ELEN90055 · Control Systems
Poles, zeros, stability and frequency response
This chapter is the analysis core of ELEN90055 Control Systems at the University of Melbourne: given a transfer function G(s), it reads the whole behaviour of a linear time-invariant system straight off its poles (denominator roots) and zeros (numerator roots), without ever solving a differential equation. It covers splitting G(s) by partial fractions into first-order modes, the canonical second-order response (damping ratio ζ, natural frequency ωn, overshoot Mp, settling time ts), the BIBO stability rule, and the sinusoidal steady state and Bode plot obtained by setting s = jω.
It sits in Part II of the subject (dynamical system models), supplies the ending of the opening exam question — locate the poles, comment on stability and steady state — and fixes the frequency-response language every later design chapter reuses.
What this chapter covers
- 01Split a transfer function by partial fractions and find each residue with the cover-up rule B_k = lim (s - alpha_k) G(s)
- 02Write the impulse response as a sum of modes: a real pole gives e^(alpha t), a complex pair gives a decaying cosine
- 03Recognise the canonical second-order form G(s) = wn^2 / (s^2 + 2*zeta*wn*s + wn^2) and place its poles
- 04Compute the transient specs: damped frequency wd = wn*sqrt(1 - zeta^2), overshoot Mp = exp(-pi*zeta/sqrt(1 - zeta^2)), settling time ts about 4/(zeta*wn)
- 05State the BIBO stability rule: stable if and only if every pole has strictly negative real part (open left-half plane)
- 06Tell stable from marginally stable from unstable, and know that zeros never change stability
- 07Read the sinusoidal steady state off G(j-omega): gain |G(j-omega)|, phase angle (negative phase means the output lags)
- 08Draw a straight-line Bode magnitude plot in decibels, dB = 20*log10|G|, with corners and a -20 dB/decade slope
- 09Explain why a right-half-plane (non-minimum-phase) zero forces step-response undershoot bounded by Mu about 1/(c*ts)
Second-order transient specs from (zeta, wn)
- +1Transfer function. G(s) = wn^2 / (s^2 + 2*zeta*wn*s + wn^2). With wn^2 = 25 and 2*zeta*wn = 2(0.6)(5) = 6, G(s) = 25 / (s^2 + 6s + 25). DC gain G(0) = 25/25 = 1 (a unit step settles to 1).
- +1Poles. s = -zeta*wn ± j*wn*sqrt(1 - zeta^2) = -3 ± j*5*sqrt(1 - 0.36) = -3 ± j*5(0.8) = -3 ± j4.
- +1Damped frequency. wd = wn*sqrt(1 - zeta^2) = 5(0.8) = 4 rad/s (the imaginary part of the poles).
- +1Overshoot. Mp = exp(-pi*zeta/sqrt(1 - zeta^2)) = exp(-pi(0.6)/0.8) = exp(-2.356) = 0.0948, i.e. about 9.5% overshoot (it depends on zeta only).
- +1Settling time (2% band). ts is about 4/(zeta*wn) = 4/3 = 1.33 s.
- +1Stability. Both poles have real part -3 < 0, so they lie in the open left-half plane and the system is BIBO stable; with zeta = 0.6 it overshoots modestly then settles.
Key terms
- Pole
- A root of the denominator of the transfer function G(s). Each pole contributes one time-domain mode; a real pole alpha gives e^(alpha t), a complex pair sigma +/- j*omega gives an exponentially enveloped cosine. Pole locations decide the response shape and stability.
- Zero
- A root of the numerator of G(s). Zeros and the gain set how strongly each mode is excited (the residues) but do not change stability. A zero in the right-half plane makes the system non-minimum-phase and causes step-response undershoot.
- Partial-fraction expansion
- Rewriting a strictly proper G(s) with distinct poles as a sum of first-order terms B_k/(s - alpha_k). The residue is found by the cover-up rule B_k = lim as s tends to alpha_k of (s - alpha_k)G(s); inverse-transforming gives the impulse response as a sum of modes.
