GATE Chemical Engineering Instrumentation and Process Control Syllabus

What is a deviation (perturbation) variable and why is it used?

A deviation variable is the difference between a variable and its steady-state value, e.g., $x' = x - x_s$. It is used because at steady state the deviation and its derivatives are zero, eliminating constant terms and giving transfer functions with zero initial conditions.

As an example, linearize the efflux term $F = c\sqrt{h}$ (tank outflow) about $h_s$.

$$F \approx c\sqrt{h_s} + \frac{c}{2\sqrt{h_s}}(h - h_s)$$ The linearized resistance/slope is $\dfrac{c}{2\sqrt{h_s}}$, valid for small deviations of $h$ about $h_s$.

What is inverse response in a process system?

Inverse response occurs when the process output initially moves in the opposite direction to its eventual steady-state direction following an input change. It is caused by competing fast and slow effects of opposite sign.

What feature in the transfer function causes inverse response?

Inverse response is caused by a right-half-plane (positive) zero in the transfer function, i.e., a numerator zero with positive real part. Example: $$G(s) = \frac{-\tau_a s + 1}{(\tau_1 s + 1)(\tau_2 s + 1)}, \quad \tau_a > 0.$$

For two opposing first-order responses $G(s)=\frac{K_1}{\tau_1 s+1}-\frac{K_2}{\tau_2 s+1}$, what condition gives inverse response?

Inverse response occurs when the steady-state gains satisfy $K_1 - K_2 > 0$ but the initial slope is negative, which requires $$\frac{K_1}{\tau_1} < \frac{K_2}{\tau_2}.$$ The faster (smaller $\tau$) effect dominates initially and acts opposite to the steady state.

Give a physical example of inverse response in chemical engineering.

Drum-boiler level: a sudden increase in cold feedwater first collapses bubbles and lowers the level (inverse) before the added mass eventually raises it. Also seen in some distillation tray temperatures and exothermic reactors.

Why does inverse response make control difficult, and how is it tied to RHP zeros?

The initial 'wrong way' movement misleads feedback controllers and limits achievable closed-loop speed. RHP zeros impose a fundamental bandwidth limitation; aggressive tuning destabilizes the loop, so controllers must be detuned.

What is the characteristic equation of a feedback control loop?

$$1 + G_{OL}(s) = 0 \quad\text{i.e.}\quad 1 + G_c G_v G_p G_m = 0$$ where $G_c, G_v, G_p, G_m$ are controller, valve, process, and measurement transfer functions. Its roots determine closed-loop stability.

State the fundamental stability criterion in terms of characteristic-equation roots.

A linear system is stable if and only if all roots of the characteristic equation (closed-loop poles) have negative real parts, i.e., lie in the left half of the complex plane. Any root in the right half plane makes the system unstable.

What does a pole on the imaginary axis (zero real part) indicate?

A pair of purely imaginary roots ($\pm j\omega$) corresponds to marginal (sustained, undamped) oscillation — the boundary between stable and unstable behavior. This condition defines the ultimate gain and ultimate period.

State the Routh-Hurwitz stability criterion and its necessary condition.

For the characteristic polynomial, a system is stable iff all entries in the first column of the Routh array have the same sign (no sign changes). The number of sign changes equals the number of RHP roots. Necessary condition: all polynomial coefficients present and same sign.

In the Routh test, what does a sign change in the first column tell you?

Each sign change in the first column of the Routh array corresponds to one root with a positive real part (in the right half plane). Any sign change means the closed-loop system is unstable.

Define ultimate gain $K_{cu}$ and ultimate period $P_u$.

$K_{cu}$ is the proportional controller gain at which the closed loop sustains continuous oscillation (marginal stability). $P_u$ is the period of that oscillation. They are found where the characteristic roots are purely imaginary and form the basis of Ziegler-Nichols tuning.

What is the frequency response of a system?

The steady-state response of a linear system to a sinusoidal input. For input $\sin(\omega t)$, the output is a sinusoid of the same frequency, scaled by the amplitude ratio $|G(j\omega)|$ and shifted by phase angle $\angle G(j\omega)$, obtained by substituting $s = j\omega$.

How are amplitude ratio (AR) and phase angle obtained from a transfer function?

Substitute $s = j\omega$ into $G(s)$. Then $$AR = |G(j\omega)|, \qquad \phi = \angle G(j\omega) = \tan^{-1}\!\left(\frac{\text{Im}}{\text{Re}}\right).$$ AR is the magnitude and $\phi$ the argument of the complex number $G(j\omega)$.

