Control Systems
Transfer functions, transient specifications, Routh-Hurwitz, root locus, Bode and Nyquist, and steady-state error.
About this subject
Control Systems studies how feedback makes a system behave the way we want. It covers transfer functions and block diagrams, transient and steady-state response, stability (Routh-Hurwitz, root locus) and frequency-domain methods (Bode, Nyquist, gain and phase margins).
Most second-order questions depend on just two numbers, the damping ratio $\zeta$ and the natural frequency $\omega_n$, and the skill is knowing which quantities depend on which. Most stability questions reduce to a Routh table or a crossover frequency.
Solved questions
Each question is a practice problem written for this portal at GATE standard. Try it first, then open the full derivation, and read the note on the idea behind it.
- NAT Peak overshoot of an underdamped second-order system Transient response · 2 marks
- MCQ Range of gain K for stability by the Routh-Hurwitz criterion Stability · 2 marks
- NAT Steady-state error of a type-1 system to a unit-ramp input Steady-state error · 1 mark
- NAT Gain margin of a third-order open-loop transfer function Frequency response · 2 marks
Key formulas
The results these questions lean on most. Know what each symbol means, and when each one applies.
| What | Formula |
|---|---|
| Standard second-order system | $T(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$ |
| Peak overshoot | $M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ - depends on $\zeta$ only |
| Peak time | $t_p = \pi / (\omega_n\sqrt{1-\zeta^2})$ |
| Settling time (2%) | $t_s \approx 4 / (\zeta\omega_n)$ |
| Error constants | $K_p = \lim_{s\to 0} GH$, $\;K_v = \lim_{s\to 0} sGH$, $\;K_a = \lim_{s\to 0} s^2GH$ |
| Steady-state errors | step $1/(1+K_p)$, ramp $1/K_v$, parabola $1/K_a$ |
| Gain margin | $GM = -20\log_{10}|G(j\omega_{pc})|$ dB at the phase crossover, where $\angle G = -180^\circ$ |
How to prepare
- Match the error constant to the input: $K_p$ for a step, $K_v$ for a ramp, $K_a$ for a parabola.
- In a Routh table, check every entry of the first column - including the last one.
- Remember which responses depend on $\zeta$ alone (overshoot) and which also on $\omega_n$ (peak and settling times).
- For margins, find the phase crossover first for the gain margin, and the gain crossover first for the phase margin.
Common traps
Each of these costs marks every year. The questions above are built to catch them.
- Putting $\omega_n$ into the overshoot formula.
- Ignoring the lower bound on gain from the last row of the Routh table.
- Using the position error constant for a ramp input.
- Forgetting the fixed $-90^\circ$ from a pole at the origin when finding a crossover.