mathematics//dynamical systems//model and representation

What kind of object am I reading, and what does it claim about the system?

What kind of object am I reading, and what does it claim about the system? A model gathers variables, relations, parameters, domain and hypotheses for a purpose. A representation decides how to express it: differential equation, state space, transfer function or matrix. An equation is one relation inside that representation; it is not the whole physical system.

Name before operating. In

x˙=f(x,u,t;θ),y=h(x,u,t;θ),\dot x=f(x,u,t;\theta),\qquad y=h(x,u,t;\theta),x˙=f(x,u,t;θ),y=h(x,u,t;θ),

xxx is the state vector, uuu the input, yyy the output, ttt time and θ\thetaθ the parameters. fff describes the dynamics and hhh the observation. An input can be a chosen actuation or a disturbance; not every input is control.

"Vector" describes the shape of an object. "State" describes its predictive role. "Parameter" means it stays fixed during the run considered. "Coefficient" means it multiplies a term: a coefficient can be constant or depend on the state. These categories overlap but are not synonyms.

A measurement is not necessarily a sufficient state. Measuring a drone's position does not determine its future without also knowing its velocity and the future inputs.

Engineering example. In mq¨=−cq˙−kq+um\ddot q=-c\dot q-kq+umq¨​=−cq˙​−kq+u, qqq is position, mmm mass, ccc damping, kkk stiffness and uuu the applied force. The state can be x=(q,q˙)Tx=(q,\dot q)^{\mathsf T}x=(q,q˙​)T. Every term of the balance is a force; after dividing by mmm, an acceleration. The viscous force −cq˙-c\dot q−cq˙​ is an approximate constitutive relation: the force balance and the choice of that relation do not have the same provenance. A controller that picks uuu belongs to another layer (feedback control).

What a formula does not prove. A coherent equation does not show that it represents the phenomenon. Keep apart mathematical consequence, modeling hypothesis, physical interpretation and validation against data. A matrix can represent a physical system, a statistic or a geometric transformation; it is not physically all of those at once.

Convention in these notes: FFF for the dynamics matrix in x˙=Fx+Bu\dot x=Fx+Bux˙=Fx+Bu, MMM for nodal storage, WWW for graph weights, DDD for degree, LLL for the Laplacian, PPP for Markov transitions and Σ\SigmaΣ for covariance. See dynamical systems.