Nonlinear Dynamics

, Volume 73, Issue 1, pp 583–592

Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle

Authors

  • Peipei Hu
    • Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education)Jiangnan University
    • Control Science and Engineering Research CenterJiangnan University
Original Paper

DOI: 10.1007/s11071-013-0812-0

Cite this article as:
Hu, P. & Ding, F. Nonlinear Dyn (2013) 73: 583. doi:10.1007/s11071-013-0812-0

Abstract

This paper develops a multistage least squares based iterative algorithm to estimate the parameters of feedback nonlinear systems with moving average noise from input–output data. Since that the identification model is bilinear on the unknown parameter space, the solution is to decompose a system into several subsystems with each of which is linear about its parameter vector, then to replace the unknown noise terms in the information vectors with their corresponding estimates at the previous iteration of each subsystem, and estimate each subsystem, respectively. The simulation results show that the proposed algorithm can work well.

Keywords

Parameter estimationIterative identificationLeast squares algorithmHierarchical identification principleNonlinear systemFeedback system

1 Introduction

System identification can construct mathematical models of dynamic systems from the given input-output data [14]. The parameter estimation methods have received much attention in system identification [57]. For example, Ho et al. offered an improved differential evolution algorithm for parameter identification of chaotic systems [8]; Wang et al. presented the maximum likelihood recursive least squares identification methods for linear controlled autoregressive models [9] and for controlled autoregressive moving average systems [10]; Li, et al. presented a maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems [11].

Nonlinear systems (e.g., the Hammerstein models and the Wiener models) can model block structure nonlinear dynamic processes. A Hammerstein model is a static nonlinear block followed by a linear dynamic subsystem and a Wiener model is a linear subsystem with a static nonlinearity [1215]. Recently, several estimation algorithms have been developed, such as the projection algorithm, the stochastic gradient identification algorithm, and the Newton recursive algorithm and the Newton iterative algorithm for Hammerstein nonlinear systems [16], a multiple-trial harmonic balance scheme for nonlinear systems [17], the identification algorithm of nonlinear aeroelastic systems based on the Volterra theory [18], the least squares based and the gradient based iterative identification algorithms for Wiener nonlinear systems [19], the recursive algorithm for disturbed multiinput multioutput Wiener systems [20], and the extended stochastic gradient algorithm for the Hammerstein nonlinear ARMAX model [21].

Recently, iterative methods are effective for solving matrix equations [2224] and may be applied to estimate the parameters of systems. Liu et al. developed a least squares based iterative identification approach for a class of multirate systems [25]; Zhang et al. presented a hierarchical least squares iterative estimation algorithm for multivariable Box–Jenkins-like systems using the auxiliary model [26]; Bao et al. proposed a least squares based iterative identification method for multivariable controlled ARMA systems [27]; Ding et al. presented the least squares based iterative algorithms for Hammerstein nonlinear ARMAX systems [28]. Also, a two-stage recursive least squares parameter estimation algorithm was presented for output error models [29] and a two-stage least squares based iterative identification algorithm was presented for controlled autoregressive moving average (CARMA) systems [30].

This paper studies the multistage identification method for feedback nonlinear systems using the hierarchical identification principle. By means of the decomposition technique, a nonlinear feedback system is decomposed into three subsystems and a hierarchical least squares algorithm is derived.

The paper is organized as follows. Section 2 describes the identification problem formulation for feedback nonlinear systems. Section 3 derives a hierarchical least squares based iterative algorithm for feedback nonlinear systems. Section 4 presents the least squares based iterative estimation algorithm. Section 5 provides an illustrative example to show the effectiveness of the proposed algorithm. Finally, we offer some concluding remarks in Sect. 6.

2 System description and identification model

Let us introduce some notation first. The symbol \(\mbox{\boldmath $I$}_{n}\) stands for an identity matrix of order n; the superscript T denotes the matrix transpose; \({\bf1}_{n}\) represents an n-dimensional column vector whose elements are 1; the norm of a matrix \(\mbox{\boldmath $X$}\) is defined as https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq4_HTML.gif; \(\widehat{\mbox{\boldmath $X$}}(t)\) denotes the estimate of \(\mbox{\boldmath$X$}\) at time t.

