ELEN90055 · Control Systems
ODE models and linearization
This chapter of ELEN90055 Control Systems at the University of Melbourne takes the first and most examined step of classical control: turning a real, nonlinear physical plant into a linear model that transfer-function methods can handle. It writes the plant as an implicit / state-space ordinary differential equation, finds an equilibrium operating point, and keeps only the first-order Taylor terms to build an incremental (small-signal) model in the deviations δu and δy.
It sits in Part II of the subject (dynamical system models) and underpins the opening exam question, where a nonlinear system must be linearised, converted to a transfer function, and its poles used to comment on stability.
What this chapter covers
- 01Write a causal plant as an implicit ODE and as a linear state-space model x-dot = Ax + Bu, y = Cx + Du
- 02Find an equilibrium (static) operating point by setting all derivatives to zero
- 03Define the incremental variables du = u - u-bar and dy = y - y-bar about that point
- 04Compute the Taylor coefficients a_k and b_k as partial derivatives of the ODE at the equilibrium
- 05Assemble the small-signal linear ODE with the correct output-left, input-right minus-sign convention
- 06Handle the static-map and Jacobian special cases of linearisation
- 07Laplace-transform the small-signal ODE to the incremental transfer function and locate its poles
- 08Read stability off the pole signs and explain why the operating point changes the model
- 09Apply the recipe to standard plants: tanks, pendulums and magnetic suspensions
Linearise a water tank and find its pole
- +1Balance law. Conservation of volume gives A*h-dot = u - k*sqrt(h), i.e. the implicit ODE l = A*h-dot - u + k*sqrt(h) = 0.
- +1Equilibrium. Set h-dot = 0 with u = u-bar: 0 = u-bar - k*sqrt(h-bar), so h-bar = (u-bar/k)^2 = (6/2)^2 = 9 m.
- +2Partials at the equilibrium. a_1 = dl/d(h-dot) = A = 1 m-squared; a_0 = dl/dh = k/(2*sqrt(h-bar)) = 2/(2*sqrt(9)) = 1/3 (units m-squared/s); b_0 = dl/du = -1.
- +1Small-signal ODE. With output-left, input-right (and its minus sign): A*dh-dot + (k/(2*sqrt(h-bar)))*dh = du, i.e. dh-dot + (1/3)*dh = du.
- +1Transfer function. Laplace with zero initial conditions: (s + 1/3)*dH = dU, so dH(s)/dU(s) = 1/(s + 1/3) = 3/(3s + 1).
- +2Pole and verdict. Single pole at s = -1/3 rad/s (time constant tau = 3 s, steady-state gain 1/a_0 = 3). The pole is in the left-half plane, so the linearised tank is stable and settles to a new level in about 4*tau = 12 s.
Key terms
- Implicit ODE model
- A single equation l(y^(n),...,y, u^(n),...,u) = 0 relating a plant's output y and input u and their time derivatives. It is linear exactly when l is affine (a constant-coefficient sum) in its arguments.
- State-space model
- The first-order form x-dot = Ax + Bu, y = Cx + Du, where x is the n-by-1 state vector, A the system matrix, B the input matrix, C the output matrix and D the feedthrough. The realisation of a given transfer function is not unique.
- Equilibrium (operating point)
- A pair of constant signals (u-bar, y-bar) with all derivatives zero that satisfies the ODE; for a state model it solves f(x-bar, u-bar) = 0. A plant may have several equilibria or none, and you linearise about one of them.
- Incremental variables
- The small deviations du = u - u-bar and dy = y - y-bar from the chosen equilibrium. The linearised (small-signal) model is written entirely in these deviations, with the equilibrium subtracted out.
- Taylor-series linearisation
- Expanding the ODE about the equilibrium and discarding second-order and higher terms. The coefficients a_k = dl/dy^(k) and b_k = dl/du^(k) are the partial derivatives evaluated at the operating point.
- Small-signal transfer function
- The Laplace-domain ratio dY(s)/dU(s) obtained from the small-signal ODE with zero initial conditions. Its denominator roots are the poles that determine stability of the incremental model.
- Jacobian linearisation
- For a nonlinear state model x-dot = f(x,u), the linear model uses A = df/dx and B = df/du evaluated at the equilibrium (the Jacobian matrices).
- Open-loop stability
- The linearised plant is stable when every pole lies strictly in the open left-half plane (Re < 0). A single right-half-plane pole, as in a magnetic suspension, makes the plant unstable and forces the use of feedback.
ODE models and linearization FAQ
Why linearise at all instead of solving the nonlinear ODE directly?
Almost every classical-control tool - transfer functions, poles and zeros, Bode, root locus, Nyquist, and the PID / lead-lag design procedures - is defined only for linear time-invariant systems. Linearising about an operating point produces a small-signal model those tools can act on, and it is accurate for the small perturbations feedback loops are designed to keep the plant near. The price is that the model is local: a different operating point gives different coefficients and can even change the stability verdict, so you must state the point you linearised at.
How do I know whether my linearised plant is stable?
Form the incremental transfer function and look at the roots of its denominator. If every pole has a strictly negative real part the small-signal model is stable; a pole on the imaginary axis is marginally stable (undamped), and any pole with a positive real part is unstable. For a second-order model a-2*s^2 + a-0 = 0, the sign of a-0/a-2 decides it: positive gives imaginary-axis poles, negative gives a right-half-plane pole. Getting the sign of that coefficient right is where the marks are.
Can AI help me with ODE models and linearization in ELEN90055?
Yes, as a study aid. An AI tutor such as Sia can explain the linearisation recipe step by step, walk you through finding an equilibrium, writing the partial-derivative coefficients, and Laplace-transforming to a transfer function, and it can check your algebra and sign conventions on practice problems. Use it to understand the method and to rehearse, not to obtain answers for assessed work: it does not sit your open-book exam or guarantee a grade, 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 five-step recipe automatic, because the opening exam question almost always follows it: write the balance law as an ODE, set the derivatives to zero for the equilibrium, list the a_k and b_k as partial derivatives evaluated there, assemble the small-signal ODE with the output-left, input-right minus sign, Laplace it to the transfer function, and read stability off the poles. Drill the two failure modes the markers look for - linearising in the raw variables instead of the deviations, and dropping or flipping a sign so the coefficient a_0 comes out with the wrong sign and the stability verdict inverts. Practise on the standard plants (a tank, a pendulum, a magnetic suspension) so both the stable and the unstable outcome feel familiar. The final examination is 3 hours, open book, worth 70% with a hurdle you must pass, and you answer four questions summing to 50 marks, so budget about 3.6 minutes per mark and give the plant-modelling question roughly its share. Because it is open book, marks reward the method shown with correct units and sign conventions rather than a memorised result, so define every symbol, carry SI units, and finish with a one-line pole-and-stability comment. Confirm the exact date of the next (Semester 1, ~June 2027) sitting on Canvas.