Give the amplitude ratio and phase angle for a first-order system $G(s)=\frac{K}{\tau s+1}$.

$$AR = \frac{K}{\sqrt{1 + (\omega\tau)^{2}}}, \qquad \phi = -\tan^{-1}(\omega\tau).$$ As $\omega\to\infty$, AR$\to 0$ and $\phi \to -90^\circ$. The corner frequency is $\omega = 1/\tau$.

What is the amplitude ratio and phase contribution of a pure time delay $e^{-\theta s}$?

$$AR = 1, \qquad \phi = -\omega\theta \ \text{(radians)}.$$ A dead time does not change amplitude but adds phase lag that grows without bound with frequency, which strongly degrades closed-loop stability.

State the Bode stability criterion.

A closed loop is stable if the open-loop amplitude ratio is less than 1 (0 dB) at the crossover frequency where the open-loop phase angle equals $-180^\circ$. If $AR > 1$ at $\phi = -180^\circ$, the loop is unstable.

Define gain margin and phase margin.

Gain margin $= \dfrac{1}{AR}$ evaluated at the phase-crossover frequency (where $\phi=-180^\circ$). Phase margin $= 180^\circ + \phi$ evaluated at the gain-crossover frequency (where $AR=1$). Both measure relative stability; typical design: GM $\approx 1.7$–$2$, PM $\approx 30^\circ$–$45^\circ$.

What is the Nyquist stability criterion (statement)?

Plot the open-loop $G(j\omega)$ for $\omega$ from $-\infty$ to $\infty$ (Nyquist plot). The closed loop is stable if the number of encirclements of the $(-1, 0)$ point (clockwise) equals the number of open-loop RHP poles, i.e., $Z = N + P$ with $Z=0$ for stability.

Write the ideal (parallel) form of a PID controller.

$$G_c(s) = K_c\left(1 + \frac{1}{\tau_I s} + \tau_D s\right)$$ where $K_c$ is the proportional gain, $\tau_I$ the integral (reset) time, and $\tau_D$ the derivative time.

What are the effects of proportional, integral, and derivative actions in a PID controller?

Proportional: speeds response, reduces but leaves offset. Integral: eliminates steady-state offset but can add lag/oscillation. Derivative: anticipates error trend, adds damping and stability, but amplifies measurement noise.

What causes offset with a pure proportional controller, and how is it removed?

A P-only controller needs a nonzero error to produce a control action that holds a load, leaving a steady-state offset $\propto 1/(1+K_c K_p)$. Adding integral action drives the offset to zero because integration continues until error is exactly zero.

State the Ziegler-Nichols continuous-cycling tuning rules for P, PI, and PID controllers.

Using ultimate gain $K_{cu}$ and ultimate period $P_u$: P: $K_c=0.5K_{cu}$. PI: $K_c=0.45K_{cu},\ \tau_I = P_u/1.2$. PID: $K_c=0.6K_{cu},\ \tau_I = P_u/2,\ \tau_D = P_u/8$.

What is the basis of the Cohen-Coon tuning method?

Cohen-Coon uses an open-loop step (process reaction curve) fit to a First-Order-Plus-Dead-Time model $G(s)=\frac{K e^{-\theta s}}{\tau s+1}$. Tuning formulas use $K$, $\tau$, $\theta$ to give roughly quarter-decay response; it works better than Z-N for large dead-time/lag ratios.

Name common time-integral performance criteria used for controller tuning.

IAE $=\int_0^\infty |e|\,dt$; ISE $=\int_0^\infty e^{2}\,dt$; ITAE $=\int_0^\infty t|e|\,dt$. ISE penalizes large errors, ITAE penalizes long-persisting errors and gives the least oscillatory, well-damped tuning.

What is cascade control and when is it advantageous?

Cascade control uses two nested loops: a master (primary, outer) controller sets the setpoint of a slave (secondary, inner) controller that acts on the valve. It is advantageous when a measurable secondary disturbance affects an intermediate variable faster than the primary output, allowing early correction.

What is the key requirement on the inner versus outer loop dynamics in cascade control?

The inner (secondary) loop must be significantly faster than the outer (primary) loop — typically at least 3–5 times faster. This lets the inner loop reject disturbances before they propagate to the slow primary variable.

What is feedforward control and how does it differ from feedback?

Feedforward control measures a disturbance and computes a corrective action before the disturbance affects the controlled variable. Unlike feedback (which acts only after an error appears), feedforward is proactive but requires an accurate process model and cannot correct unmeasured disturbances.

Give the ideal feedforward controller transfer function for a measured disturbance.

For load $G_L$ and manipulated-path $G_p$, the ideal feedforward controller is $$G_{ff}(s) = -\frac{G_L(s)}{G_p(s)}$$ so that the disturbance effect is exactly cancelled. In practice it is combined with feedback (feedforward-feedback control) to handle model error and unmeasured disturbances.