Consider the following feedback nonlinear system, depicted in Fig. 1:
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ1_HTML.gif
(1)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ2_HTML.gif
(2)
where the open-loop part is a controlled autoregressive moving average (CARMA) subsystem, {r(t)} and {y(t)} are the reference input and output sequences of the system, {v(t)} is a stochastic white noise sequence with zero mean and variance σ2, and A(z), B(z), and D(z) are polynomials in the unit backward shift operator z−1 [i.e., z−1y(t)=y(t−1)] with the known orders na, nb, and nd, and r(t)=0, y(t)=0, v(t)=0 for t≤0.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Fig1_HTML.gif
Fig. 1

The feedback nonlinear system

Suppose that the nonlinearity \(\bar{y}=f(y)\) is a linear combination of a known basis \(\mbox{\boldmath$f$}:=(f_{1}, f_{2}, \ldots, f_{m})\) in the system output y(t), with the unknown coefficients ci [5, 31, 32]:
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ3_HTML.gif
(3)
where \(\mbox{\boldmath$c$}:=[c_{1}, c_{2}, \ldots, c_{m}]^{\mathrm{T}}\in{\mathbb{R}}^{m}\) is the parameter vector of the nonlinear block.
Define the parameter vectors \(\mbox{\boldmath$a$}\), \(\mbox{\boldmath $b$}\), and \(\mbox{\boldmath$d$}\) of the linear part, and \(\mbox{\boldmath$c$}\) of the nonlinear part as
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equa_HTML.gif
and the output information vector \(\mbox{\boldmath${\varphi }$}_{y}(t)\), the input information vector \(\mbox{\boldmath${\varphi}$}_{r}(t)\), the noise information vector \(\mbox{\boldmath${\varphi}$}_{v}(t)\), and the feedback information matrix \(\mbox{\boldmath$F$}(t)\) as
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equb_HTML.gif
From Eqs. (1)–(3), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ4_HTML.gif
(4)
For the pair \(\lambda\mbox{\boldmath$b$}\) and \(\mbox{\boldmath $c$}/\lambda\) with any nonzero constant λ, the model in (4) keeps the identical input-output relationship. Therefore, to get a unique parameterization, without loss of generality, we adopt the normalization constraint on \(\mbox{\boldmath$c$}\) with \(\|\mbox {\boldmath$c$}\|=1\) and its first positive entry of \(\mbox{\boldmath$c$}\), i.e., c1>0 [16, 33].

The objective of this paper is to derive a novel least squares based iterative identification algorithm for estimating the system parameters ai, bi, di of linear part and ci of nonlinear block using the hierarchical identification principle [3437].

3 The multistage least squares based iterative algorithm

In this section, the feedback nonlinear system is decomposed into three subsystems by means of the hierarchical identification principle: They contain the parameter vectors \(\mbox{\boldmath${\theta}$}_{1}:=\mbox{\boldmath$c$}\in{\mathbb{R}}^{m}\), https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq24_HTML.gif, and \(\mbox{\boldmath${\theta}$}_{3}:=\mbox{\boldmath$d$}\in{\mathbb{R}}^{n_{d}}\). Considering that there exist the associate terms between two subsystems, such as \(\mbox{\boldmath$b$}\) in \(\mbox{\boldmath${\varphi}$}_{1}(t)\) and \(\mbox{\boldmath$c$}\) in \(\mbox{\boldmath${\varphi}$}_{2}(t)\). The basic idea is to replace the unknown vectors with their estimates at the previous time, and to identify the subsystems using the least squares based iterative method.

