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ENGR5001

ENGR5001 must memorise

Everything you need in your head for the exam: formulas that are not on the formula sheet, and the definitions examiners ask for. Print it, or cover the right-hand side and test yourself.

Formulas not on the formula sheet

Needed in past-paper solutions

Generalised first order form τdxdt+x=Ku(t)\tau\dfrac{dx}{dt} + x = Ku(t)
used in 4 questions, e.g. Lecture 2 example, thermal system (liquid cooling), Lecture 2 example, single tank hydraulic system, ENGR202 2022 Q B1
Steady state gain: set s=0s = 0
used in 4 questions, e.g. Old spec 3.3 exercise, Transfer Function of a 4th order equation, Old spec 3.4 example, dominant poles of a third order system, ENGR202 2017 Q A1
Series rule and negative feedback rule G11+G1G2\dfrac{G_1}{1 + G_1G_2}
used in 4 questions, e.g. Old spec 4.4 and 4.5 examples, proportional and integral action on a second order plant, ENGR202 2017 Q A3, ENGR202 2021 Q1
Negative feedback rule: G11+G1G2\dfrac{G_1}{1 + G_1 G_2}; series rule: G1G2G_1 G_2
used in 4 questions, e.g. ENGR202 2022 Q B2, ENGR202 2023 Q B2, ENGR202 2024 Q B1
Generalised second order form d2xdt2+2ζωndxdt+ωn2x=Ku(t)\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t), steady state gain K/ωn2K/\omega_n^2
used in 3 questions, e.g. Lecture 2 example, harmonic oscillator with external force, Lecture 2 example, mass-spring-damper, Part 1 SAQ 1
Generalised second order form d2xdt2+2ζωndxdt+ωn2x=Ku(t)\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t)
used in 3 questions, e.g. Lecture 2 example, LC circuit, Old spec 1.2 example, two tank hydraulic system, Part 1 SAQ 2
Third order: stable if all coefficients are positive and a1a2>a0a3a_1a_2 > a_0a_3
used in 3 questions, e.g. Old spec 3.5 example, Hurwitz shortcut for a third order system, Old spec 5.2 example, PID control stability, Old spec 5.4 example, Nyquist stability and Hurwitz limit for a third order loop
∣jω+a∣=ω2+a2|j\omega + a| = \sqrt{\omega^2 + a^2}, Arg(jω+a)=tan⁡−1(ω/a)\mathrm{Arg}(j\omega + a) = \tan^{-1}(\omega/a)
used in 3 questions, e.g. ENGR202 2019 Q B2, ENGR202 2021 Q2, Old spec 5.3 exercise
Conservation of volume (mass) for an incompressible fluid: Adhdt=Qi−QoA\dfrac{dh}{dt} = Q_i - Q_o
used in 2 questions, e.g. Lecture 2 example, single tank hydraulic system, ENGR202 2019 Q B3
Final Value Theorem lim⁡t→∞f(t)=lim⁡s→0sF(s)\lim_{t\to\infty} f(t) = \lim_{s\to 0} sF(s)
used in 2 questions, e.g. Old spec 3.2 worked example, mass-spring-damper step response by Laplace, ENGR201 2017 B3
Characteristic equation: denominator of the Transfer Function =0= 0; its roots are the poles
used in 2 questions, e.g. Old spec 3.4 example, pole of a first order system, ENGR202 2021 Q1
L−1[s−a(s−a)2+b2]=eatcos⁡bt\mathcal{L}^{-1}\left[\dfrac{s - a}{(s - a)^2 + b^2}\right] = e^{at}\cos bt, L−1[b(s−a)2+b2]=eatsin⁡bt\mathcal{L}^{-1}\left[\dfrac{b}{(s - a)^2 + b^2}\right] = e^{at}\sin bt
used in 2 questions, e.g. ENGR201 Laplace deck example, partial fractions with a quadratic factor, ENGR201 Laplace deck example, railway buffer hit by a 5 s pulse
eatcos⁡ωt↔s−a(s−a)2+ω2e^{at}\cos\omega t \leftrightarrow \dfrac{s - a}{(s - a)^2 + \omega^2}, eatsin⁡ωt↔ω(s−a)2+ω2e^{at}\sin\omega t \leftrightarrow \dfrac{\omega}{(s - a)^2 + \omega^2}
used in 2 questions, e.g. ENGR201 2022 A2, ENGR201 2024 A2(a)-(b)
Pole placement: match the closed-loop characteristic equation to s2+2ζωns+ωn2s^2 + 2\zeta\omega_n s + \omega_n^2
used in 2 questions, e.g. ENGR202 2017 Q A1, Old spec 4.6 exercise
Stability shortcuts: all coefficients must exist and have the same sign; an=0a_n = 0 gives a pole at the origin; for a second order equation, all coefficients positive is necessary AND sufficient
used in 2 questions, e.g. ENGR202 2017 Q A3, ENGR270 2026 Q1
Unity feedback: CG1+CG\dfrac{CG}{1 + CG}
used in 2 questions, e.g. ENGR202 2018 Q A2, ENGR202 2019 Q B1