- Damping ratio (zeta)
- The dimensionless parameter in the canonical second-order form that sets how oscillatory the response is. zeta = 0 gives sustained oscillation; 0 < zeta < 1 is underdamped (overshoot); zeta = 1 is critically damped; zeta > 1 is overdamped. Percentage overshoot depends on zeta alone.
- Natural frequency (wn)
- The undamped natural frequency in rad/s of the canonical second-order system; it equals the distance of the complex poles from the origin. Raising wn with zeta fixed speeds up the response (shorter settling and rise time) without changing the overshoot.
- BIBO stability
- Bounded-input bounded-output stability. A proper rational LTI system is BIBO stable if and only if every pole lies strictly in the open left-half plane (real part < 0), equivalently its impulse response is absolutely integrable. Distinct imaginary-axis poles are marginally stable; any right-half-plane or repeated imaginary-axis pole is unstable.
- Frequency response G(j-omega)
- The transfer function evaluated at s = j*omega. For a stable system a persistent sinusoid input sin(omega t) gives a steady-state output |G(j-omega)| sin(omega t + angle G(j-omega)); the magnitude is the gain and a negative phase means the output lags the input.
- Decibel (Bode magnitude)
- The logarithmic magnitude unit used on a Bode plot, dB = 20*log10|G(j-omega)| (the factor is 20 for amplitude ratios, not 10). Because logarithms and angles add, a Bode plot is the sum of the plots of the individual factors, which makes it sketchable by hand.
Poles, zeros, stability and frequency response FAQ
Do zeros affect whether a system is stable?
No. Stability is decided by the poles, the denominator roots: a system is BIBO stable exactly when every pole has a negative real part. Zeros and the gain reshape the transient and the frequency response, and a right-half-plane zero causes undershoot, but moving or adding a zero can never stabilise or destabilise the system. When you are asked about stability, look only at the pole locations.
What is the difference between marginally stable and stable?
A system is stable when all poles are strictly in the left-half plane, so every mode decays and the response settles. It is marginally stable when it has distinct poles exactly on the imaginary axis: the response stays bounded but does not decay, giving a sustained oscillation (for example the undamped linearised pendulum with poles at plus and minus j*sqrt(g/L)). Repeated imaginary-axis poles are a different case and are unstable because a polynomial-times-oscillation term grows without bound.
Can AI help me with poles, zeros, stability and frequency response in ELEN90055?
Yes, as a study aid. An AI tutor like Sia can explain the method step by step: how to do a partial-fraction expansion, how to read overshoot and settling time from zeta and wn, how to test stability from pole locations, and how to evaluate gain and phase from G(j-omega) for a Bode sketch. Use it to understand the conventions (the 20 in the decibel, the negative phase for a lag) and to practise; it does not sit your open-book exam or guarantee a mark, and you should confirm every formula against the unit notes on Canvas.
Studying with AI? Sia — free AI electrical engineering tutor works through ELEN90055 step by step.
Exam move
Make the pole-to-behaviour link automatic, because it is the payoff of the whole first half of the subject. First, practise partial fractions until residues by cover-up are quick, and be able to write the impulse response as a sum of modes just by looking at the poles. Second, memorise the canonical second-order results by meaning, not rote: poles at minus zeta*wn plus or minus j*wn*sqrt(1 - zeta^2), overshoot Mp = exp(-pi*zeta/sqrt(1 - zeta^2)) which depends on zeta only, and settling time about 4/(zeta*wn) which you shorten by raising wn. Third, drill the BIBO rule (all poles with negative real part) and the three verdicts stable, marginally stable, unstable, remembering that zeros never change stability. Fourth, get fluent at setting s = j*omega to read gain, phase and decibels, keeping the 20 in dB = 20*log10|G| and the sign convention that negative phase is a lag. The final exam is 3 hours, open book, worth 70% with a hurdle, and you answer four questions summing to 50 marks, so budget about 3.6 minutes per mark. Because it is open book, marks reward method shown with correct sign and unit conventions rather than memorised formulae, so define your symbols, show each step, and sanity-check the DC value and the units at the end. Confirm the exact date of the next (Semester 1, about June 2027) sitting on Canvas.
