G and letting the elements of n-th row be equal, we have: According to the definitions of ϑ g and ϑ h , we have: Once the estimate of the parameter vector ϑ is computed by the BC-PE algorithm in Equations ( 26)-( 40), we can extract the estimates θg (k) and θh (k) of the parameter vectors ϑ g and ϑ h .Then, the estimates ĝi (k) and ĥi (k) can be computed by: From Equation ( 11), we have: Replacing M and φ e (k) with their corresponding estimates M(k) and φe (k), we have: Equations ( 41)-( 47) form the state estimation algorithm for the state-space system in Equation ( 1).
By interactively implementing the parameter estimation algorithm in Equations ( 26)-( 40) and the state estimation algorithm in Equations ( 41)-( 47), the estimation of the system parameter vector θ and the state x(k) can be obtained.
The steps for implementing the bias compensation-based parameter and state estimation (BC-PSE) algorithm in Equations ( 26) and (47) for state-space systems with colored noise are listed as follows.
Compute the state estimate x(k) using Equation ( .If k < L, increase k by one, and go to Step 2; otherwise, stop, and obtain the parameter estimation vector θ(L).

Examples
Example 1.Consider the following state-space system with moving average noise, The parameter vector to be estimated is: In simulation, the input {u(k)} is set as a persistent excitation sequence and {v(k)} as a zero-mean noise sequence with variance δ = 0.25.Take the data length L = 1200.Use the first 1000 data and apply the BC-PSE algorithm to get the estimates of the parameter vector θ and the system state x(k).The parameter estimates and their errors under different noise variances are displayed in Table 2 and Figure 1, where the estimation error τ := θ(k) − θ / θ × 100%.The parameter estimate θ(k) versus k is plotted in Figure 2, and the state estimate x(k) computed by the BC-PSE algorithm is illustrated in Figures 3 and 4.
The remaining 200 data from k = 1001-1200 are taken to test the effectiveness of the estimated model.As a comparison, the curves of the estimated output ŷ(k) and the predicted output y(k) are depicted in Figure 5.     Example 2. Consider a third-order observability canonical state-space system, The parameter vector to be estimated is: The simulation conditions are the same as those in Example 1. Table 3 and Figure 6 compare the bias compensation-based parameter estimates and errors under noise variances δ = 0.25 and δ = 1.00.Figures 7 and 8 depict the curves of the parameter estimates θ(k) versus the instant k.From Tables 2 and 3 and Figures 1-12, we can draw some conclusions as follows.

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The parameter estimation errors given by the BC-PSE algorithm become small with k increasing; see Tables 2 and 3 and Figures 1 and 6.

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A low noise variance leads to higher accuracy of parameter estimates; see Figures 1 and 6.

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As the data length k increases, the parameter estimates approach their true values; see Figures 2, 7 and 8.

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The estimated states are close to the actual states of the system model; see Figures 3, 4 and 9-11.

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The predicted outputs match well with the actual outputs; see Figures 5 and 12.

Conclusions
This paper discusses the identification of the observability canonical state-space system with colored noise via our proposed bias compensation-based parameter and state estimation algorithm.The numerical results indicate that the algorithm can effectively estimate the system states and parameters.The advantage of this algorithm is that the parameter estimates are unbiased.The algorithm can be combined with other recursive algorithms, such as the multi-innovation algorithm, to study the identification of nonlinear state space systems [39], dual-rate systems [40], signal modeling [41] and time series analysis [42,43].

Figure 1 .Figure 2 .
Figure 1.The estimation errors τ versus k with different noise variances for Example 1.

Figure 5 .
Figure 5.The actual output y(k) and the estimated output ŷ(k) versus k for Example 1.

Figure 6 .
Figure 6.The estimation errors τ versus k with different noise variances for Example 2.

Figure 12 .
Figure 12.The actual output y(k) and the estimated output ŷ(k) versus k for Example 2.

Table 1 .
The nomenclature for symbols.

Table 2 .
The bias compensation-based parameter estimates and errors for Example 1.

Table 3 .
The bias compensation-based parameter estimates and errors for Example 2.