Define three intermediate variables,
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ5_HTML.gif
(5)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ6_HTML.gif
(6)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ7_HTML.gif
(7)
and the information vectors as
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equc_HTML.gif
The system in (4) can be decomposed into the following three fictitious subsystems (i.e., subidentification models) and the fictitious outputs yi(t) are linear about the parameter vectors \(\mbox{\boldmath${\theta}$}_{i}\),
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ8_HTML.gif
(8)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ10_HTML.gif
(10)
Subsystem (8) contains m parameters in \(\mbox{\boldmath ${\theta}$}_{1}\), Subsystem (9) contains (na+nb) parameters in \(\mbox{\boldmath${\theta}$}_{2}\), and Subsystem (10) contains nd parameters in \(\mbox{\boldmath${\theta}$}_{3}\).
Consider the data from t=1 to t=L (L is the data length and Ln) and define the stacked output vectors \(\mbox{\boldmath $Y$}\), \(\mbox{\boldmath$Y$}_{1}\), \(\mbox{\boldmath$Y$}_{2}\), and \(\mbox{\boldmath$Y$}_{3}\), the stacked information matrices \(\mbox{\boldmath$\Phi$}_{1}\), \(\mbox{\boldmath$\Phi$}_{2}\), \(\mbox{\boldmath$\Phi$}_{3}\), and \(\mbox{\boldmath$\Phi$}_{4}\), and the stacked white noise vector \(\mbox{\boldmath$V$}\) as
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equd_HTML.gif
From (8)–(10), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Eque_HTML.gif
Define three criterion functions,
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equf_HTML.gif
Letting the partial derivatives of Ji(θi) with respect to \(\mbox{\boldmath${\theta}$}_{i}\) be zero, respectively, gives
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equg_HTML.gif
Suppose that the stacked information matrices \(\mbox{\boldmath$\Phi$}_{i}\) (i=1,2,3) are persistently exciting, that is, the inverses of matrices https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq45_HTML.gif exist. Using the definitions of \(\mbox{\boldmath$Y$}_{1}\), \(\mbox{\boldmath$Y$}_{2}\), and \(\mbox{\boldmath$Y$}_{3}\) and from the above equations, we can obtain the following least squares estimates \(\hat{\mbox{\boldmath${\theta}$}}_{i}\) of the parameter vectors \(\mbox{\boldmath${\theta}$}_{i}\) (i=1,2,3):
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ11_HTML.gif
(11)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ12_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ13_HTML.gif
(13)
Noting that the right-hand sides of (11)–(13) contain the unknown parameter vectors \(\mbox{\boldmath${\theta}$}_{2}\) and/or \(\mbox{\boldmath${\theta}$}_{3}\), we cannot compute the estimates \(\hat{\mbox{\boldmath${\theta}$}}_{1}\), \(\hat{\mbox{\boldmath${\theta}$}}_{2}\), and \(\hat{\mbox{\boldmath ${\theta}$}}_{3}\) by using (11)–(13). The solution is based on the hierarchical identification principle: let k=1,2,3,… be an iteration variable, \(\hat{\mbox{\boldmath${\theta}$}}_{1,k}:=\hat {\mbox{\boldmath$c$}}_{k}\), https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq57_HTML.gif and \(\hat{\mbox{\boldmath${\theta}$}}_{3,k}:=\hat{\mbox{\boldmath $d$}}_{k}\) be the iterative estimates of \(\mbox{\boldmath${\theta}$}_{1}=\mbox{\boldmath$c$}\), https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq60_HTML.gif and \(\mbox{\boldmath${\theta}$}_{3}=\mbox{\boldmath$d$}\) at iteration k, and \(\hat{v}_{k}(t)\) be the estimate of v(t) at iteration k, and define
