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6: Continuous-Time Models I - Modeling
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- 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:
- 6.2: Classiﬁcations of Model Equation
- Distinctions between linear and nonlinear systems as well as autonomous and non-autonomous systems apply to continuous-time models. But the distinction between ﬁrst-order and higher-order systems are slightly different.
- 6.3 Connecting Continuous-Time Models with DiscreteTime Models
- Continuous-time models and discrete-time models are different mathematical models with different mathematical properties. But it is still possible to develop a “similar” continuous-time model from a discrete-time model, and vice versa.
- 6.4 Simulating Continuous-Time Models
- Simulation of a continuous-time model is equivalent to the numerical integration of differential equations, which, by itself, is a major research area in applied mathematics and computational science with more than a century of history.
- 6.5: Building Your Own Model Equation
- Principles and best practices of building your own equations for a continuous-time model are very much the same as those we discussed for discrete-time models in Sections 4.5 and 4.6. The only difference is that, in differential equations, you need to describe time derivatives, i.e., instantaneous rates of change of the system’s state variables, instead of their actual values in the next time step.