GATE EC Control Systems Formula Sheet

Organised by the official GATE 2027 EC syllabus, Section 6: Control Systems. Each worked example was checked in code. Practise these: GATE EC Control Systems previous year questions.

Feedback, transfer functions and block diagrams

Syllabus: Basic control system components; feedback principle; transfer function; block diagram representation; signal flow graph.

Closed-loop transfer function

T(s) = \frac{G(s)}{1 + G(s)H(s)}\ \ (\text{negative feedback})

\begin{gathered} T(s) = \frac{G(s)}{1 + G(s)H(s)} \\ \text{(negative feedback)} \end{gathered}

Symbols

$G(s)$ : forward-path transfer function
$H(s)$ : feedback-path transfer function
$G(s)H(s)$ : loop transfer function

Applies when

Single loop, negative feedback. With positive feedback the denominator is 1 − GH.

Common mistake

Using 1 + G in the denominator when H ≠ 1.

Worked example

G = 10/(s + 1), H = 1: the DC gain: 0.909 (10/11)

Sensitivity to the plant

S_G^T = \frac{\partial T / T}{\partial G / G} = \frac{1}{1 + GH}

\begin{aligned} S_G^T &= \frac{\partial T / T}{\partial G / G} \\ &= \frac{1}{1 + GH} \end{aligned}

Symbols

$S_G^T$ : fractional change in T per fractional change in G

Applies when

Evaluated at a frequency (often DC). Feedback with large loop gain makes T insensitive to G; sensitivity to H approaches −1.

Common mistake

Thinking feedback reduces sensitivity to the sensor H as well. It doesn't: T ≈ 1/H.

Worked example

Loop gain GH = 99: 0.01

Mason's gain formula

T = \frac{\sum_k P_k \Delta_k}{\Delta}, \qquad \Delta = 1 - \sum L_i + \sum L_i L_j - \cdots

\begin{gathered} T = \frac{\sum_k P_k \Delta_k}{\Delta} \\ \Delta = 1 - \sum L_i \\ + \sum L_i L_j - \cdots \end{gathered}

Symbols

$P_k$ : gain of the k-th forward path
$L_i$ : loop gains; the sums of products are over non-touching loops
$\Delta_k$ : Δ with the loops touching path k removed

Applies when

Any linear signal flow graph. Products in Δ are only over loops that share no node.

Common mistake

Including touching loops in the product terms of Δ.

Worked example

Forward path G₁G₂; loops −G₁H₁ and −G₂H₂, not touching; G₁ = 2, G₂ = 3, H₁ = H₂ = 1: 0.5 (6/12)

Transient and steady-state analysis

Syllabus: Transient and steady-state analysis of LTI systems.

Second-order step response

M_p = e^{-\zeta\pi/\sqrt{1 - \zeta^2}}, \qquad t_p = \frac{\pi}{\omega_n\sqrt{1 - \zeta^2}}, \qquad t_s \approx \frac{4}{\zeta\omega_n}\ (2\%)

\begin{aligned} M_p &= e^{-\zeta\pi/\sqrt{1 - \zeta^2}}, \\ t_p &= \frac{\pi}{\omega_n\sqrt{1 - \zeta^2}}, \\ t_s &\approx \frac{4}{\zeta\omega_n}\ (2\%) \end{aligned}

Symbols

$\zeta$ : damping ratio, between 0 and 1
$\omega_n$ : natural frequency (rad/s)
$M_p$ : peak overshoot, as a fraction

Applies when

Standard form

ω_{n}

²/(s² + 2ζ

ω_{n}

s +

ω_{n}

²), underdamped, no zeros. Rise time and settling time are approximations.

Common mistake

Using the 5% settling rule (3/ζ

ω_{n}

) when the question asks for 2%.

Worked example

ζ = 0.5,

ω_{n}

= 10 rad/s: overshoot: 16.3 % (

t_{p}

= 0.363 s,

t_{s}

≈ 0.8 s)

Steady-state error and error constants

K_p = \lim_{s\to0} G,\ K_v = \lim_{s\to0} sG,\ K_a = \lim_{s\to0} s^2 G;\quad e_{ss} = \frac{1}{1 + K_p},\ \frac{1}{K_v},\ \frac{1}{K_a}

\begin{aligned} K_p &= \lim_{s\to0} G,\ K_v \\ &= \lim_{s\to0} sG,\ K_a \\ &= \lim_{s\to0} s^2 G; \\ e_{ss} &= \frac{1}{1 + K_p},\ \frac{1}{K_v},\ \frac{1}{K_a} \end{aligned}

Symbols

$G$ : open-loop transfer function with unity feedback
$K_p, K_v, K_a$ : position, velocity and acceleration constants

Applies when

Unit step, unit ramp and unit parabola inputs, unity negative feedback, and a stable closed loop (check that first).

