# Additive perturbation

consider the following transfer functions: $$G(s)=\frac{1}{\tau s+1}G_0 \quad \tau_{min}<\tau<\tau_{max}$$ show that this system equal to model with additive perturbation as follow $$G(s)=G_1(s)+W(s)\Delta \qquad ||\Delta||\le1$$ $$\Delta$$ is perturbation and where $$W(s)=-\frac{r_{\tau}\tau_ms}{(1+\tau_m s)^2}\qquad G_1(s)=\frac{G_0}{(1+\tau_m s)}$$ in this $$\tau_m=\frac{\tau_{max}+\tau_{min}}{2}\quad r_{\tau}=\frac{\tau_{max}-\tau_{min}}{\tau_{max}+\tau_{min}}$$ and

• @AJN I edited the question. thank you for your attention Nov 28 '21 at 5:38
• Is there any assumption on $G_0$ ? Like its order or if it a proper transfer function etc. ? I think that there probly exists some assumption that order of $G_0$ denominator is one more than $G_0$ numerator.
– AJN
Nov 28 '21 at 16:55
• yes, $G_0$ is a proper transfer function. Nov 28 '21 at 19:04