6.1: Continuous-Time Models with Differential Equations

Continuous-time models are written in differential equations. They are probably more mainstream in science and engineering, and studied more extensively, than discrete-time models, because various natural phenomena (e.g., motion of objects, flow of electric current) take place smoothly over continuous time. A general mathematical formulation of a first-order continuous-time model is given by this:

\[\frac{dx}{dt} =F(x,t)\label{(6.1)}\]

Just like in discrete-time models, x is the state of a system (which may be a scalar or vector variable). The left hand side is the time derivative of \(x\), which is formally defined as

\[\frac{dx}{dt} =\lim_{\delta{t}\rightarrow{0}}\frac{x(t+\delta{t}) -x(t)}{\delta{t}}. \label{(6.2)}\]

Integrating a continuous-time model over \(t\) gives a trajectory of the system’s state over time. While integration could be done algebraically in some cases, computational simulation (= numerical integration) is always possible in general and often used as the primary means of studying these models. One fundamental assumption made in continuous-time models is that the trajectories of the system’s state are smooth everywhere in the phase space, i.e., the limit in the definition above always converges to a well-defined value. Therefore, continuous-time models don’t show instantaneous abrupt changes, which could happen in discrete-time models.