A Unifying Method to Construct Rational Basis Functions for Linear and Nonlinear Systems

Abstract

This paper proposes a general and unifying method for constructing rational basis functions (RBFs). The proposal is specifically designed for finding characterizing matrices of linear RBF state equations. Analytic expressions for the realization of such state-space matrices are also developed. As will be shown, the construction method here proposed can be easily applied (but is not limited) to several RBF sets which have been successfully adopted in both linear and nonlinear system identification areas.

Appendix A: State-Space Realizations of REFs

In this appendix, we demonstrate how the elements of the two-dimensional state-space realization ($\widehat{\hbox {A}}_{i},\widehat{\hbox {B}}_{i},\widehat{\hbox {C}}_{i},\widehat{\hbox {D}}_{i}$) can be obtained in order to satisfy conditions (48) and (49).

Firstly, from (40) and (47), note that we can rewrite condition (48) as

$$\begin{aligned} \left[ \begin{array}{c} \displaystyle \frac{(\alpha -\hbox {A}_{22}^i)\hbox {B}_1^i+\hbox {A}_{12}^i\hbox {B}_2^i}{(\alpha -\hbox {A}_{11}^i)(\alpha -\hbox {A}_{22}^i)-\hbox {A}_{12}^i\hbox {A}_{21}^i}\\ \displaystyle \frac{(\alpha -\hbox {A}_{11}^i)\hbox {B}_2^i+\hbox {A}_{21}^i\hbox {B}_1^i}{(\alpha -\hbox {A}_{11}^i)(\alpha -\hbox {A}_{22}^i)-\hbox {A}_{12}^i\hbox {A}_{21}^i} \end{array} \right] = \left[ \begin{array}{c} \displaystyle \frac{m_{1}^i\alpha + m_{0}^i}{(\alpha -a_i)(\alpha -a_i^*)} \\ \displaystyle \frac{n_{1}^i\alpha + n_{0}^i}{(\alpha -a_i)(\alpha -a_i^*)} \end{array} \right] . \end{aligned}$$

(60)

By equating the numerators and denominators of (60), it follows that

$$\begin{aligned} \alpha \hbox {B}_{1}^i - \hbox {A}_{22}^i \hbox {B}_{1}^i + \hbox {A}_{12}^i \hbox {B}_{2}^i= & {} m_1^i \alpha + m_0^i, \end{aligned}$$

(61)

$$\begin{aligned} \alpha \hbox {B}_{2}^i - \hbox {A}_{11}^i \hbox {B}_{2}^i + \hbox {A}_{21}^i \hbox {B}_{1}^i= & {} n_1^i \alpha + n_0^i, \end{aligned}$$

(62)

$$\begin{aligned} \alpha ^2 + (-\hbox {A}_{11}^i-\hbox {A}_{22}^i) \alpha + \hbox {A}_{11}^i \hbox {A}_{22}^i - \hbox {A}_{12}^i\hbox {A}_{21}^i= & {} \alpha ^2 -2 \mathfrak {R}\hbox {e} (a_i) \alpha + |a_i|^2. \end{aligned}$$

(63)

From (61) and (62) we have

$$\begin{aligned} \hbox {B}_{1}^i= & {} m_1^i, \end{aligned}$$

(64)

$$\begin{aligned} \hbox {B}_{2}^i= & {} n_1^i. \end{aligned}$$

(65)

Using this result in (61), (62) and (63), and, the definitions of $d_1^i$ and $d_0^i$ given by (42) and (43), we can then write the following system of nonlinear equations:

$$\begin{aligned} - m_1^i \hbox {A}_{22}^i + n_1^i \hbox {A}_{12}^i= & {} m_0^i, \end{aligned}$$

(66)

$$\begin{aligned} - n_1^i \hbox {A}_{11}^i + m_1^i \hbox {A}_{21}^i= & {} n_0^i, \end{aligned}$$

(67)

$$\begin{aligned} -\hbox {A}_{11}^i - \hbox {A}_{22}^i= & {} d_1^i, \end{aligned}$$

(68)

$$\begin{aligned} \hbox {A}_{11}^i\hbox {A}_{22}^i - \hbox {A}_{12}^i\hbox {A}_{21}^i= & {} d_0^i. \end{aligned}$$

(69)

Such a nonlinear system is solved as follows. From (66) and (67), it results that

$$\begin{aligned} \hbox {A}_{22}^i= & {} \frac{-m_0^i+n_1^i \hbox {A}_{12}^i}{ m_1^i }, \end{aligned}$$

(70)

$$\begin{aligned} \hbox {A}_{21}^i= & {} \frac{n_0^i+n_1^i \hbox {A}_{11}^i}{ m_1^i }. \end{aligned}$$

(71)

Direct substitution of (70) and (71) into (69) leads us to

$$\begin{aligned} \hbox {A}_{12}^i = \frac{-d_0^i m_1^i - m_0^i \hbox {A}_{11}^i}{ n_0^i }. \end{aligned}$$

(72)

