In an industrial control system, the output scaling factor, FFF, depends on the input voltage, vvv. The voltage is regulated by a controller setting, sss. The relationship is defined by the following functions:
F(v)=v+3v−2,v∈R,v≠2 F(v) = \frac{v + 3}{v - 2}, \quad v \in \mathbb{R}, v \neq 2 F(v)=v−2v+3,v∈R,v=2 V(s)=2s+1,s∈R V(s) = 2s + 1, \quad s \in \mathbb{R} V(s)=2s+1,s∈RDetermine the output scaling factor when the controller is set to 1.51.51.5, by finding the value of FV(1.5)FV(1.5)FV(1.5).
Derive an expression for the inverse scaling function, F−1(x)F^{-1}(x)F−1(x).
Solve the equation F(1k)=V(k+1)F\left(\frac{1}{k}\right) = V(k + 1)F(k1)=V(k+1) for kkk.