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equh_HTML.gif
From (4), we have
$$v(t)=y(t)-\mbox{\boldmath${\varphi}$}^{\mathrm{T}}_4(t)\mbox{ \boldmath${\theta}$}_2-\mbox{\boldmath${\varphi }$}^{\mathrm{T} }_3(t) \mbox{\boldmath${\theta}$}_3-\mbox{\boldmath${\varphi }$}^{\mathrm{T}}_1(t)\mbox{\boldmath${\theta}$}_1. $$
Replacing \(\mbox{\boldmath${\theta}$}_{2}\), \(\mbox{\boldmath${\theta }$}_{3}\), \(\mbox{\boldmath${\theta}$}_{1}\), \(\mbox{\boldmath${\varphi }$}_{3}(t)\), and \(\mbox{\boldmath${\varphi}$}_{1}(t)\) with their estimates \(\hat {\mbox{\boldmath${\theta}$}}_{2,k}\), \(\hat{\mbox{\boldmath${\theta}$}}_{3,k}\), \(\hat{\mbox{\boldmath ${\theta}$}}_{1,k}\), \(\hat{\mbox{\boldmath${\varphi}$}}_{3,k}(t)\), and \(\hat{\mbox {\boldmath${\varphi}$}}_{1,k}(t)\), respectively, the estimate \(\hat{\mbox{\boldmath$v$}}_{k}(t)\) of v(t) can be computed by
$$\hat{v}_k(t)=y(t)-\mbox{\boldmath${\varphi}$}^{\mathrm{T}}_4(t) \hat{\mbox{\boldmath${\theta}$}}_{2,k}-\hat {\mbox{\boldmath${ \varphi}$}}^{\mathrm{T}}_{3,k}(t)\hat {\mbox{\boldmath${ \theta}$}}_{3,k}- \hat{\mbox{\boldmath${\varphi}$}}^{\mathrm{T}}_{1,k}(t) \hat{\mbox{\boldmath${\theta}$}}_{1,k}. $$
Replacing \(\mbox{\boldmath${\theta}$}_{2}\), \(\mbox{\boldmath${\theta }$}_{3}\), \(\mbox{\boldmath$\Phi$}_{1}\), and \(\mbox{\boldmath$\Phi$}_{3}\) in (11) with their estimates \(\hat{\mbox{\boldmath${\theta }$}}_{2,k-1}\), \(\hat{\mbox{\boldmath${\theta}$}}_{3,k-1}\), \(\hat{\mbox{\boldmath $\Phi$}}_{1,k,}\) and \(\hat{\mbox{\boldmath$\Phi$}}_{3,k}\), replacing \(\mbox{\boldmath ${\theta}$}_{3}\), \(\mbox{\boldmath$\Phi$}_{2}\), and \(\mbox{\boldmath$\Phi$}_{3}\) in (12) with their estimates \(\hat{\mbox{\boldmath${\theta}$}}_{3,k-1}\), \(\hat{\mbox{\boldmath $\Phi$}}_{2,k}\), and \(\hat{\mbox{\boldmath$\Phi$}}_{3,k}\), and replacing \(\mbox {\boldmath${\theta}$}_{2}\), \(\mbox{\boldmath$\Phi$}_{3}\), and \(\mbox{\boldmath$\Phi$}_{2}\) in (13) with their estimates \(\hat{\mbox{\boldmath${\theta}$}}_{2,k}\), \(\hat{\mbox{\boldmath $\Phi$}}_{3,k}\), \(\hat{\mbox{\boldmath$\Phi$}}_{2,k}\), we can summarize the hierarchical least squares based iterative (H-LSI) identification algorithm for estimating \(\mbox{\boldmath ${\theta}$}_{1}\), \(\mbox{\boldmath${\theta}$}_{2}\), and \(\mbox{\boldmath${\theta}$}_{3}\):
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ14_HTML.gif
(14)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ15_HTML.gif
(15)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ16_HTML.gif
(16)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ19_HTML.gif
(19)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ20_HTML.gif
(20)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ21_HTML.gif
(21)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ22_HTML.gif
(22)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ23_HTML.gif
(23)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ24_HTML.gif
(24)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ25_HTML.gif
(25)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ26_HTML.gif
(26)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equ27_HTML.gif
(27)
The steps of computing \(\hat{\mbox{\boldmath${\theta}$}}_{1,k}\), \(\hat{\mbox{\boldmath${\theta}$}}_{2,k,}\) and \(\hat{\mbox {\boldmath${\theta}$}}_{3,k}\) in the H-LSI algorithm in (14)–(27) are listed in the following:
  1. 1.

