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

The controlled system has the transfer function G(s)=K/(s-¿1)(s-¿2). The control input is u(t), |u(t)|¿1, and the out put is y1(t). the control which forces any initial state y1(0), ¿1(0) to the terminal state y1(T), ¿1,(T), such that -K/¿1¿2¿y1(T)¿K/¿1¿2, ¿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 G0; 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 G0, then u(t)=0 is used.