Laplace with zero initial conditions: x˙→sX\dot x \to sX, x¨→s2X\ddot x \to s^2X
used in 2 questions, e.g. ENGR202 2019 Q B1, ENGR202 2024 Q B2
Poles, stability, steady state gain G(0)G(0)
used in 2 questions, e.g. ENGR202 2019 Q B2, ENGR270 2026 Q1
Steady state gain: G(0)G(0), i.e. set s=0s = 0 in the Transfer Function
used in 2 questions, e.g. ENGR202 2021 Q1, ENGR202 2025 Q2
Second order steady state gain: K/ωn2K/\omega_n^2
used in 2 questions, e.g. ENGR202 2023 Q B1, ENGR202 2024 Q B2
Poles: roots of the denominator; zeros: roots of the numerator; stable if all poles have negative real parts
used in 2 questions, e.g. ENGR202 2025 Q2, Old spec 3.4 exercise 1
Newton's second law ∑F=Mx¨\sum F = M\ddot x; linear spring force K′xK'x
used in 1 question, e.g. Lecture 2 example, harmonic oscillator with external force
Inductor VL=LdidtV_L = L\dfrac{di}{dt}; capacitor VC=qC=1C∫i dtV_C = \dfrac{q}{C} = \dfrac{1}{C}\displaystyle\int i\,dt; Kirchhoff's voltage law
used in 1 question, e.g. Lecture 2 example, LC circuit
Newton's second law; spring force K′xK'x; viscous damper force CdxdtC\dfrac{dx}{dt}
used in 1 question, e.g. Lecture 2 example, mass-spring-damper
Newton's law of cooling: rate of change of temperature proportional to the excess temperature
used in 1 question, e.g. Lecture 2 example, thermal system (liquid cooling)
Linear outflow assumption Qo=K′hQ_o = K'h
used in 1 question, e.g. Lecture 2 example, single tank hydraulic system
Volume balance Adhdt=Qin−QoutA\dfrac{dh}{dt} = Q_{in} - Q_{out}; linear orifice Q=KhQ = Kh
used in 1 question, e.g. Old spec 1.2 example, two tank hydraulic system
Linearisation about an operating point x0x_0: g(x)≈g(x0)+dgdx∣x=x0(x−x0)g(x) \approx g(x_0) + \left.\dfrac{dg}{dx}\right|_{x = x_0}(x - x_0) (first order Taylor series)
used in 1 question, e.g. Old spec 1.4 example, linearised free pendulum
Unit impulse response of τx˙+x=Ku\tau\dot x + x = Ku (zero initial conditions): x(t)=Kτe−t/τx(t) = \dfrac{K}{\tau}e^{-t/\tau}
used in 1 question, e.g. Old spec 2.4 numerical example, first order impulse response
First order unit step response x(t)=K(1−e−t/τ)x(t) = K\left(1 - e^{-t/\tau}\right)
used in 1 question, e.g. Old spec 2.4 exercise, key points of the first order step response
K=x(t→∞)u(t→∞)K = \dfrac{x(t \to \infty)}{u(t \to \infty)} (for a step from zero)
used in 1 question, e.g. Old spec 2.4 worked example, axial fan model from a step test
At t=τt = \tau after the step: x=0.632 x(t→∞)x = 0.632\,x(t \to \infty)
used in 1 question, e.g. Old spec 2.4 worked example, axial fan model from a step test
L[x˙]=sX(s)−x(0)\mathcal{L}[\dot x] = sX(s) - x(0), L[x¨]=s2X(s)−sx(0)−x˙(0)\mathcal{L}[\ddot x] = s^2X(s) - sx(0) - \dot x(0); unit step →1/s\to 1/s; e−αt↔1s+αe^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha}
used in 1 question, e.g. Old spec 3.2 worked example, mass-spring-damper step response by Laplace
Steady state gain: set s=0s = 0 in the Transfer Function; x(t→∞)=gain×u0x(t \to \infty) = \text{gain} \times u_0 for a constant input u0u_0 (stable system)
used in 1 question, e.g. Old spec 3.3 example, steady state gain of a Transfer Function
x(t→∞)=steady state gain×u0x(t \to \infty) = \text{steady state gain} \times u_0
used in 1 question, e.g. Old spec 3.3 "exam question (1 mark)", steady state displacement
Zero initial conditions: dnxdtn→snX\dfrac{d^n x}{dt^n} \to s^nX
used in 1 question, e.g. Old spec 3.3 exercise, Transfer Function of a 4th order equation
Stable if all poles have negative real parts
used in 1 question, e.g. Old spec 3.4 example, pole of a first order system
Zeros: roots of the numerator; poles: roots of the denominator
used in 1 question, e.g. Old spec 3.4 example, poles and zeros of a second order system
Quadratic formula s=−b±b2−4ac2as = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
used in 1 question, e.g. Old spec 3.4 example, poles and zeros of a second order system