    Collect the input-output data {r(t),y(t),i=1,2,…,L} (L is the data length), form \(\mbox{\boldmath${\varphi}$}_{y}(t)\) by (15), \(\mbox{\boldmath${\varphi}$}_{r}(t)\) by (16), \(\mbox {\boldmath$F$}(t)\) by (17), \(\mbox{\boldmath$\Phi$}_{4}\) by (20), and \(\mbox{\boldmath $Y$}\) by (18), and give a positive number ε.

     
  2. 2.

    To initialize, let k=1, https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq105_HTML.gif and \(\hat{\mbox{\boldmath${\theta}$}}_{3,0}=\hat{\mbox{\boldmath $d$}}_{0}\) be an arbitrary real vector, and \(\hat{v}_{0}(t)\) be a random number.

     
  3. 3.

    Form \(\hat{\mbox{\boldmath$\Phi$}}_{1,k}\) by (19), \(\hat{\mbox{\boldmath${\varphi}$}}_{3,k}(t)\) by (21), and \(\hat{\mbox{\boldmath$\Phi$}}_{3,k}\) by (22).

     
  4. 4.
    Update \(\hat{\mbox{\boldmath${\theta}$}}_{1,k}\) by (14) and normalize \(\hat{\mbox{\boldmath$c$}}_{k}\) with the first positive element. i.e.,
    $$\hat{\mbox{\boldmath$c$}}_k={\rm sgn}\bigl[\hat{\mbox{\boldmath ${ \theta}$}}_{1,k}(1)\bigr]\frac{\hat{\mbox{\boldmath${\theta}$} }_{1,k}}{\|\hat{\mbox{\boldmath${\theta}$}}_{1,k}\|}, \nonumber $$
    where https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq113_HTML.gif represents the sign of the first element of https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq114_HTML.gif and we set https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_IEq115_HTML.gif.
     
  5. 5.

    Form \(\hat{\mbox{\boldmath${\varphi}$}}_{2,k}(t)\) by (24) and \(\hat{\mbox{\boldmath$\Phi$}}_{2,k}\) by (25), and update \(\hat{\mbox{\boldmath${\theta}$}}_{2,k}\) by (23) and \(\hat{\mbox{\boldmath${\theta}$}}_{3,k}\) in (26).

     
  6. 6.

    Compute \(\hat{v}_{k}(t)\) by (27).

     
  7. 7.

    If \(\mathit{Error}:=\|\hat{\mbox{\boldmath${\theta}$}}_{1,k}-\hat{\mbox {\boldmath${\theta}$}}_{1,k-1}\|+\|\hat {\mbox{\boldmath${\theta}$}}_{2,k}-\hat{\mbox{\boldmath${\theta }$}}_{2,k-1}\|+ \|\hat{\mbox{\boldmath${\theta}$}}_{3,k}-\hat{\mbox{\boldmath ${\theta}$}}_{3,k-1}\|\leq \varepsilon\), we obtain the parameter estimates \(\hat{\mbox{\boldmath${\theta}$}}_{1,k}\), \(\hat{\mbox{\boldmath${\theta}$}}_{2,k}\) and \(\hat{\mbox {\boldmath${\theta}$}}_{3,k}\); otherwise, increase k by 1 and go to step 3.

     
The flowchart of computing the parameter estimate \(\hat{\mbox{\boldmath${\theta}$}}_{1,k}\), \(\hat{\mbox{\boldmath ${\theta}$}}_{2,k}\), and \(\hat{\mbox{\boldmath${\theta}$}}_{3,k}\) is shown in Fig. 2.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Fig2_HTML.gif
Fig. 2

The flowchart for computing the H-LSI estimates \(\hat {\mbox{\boldmath${\theta}$}}_{1}\), \(\hat{\mbox{\boldmath${\theta }$}}_{2}\), and \(\hat{\mbox{\boldmath${\theta}$}}_{3}\)

4 The least squares based iterative estimation algorithm

To show the advantages of the proposed H-LSI algorithm for feedback nonlinear systems, the least squares based iterative (LSI) algorithm is given for comparison in the following.

