IntermediateTopic 67 of 101Signals, Control & Comms
Linearisation
Approximating a nonlinear system near an operating point.
In depth
Near an equilibrium (x₀,u₀) approximate f(x,u)≈f(x₀,u₀)+A δx+B δu with Jacobians A=∂f/∂x, B=∂f/∂u. Valid for small deviations; allows linear control design and stability analysis (Lyapunov's indirect method). Gain scheduling uses a family of linearisations.
Key points
- Taylor expansion
- Operating point
- Hartman-Grobman theorem
ExampleA pendulum linearised at the bottom behaves as an oscillator, at the top as unstable.
What is Linearisation?
Approximating a nonlinear system near an operating point.
Why is it important to understand linearisation?
Linearisation gives a simple linear model near an operating point, so the tools of linear control can be applied to nonlinear plants.
How do I linearise a system?
Take the Jacobian of the nonlinear equations at the operating point and use it as the A matrix of a local linear model.
When does linearisation fail?
When the system moves far from the operating point, so the linear model is no longer accurate and a nonlinear method may be needed.
How is Linearisation used in practice?
Typical applications include designing a controller for a robot arm around its normal working posture, linearising a tank outflow equation around the normal level and estimating the small-signal gain of a transistor amplifier at its bias point.
What should I learn before Linearisation?
Study "Nonlinear systems" first (topic 66 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Linearisation?
Taylor expansion. Operating point. Hartman-Grobman theorem.
Can you give an example of Linearisation?
A pendulum linearised at the bottom behaves as an oscillator, at the top as unstable.
What comes after Linearisation?
The next topic is "Lyapunov stability", which builds directly on this one.
How difficult is Linearisation?
It is classed as Intermediate in Signals, Control & Comms (topic 67 of 101). It is an intermediate topic. Work through the beginner topics in Signals, Control & Comms first.