Complex poles σ±jω\sigma \pm j\omega give a mode eσtsin⁡(ωt+ϕ)e^{\sigma t}\sin(\omega t + \phi)
used in 1 question, e.g. Old spec 3.4 example, poles and zeros of a second order system
Dominant pole(s): the pole(s) closest to the imaginary axis, which decay most slowly
used in 1 question, e.g. Old spec 3.4 example, dominant poles of a third order system
Zeros do not affect stability; a zero with positive real part makes the system non-minimum phase
used in 1 question, e.g. Old spec 3.4 example, effect of a zero
Unit impulse δ(t)\delta(t) is the derivative of the unit step
used in 1 question, e.g. Old spec 3.4 example, effect of a zero
Poles on the imaginary axis (non-repeated), the rest in the left half plane: marginally stable
used in 1 question, e.g. Old spec 3.4 example, third order system with poles on the imaginary axis
Hurwitz determinant for a0sn+a1sn−1+⋯+an=0a_0s^n + a_1s^{n-1} + \dots + a_n = 0: rows (a1,a3,a5,… )(a_1, a_3, a_5, \dots), (a0,a2,a4,… )(a_0, a_2, a_4, \dots), then the same pair shifted one column right, and so on; stable if all principal minors Δ1,…,Δn>0\Delta_1, \dots, \Delta_n > 0
used in 1 question, e.g. Old spec 3.5 example, fourth order Hurwitz determinant
Necessary condition: all coefficients present and of the same sign
used in 1 question, e.g. Old spec 3.5 example, fourth order Hurwitz determinant
Hurwitz criterion; shortcut: it is enough that either Δ1,Δ3,Δ5,⋯>0\Delta_1, \Delta_3, \Delta_5, \dots > 0 or Δ2,Δ4,⋯>0\Delta_2, \Delta_4, \dots > 0 (with all coefficients positive)
used in 1 question, e.g. Old spec 3.5 example, Hurwitz shortcut for a third order system
Necessary conditions for stability: all coefficients exist (non-zero) and have the same sign
used in 1 question, e.g. Old spec 3.5 examples, stability short cuts
If these hold, the Hurwitz criterion (or the poles) is still needed
used in 1 question, e.g. Old spec 3.5 examples, stability short cuts
Hurwitz determinant and principal minors
used in 1 question, e.g. Old spec 3.5 example, Hurwitz conditions for the mass-spring-damper
Series rule: X=G1(s)G2(s)UX = G_1(s)G_2(s)U
used in 1 question, e.g. Old spec 3.6 examples, blocks in series
Series rule: X=G1(s)G2(s)G3(s)UX = G_1(s)G_2(s)G_3(s)U
used in 1 question, e.g. Old spec 3.6 example, wind turbine blocks in series
Unity negative feedback: X=C(s)G(s)1+C(s)G(s)DX = \dfrac{C(s)G(s)}{1 + C(s)G(s)}D
used in 1 question, e.g. Old spec 3.6 example, closed-loop control of vehicle speed
Definition F(s)=L[f(t)]=∫0∞e−stf(t) dtF(s) = \mathcal{L}[f(t)] = \displaystyle\int_0^\infty e^{-st}f(t)\,dt
used in 1 question, e.g. ENGR201 Laplace deck examples, transforms from the definition
∫eβtdt=1βeβt\displaystyle\int e^{\beta t}dt = \dfrac{1}{\beta}e^{\beta t}; cosh⁡at=eat+e−at2\cosh at = \dfrac{e^{at} + e^{-at}}{2}, sinh⁡at=eat−e−at2\sinh at = \dfrac{e^{at} - e^{-at}}{2}
used in 1 question, e.g. ENGR201 Laplace deck examples, transforms from the definition
Linearity L[αf+βg]=αL[f]+βL[g]\mathcal{L}[\alpha f + \beta g] = \alpha\mathcal{L}[f] + \beta\mathcal{L}[g]
used in 1 question, e.g. ENGR201 Laplace deck examples, transforms from the definition
L[eat]=1s−a\mathcal{L}[e^{at}] = \dfrac{1}{s - a}, L[sin⁡ωt]=ωs2+ω2\mathcal{L}[\sin\omega t] = \dfrac{\omega}{s^2 + \omega^2}; linearity
used in 1 question, e.g. ENGR201 Laplace deck example, transform by linearity
L[u(t−c)]=e−css\mathcal{L}[u(t - c)] = \dfrac{e^{-cs}}{s}
used in 1 question, e.g. ENGR201 Laplace deck example 1, piecewise signal built from steps
L[f′(t)]=sF(s)−f(0)\mathcal{L}[f'(t)] = sF(s) - f(0); L[f′′(t)]=s2F(s)−sf(0)−f′(0)\mathcal{L}[f''(t)] = s^2F(s) - sf(0) - f'(0)
used in 1 question, e.g. ENGR201 Laplace deck examples, Laplace transform of derivatives
L[sin⁡t]=1s2+1\mathcal{L}[\sin t] = \dfrac{1}{s^2 + 1}, L[cos⁡t]=ss2+1\mathcal{L}[\cos t] = \dfrac{s}{s^2 + 1}, L[tn]=n!sn+1\mathcal{L}[t^n] = \dfrac{n!}{s^{n+1}}
used in 1 question, e.g. ENGR201 Laplace deck examples, Laplace transform of derivatives
