A useful model near one operating point
Linearization is a local approximation, not a claim that a nonlinear machine has become linear.
Find a local slope and state clearly where the approximation is meant to work.
State equations can be nonlinear. We can make a simpler local model by expanding around one resting operating point.
Review the previous lesson: States and the cart-pole modelFirst, picture it
A curved road looks almost straight when you inspect a tiny stretch. A tangent tells you how the road changes right here, but it may point the wrong way farther away. A linearized model does the same for a moving system near a chosen condition.
Build the idea step by step
An equilibrium is a state that stays still under a constant input. First choose that state and input. Then describe only the small changes away from them. Replace each curved relation by its slope at the equilibrium, just as a tangent approximates a curve. Those slopes form A and B in the linear state equation. Changing the equilibrium changes the slopes, so the same approximation need not work elsewhere.
Read the equations
Small changes in state and input are linked by slopes evaluated at the chosen equilibrium.
What the symbols mean
- state change from the equilibrium, in the state's units
- input change from the equilibrium, in the input's units
- slope of the state rate with respect to state at the equilibrium
- slope of the state rate with respect to input at the equilibrium
- rate of the state deviation; the dot is a time derivative
The symbol δ means a difference from the chosen equilibrium, not a new physical variable. The approximation sign reminds us that curved, higher-order effects were left out. Small angle means radians here.
The tangent is useful locally. This sketch is not a measured error curve from the cart-pole Lab.
Approximate a small pole tilt
A pendulum model contains the nonlinear term sin of its angle. Around upright, the equilibrium angle is zero.
The tangent model replaces sine of the angle by the angle in radians. At 10 degrees, the angle is about 0.1745 radians.
The true sine is about 0.1736, so this term differs by about 0.5%. A larger angle gives a larger error.
More detail and model assumptions
Why the upright pole is a local problem
Near upright, sin θ is close to θ and cos θ is close to 1 when θ is measured in radians. This simplifies the equations used for LQR and the Kalman filter. At a large tilt those replacements and the equilibrium assumption become less accurate, even if the simulator still integrates its nonlinear equations.
Actuator saturation and the rail end are separate nonlinear constraints. Even a good small-angle model does not make a controller capable of producing unlimited force or keeping a cart on an infinite rail.
How to test the approximation
Keep controller, disturbance, seed and run duration fixed. Change only the initial pole angle, then compare angle, cart travel, applied force and failure state. A worsening run is evidence about this controller on the nonlinear plant; without plotting the linear model beside it, it is not a measured linearization error.
Connect the example to the lab
Begin with a small upright cart-pole tilt under LQR, pin the run, and increase only the starting angle. Record where recovery or rail travel changes markedly.
Compare two starting anglesMake a prediction before moving a control.
- PREDICTWill a larger initial tilt be easier to recover? Why?
- CHANGE ONE THINGKeep LQR and the kick fixed; change only initial angle.
- OBSERVECompare angle, cart position, applied force and rail status.
- EXPLAINName the local-model and physical-limit assumptions separately.
Do not announce one universal 'valid angle'. Validity depends on model, operating point, input, accuracy required and constraints.
Check your understanding+
A small-angle model fits a 5-degree tilt but not a 60-degree tilt. What should you conclude?
The local approximation was used too far from its operating point. This does not by itself show that the underlying nonlinear model is wrong.Why the next lesson follows
A draining tank lets us compute such a slope from a physical flow law and see how its operating level matters.