ELEN90055 · Control Systems
System models and block diagrams
This chapter opens Part II of ELEN90055 Control Systems at the University of Melbourne by fixing how we describe a system before we analyse it. A signal is a function of time, a system maps an input signal to an output signal, and a block diagram wires systems together; a static system is memoryless while a dynamic system carries state and obeys a differential equation.
It shows that the same linear time-invariant system has two equivalent descriptions — a time-domain ODE (or convolution with the impulse response) and a frequency-domain transfer function G(s) — and it introduces the model-fidelity trade-off (nonlinear to linearised, delay to Pade, high-order to dominant-pole). This is exactly the toolkit the opening exam question is built from: model a plant, linearise, write the transfer function, and locate its poles.
What this chapter covers
- 01Define a signal, a system (block) and the LTI (linear + time-invariant) assumption that makes transfer functions legal
- 02Tell a static (memoryless) system from a dynamic one by whether the model contains a time derivative
- 03Write the linear state-space model x-dot = Ax + Bu, y = Cx + Du and name each matrix and its role
- 04Reduce series, parallel and negative-feedback blocks, and derive the loop rule Y/R = GC/(1 + GCH)
- 05Move between the two equivalent descriptions of an LTI system: time-domain ODE / convolution and frequency-domain transfer function G(s) = B(s)/A(s)
- 06Read the frequency response off G(jw): gain |G(jw)|, phase angle G(jw) (with dB = 20 log10), and negative phase = lag
- 07Linearise a nonlinear model about an equilibrium and get the small-signal transfer function, its pole and time constant
- 08Judge the model-fidelity trade-off: choose the simplest description that still answers the question, and state its region of validity
Linearise a nonlinear plant and find its transfer function
- +2Equilibrium (the static picture). Set v-dot = 0: F-bar = b v-bar^2, so v-bar = sqrt(F-bar/b) = sqrt(4/1) = 2 m/s. This is the operating point.
- +1Define increments. Let dF = F - F-bar and dv = v - v-bar be small perturbations about the operating point.
- +2Linearise (Taylor, drop second order). The nonlinear term b v^2 has slope d(b v^2)/dv = 2 b v-bar at the operating point. So m dv-dot = dF - 2 b v-bar dv.
- +1Substitute numbers. 2 b v-bar = 2 x 1 x 2 = 4, and m = 1, giving dv-dot + 4 dv = dF.
- +2Transfer function (zero initial conditions). Laplace-transform: (s + 4) dV(s) = dF(s), so G(s) = dV/dF = 1/(s + 4), in units (m/s)/N.
- +2Pole, time constant, and character. One pole at s = -4 (open left-half plane, so stable); time constant tau = 1/|-4| = 0.25 s; DC gain G(0) = 0.25 (m/s)/N. The equilibrium relation is static, but the speed responds through a first-order ODE, so the system is dynamic (one state).
Key terms
- Signal
- A real-valued function of time, such as an input u(t) or output y(t). Its physical unit (newtons, volts, m/s) is carried through the whole model.
- System (block)
- An operator that maps an input signal to an output signal, drawn as a box labelled with its transfer function. Arrows between blocks are signals; a circle is a summing junction.
- LTI system
- A system that is Linear (superposition holds) and Time-Invariant (delaying the input only delays the output). Linearity and time-invariance are what make convolution, transfer functions and frequency response valid.
- Static vs dynamic
- A static (memoryless) system is an algebraic map y = f(u) with no state; a dynamic system carries state and obeys a differential equation. The tell is whether the governing equation contains a time derivative.
- State-space model
- The linear form x-dot = Ax + Bu, y = Cx + Du, where x is the state vector (the memory), A the system matrix, B the input matrix, C the output matrix and D the feedthrough. It is not unique: a change of state coordinates gives a different (A,B,C,D) for the same behaviour.
- Transfer function G(s)
- For an LTI system with zero initial conditions, G(s) = Y(s)/U(s) = B(s)/A(s). The poles are the roots of the denominator A(s), the zeros the roots of B(s). A proper (realisable) transfer function has denominator degree at least the numerator degree.
- Frequency response
- The value of G(jw) as w varies. For a stable G, a sinusoid sin(wt) in gives |G(jw)| sin(wt + angle G(jw)) out; the gain is |G(jw)|, the phase is angle G(jw) (negative phase = the output lags), and magnitude in decibels is 20 log10 |G(jw)|.
- Model fidelity
- How faithfully a model matches reality, traded against how easily it can be analysed. Standard simplifications are linearising a nonlinear ODE about an operating point, approximating a delay by a Pade rational, and keeping only the dominant (slowest) poles of a high-order model.
System models and block diagrams FAQ
What is the difference between the time-domain and frequency-domain descriptions of a system?
They describe the same LTI system in two equivalent ways. In the time domain the system is a differential equation, or equivalently the output is the convolution of the input with the impulse response g(t). The Laplace transform turns this into algebra: the transfer function G(s) = Y(s)/U(s), where convolution becomes multiplication Y(s) = G(s)U(s). Transient shape and initial-value behaviour are clearest in the time domain, while interconnection of blocks and steady-state response to sinusoids are trivial in the s-domain, so you switch to whichever makes the question easy.
Why do we linearise, and when is the linear model valid?
Most physical plants are nonlinear, but the classical analysis and design tools (transfer functions, poles, Bode, root locus) apply to linear systems. Linearising about an equilibrium replaces the nonlinear model by a linear small-signal model that is tractable. The cost is that it is only valid for small perturbations about that one operating point; a different operating point gives a different linear model, and large excursions leave its region of validity. Always state the operating point you linearised about.
Can AI help me with system models and block diagrams in ELEN90055?
Yes, as a study aid. An AI tutor such as Sia can explain the difference between static and dynamic systems step by step, walk you through linearising a nonlinear model and forming its transfer function, and check your block-diagram algebra and sign conventions. Use it to understand method 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
Treat this chapter as the language every later part of the subject speaks. First, be fluent in the vocabulary: signal, system, static vs dynamic, and the LTI assumption, and be able to write the state-space form x-dot = Ax + Bu, y = Cx + Du and the transfer function G(s) = B(s)/A(s) from memory. Second, drill the block-diagram algebra until series (G1 G2), parallel (G1 + G2) and the negative-feedback rule Y/R = GC/(1 + GCH) are automatic, and derive the loop rule once so the plus sign in the denominator is never guessed. Third, practise moving between the time-domain (ODE, convolution) and frequency-domain (transfer function, frequency response) descriptions, and reading gain and phase off G(jw) with dB = 20 log10. Finally, rehearse the exam-signature pipeline on a physical plant: find the equilibrium, linearise term by term, transform with zero initial conditions, locate the poles and comment on stability. 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, state the operating point, and sanity-check the units and the size of the response at the end. Confirm the exact date of the next (Semester 1, ~June 2027) sitting on Canvas.