L[y′]=sY−y(0)\mathcal{L}[y'] = sY - y(0), L[y′′]=s2Y−sy(0)−y′(0)\mathcal{L}[y''] = s^2Y - sy(0) - y'(0); constant c→c/sc \to c/s; e−at↔1s+ae^{-at} \leftrightarrow \dfrac{1}{s + a}
used in 1 question, e.g. ENGR201 Laplace deck example, solving a second order ODE
L[δ(t−c)]=e−cs\mathcal{L}[\delta(t - c)] = e^{-cs}; L[x′′]=s2X−sx(0)−x′(0)\mathcal{L}[x''] = s^2X - sx(0) - x'(0)
used in 1 question, e.g. ENGR201 Laplace deck tutorial Q7, water tank struck by a gust (impulse)
Shift: L−1[e−csF(s)]=f(t−c)u(t−c)\mathcal{L}^{-1}[e^{-cs}F(s)] = f(t - c)u(t - c); L−1[b(s−a)2+b2]=eatsin⁡bt\mathcal{L}^{-1}\left[\dfrac{b}{(s - a)^2 + b^2}\right] = e^{at}\sin bt
used in 1 question, e.g. ENGR201 Laplace deck tutorial Q7, water tank struck by a gust (impulse)
L−1[1s+a]=e−at\mathcal{L}^{-1}\left[\dfrac{1}{s + a}\right] = e^{-at}; linearity of L−1\mathcal{L}^{-1}
used in 1 question, e.g. ENGR201 Laplace deck examples, inverse transforms with distinct real poles
Partial fractions: one term As+p\dfrac{A}{s + p} per distinct linear factor
used in 1 question, e.g. ENGR201 Laplace deck examples, inverse transforms with distinct real poles
Repeated factor (s+1)3(s + 1)^3: terms As+1+B(s+1)2+C(s+1)3\dfrac{A}{s + 1} + \dfrac{B}{(s + 1)^2} + \dfrac{C}{(s + 1)^3}
used in 1 question, e.g. ENGR201 Laplace deck example, partial fractions with a repeated pole
L−1[n!(s−a)n+1]=tneat\mathcal{L}^{-1}\left[\dfrac{n!}{(s - a)^{n+1}}\right] = t^ne^{at}
used in 1 question, e.g. ENGR201 Laplace deck example, partial fractions with a repeated pole
Quadratic factor with no real roots: term Bs+Cs2+ps+q\dfrac{Bs + C}{s^2 + ps + q}; complete the square
used in 1 question, e.g. ENGR201 Laplace deck example, partial fractions with a quadratic factor
L[u(t−c)]=e−css\mathcal{L}[u(t - c)] = \dfrac{e^{-cs}}{s}; shift theorem L−1[e−csF(s)]=f(t−c)u(t−c)\mathcal{L}^{-1}[e^{-cs}F(s)] = f(t - c)u(t - c)
used in 1 question, e.g. ENGR201 Laplace deck example, railway buffer hit by a 5 s pulse
L[y′′]=s2Y−sy(0)−y′(0)\mathcal{L}[y''] = s^2Y - sy(0) - y'(0)
used in 1 question, e.g. ENGR201 Laplace deck example, free vibration of a two-mass system
L−1[αs2+α2]=sin⁡αt\mathcal{L}^{-1}\left[\dfrac{\alpha}{s^2 + \alpha^2}\right] = \sin\alpha t, L−1[ss2+α2]=cos⁡αt\mathcal{L}^{-1}\left[\dfrac{s}{s^2 + \alpha^2}\right] = \cos\alpha t
used in 1 question, e.g. ENGR201 Laplace deck example, free vibration of a two-mass system
x(t→∞)=K×u0x(t \to \infty) = K \times u_0; open-loop proportional control U=KpDU = K_pD
used in 1 question, e.g. Old spec 4.2 worked example, open and closed loop control of a DC motor
Closed loop with U=Kp(D−X)U = K_p(D - X): X=HKp1+HKpDX = \dfrac{HK_p}{1 + HK_p}D
used in 1 question, e.g. Old spec 4.2 worked example, open and closed loop control of a DC motor
Final Value Theorem lim⁡t→∞x(t)=lim⁡s→0sX(s)\lim_{t\to\infty}x(t) = \lim_{s\to 0}sX(s)
used in 1 question, e.g. Old spec 4.4 and 4.5 examples, proportional and integral action on a second order plant
Derivative action u(t)=kddedtu(t) = k_d\dfrac{de}{dt}, i.e. U=kds(D−X)U = k_ds(D - X)
used in 1 question, e.g. Old spec 4.6 example, derivative action on a second order plant
Negative feedback rule; Final Value Theorem
used in 1 question, e.g. Old spec 4.6 example, derivative action on a second order plant
PI control law u(t)=kpe(t)+kI∫e dtu(t) = k_pe(t) + k_I\displaystyle\int e\,dt, i.e. C(s)=kp+kIs=kI+kpssC(s) = k_p + \dfrac{k_I}{s} = \dfrac{k_I + k_ps}{s}
used in 1 question, e.g. Old spec 5.1 example, PI control stability and steady state
Third order Hurwitz: a0s3+a1s2+a2s+a3=0a_0s^3 + a_1s^2 + a_2s + a_3 = 0 stable if all ai>0a_i > 0 and a1a2>a0a3a_1a_2 > a_0a_3
used in 1 question, e.g. Old spec 5.1 example, PI control stability and steady state
PD: u=kpe+kddedtu = k_pe + k_d\dfrac{de}{dt}; PV: u=kpe−kvdxdtu = k_pe - k_v\dfrac{dx}{dt}
used in 1 question, e.g. Old spec 5.2 examples, PD and PV control
Compare the closed-loop characteristic equation with s2+2ζclωcls+ωcl2s^2 + 2\zeta_{cl}\omega_{cl}s + \omega_{cl}^2
