States based iterative parameter estimation for a state space model with multi-state delays using decomposition☆
Introduction
It is well known that time delays are frequently encountered in various practical systems such as chemical processes, communication networks and digital signal analysis [1], [2]. Such a delay is usually a source of instability and performance deterioration. The analysis and synthesis of time-delay systems have received growing interests in recent years [3], [4]. Several approaches have been proposed [5], [6], in particular using the linear matrix inequality approach which has been extensively used for design of uncertain systems [7], [8].
Parameter estimation is the eternal theme of system identification [9] and many identification methods were developed for linear systems [10], [11], [12] and errors-in-variables systems [13], [14], e.g., the iterative parameter estimation algorithm for multivariable systems [15], the hierarchical gradient based identification algorithm [16] and the hierarchical least squares identification method [17], [18] for multivariable systems, and the data filtering based recursive least squares algorithm for Hammerstein systems [19]. Recently, an auxiliary model based parameter estimation algorithm was presented for dual-rate output error systems with colored noises [20], a modified subspace identification algorithm was presented for periodically non-uniformly sampled systems by using the lifting technique [21], an efficient hierarchical identification method was proposed for general dual-rate sampled-data systems [22], and an adaptive identification problem was discussed for linear-in-the-parameters nonlinear filters using the periodic input sequences [23].
Iterative methods can be used for system identification with unknown variables in the information vector [24]. For example, Ding et al. proposed a hierarchical gradient based iterative algorithm and a hierarchical least squares based iterative algorithm for CARARMA systems [25]; Li studied the parameter estimation for Hammerstein CARARMA systems based on the Newton iteration [26]; Ding et al. presented the decomposition based Newton iterative identification method for a Hammerstein nonlinear FIR system with ARMA noise [27].
The significance and difficulty of estimating state space systems are widely admitted [28], [29]. Consequently, there is extremely large and active research effort directed towards this problem. A primary aspect is that the states in the process should be identified first or simultaneously with the system parameters since the states are usually unknown [30], [31]. Rapisarda and Trentelman studied identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data [32]; Mercère and Bako presented parameterization and identification of multivariable state space systems using a canonical approach [33].
This paper develops a new system identification algorithm using the iterative technique and the hierarchical identification principle. We frame our study in the identification of state space systems with multi-state delays. The basic idea is to use the iterative technique to deal with the identification problem and to present a hierarchical gradient based iterative algorithm and a hierarchical least squares based iterative algorithm for a state space model with multi-state delays.
This paper is organized as follows. Section 2 gives the identification model of state delay systems. Section 3 introduces the hierarchical gradient based iterative algorithm. Section 4 presents the least squares based iterative estimation algorithm. Section 5 provides an illustrative example for the results in this paper. Finally, we offer some concluding remarks in Section 6.
Section snippets
System description and identification model
Let us introduce the necessary notation. “” or “” stands for “A is defined as X”; the symbol stands for an identity matrix of appropriate size (n×n); z represents a unit forward shift operator: and ; the superscript T denotes the matrix/vector transpose; the norm of a matrix is defined as ; represents the maximum eigenvalue of the non-negative definite matrix ; denotes the estimate of ϑat time t; represents an m×n matrix
The hierarchical gradient based iterative algorithm
For two models in (7), (8), we derive a hierarchical identification algorithm to estimate the parameter matrix in S1 and the parameter vector in S2.
Suppose that the data length and define the stack information matrices and , the stacked state matrices and and vector asFrom (7), (8), we have
The hierarchical least squares based iterative algorithm
The hierarchical gradient based iterative algorithm converges slowly. To improve the convergence rate, here derives a hierarchical least squares based iterative algorithm.
Referring to the method in [16], [17] and minimizing (14), (15), we can obtain the least squares based iterative solutions and of and :where is the time-varying step-size or time-varying convergence factor.
Thus, we can obtain the
Example
Consider the following state space system with 2-step state-delay:In simulation, the input is taken as an uncorrelated persistent excitation signal vector sequence with zero mean and unit variance. Taking the data
Conclusion
This paper studies parameter estimation problems for a class of linear dynamic systems. Compared with the stochastic gradient algorithm and the recursive least squares algorithms, the proposed iterative algorithms can generate more accurate parameter estimates. The convergence properties of the proposed algorithm can be analyzed by means of the martingale convergence theorem. The methods in this paper can be extended to study identification problems of other linear systems [34] or nonlinear
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