Controls: Completely reconstructible system?

[Kaydee371]

Consider the following model of a car, where v(t) is the speed and θ(t) is the throttle
angle

v(dot) ̇= -av+bθ

where a and b are positive constants. Now express the model in state space form with the states as position and velocity, assuming the measured output is the velocity. Is (A,C) completely reconstructible? Interpret your answer.

This is a confusing area of controls for me. Any help would be greatly appreciated, thanks!

 

[saltine]

Hi, were you able to get the state-space representation?

Is "completely reconstructible" the same as "completely observable"?

 

[Mene]

I would like to clarify that the term "completely reconstructible system" is often used in the field of control theory to describe a system that can be fully determined or reconstructed from its initial conditions and inputs. In other words, it means that the system can be completely characterized and its behavior predicted without any uncertainty.

In the given model of a car, the state space form would be:

ẋ = Ax + Bu
y = Cx

where x is the state vector, u is the input vector, and y is the output vector. In this case, the state vector would consist of the position and velocity of the car, and the input vector would be the throttle angle. The output vector would be the measured velocity.

To determine if the system is completely reconstructible, we need to check if the pair (A,C) is completely observable. This means that all the states of the system can be determined from the output measurements.

In this case, the matrix C would be [0 1], and the matrix A would be [-a 0; b 0]. As both of these matrices have full rank, the pair (A,C) is completely observable and hence, the system is completely reconstructible.

Interpreting this answer, it means that with the given model and measurements, we can accurately predict the behavior of the car and determine its states at any given time without any uncertainty. This is important in control systems as it allows for precise control and optimization of the system's performance. I hope this helps clarify the concept of "completely reconstructible system" in the context of control theory.