A useful model near one operating point
Linearization is a local approximation, not a claim that a nonlinear machine has become linear.
First, 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.
What the model says
Start with a nonlinear state equation ẋ=f(x,u). Choose an equilibrium (x₀,u₀) where f(x₀,u₀)=0. Describe small departures as δx=x−x₀ and δu=u−u₀. The first derivatives of f at that equilibrium form matrices A and B, giving δẋ≈Aδx+Bδu. The approximation omits higher-order terms; changing the operating point changes A and B.
δẋ ≈ Aδx+Bδu; A=∂f/∂x|₀, B=∂f/∂u|₀The subscript 0 means evaluate the derivatives at the stated equilibrium, not at every point of a run.
The tangent is useful locally. This sketch is not a measured error curve from the cart-pole Lab.
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.
Make it concrete
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+
If an LQR run fails at a large tilt, has the nonlinear simulator become wrong?
Not necessarily. The gain was designed from a local linear model and may lack enough force or rail room outside that neighborhood.