States, cart-pole and LQR
When one output hides too much, track the system's internal state.
First, picture it
A pole can be upright while its cart rolls rapidly toward the rail edge. Angle alone says 'balanced'; position and velocity say 'not safe for long'. State feedback can use all of them.
What the model says
A linearized state-space model is ẋ=Ax+Bu, y=Cx+Du. For cart-pole, a common state vector is [cart position, cart velocity, pole angle, angular velocity]. LQR chooses a feedback gain K by balancing a quadratic state cost Q against input cost R for a specified linear model. It does not by itself solve nonlinear swing-up, hard rail limits or unmeasured-state estimation. SISO describes an input/output arrangement; LQR describes a design method, so they are not alternative categories.
ẋ=Ax+Bu; u=−Kx; J=∫(xᵀQx+uᵀRu)dtThe operating point, linearization and chosen Q/R must be stated before interpreting a design.
Make it concrete
In the cart-pole lab, use angle-only control and watch cart drift. Then use the position-return preset to see the extra objective. This is a conceptual bridge to state feedback, not an LQR implementation.
Try the cart-poleDo not describe a cart-pole as a scalar SISO transfer-function problem without saying which output and operating point you selected.
Check your understanding+
Can an upright pole still require urgent control action?
Yes. The cart may be moving toward the rail end, even while angle error is near zero.