used in 1 question, e.g. Old spec 5.2 examples, PD and PV control
PID: C(s)=kp+kIs+kdsC(s) = k_p + \dfrac{k_I}{s} + k_ds
used in 1 question, e.g. Old spec 5.2 example, PID control stability
L[sin⁡ωt]=ωs2+ω2\mathcal{L}[\sin\omega t] = \dfrac{\omega}{s^2 + \omega^2}
used in 1 question, e.g. Old spec 5.3 worked example, first order response to a harmonic input by Laplace
First order frequency response M=K1+τ2ω2M = \dfrac{K}{\sqrt{1 + \tau^2\omega^2}}, ϕ=−tan⁡−1(ωτ)\phi = -\tan^{-1}(\omega\tau)
used in 1 question, e.g. Old spec 5.3 worked example, first order response to a harmonic input by Laplace
M=∣G(jω)∣M = |G(j\omega)|, ϕ=Arg G(jω)\phi = \mathrm{Arg}\,G(j\omega); ∣z1/z2∣=∣z1∣/∣z2∣|z_1/z_2| = |z_1|/|z_2|, Arg(z1/z2)=Arg z1−Arg z2\mathrm{Arg}(z_1/z_2) = \mathrm{Arg}\,z_1 - \mathrm{Arg}\,z_2
used in 1 question, e.g. Old spec 5.3 numerical example, Bode diagram of a first order system
Corner frequency ω=1/τ\omega = 1/\tau, where log⁡10M=log⁡10K−0.15\log_{10}M = \log_{10}K - 0.15 and ϕ=−45∘\phi = -45^\circ
used in 1 question, e.g. Old spec 5.3 numerical example, Bode diagram of a first order system
Unity feedback around KG(s)KG(s): closed-loop characteristic equation 1+KG(s)=01 + KG(s) = 0; marginal stability when ∣KG(jω)∣=1|KG(j\omega)| = 1 at the frequency where Arg KG(jω)=−180∘\mathrm{Arg}\,KG(j\omega) = -180^\circ
used in 1 question, e.g. Old spec 5.4 example, closed-loop stability from the Bode diagram
Gain margin =1/∣KG(jωpc)∣= 1/|KG(j\omega_{pc})| at the phase crossover; phase margin =180∘+Arg KG(jωgc)= 180^\circ + \mathrm{Arg}\,KG(j\omega_{gc}) at the gain crossover
used in 1 question, e.g. Old spec 5.4 example, closed-loop stability from the Bode diagram
Critical point for neutral stability on the Nyquist diagram: −1+0j-1 + 0j
used in 1 question, e.g. Old spec 5.4 example, Nyquist stability and Hurwitz limit for a third order loop
Newton's second law mdvdt=∑Fm\dfrac{dv}{dt} = \sum F, weight W=mgW = mg
used in 1 question, e.g. ENGR201 2017 B3
Laplace of a derivative L[dvdt]=sV(s)−v(0)\mathcal{L}\left[\dfrac{dv}{dt}\right] = sV(s) - v(0)
used in 1 question, e.g. ENGR201 2017 B3
Pairs: 1↔1s1 \leftrightarrow \dfrac{1}{s}, e−αt↔1s+αe^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha}
used in 1 question, e.g. ENGR201 2017 B3
Partial fractions 1s(s+a)=1a(1s−1s+a)\dfrac{1}{s(s + a)} = \dfrac{1}{a}\left(\dfrac{1}{s} - \dfrac{1}{s + a}\right)
used in 1 question, e.g. ENGR201 2017 B3
Component laws vR=Riv_R = Ri, vL=Ldidtv_L = L\dfrac{di}{dt}, vC=1C∫i dtv_C = \dfrac{1}{C}\displaystyle\int i\,dt; Kirchhoff's voltage law
used in 1 question, e.g. ENGR201 2018 B3
Laplace with zero initial conditions: didt→sI\dfrac{di}{dt} \to sI, ∫0ti dt→Is\displaystyle\int_0^t i\,dt \to \dfrac{I}{s}; unit step →1s\to \dfrac{1}{s}
used in 1 question, e.g. ENGR201 2018 B3
e−αtsin⁡ωt↔ω(s+α)2+ω2e^{-\alpha t}\sin\omega t \leftrightarrow \dfrac{\omega}{(s + \alpha)^2 + \omega^2}
used in 1 question, e.g. ENGR201 2018 B3
Second order form s2+2ζωns+ωn2s^2 + 2\zeta\omega_n s + \omega_n^2, damped frequency ωd=ωn1−ζ2\omega_d = \omega_n\sqrt{1 - \zeta^2}
used in 1 question, e.g. ENGR201 2018 B3
vR=Riv_R = Ri, vL=Ldidtv_L = L\dfrac{di}{dt}, Kirchhoff's voltage law
used in 1 question, e.g. ENGR201 2019 B3
L[didt]=sI(s)−i(0)\mathcal{L}\left[\dfrac{di}{dt}\right] = sI(s) - i(0)
used in 1 question, e.g. ENGR201 2019 B3
sin⁡ωt↔ωs2+ω2\sin\omega t \leftrightarrow \dfrac{\omega}{s^2 + \omega^2}, cos⁡ωt↔ss2+ω2\cos\omega t \leftrightarrow \dfrac{s}{s^2 + \omega^2}, e−αt↔1s+αe^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha}
used in 1 question, e.g. ENGR201 2019 B3
Partial fractions with an irreducible quadratic: ALs+R+Bs+Cs2+1\dfrac{A}{Ls + R} + \dfrac{Bs + C}{s^2 + 1}
used in 1 question, e.g. ENGR201 2019 B3
L[y′]=sY−y(0)\mathcal{L}[y'] = sY - y(0), L[y′′]=s2Y−sy(0)−y′(0)\mathcal{L}[y''] = s^2Y - sy(0) - y'(0)
used in 1 question, e.g. ENGR201 2021 Q2