Common mistake

Quoting an error constant for an unstable closed loop, where there is no steady state.

Worked example

G = 10/(s(s + 2)), unit ramp: 0.2 (

K_{v}

= 5)

Stability, frequency response, Bode and root locus

Syllabus: Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots.

Routh-Hurwitz for a cubic

s^3 + a_2 s^2 + a_1 s + a_0:\ \text{stable} \iff a_2, a_1, a_0 > 0 \text{ and } a_2 a_1 > a_0

\begin{gathered} s^3 + a_2 s^2 + a_1 s + a_0 \\ \text{stable} \iff a_2, a_1, a_0 > 0 \\ \text{and } a_2 a_1 > a_0 \end{gathered}

Symbols

$a_2, a_1, a_0$ : coefficients of the characteristic polynomial

Applies when

A third-order characteristic equation. When

a_{2}

a_{1}

a_{0}

exactly, a pair of roots sits on the jω axis at ω = √

a_{1}

Common mistake

Checking only that all coefficients are positive, which is necessary but not sufficient from third order on.

Worked example

s³ + 2s² + 3s + K: the largest K for stability: 6 (oscillates at √3 rad/s when K = 6)

Nyquist criterion

Z = N + P

Symbols

$Z$ : closed-loop poles in the right half-plane
$N$ : clockwise encirclements of −1 by G(jω)H(jω)
$P$ : open-loop poles in the right half-plane

Applies when

The Nyquist contour encloses the whole right half-plane, detouring round open-loop poles on the jω axis.

Common mistake

Counting counter-clockwise encirclements as positive: with Z = N + P, N counts clockwise ones.

Worked example

GH = 10/(s + 1)³: unstable closed-loop poles: 2 (P = 0, N = 2)

Gain and phase margins

GM = -20\log_{10}|GH(j\omega_{pc})|\ \text{dB}, \qquad PM = 180^\circ + \angle GH(j\omega_{gc})

\begin{gathered} GM = -20\log_{10}|GH(j\omega_{pc})| \\ \text{(in dB)} \\ PM = 180^\circ + \angle GH(j\omega_{gc}) \end{gathered}

Symbols

$\omega_{pc}$ : phase crossover: where the phase is −180°
$\omega_{gc}$ : gain crossover: where |GH| = 1

Applies when

Minimum-phase systems: stable when both margins are positive.

Common mistake

Reading the gain margin at the gain crossover frequency instead of the phase crossover.

Worked example

GH = 2/(s(s + 1)(s + 2)): gain margin: 9.54 dB (

ω_{pc}

= √2, |GH| = 1/3)

Bode asymptotes

\pm 20\ \text{dB/decade per zero or pole}, \qquad \left|\frac{1}{1 + j\omega T}\right|_{\omega = 1/T} = -3.01\ \text{dB}

\begin{gathered} \pm 20\ \text{dB/decade} \\ \text{per zero or pole} \\ \left|\frac{1}{1 + j\omega T}\right|_{\omega = 1/T} = -3.01\ \text{dB} \end{gathered}

Symbols

$T$ : time constant; the corner is at ω = 1/T

Applies when

Real poles and zeros. The asymptote is 3 dB off at the corner; a pole at the origin gives −20 dB/decade from the start.

Common mistake

Adding the slope change at the wrong corner, or forgetting a pole at the origin.

Worked example

1/(1 + jω) at ω = 1: -3.01 dB

Root locus asymptotes and breakaway

\sigma_A = \frac{\sum p_i - \sum z_i}{n - m}, \qquad \theta_A = \frac{(2k + 1)180^\circ}{n - m}, \qquad \frac{dK}{ds} = 0 \text{ at breakaway}

\begin{aligned} \sigma_A &= \frac{\sum p_i - \sum z_i}{n - m}, \\ \theta_A &= \frac{(2k + 1)180^\circ}{n - m}, \\ \frac{dK}{ds} &= 0 \text{ at breakaway} \end{aligned}

Symbols

$p_i, z_i$ : open-loop poles and zeros
$n, m$ : numbers of poles and zeros
$K$ : gain, from 1 + KG(s)H(s) = 0

Applies when

For K from 0 to ∞ (negative feedback). Branches start at the poles and end at the zeros or along the asymptotes.

Common mistake

Taking any root of dK/ds = 0 as a breakaway point: it must lie on a part of the real axis that belongs to the locus.