Define the information vector \(\mbox{\boldmath${\varphi}$}(t)\), parameter vector \(\mbox{\boldmath${\vartheta}$}\) and the stacked information matrix \(\mbox{\boldmath$H$}\) as
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equl_HTML.gif
From (4), we have
$$ y(t)=\mbox{\boldmath${\varphi}$}^{\mathrm{T}}(t) \mbox {\boldmath${\vartheta}$}+v(t). $$
(28)
Using the definitions of \(\mbox{\boldmath$Y$}\), \(\mbox{\boldmath $V$}\), and \(\mbox{\boldmath$H$}\), Eq. (28) can be written as
$$\mbox{\boldmath$Y$}=\mbox{\boldmath$H$}\mbox{\boldmath${\vartheta }$}+\mbox{ \boldmath$V$}. $$
Define the quadratic criterion function,
$$J(\mbox{\boldmath${\vartheta}$}):=\|\mbox{\boldmath$Y$}-\mbox {\boldmath$H$} \mbox{\boldmath${\vartheta}$}\|^2. $$
Minimizing the criterion function and letting the partial derivative of J(ϑ) with respect to \(\mbox {\boldmath${\vartheta}$}\) be zero, we can obtain the least squares estimate for estimating the parameter vector \(\mbox{\boldmath${\vartheta}$}\):
$$ \hat{\mbox{\boldmath${\vartheta}$}}=\bigl[\mbox{ \boldmath$H$}^{\mathrm{T}}\mbox{\boldmath$H$}\bigr]^{-1}\mbox{ \boldmath$H$}^{\mathrm{T}}\mbox{\boldmath$Y$}. $$
(29)
The matrix \(\mbox{\boldmath$H$}\) in (29) (that is \(\mbox {\boldmath${\varphi}$}(t)\)) contains the unknown noise terms v(ti), we cannot compute the estimate \(\hat{\mbox{\boldmath${\vartheta}$}}\) directly. The approach is based on the hierarchical identification principle: let \(\hat{\mbox{\boldmath ${\varphi}$}}_{k}(t)\) denote the information vector by replacing the v(ti) with the estimates \(\hat{v}_{k-1}(t-i)\) at iterative k−1 and \(\hat{\mbox {\boldmath$H$}}_{k}\) denote the information matrix by replacing the \(\mbox{\boldmath ${\varphi}$}(t)\) with its estimates \(\hat{\mbox{\boldmath${\varphi}$}}_{k}(t)\), and we have
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equo_HTML.gif
Replacing \(\mbox{\boldmath$H$}\) in (29) with its estimate \(\hat{\mbox {\boldmath$H$}}_{k}\), we obtain the least squares based iterative (LSI) algorithm for estimating \(\mbox{\boldmath${\vartheta}$}\) of feedback nonlinear systems as follows:
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equp_HTML.gif
Suppose that the initial value of \(\hat{\mbox{\boldmath${\vartheta }$}}_{0}\) be a random number such that the matrix \(\hat{\mbox{\boldmath$H$}}^{\mathrm{T}}_{0}\hat{\mbox{\boldmath$H$}}_{0}\) is nonsingular.

In order to guarantee the system identifiability, the input signal must be persistently exciting or sufficiently rich, e.g., the pseudo-random binary sequence. The convergence of the identification algorithms requires the persistent excitation assumption regardless of open-loop or feedback nonlinear systems.

5 Examples

Consider the following feedback nonlinear system:
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-013-0812-0/MediaObjects/11071_2013_812_Equq_HTML.gif
In simulation, the reference input {r(t)} is taken as a persistent excitation signal sequence with zero mean and unit variance, and {v(t)} as a white noise sequence with zero mean and variances σ2=0.052 and σ2=0.102, respectively. Using the H-LSI algorithm and LSI algorithm to estimate the parameters of this example system, the parameter estimates and errors \(\delta:=\|\hat{\mbox{\boldmath${\vartheta}$}}_{k}-\mbox{\boldmath ${\vartheta}$}\|/\|\mbox{\boldmath${\vartheta}$}\|\) with different noise variance σ2 and different data length L are shown in Tables 1, 2, 3, 4 with different variances.
Table 1

The parameter estimates and errors with σ2=0.052

Algorithms

t=L

a1

a2

b1

b2

d1

d2

c1

c2

δ (%)

H-LSI

1000

0.34997

0.04998

0.68303

0.02126

0.79998

0.29999

0.60303

0.79772

0.32961

2000

0.34999

0.04999

0.68144

0.02076

0.79999

0.30000

0.60017

0.79988

0.10783

3000

0.35000

0.05000

0.68026

0.01884

0.80000

0.30000

0.59770

0.80172

0.20395

LSI

1000

0.67324

0.14482

0.68165

0.24322

0.59446

0.80413

0.93614

0.37885

28.48840

2000

0.57147

0.11573

0.68129

0.17249

0.59524

0.80355

0.95756

0.41776

22.30847

3000

0.20322

0.02048

0.72440

−0.03307

0.59550

0.80335

0.69346

0.22723

13.75984

True values

0.35000

0.05000

0.68000

0.02000

0.80000

0.30000

0.60000

0.80000

 
Table 2

The parameter estimates and errors with σ2=0.102

Algorithms

t=L

a1

a2

b1

b2

d1

d2

c1

c2

δ (%)