eat↔1s−ae^{at} \leftrightarrow \dfrac{1}{s - a}, t eat↔1(s−a)2t\,e^{at} \leftrightarrow \dfrac{1}{(s - a)^2}
used in 1 question, e.g. ENGR201 2021 Q2
Impulse δ(t−a)↔e−as\delta(t - a) \leftrightarrow e^{-as}; delay Ha(t)g(t−a)↔e−asG(s)H_a(t)g(t - a) \leftrightarrow e^{-as}G(s)
used in 1 question, e.g. ENGR201 2021 Q2
Repeated-root partial fractions As−2+Bs−1+C(s−1)2\dfrac{A}{s - 2} + \dfrac{B}{s - 1} + \dfrac{C}{(s - 1)^2}
used in 1 question, e.g. ENGR201 2021 Q2
L[x′]=sX−x(0)\mathcal{L}[x'] = sX - x(0), L[x′′]=s2X−sx(0)−x′(0)\mathcal{L}[x''] = s^2X - sx(0) - x'(0); unit step →1/s\to 1/s
used in 1 question, e.g. ENGR201 2022 A2
Step function Ha(t)↔e−as/sH_a(t) \leftrightarrow e^{-as}/s
used in 1 question, e.g. ENGR201 2022 A2
Generalised second order form x¨+2ζωnx˙+ωn2x=Ku\ddot x + 2\zeta\omega_n\dot x + \omega_n^2 x = Ku
used in 1 question, e.g. ENGR201 2022 A2
L[didt]=sI(s)−i(0)\mathcal{L}\left[\dfrac{di}{dt}\right] = sI(s) - i(0); constant 100→100s100 \to \dfrac{100}{s}
used in 1 question, e.g. ENGR201 2023 A2
eat↔1s−ae^{at} \leftrightarrow \dfrac{1}{s - a}
used in 1 question, e.g. ENGR201 2023 A2
Cover-up rule for distinct poles: residue at s=ps = p is [(s−p)F(s)]s=p\left[(s - p)F(s)\right]_{s = p}
used in 1 question, e.g. ENGR201 2023 A2
Final value theorem lim⁡t→∞i(t)=lim⁡s→0sI(s)\lim_{t \to \infty} i(t) = \lim_{s \to 0} sI(s)
used in 1 question, e.g. ENGR201 2023 A2
L[q′]=sQ−q(0)\mathcal{L}[q'] = sQ - q(0), L[q′′]=s2Q−sq(0)−q′(0)\mathcal{L}[q''] = s^2Q - sq(0) - q'(0); constant c→c/sc \to c/s
used in 1 question, e.g. ENGR201 2024 A2(a)-(b)
Current I=dqdtI = \dfrac{dq}{dt}, so I(s)=sQ(s)−q(0)I(s) = sQ(s) - q(0)
used in 1 question, e.g. ENGR201 2024 A2(a)-(b)
L[y′]=sY−y(0)\mathcal{L}[y'] = sY - y(0), L[y′′]=s2Y−sy(0)−y′(0)\mathcal{L}[y''] = s^2Y - sy(0) - y'(0); unit step →1/s\to 1/s
used in 1 question, e.g. ENGR201 2025 Q2(c)
e−αt↔1s+αe^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha}
used in 1 question, e.g. ENGR201 2025 Q2(c)
Final value theorem lim⁡t→∞y(t)=lim⁡s→0sY(s)\lim_{t \to \infty} y(t) = \lim_{s \to 0} sY(s), or steady state gain = TF at s=0s = 0
used in 1 question, e.g. ENGR201 2025 Q2(c)
Laplace with zero initial conditions: dxdt→sX(s)\dfrac{dx}{dt} \to sX(s), d2xdt2→s2X(s)\dfrac{d^2x}{dt^2} \to s^2X(s)
used in 1 question, e.g. ENGR202 2017 Q A1
Poles: roots of the characteristic equation; stable if all have negative real parts; dominant = closest to the imaginary axis
used in 1 question, e.g. ENGR202 2017 Q A2
Closed loop with feedback element: XV=CH1+CHF\dfrac{X}{V} = \dfrac{C H}{1 + C H F}
used in 1 question, e.g. ENGR202 2017 Q A2
PI controller C(s)=K1+K2sC(s) = K_1 + \dfrac{K_2}{s}
used in 1 question, e.g. ENGR202 2017 Q A2
Gain margin and phase margin read from the Bode diagram of the open-loop C(s)H(s)C(s)H(s)
used in 1 question, e.g. ENGR202 2017 Q A3
Newton's second law ∑F=Mx¨\sum F = M\ddot x; spring force KxKx; viscous damper force Cx˙C\dot x
used in 1 question, e.g. ENGR202 2018 Q A1
Mass-spring-damper: ωn=K/M\omega_n = \sqrt{K/M}, ζ=C2MK\zeta = \dfrac{C}{2\sqrt{MK}}, steady state gain 1/K1/K
used in 1 question, e.g. ENGR202 2018 Q A1
Critical damping ζ=1\zeta = 1
used in 1 question, e.g. ENGR202 2018 Q A1
Laplace with zero initial conditions: dx/dt→sXdx/dt \to sX
used in 1 question, e.g. ENGR202 2018 Q A2
Stable if all poles have negative real parts; a single pole at the origin is marginally stable; repeated poles on the imaginary axis are unstable
used in 1 question, e.g. ENGR202 2018 Q A2
Necessary conditions: all coefficients exist and have the same sign
used in 1 question, e.g. ENGR202 2018 Q A2
Steady state gain lim⁡s→0G(s)\lim_{s \to 0} G(s)
used in 1 question, e.g. ENGR202 2018 Q A2
Closed loop with feedback element: XV=CH1+CHF\dfrac{X}{V} = \dfrac{CH}{1 + CHF}; characteristic equation 1+CHF=01 + CHF = 0
used in 1 question, e.g. ENGR202 2018 Q A3
Partial fractions: As(s+a)=A/as−A/as+a\dfrac{A}{s(s + a)} = \dfrac{A/a}{s} - \dfrac{A/a}{s + a}