Worked example

Poles 0, −1, −2, no zeros: the centroid: -1 (asymptotes at 60°, 180° and 300°; breakaway at −0.423)

Compensators, PID and state variables

Syllabus: Compensators, PID controller; state variable model and solution of state equation of LTI systems.

Lead compensator

C(s) = \frac{1 + \alpha T s}{1 + T s}\ (\alpha > 1), \qquad \sin\phi_m = \frac{\alpha - 1}{\alpha + 1}\ \text{at}\ \omega_m = \frac{1}{T\sqrt{\alpha}}

\begin{aligned} C(s) &= \frac{1 + \alpha T s}{1 + T s}\ (\alpha > 1), \\ \sin\phi_m &= \frac{\alpha - 1}{\alpha + 1}\ \text{at}\ \omega_m \\ &= \frac{1}{T\sqrt{\alpha}} \end{aligned}

Symbols

$\alpha$ : ratio of the pole to the zero frequency
$\phi_m$ : maximum phase lead
$\omega_m$ : where it occurs (rad/s)

Applies when

Lead: zero below the pole, adding phase near crossover. A lag compensator has α < 1. A PID controller is

K_{p}

K_{i}

/s +

K_{d}

Common mistake

Putting the maximum phase at the arithmetic mean of the corners: it is at their geometric mean.

Worked example

α = 10: 54.9 ° of maximum lead

Transfer function from a state model

\dot{x} = Ax + Bu,\ y = Cx + Du \;\Rightarrow\; G(s) = C(sI - A)^{-1}B + D

\begin{aligned} \dot{x} &= Ax + Bu,\ y \\ &= Cx + Du \\ \Rightarrow\; G(s) &= C(sI - A)^{-1}B + D \end{aligned}

Symbols

$A, B, C, D$ : state, input, output and feed-through matrices

Applies when

LTI state model. The poles of G(s) are among the eigenvalues of A.

Common mistake

Writing B(sI − A)⁻¹C: the order is C, then the inverse, then B.

Worked example

A = [[0, 1], [−2, −3]], B = [0; 1], C = [1, 0], D = 0: G(0): 0.5 (G = 1/(s² + 3s + 2))

State transition matrix

\Phi(t) = e^{At} = \mathcal{L}^{-1}\!\left[(sI - A)^{-1}\right], \qquad x(t) = \Phi(t)\, x(0)\ \ (u = 0)

\begin{aligned} \Phi(t) &= e^{At} \\ &= \mathcal{L}^{-1}\!\left[(sI - A)^{-1}\right], \\ x(t) &= \Phi(t)\, x(0)\ \ (u = 0) \end{aligned}

Symbols

$\Phi(t)$ : state transition matrix

Applies when

LTI systems. Φ(0) = I and Φ(t₁ + t₂) = Φ(t₁)Φ(t₂).

Common mistake

Taking e^(At) entry by entry: the exponential of a matrix is not the matrix of exponentials.

Worked example

A = [[0, 1], [−2, −3]]: Φ₁₁ at t = 1: 0.6004 (2e^(−t) − e^(−2t))

Controllability and observability

\operatorname{rank}\,[B\ \ AB\ \cdots\ A^{n-1}B] = n, \qquad \operatorname{rank}\begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1}\end{bmatrix} = n

\begin{aligned} &\operatorname{rank}\,[B\ \ AB\ \cdots\ A^{n-1}B] \\ &\quad = n, \\ &\operatorname{rank}\begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1}\end{bmatrix} \\ &\quad = n \end{aligned}

Symbols

$n$ : number of states

Applies when

LTI state models. Full rank of the first matrix means controllable; of the second, observable.

Common mistake

Building the controllability matrix with powers of A on the wrong side: it is AB, not BA.

Worked example

A = [[0, 1], [−2, −3]], B = [0; 1]: rank of [B AB]: 2 (controllable)

Practise these: GATE EC Control Systems previous year questions

Source for the section headings: the official GATE 2027 EC syllabus, IIT Madras. The formulas, examples and notes are Studyloaf's own.

GATE EC Control Systems Formula Sheet

Feedback, transfer functions and block diagrams

Closed-loop transfer function

Sensitivity to the plant

Mason's gain formula

Transient and steady-state analysis

Second-order step response

Steady-state error and error constants

Stability, frequency response, Bode and root locus

Routh-Hurwitz for a cubic

Nyquist criterion

Gain and phase margins

Bode asymptotes

Root locus asymptotes and breakaway

Compensators, PID and state variables

Lead compensator

Transfer function from a state model

State transition matrix

Controllability and observability

Other GATE EC formula sheets