An improved relay feedback identification technique for Hammerstein model

Abstract

This research work is motivated by the success of relay feedback in linear system domain thereby an attempt has been made at extending it to nonlinear Hammerstein models. The identification is carried out using a relay with hysteresis in feedback, the parameter estimation is done solely by solving state-space equations at various points on one cycle of the input and output signals. The work considers not all but wider probable cases of the linear subsystem-real roots, complex conjugate roots, first order plus dead time systems, integrating second order plus dead time and repeated roots. The proposed method is simple compared to other works in literature as well as applicable to unstable systems. Simulations of various examples are carried out to demonstrate the efficacy of the proposed identification scheme.

Appendix

For the critically damped case the values of the matrices are given as,

$$\begin{aligned} \mathbf{A }&= \left[ { \begin{array}{cc} -a &{} 1 \\ 0 &{} -a \end{array} }\right] ;&\mathbf{B }= \left[ { \begin{array}{c} 0 \\ 1 \end{array} }\right] ;&\mathbf{C }=a^2 \left[ { \begin{array}{cc} 1&0 \end{array} }\right] \end{aligned}$$

where $a=\frac{1}{T_1}$. Using these state matrices in the state Eqs. (3) and (4) the following expressions are obtained,

$$\begin{aligned} y(t)&=a^2x_1(t) \end{aligned}$$

(43)

$$\begin{aligned} {\dot{x}}_1(t)&=-ax_1(t)+x_2(t) \end{aligned}$$

(44)

$$\begin{aligned} {\dot{x}}_2(t)&=-ax_2(t)+v(t-\theta ) \end{aligned}$$

(45)

Using Eqs. (43) and (44) the state vector $\mathbf{X }(t)$ in terms of the output can be expressed as,

$$\begin{aligned} \mathbf{X }(t)&= \left[ { \begin{array}{c} x_{1}(t) \\ x_{2}(t) \end{array} }\right] =\frac{1}{a^2} \left[ { \begin{array}{c} y(t) \\ {\dot{y}}(t)+ay(t) \end{array} }\right] \end{aligned}$$

(46)

Solving Eq. (4) for time range $0\leqslant t \leqslant t_1$,

$$\begin{aligned} \mathbf{X }(t)&=\left[ { \begin{array}{c} x_{1}(t) \\ x_{2}(t) \end{array} }\right] \nonumber \\&=\frac{1}{a^2} \left[ { \begin{array}{c} {\dot{y}}(0)te^{-at}+v_1\Big (1-e^{-at}-ate^{-at}\Big ) \\ {\dot{y}}(0)e^{-at}+av_1\Big (1-e^{-at}\Big ) \end{array} }\right] \end{aligned}$$

(47)

Equating the values of $\mathbf{X }(t)$ in Eq. (46) to that in Eq. (47) for $t=t_{ep}$,

$$\begin{aligned}&\varepsilon -{\dot{y}}(0)t_{ep}e^{-at_{ep}}=v_1\Big (1-e^{-at}-ate^{-at}\Big ) \end{aligned}$$

(48)

$$\begin{aligned}&-a\varepsilon +{\dot{y}}(0)e^{-at_{ep}}-{\dot{y}}(t_{ep})=-av_1\Big (1-e^{-at_{ep}}\Big ) \end{aligned}$$

(49)

Dividing Eq. (48) by (49) and solving gives,

$$\begin{aligned}&a^2\varepsilon t_{ep}+{\dot{y}}(0)\Big (1-at_{ep}-e^{-at_{ep}}\Big )\nonumber \\&\quad ={\dot{y}}(t_{ep})\Big (e^{at_{ep}}-at_{ep}-1\Big ) \end{aligned}$$

(50)

The above Eq. (50) is same as the one in Eq. (29), which is derived by imparting limiting conditions on Eq. (14), found for all the non-critical cases. From Eq. (49) the equation for finding $v_1$ can be derived as the following,

$$\begin{aligned} v_1=\dfrac{\varepsilon a+{\dot{y}}(t_{ep})-{\dot{y}}(0)e^{-at_{ep}}}{a\Big (1-e^{-at_{ep}}\Big )} \end{aligned}$$

(51)

Solving Eq. (4) for time range $t_1\leqslant t \leqslant t_3$ ,

$$\begin{aligned}&\mathbf{X }(t)=\frac{1}{a^2} \left[ { \begin{array}{c} -{\dot{y}}(t_2)(t-t_2)e^{a(t_2-t)}-v_2\Big (1-e^{a(t_2-t)}-a(t-t_2)e^{a(t_2-t)}\Big ) \\ -{\dot{y}}(t_2)e^{-at}+av_2\Big (1-e^{a(t_2-t)}\Big ) \end{array} }\right] \end{aligned}$$

(52)

Equating $x_2(t_{em})$ from Eq. (46) to that in Eq. (52) gives the value of $v_2$ as the following,

$$\begin{aligned} v_2=\dfrac{a\varepsilon +{\dot{y}}(t_{em}) -{\dot{y}}(t_2)e^{a(t_2-t_{em})}}{a\Big (1-e^{a(t_2-t_{em})}\Big )} \end{aligned}$$

(53)

It is worthy to note that the expression for $v_1$, $v_2$ in Eqs. (40), (41) are the same as that in Eqs. (51), (53). Hence, the equations of various parameters for critically damped case can be derived from the general non-critical case by imparting the limiting condition.