used in 1 question, e.g. ENGR202 2018 Q A3
Nyquist diagram: Im G(jω)\mathrm{Im}\,G(j\omega) against Re G(jω)\mathrm{Re}\,G(j\omega) as ω\omega goes from 0 to ∞\infty; critical point −1+0j-1 + 0j
used in 1 question, e.g. ENGR202 2019 Q B2
Gain margin and phase margin
used in 1 question, e.g. ENGR202 2019 Q B2
Inductor VL=LdidtV_L = L\dfrac{di}{dt}; capacitor i=CdVCdti = C\dfrac{dV_C}{dt}; Kirchhoff's voltage law
used in 1 question, e.g. ENGR202 2019 Q B3
Steady state: x(t→∞)=K ux(t \to \infty) = K\,u (first order); K/ωn2K/\omega_n^2 times the input (second order)
used in 1 question, e.g. ENGR202 2019 Q B3
Stable if ALL poles have negative real parts; dominant pole = the one closest to the imaginary axis
used in 1 question, e.g. ENGR202 2021 Q1
Second order characteristic equation: s2+2ζωns+ωn2=0s^2 + 2\zeta\omega_n s + \omega_n^2 = 0
used in 1 question, e.g. ENGR202 2021 Q1
PI control: U(s)=(kP+kIs)(V(s)−X(s))U(s) = \left(k_P + \dfrac{k_I}{s}\right)\left(V(s) - X(s)\right)
used in 1 question, e.g. ENGR202 2021 Q1
Hurwitz: for a0s3+a1s2+a2s+a3=0a_0 s^3 + a_1 s^2 + a_2 s + a_3 = 0, stable if Δ1=a1>0\Delta_1 = a_1 > 0, Δ2=a1a2−a0a3>0\Delta_2 = a_1 a_2 - a_0 a_3 > 0, Δ3=a3Δ2>0\Delta_3 = a_3\Delta_2 > 0
used in 1 question, e.g. ENGR202 2021 Q2
Frequency response: M=∣G(s)∣s=jωM = |G(s)|_{s = j\omega}, ϕ=Arg(G(s))∣s=jω\phi = \mathrm{Arg}(G(s))|_{s = j\omega}
used in 1 question, e.g. ENGR202 2021 Q2
First order: τdxdt+x=Ku\tau\dfrac{dx}{dt} + x = Ku, X(s)=Kτs+1U(s)X(s) = \dfrac{K}{\tau s + 1}U(s)
used in 1 question, e.g. ENGR202 2021 Q2
Proportional control: U=kP(V−X)U = k_P(V - X); closed loop kPG1+kPG\dfrac{k_P G}{1 + k_P G}
used in 1 question, e.g. ENGR202 2021 Q2
First order model: τdxdt+x=Ku(t)\tau\dfrac{dx}{dt} + x = Ku(t); unit step response x(t)=K(1−e−t/τ)x(t) = K\left(1 - e^{-t/\tau}\right)
used in 1 question, e.g. ENGR202 2021 Q3
Graphical estimates: K=x(t→∞)u(t→∞)K = \dfrac{x(t\to\infty)}{u(t\to\infty)}; τ\tau = time to reach 63% of the final change
used in 1 question, e.g. ENGR202 2021 Q3
Laplace transform with zero initial conditions: dxdt→sX(s)\dfrac{dx}{dt} \to sX(s)
used in 1 question, e.g. ENGR202 2022 Q B1
First order frequency response: M=K1+ω2τ2M = \dfrac{K}{\sqrt{1 + \omega^2\tau^2}}, ϕ=−tan⁡−1(ωτ)\phi = -\tan^{-1}(\omega\tau); corner (break) frequency ω=1/τ\omega = 1/\tau
used in 1 question, e.g. ENGR202 2022 Q B1
Decibels: 20log⁡10M20\log_{10}M
used in 1 question, e.g. ENGR202 2022 Q B1
Four control objectives: stability, tracking, transient behaviour, disturbance rejection
used in 1 question, e.g. ENGR202 2022 Q B1
Steady state gain G(0)G(0); poles from the characteristic equation; quadratic formula s=−a1±a12−4a22s = \dfrac{-a_1 \pm \sqrt{a_1^2 - 4a_2}}{2}
used in 1 question, e.g. ENGR202 2022 Q B2
Pole placement: divide the characteristic equation by the s2s^2 coefficient and match to s2+2ζωns+ωn2s^2 + 2\zeta\omega_n s + \omega_n^2
used in 1 question, e.g. ENGR202 2022 Q B2
Hurwitz for s3+a1s2+a2s+a3s^3 + a_1 s^2 + a_2 s + a_3: Δ1=a1\Delta_1 = a_1, Δ2=a1a2−a3\Delta_2 = a_1 a_2 - a_3, Δ3=a3Δ2\Delta_3 = a_3\Delta_2, all >0> 0
used in 1 question, e.g. ENGR202 2022 Q B2
Complex poles: s=−ζωn±jωn1−ζ2s = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2}
used in 1 question, e.g. ENGR202 2023 Q B1
Stability needs Δ1=a1>0\Delta_1 = a_1 > 0, Δ2=a1a2−a3>0\Delta_2 = a_1 a_2 - a_3 > 0, Δ3=a3Δ2>0\Delta_3 = a_3\Delta_2 > 0
used in 1 question, e.g. ENGR202 2023 Q B2
PI controller: k1s+k2s=k1+k2s\dfrac{k_1 s + k_2}{s} = k_1 + \dfrac{k_2}{s}
used in 1 question, e.g. ENGR202 2023 Q B2
Pole placement: desired poles p1,2p_{1,2} give the characteristic polynomial (s−p1)(s−p2)(s - p_1)(s - p_2); match coefficients
used in 1 question, e.g. ENGR202 2024 Q B1
s=−ζωn±jωn1−ζ2s = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2}; ωn=∣s∣\omega_n = |s|
used in 1 question, e.g. ENGR202 2024 Q B1
