Parameter and State Estimator for State Space Models

This paper proposes a parameter and state estimator for canonical state space systems from measured input-output data. The key is to solve the system state from the state equation and to substitute it into the output equation, eliminating the state variables, and the resulting equation contains only the system inputs and outputs, and to derive a least squares parameter identification algorithm. Furthermore, the system states are computed from the estimated parameters and the input-output data. Convergence analysis using the martingale convergence theorem indicates that the parameter estimates converge to their true values. Finally, an illustrative example is provided to show that the proposed algorithm is effective.


Introduction
Parameter estimation and identification have had important applications in system modelling, system control, and system analysis [1][2][3][4][5] and thus have received much research attention in recent decades [6][7][8][9][10][11]. Several identification methods have been developed for state space models, for example, the subspace identification methods [12]. Gibson and Ninness presented a robust maximum-likelihood estimation for fully parameterized linear time-invariant (LTI) state space models; the idea is to use the expectation maximization (EM) algorithm to estimate maximum-likelihood degrees [13]. Raghavan et al. studied the EM-based state space model identification problems with irregular output sampling [14].
The state space model includes not only the unknown parameter matrices/vectors, but also the unknown noise terms in the formation vector and unmeasurable state vector. Many algorithms can estimate the system states assuming that the system parameter matrices/vectors are available but such state estimation algorithm cannot work if the system parameters are unknown [15]. Recently, Ding presented a combined state and least squares parameter estimation algorithm for dynamic systems [16].
In the area of state space model identification, Ding and Chen proposed a hierarchical identification estimation algorithm for estimating the system parameters and states [17]. Li et al. assumed that the system states were available and used the measurable states and input-output data to estimate the parameters of lifted state space models for general dual-rate systems [18]. Recently, some identification methods have been developed, for example, the least squares methods [19,20], the gradient-based methods [21,22], the bias compensation methods [23,24], and the maximum likelihood methods [25][26][27][28][29][30]. The objective of this paper is to present a new parameter and state estimation-based residual algorithm from the given input-output data and further to analyze the convergence of the proposed algorithm.
This paper is organized as follows. Section 2 introduces the system description and its identification model paper. Section 3 derives a basic parameter identification algorithm 2 The Scientific World Journal for canonical state space systems and analyzes the performance of the proposed algorithm. Section 4 gives a state estimation algorithm. Section 5 provides an example for the proposed algorithm. Finally, concluding remarks are given in Section 6.

System Description and Identification Model
Let us introduce some notation [15]. " =: " or " := " stands for " is defined as "; the symbol I(I ) stands for an identity matrix of appropriate size ( × ); the superscript T denotes the matrix transpose; |X| = det[X] represents the determinant of a square matrix X; the norm of a matrix X is defined by ‖X‖ 2 = tr[XX T ]; 1 := 1 ×1 represents an × 1 vector whose elements are all 1; min [X] represents the minimum eigenvalues of X; for ( ) ⩾ 0, we write ( ) = ( ( )) if there exists a positive constant 1 such that | ( )| ⩽ 1 ( ).
In order to study the convergence of the algorithm proposed in [15], here we simply give that algorithm in [15]. Consider a linear system described by the following observability canonical state space model [15]: where x( ) ∈ R is the state vector, ( ) ∈ R is the system input, ( ) ∈ R is the system output, and V( ) ∈ R is a random noise with zero mean. Assume that the order is known, and ( ) = 0, ( ) = 0 and V( ) = 0 for ⩽ 0.
The system in (1) is an observability canonical form, and its observability matrix Q is an identity matrix; that is, . . .
For the system in (1), the objective of this paper is to develop a new algorithm to estimate the parameter matrix/vector A and b (i.e., the parameters and ) and the system state vector x( ) from the available measurement input-output data { ( ), ( )}.
Since the available measurement input-output data { ( ), ( )} are known but the state vector x( ) is unknown, it is required to eliminate the state vector from (1) and obtain a new expression which only involves the input and output, in order to obtain the estimates of the parameters in (1). The following derives the identification model based on the method in [15].
Replacing in (13) with − yields which is called the identification model or identification expression of the state-space model.

The Parameter Estimation Algorithm and Its Convergence
The recursive least squares algorithm for estimating is expressed aŝ This algorithm is commonly used for convergence analysis.
To avoid computing the matrix inversion, this algorithm is equivalently expressed aŝ Then the following inequality holds: The Scientific World Journal Proof. Define the innovation vector ( ) := ( ) −̂T( )̂( − 1). Using (17), it follows that Subtracting from both sides of (15) and using (14), we havẽ According to the definition of ( ) and using (16) and (29), we have Using (26) The state space model in (1) can be transformed into an inputoutput representation, Referring to the proof of Lemma 3 in [43], using (33), we have Using (17) Since ( ) is a strictly positive real function, referring to Appendix C in [79], we can obtain the conclusion ( ) ⩾ 0.
Adding both sides of (32) by ( ) gives the conclusion of Theorem 1.
Theorem 2. For the system in (1) and the algorithm in (15)-(18), assume that (A1)-(A3) hold and that ( ) is stable; that is, all zeros of ( ) are inside the unit circle; then the parameter estimation error satisfieŝ Proof. Using the formula min [Q]‖x‖ 2 ⩽ x T Qx ⩽ max [Q]‖x‖ 2 , and from the definition of ( ), we havẽ Let The Since ln |P −1 ( )| is nondecreasing, using Theorem 1 yields Referring to the proof of Theorem 2 in [43], we havẽ ) , a.s. for any > 1.
Assume that there exist positive constants , 1 , 2 , and 0 such that the following generalized persistent excitation condition (unbounded condition number) holds: Then for any > 1, we havê

Example
Consider the following single-input single-output secondorder system in canonical form: The simulation conditions are the same as in [15]. That is, the input { ( )} is taken as an independent persistent excitation signal sequence with zero mean and unit variances and {V( )} as a white noise sequence with zero mean and variances 2 = 0.20 2 and 2 = 1.00 2 , respectively. Apply the proposed parameter and state estimation algorithm in (19)-(23) and (44)- (51) to estimate the parameters and states of this example system; the parameter estimates and their estimation errors are shown in Tables 1 and 2; the parameter estimation errors versus are shown in Figure 1; the states ( ) and their estimateŝ( ) versus are shown in Figures 2  and 3, where := ‖̂( )− ‖/‖ ‖ (‖x‖ 2 = x T x) is the parameter estimation error.
From the simulation results of Tables 1 and 2 and Figures  1-3, we can draw the following conclusions.
(1) A lower noise level leads to a faster rate of convergence of the parameter estimates to the true parameters.
(2) The parameter estimation errors become smaller (in general) as the data length increases; see 6 The Scientific World Journal