H-LSI

1000

0.34996

0.04997

0.68478

0.02050

0.79997

0.30000

0.60300

0.79774

0.40063

2000

0.34996

0.04997

0.68438

0.02192

0.79998

0.29999

0.60518

0.79609

0.52938

3000

0.34997

0.04997

0.68013

0.01962

0.79998

0.29999

0.60095

0.79928

0.08256

LSI

1000

0.40901

0.04610

0.74814

0.13483

0.58329

0.81226

0.87341

0.32864

10.98390

2000

0.43491

0.08059

0.70192

0.10178

0.59331

0.80498

0.92106

0.38687

12.73406

3000

0.41171

0.06635

0.70197

0.8552

0.59194

0.80598

0.89851

0.36811

10.02535

True values

0.35000

0.05000

0.68000

0.02000

0.80000

0.30000

0.60000

0.80000

 
Table 3

The H-LSI estimates and errors versus iteration k (L=2000)

σ2

k

a1

a2

b1

b2

d1

d2

c1

c2

δ (%)

0.052

1

0.34831

0.05013

0.68157

0.02075

0.80231

0.30020

0.58617

0.81019

1.14973

5

0.35077

0.05072

0.68178

0.02049

0.79848

0.29802

0.60291

0.79781

0.32173

10

0.34931

0.04999

0.68112

0.01986

0.79976

0.30018

0.60409

0.79692

0.34797

20

0.34997

0.05000

0.68155

0.02035

0.80000

0.30002

0.59715

0.80213

0.25609

0.102

1

0.34642

0.05116

0.68427

0.02122

0.80441

0.29983

0.58515

0.81093

1.30286

5

0.35077

0.05072

0.68178

0.02049

0.79848

0.29802

0.60291

0.79781

0.32173

10

0.34879

0.05032

0.68450

0.02086

0.80017

0.30087

0.60387

0.79708

0.44947

20

0.35004

0.04999

0.68246

0.02180

0.79999

0.29995

0.59474

0.80392

0.47487

True values

0.35000

0.05000

0.68000

0.02000

0.80000

0.30000

0.60000

0.80000

 
Table 4

The H-LSI estimates and errors versus iteration k (L=3000)

σ2

k

a1

a2

b1

b2

d1

d2

c1

c2

δ (%)

0.052

1

0.34642

0.05116

0.68427

0.02122

0.80441

0.29983

0.58515

0.81093

1.30286

5

0.35055

0.05044

0.68633

0.02198

0.79649

0.29546

0.59603

0.80296

0.66348

10

0.34879

0.05032

0.68450

0.02086

0.80017

0.30087

0.60387

0.79708

0.44947

20

0.35004

0.04999

0.68246

0.02180

0.79999

0.29995

0.59474

0.80392

0.47487

0.102

1

0.34554

0.05071

0.68076

0.01778

0.80467

0.30201

0.58481

0.81117

1.32524

5

0.34825

0.04927

0.68168

0.01668

0.79765

0.29777

0.59931

0.80052

0.35190

10

0.34937

0.05019

0.68157

0.01986

0.80024

0.30056

0.59963

0.80028

0.12270

20

0.35009

0.05015

0.67921

0.01819

0.80015

0.30012

0.59933

0.80050

0.14190

True values

0.35000

0.05000

0.68000

0.02000

0.80000

0.30000

0.60000

0.80000

 
From Tables 14, we can draw the following conclusions.
  • The estimation errors δ given by the H-LSI algorithm become smaller with the data length L increasing.

  • For the same variances, the H-LSI algorithm gives more accurate parameter estimates compared with the LSI algorithm.

  • The proposed H-LSI algorithm has a high computational efficiency since the dimensions of the involved covariance matrices in each subsystem become small,

6 Conclusions

In this paper, a least squares based iterative algorithm is derived for feedback nonlinear systems by means of the hierarchical identification principle. The simulation results show that the proposed algorithm generate more accurate parameter estimates after only several iterations. The proposed H-LSI method can combine the multiinnovation identification theory [38] to study identification problems of nonlinear systems with colored noises, dual-rate systems, nonuniformly sampled data systems or missing-data systems [3942] and other linear or nonlinear systems with colored noises [4347]. The convergence properties of the proposed algorithm is worth further studying.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 61273194, 61203111), the Natural Science Foundation of Jiangsu Province (China, BK2012549) and the 111 Project (B12018).

Copyright information

© Springer Science+Business Media Dordrecht 2013