Hurwitz for a0s4+a1s3+a2s2+a3s+a4a_0 s^4 + a_1 s^3 + a_2 s^2 + a_3 s + a_4: Δ1=a1\Delta_1 = a_1, Δ2=a1a2−a0a3\Delta_2 = a_1 a_2 - a_0 a_3, Δ3=a3Δ2−a12a4\Delta_3 = a_3\Delta_2 - a_1^2 a_4, Δ4=a4Δ3\Delta_4 = a_4\Delta_3
used in 1 question, e.g. ENGR202 2024 Q B2
Necessary conditions for stability: all coefficients exist and have the same sign
used in 1 question, e.g. ENGR202 2024 Q B2
s2+2ζωns+ωn2s^2 + 2\zeta\omega_n s + \omega_n^2: ωn=a2\omega_n = \sqrt{a_2}, ζ=a1/(2ωn)\zeta = a_1/(2\omega_n)
used in 1 question, e.g. ENGR202 2025 Q2
Generalised second order form: x¨+2ζωnx˙+ωn2x=Ku(t)\ddot x + 2\zeta\omega_n\dot x + \omega_n^2 x = Ku(t)
used in 1 question, e.g. ENGR270 2026 Q1
For mx¨+bx˙+kx=Fm\ddot x + b\dot x + kx = F: ωn=k/m\omega_n = \sqrt{k/m}, ζ=b2km\zeta = \dfrac{b}{2\sqrt{km}}, steady state gain 1/k1/k
used in 1 question, e.g. ENGR270 2026 Q1
A linear system driven by a sine at frequency ω\omega responds at steady state with a sine of the SAME frequency, scaled by M=∣G(jω)∣M = |G(j\omega)| and shifted by ϕ=Arg G(jω)\phi = \mathrm{Arg}\,G(j\omega)
used in 1 question, e.g. ENGR270 2026 Q1
ω=2πf\omega = 2\pi f
used in 1 question, e.g. ENGR270 2026 Q1
First order G(s)=Kτs+1G(s) = \dfrac{K}{\tau s + 1}: unit step response K(1−e−t/τ)K(1 - e^{-t/\tau}), 63.2% at t=τt = \tau
used in 1 question, e.g. ENGR270 2026 Q2
Time delay TT: the same response shifted right by TT (Transfer Function factor e−sTe^{-sT})
used in 1 question, e.g. ENGR270 2026 Q2
Unity feedback: YR=CG1+CG\dfrac{Y}{R} = \dfrac{CG}{1 + CG}
used in 1 question, e.g. ENGR270 2026 Q2
Final value theorem: y(∞)=lim⁡s→0sY(s)y(\infty) = \lim_{s \to 0} sY(s); a constant dd has transform d/sd/s
used in 1 question, e.g. ENGR270 2026 Q2
Integral action C(s)=K/sC(s) = K/s; PID C(s)=KP+KIs+KDsC(s) = K_P + \dfrac{K_I}{s} + K_D s
used in 1 question, e.g. ENGR270 2026 Q2
Inductor VL=LdidtV_L = L\dfrac{di}{dt}; resistor VR=RiV_R = Ri; capacitor i=CdVCdti = C\dfrac{dV_C}{dt}; Kirchhoff's voltage law
used in 1 question, e.g. Part 1 SAQ 1
Newton's second law ∑F=mx¨\sum F = m\ddot x; spring force k×k \times (extension); viscous damper force c×c \times (relative velocity)
used in 1 question, e.g. Part 1 SAQ 2
Mass-spring-damper: ωn=k/m\omega_n = \sqrt{k/m}, ζ=c2km\zeta = \dfrac{c}{2\sqrt{km}}
used in 1 question, e.g. Part 1 SAQ 2
Newton's second law; friction force by˙b\dot y proportional to velocity
used in 1 question, e.g. Old spec 2.4 exercise
Generalised first order form τx˙+x=Ku(t)\tau\dot x + x = Ku(t)
used in 1 question, e.g. Old spec 2.4 exercise
K=x(t→∞)u(t→∞)K = \dfrac{x(t \to \infty)}{u(t \to \infty)}; at t=τt = \tau the output has completed 63% of its change
used in 1 question, e.g. Old spec 2.4 exercise
Laplace with zero initial conditions: x˙→sX\dot x \to sX, x¨→s2X\ddot x \to s^2X, u˙→sU\dot u \to sU
used in 1 question, e.g. Old spec 3.4 exercise 1
Dominant pole: the one closest to the imaginary axis
used in 1 question, e.g. Old spec 3.4 exercise 1
Laplace with zero initial conditions: x˙→sX\dot x \to sX
used in 1 question, e.g. Old spec 3.4 exercise 2
Pole at the origin (single): marginally stable
used in 1 question, e.g. Old spec 3.4 exercise 2
Open loop: input set without measuring the output; closed loop: output measured and fed back to correct the input
used in 1 question, e.g. Old spec 4.1 toaster exercise
Closed-loop block diagram: set point, summing junction, controller, control input, plant, output, feedback sensor; disturbances enter at the plant
used in 1 question, e.g. Old spec 4.1 robot exercise
Series rule: G1G2G_1 G_2; negative feedback rule: G11+G1G2\dfrac{G_1}{1 + G_1 G_2}
used in 1 question, e.g. Old spec 4.3 exercise
Steady state gain: set s=0s = 0; characteristic equation: denominator =0= 0
used in 1 question, e.g. Old spec 4.3 exercise
Frequency response: M=∣G(jω)∣M = |G(j\omega)|, ϕ=Arg G(jω)\phi = \mathrm{Arg}\,G(j\omega)
used in 1 question, e.g. Old spec 5.3 exercise
Arguments of a product add; of a quotient subtract
used in 1 question, e.g. Old spec 5.3 exercise