The figure below shows the system block diagram of a rather

The figure below shows the system block diagram of a rather sophisticated bioengineering device for regulating the dosage of aesthetic gases being delivered to a patient during surgery. Note that the plant and controller are themselves feedback control systems. a) Derive an expression for the open-loop gain of the overall control system. b) Derive an expression for the closed-loop gain of the overall control system. c) If G_1 = 1, G_2 = 2, H_1 = 1, and H_2 = 2, what is the loop-gain of the overall system?

Solution

In open loop gain there is no feedback, so if x is inlet and y is outlet, then the equation will be:

Open loop gain G, G_out/G_in = G₁ +G₂

Gy/x = G₁ +G₂

Closed loop gain G’ is G_out/G_in where G_out is y and G_in is x

G_out is count for gain from both controller and plant so, G_out = G_out1 + G_out2

In plant, G₁ and H₁ will count for G_out1 = 1/G₁ +1/H₁ =G₁+ H₁/ G₁ H₁

In controller G_out2 = 1/G₂ +1/H₂ = G₂+ H₂/ G₂ H₂

So the closed loop gain G’y/x is = (G₁+ H₁/ G₁ H₁) + (G₂+ H₂/ G₂ H₂)

So overall loop gain = Open loop gain G+ Closed loop gain G’

Open loop gain G = 1+2 = 3               [from values]

Closed loop gain G’ = (1+1/1x1) + (2+2/2x2) = 2+1 = 3

Overall loop gain = 3+3 = 6