Minimum-Fuel Control of Second-Order Systems with Real Poles

The controlled system has the transfer function G(s)=K/(s-λ
₁
)(s-λ
₂
). The control input is u(t), |u(t)|≤1, and the out put is y
₁
(t). the control which forces any initial state y
₁
(0), ẏ
₁
(0) to the terminal state y
₁
(T), ẏ
₁
,(T), such that -K/Λ
₁
Λ
₂
≤y
₁
(T)≤K/λ
₁
λ
₂
, ẏ
₁
(T)=0, and which minimizes the fuel, F(T)=∫
₀
^T
|u(t)|dt is determined. The phase plane is divided into three sets G
_-
, G
₊
, and G
₀
; if the state is in G
_-
then u(t)=-1 is used; if the state is in G
₊
, then u(t)=+1 is used; if the state is in G
₀
, then u(t)=0 is used.