mathematics//dynamical systems//model and representation

A model is a compressed description of a system that answers a concrete question, almost always *what will happen if*, and telling it apart from its representation settles what kind of object is being read. Newton's equations of a drone, a thrust table measured on a bench and a neural network trained on hours of flight are three models of the same drone; they differ in where their knowledge comes from (physics or data), in what they cost and in what they do outside the conditions they know (grey-box model). 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, never the whole physical system.

A model is a compressed description of a system that answers a concrete question, almost always what will happen if, and telling it apart from its representation settles what kind of object is being read. Newton's equations of a drone, a thrust table measured on a bench and a neural network trained on hours of flight are three models of the same drone; they differ in where their knowledge comes from (physics or data), in what they cost and in what they do outside the conditions they know (grey-box model). 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, never the whole physical system.

Every symbol is named before anything is computed with it. The general form is

x˙=f(x,u,t;θ),y=h(x,u,t;θ),\dot{\ca{x}}=f(\ca{x},\cc{u},t;\cd{\theta}),\qquad \cb{y}=h(\ca{x},\cc{u},t;\cd{\theta}),x˙=f(x,u,t;θ),y=h(x,u,t;θ),

x\ca{x}x is the state vector, u\cc{u}u the input, y\cb{y}y the output, ttt time and θ\cd{\theta}θ the parameters, quantities that barely change during the run (mass, inertia, heat capacity). The state in green, the input in copper, the output in blue, the parameters in violet. An input can be a chosen actuation or a disturbance; a measurement is not necessarily a sufficient state.

The word vector describes the shape of an object and state its predictive role; a parameter stays fixed during the run considered, and a coefficient multiplies a term (it can be constant or depend on the state). These categories overlap but are not synonyms.

A mass on a spring with damping shows the layers apart. In mq¨=−cq˙−kq+um\ddot q=-c\dot q-kq+\cc{u}mq¨​=−cq˙​−kq+u every term of the balance is a force, and 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 u\cc{u}u belongs to another layer (feedback control).

A coherent equation does not show that it represents the phenomenon. Keep apart mathematical consequence, modelling hypothesis, physical interpretation and validation against data. A matrix can represent a physical system, a statistic or a geometric transformation, and it is not physically all of those at once. A model that predicts what usually happens can be useless for control, which asks what happens when the input is changed on purpose.

Model error has five homes: structure, parameters, initial conditions, discretization and numerical rounding.

The structure is the dynamics left out, the initial conditions are amplified by instability or chaos, and rounding rarely matters outside ill-conditioned problems. In an estimator the total shows up as process noise, so the filter must distrust its model by that much; in control it demands a margin of robustness. Discretization error is the most treacherous because it looks like physics: a simulated oscillator that gains energy looks like an unstable controller.

These notes write FFF for the dynamics matrix in x˙=Fx+Bu\dot{\ca{x}}=F\ca{x}+B\cc{u}x˙=Fx+Bu, LLL for a Laplacian, PPP for Markov transitions and Σ\SigmaΣ for covariance, as listed in dynamical systems; the error a model carries is budgeted in process noise and its validation against logs in simulation.