Direct synthesis

Figure 9.1: Closed loop circuit

We have

$$V(s) = \frac {G_c(s) G(s)}{1+G_c(s) G(s)}$$ (9.2)

where
V(s) : Overall closed-loop transfer function
$G_c$(s) : Controller transfer function
G(s) : System transfer function.
Therefore,

$$G_c(s) = \frac 1{G(s)} \frac {V(s)}{1-V(s)}$$ (9.3)

Let the desired closed loop transfer function be of the form

$$V(s)=\frac 1{(\tau_{cl}s+1)}$$ (9.4)

$$G(s)=\frac {K_p}{(\tau\_p s+1)}$$ (9.5)

By using the equations for G(s) and V(s), we get

$$G(c)=K_c(1 + \frac {1}{\tau s})$$ (9.6)

where,

$$K_c = \frac 1{K_p} (\tau_p / \tau_{cl} )$$ (9.7)

$$\tau_i = \tau_p$$ (9.8)

When $K_p$ and $\tau _p$ are known as a function of time, the values of $K_c$ and $\tau _i$ can be found as functions of temperature as well.