Operating characteristics of the valve is nonlinear (Fig.), a cascade controller established to reduce the effect of this.

Both controllers are proportional controllers. TC has gain K_p and FC has the gain (R). Since all signals are standard and dynamics of the FT and TT can be neglected, the transfer functions of FT and TT both equal to 1 The valve has the transfer function q (s) / u (s) = K_v / (τ_v (s) +1). The heat exchanger has transfer function
T (s) / q (s) = 3 / (10 (s) +1). τ_v is constant, all signals in the block diagram and transfer functions are deviation variables.

1 - Define K and τ ?

2 - Find (K_v_min) and (K_v_max) for the valve by using the operating characteristic curve (figure) and the highest and lowest value of K, while (R ) = 10 ?

3 - Assume that K = 0.9 0g tau = 2 SP changed abruptly 1 unit. Determine the gain Kp to step answer T (t) will have a overshoot of 20%?

4 - If the TC must extended with the I-effect, which integral time do you suggest?