The GATE Grind

GATE 2025 CH – Question 62

Instrumentation and Process Control · Cascade and feedforward control · 2 marks · Numerical answer

The block diagram of a series cascade control system (with time in minutes) is shown in the figure. For $\tau_I = 8$ min and $K_c^s = 1$, the maximum value of $K_c^m$, below which the cascade control system is stable, is ____ (rounded off to the nearest integer).

a cascade loop. The outer (master) controller is $K_c^m\left(\frac{\tau_Is + 1}{\tau_Is}\right)$ and its output is the set point of the inner loop. The inner (slave) controller is $K_c^s$ followed by a block $\frac{1}{2s + 1}$, and the inner feedback is taken from the output of that block. The block $\frac{2}{(8s + 1)(4s + 1)}$ follows, and the outer feedback is taken from its output.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 10

Explanation

The inner loop is $K_c^s\frac{1}{2s + 1}$ with unit feedback, so its closed-loop transfer function is $\frac{K_c^s}{2s + 1 + K_c^s} = \frac{1}{2s + 2}$ for $K_c^s = 1$. The master controller with $\tau_I = 8$ is $K_c^m\frac{8s + 1}{8s}$. The open loop of the outer loop is $K_c^m\frac{8s + 1}{8s} \cdot \frac{1}{2s + 2} \cdot \frac{2}{(8s + 1)(4s + 1)} = \frac{K_c^m}{8s(s + 1)(4s + 1)}$, since the factor $(8s + 1)$ cancels. The characteristic equation is $8s(s + 1)(4s + 1) + K_c^m = 0$, that is $32s^3 + 40s^2 + 8s + K_c^m = 0$. By the Routh test, the system is stable when all coefficients are positive and $40 \times 8 > 32K_c^m$, that is $K_c^m < 10$. So the maximum value is 10.