A control system has Gv = Gm = 1 and a second-order process Gp with Kp = 2, r 1 = 4 min, and r 2 = 1 min, which is to be controlled by a PI controller with Kc = 2 and r1 = r 1 = 4 min (i.e., the integral time of the controller is set equal to the dominant time constant). For a set-point change (a) Determine the closed-loop transfer function. (b) Derive the characteristic equation, which is a quadratic polynomial ins. Is it overdamped or underdamped? (c) Can a large value of Kc make the closed-loop process unstable?

Problem 11.24A control system has G = v = 1mand a second-order process transfer function with K = 2, p 1= 4 min, and 2 2 = 1 min, which is to be controlled by a PI controller with K = c and 1 = 4(i.e., the integral time of the controller is set equal to the dominant time constant). For aset-point change (a) Determine the closed-loop transfer function. (b) Derive the characteristicequation, which is a quadratic polynomial ins. Is it overdamped or underdamped (c) Can alarge value of Kc make the closed-loop process unstable Step-by-step solution Step 1 of 8 ^(a)Consider the following open-loop process transfer function: K p G = …………(1) p (1s + 1)( 2 + 1)Here,The process gain is K apTime constants are 1 and .2Consider that process gain K isp and time constants 1 and 2 are 4 and 1 respectively.Substitute the available values in equation (1). 2 Gp (4s + 1)(s + 1) ………(2)