I have a second order system, it's response to a step change can be expressed in the s-space as:
$$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$
Which can be inverse transformed to: $$y(t)=K_{2}\Bigg(1-\frac{\tau_{1}e^{\frac{-(t+\theta)}{\tau_{1}}}-\tau_{2}e^{\frac{-(t+\theta)}{\tau_{2}}}}{\tau_{1}-\tau_{2}}\Bigg)$$
I have data points ($t,y(t)$), how can I use Gauss-Newton to estimate the parameters $\tau_{1} $, $\tau_{2} $ and $\theta$ .
The model being $$Y=\frac{Ke^{-\theta s}}{s(\tau^2s^2+2\zeta\tau s + 1)}$$ it is obvious that nonlinear regression is required. But this implies to have some réasonable estimates fo parameters $K,\theta,\tau,\zeta$.
To me, the first step would be to consider that $\theta$ is given a value. Now, rewrite the model as $$\frac {e^{-\theta s}}{s Y}=\frac {\tau^2}K s^2+\frac {2\zeta \tau}K s+\frac 1K$$ So, defining for each data point $$z_i=\frac {e^{-\theta s_i}}{s_i Y_i}$$ you have to fit the model $$z=a s^2+b s+c$$ which is simple. When you have obtained parameters $a,b,c$ from the linear regression, compute the predicted $Y$'s according to $$Y_i^{calc}=\frac{e^{-\theta s_i}}{s_i(as_i^2+bs_i+c)}$$ and $$SSQ(\theta)=\sum_{i=1}^n\big(Y_i-Y_i^{calc}\big)^2$$ Now, run different values of $\theta$ and look for a point looking more or less as a minimum. At this point, you have all you estimates $$K=\frac 1c \qquad \tau=\sqrt{aK}\qquad \zeta=\frac{bK}{2\tau}$$
If parameter $K$ is imposed, the problem is simpler since the model can write $$\frac 1s\Big(\frac {Ke^{-\theta s}}{s Y}-1\Big)={\tau^2} s+ {2\zeta \tau}$$ Define $$z_i=\frac 1{s_i}\Big(\frac {Ke^{-\theta s_i}}{s_i Y_i}-1\Big)$$ and fit the model $$z=a s +b$$ and do the same as above.