Using (72) in (66), it follows that

$$\begin{aligned} \hbox {A}_{22}^i = \frac{-d_0^i m_1^i n_1^i - m_0^i n_0^i - m_0^i n_1^i \hbox {A}_{11}^i }{ m_1^i n_0^i }. \end{aligned}$$

(73)

Direct substitution of (73) into (68) finally leads us to an analytic expression for $\hbox {A}_{11}^i$,

$$\begin{aligned} \hbox {A}_{11}^i = \left( -d_{1}^i m_{1}^i n_{0}^i + m_{0}^i n_{0}^i + d_{0}^i m_{1}^i n_{1}^i\right) /\left( m_{1}^i n_{0}^i - m_{0}^i n_{1}^i\right) . \end{aligned}$$

(74)

$\hbox {A}_{12}^i$, $\hbox {A}_{21}^i$ and $\hbox {A}_{22}^i$ can then be found by combining (74) with (72), (71) and (68), respectively:

$$\begin{aligned} \hbox {A}_{12}^i= & {} \left( -d_{0}^i (m_{1}^i)^2 + d_{1}^i m_{0}^i m_{1}^i - (m_{0}^i)^2\right) /\left( m_{1}^i n_{0}^i - m_{0}^i n_{1}^i\right) , \end{aligned}$$

(75)

$$\begin{aligned} \hbox {A}_{21}^i= & {} \left( d_{0}^i (n_{1}^i)^2 - d_{1}^i n_{0}^i n_{1}^i + (n_{0}^i)^2\right) /\left( m_{1}^i n_{0}^i - m_{0}^i n_{1}^i\right) , \end{aligned}$$

(76)

$$\begin{aligned} \hbox {A}_{22}^i= & {} \left( d_{1}^i m_{0}^i n_{1}^i - m_{0}^i n_{0}^i - d_{0}^i m_{1}^i n_{1}^i\right) /(m_{1}^i n_{0}^i - m_{0}^i n_{1}^i). \end{aligned}$$

(77)

We now concentrate our attention in order to find $\widehat{\hbox {C}}_i$ and $\widehat{\hbox {D}}_i$ for which the input–output condition (49) holds.

From (40), (41) and (47), note that (49) can be rewritten as

$$\begin{aligned} \hbox {C}_{1}^i\displaystyle \frac{m_{1}^i\alpha + m_{0}^i}{\alpha ^2 + d_{1}^i \alpha + d_{0}^i} + \hbox {C}_{2}^i\displaystyle \frac{n_{1}^i\alpha + n_{0}^i}{\alpha ^2 + d_{1}^i \alpha + d_{0}^i} + \widehat{\hbox {D}}_i = \frac{\bar{l}_{2}^i\alpha ^2 + \bar{l}_{1}^i\alpha + \bar{l}_{0}^i}{\alpha ^2 + d_{1}^i \alpha + d_{0}^i}. \end{aligned}$$

(78)

By equating the numerators of both sides in (78), and applying some basic manipulations, it results that

$$\begin{aligned} \widehat{\hbox {D}}_i \alpha ^2 + \left( d_{1}^i \widehat{\hbox {D}}_i + m_{1}^i \hbox {C}_{1}^i + n_{1}^i \hbox {C}_{2}^i\right) \alpha + d_{0}^i\widehat{\hbox {D}}_i + m_{0}^i \hbox {C}_{1}^i + n_{0}^i \hbox {C}_{2}^i= \bar{l}_{2}^i\alpha ^2 + \bar{l}_{1}^i\alpha + \bar{l}_{0}^i. \end{aligned}$$

(79)

From (79) we have

$$\begin{aligned} \widehat{\hbox {D}}_i = \bar{l}_{2}^i. \end{aligned}$$

(80)

Using this result in (79), we can then write the following system of linear equations:

$$\begin{aligned} m_{1}^i \hbox {C}_{1}^i + n_{1}^i \hbox {C}_{2}^i= & {} \bar{l}_{1}^i-d_{1}^i \bar{l}_{2}^i, \end{aligned}$$

(81)

$$\begin{aligned} m_{0}^i \hbox {C}_{1}^i + n_{0}^i \hbox {C}_{2}^i= & {} \bar{l}_{0}^i - d_{0}^i \bar{l}_{2}^i, \end{aligned}$$

(82)

which has its analytic solution given by

$$\begin{aligned} \hbox {C}_{1}^i= & {} \left( ~ n_{0}^i( \bar{l}_1^i-\bar{l}_2^i d_1^i ) - n_{1}^i( \bar{l}_0^i-\bar{l}_2^i d_0^i ) ~\right) /(m_{1}^i n_{0}^i - m_{0}^i n_{1}^i), \end{aligned}$$

(83)

$$\begin{aligned} \hbox {C}_{2}^i= & {} \left( -m_{0}^i( \bar{l}_1^i-\bar{l}_2^i d_1^i ) + m_{1}^i( \bar{l}_0^i-\bar{l}_2^i d_0^i ) ~\right) /(m_{1}^i n_{0}^i - m_{0}^i n_{1}^i). \end{aligned}$$

(84)