Convergence Analysis of Forgetting Factor Least Squares Algorithm for ARMAX Time-Delay Models

Abstract

In this paper, the estimation problem is considered for both sample delay and coefficients of ARMAX model. An extended recursive least squares algorithm is derived by minimizing a quadratic cost function. However, the solution of the optimization problem returns a real value for the sample delay. To overcome this difficulty, the rounding properties are used to transform the integer nonlinear problem into a real optimization problem. In addition, consistency of the estimates with their convergence rates are established under the persistent excitation condition. Finally, experimental results on semi-batch reactor are presented to illustrate the performance of the proposed method.

Appendices

Appendix

A Round Operator and Rounding Property

Definition 1

Operator round(d) is given by

where int(d) denotes the integer part of d.

Definition 2

We say that a function $\displaystyle {f:{\mathbb {R}^n} \rightarrow \mathbb {R}}$ has the rounding property if the following holds [17].

For any optimal solution ${x^*}$ to the continuous relaxation (CP) of integer nonlinear unrestricted optimization problem: $\displaystyle { \min \left\{ {f(x):x \in {\mathbb {R}^n}} \right\} }$ there exists an optimal solution x of (IP): $\displaystyle { \min \left\{ {f(x):x \in {\mathbb {Z}^n}} \right\} }$ such that

$$\begin{aligned} x \in \mathrm{round}({x^{*}}) \end{aligned}$$

(56)

Notes that rounding the known optimal solution x of the continuous relaxation (CP) yield an optimal solution for an integer problem (IP) in case where $n = 1$ and the objective function f(.) is convex [17].

1.1 B The Computation Expression of $\bar{\rho }_k$

Let us consider

$$\begin{aligned} \bar{\rho }_k = -\hat{\theta }_k^T \varphi _k (\hat{d})+\theta ^T\varphi _k (d) \end{aligned}$$

(57)

In the same way, to appear the estimated generalized vector parameters ${{\hat{\vartheta }}}$ and the generalized vector parameters ${{\vartheta }}$, we added and subtracted each element of Eq. (57), the appropriate term

$$\begin{aligned} \begin{aligned} \bar{\rho } _k&= -{{\hat{\vartheta }_k}}\phi _k ({{\hat{\vartheta }}}) - \sum \limits _{i = 1}^{nb} {\hat{d}{{\hat{b}}_i}{q^{ - \hat{d}}}\varDelta u_{k-i}} \\&\quad +\, {\vartheta }_{k }\phi _k ({\vartheta }) + \sum \limits _{i = 1}^{nb} {d{b_i}{q^{ - d}}\varDelta u_{k-i}} \\ \end{aligned} \end{aligned}$$

(58)

Adding and subtracting the term $\displaystyle {{\vartheta }_{k}\phi _k ({{\hat{\vartheta }}})}$ from (58), we obtain

$$\begin{aligned} \begin{array}{*{20}{c}} \begin{aligned} \bar{\rho } _k = -{{\hat{\vartheta }}}_{k}\phi _k ({{\hat{\vartheta }}})+ {\vartheta }_{k}\phi _k ({\vartheta }) + \alpha + {\vartheta }_{k}\phi _k ({{\hat{\vartheta }}}) - {\vartheta }_{k}\phi _k ({{\hat{\vartheta }}}) \\ \end{aligned} \end{array} \end{aligned}$$

(59)

where

$$\begin{aligned} \alpha =-\sum \limits _{i = 1}^{nb} {\hat{d}{{\hat{b}}_i}{q^{ - \hat{d}}}\varDelta u_{k-i}}+ \sum \limits _{i = 1}^{nb} {d{b_i}{q^{ - d}}\varDelta u_{k-i}} \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} \bar{\rho } _k&= {{\bar{\vartheta }}}_{k - 1}\phi _k ({{\hat{\vartheta }}}) +\sum \limits _{i = 1}^{nb} {{b_i}{q^{ - d}}u_{k-i}} \\&\quad + \sum \limits _{i = 1}^{nb} {{b_i}{q^{ - \hat{d}}}u_{k-i}} - \sum \limits _{i = 1}^{nb} {\hat{d}{\hat{b}_i}{q^{ -\hat{d}}}\varDelta u_{k-i}} + \sum \limits _{i = 1}^{nb} {d{b_i}{q^{ - \hat{d}}}\varDelta u_{k-i}} \end{aligned} \end{aligned}$$

(60)

So,

$$\begin{aligned} \bar{\rho } _k = {{\bar{\vartheta }}}_{k - 1}\phi _k ({{\hat{\vartheta }}})+\xi _k \end{aligned}$$

(61)

where

$$\begin{aligned} \begin{aligned} \xi _k=\sum \limits _{i = 1}^{nb} {{b_i}({q^{ - d}} - {q^{ - \hat{d}}})u_{k-i}} + \sum \limits _{i = 1}^{nb} {( d{{b}_i} - \hat{d}{\hat{b}_i}){q^{ - \hat{d}}}\varDelta u_{k-i}} \end{aligned} \end{aligned}$$