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Mathematics LibreTexts

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. Here we discuss how you can jump across the border back and forth between the two time treatments.

Assume you already have an autonomous first-order discrete-time model

\[x_{t}=F(x_{t-1}), \label{6.16}\]

and you want to develop a continuous-time model analogous to it. You set up the following “container” differential equation

\[\frac{dx}{dt} =G(x),\label{6.17}\]

and try to find out how \(F\) and \(G\) are related to each other.

Here, let me introduce a very simple yet useful analogy between continuous- and discrete-time models: 

\[\frac{dx}{dt} \approx \frac{\Delta{x}}{\Delta{t}} \label{(6.18)} \]

This may look almost tautological. But the left hand side is a ratio between two infinitesimally small quantities, while the right hand side is a ratio between two quantities that are small yet have definite non-zero sizes. \(∆x\) is the difference between \(x(t + ∆t)\) and \(x(t)\), and \(∆t\) is the finite time interval between two consecutive discrete time points. Using this analogy, you can rewrite Equation \ref{6.17} as

\[\frac{Delta{x}}{\Delta{t}} = \frac{x(t+\Delta{t}) -x(t)}{\Delta{t}} \approx G(x(t)), \label{6.19}\]

\[x(t+\Delta{t}) \approx x(t) +G(x(t))\Delta{t}.\label{(6.20)}\]

By comparing this with Eq.(6.16), we notice the following analogous relationship between \(F\) and \(G\): 

\[F(x) \leftrightarrow \frac{F(x) -x}{\Delta{t}} \label{6.22}\]

For linear systems in particular, \(F(x)\) and \(G(x)\) are just the product of a coefficient matrix and a state vector. If \(F(x) = Ax\) and \(G(x) = Bx\), then the analogous relationships become 

\[Ax \leftrightarrow x+Bx\Delta{t}, \label{6.23}\]

\[A \leftrightarrow I +B\Delta{t} \label{6.24}\]


\[B \leftrightarrow \frac{A-I}{\Delta{t}}. \label{6.25}\]

I should emphasize that these analogous relationships between discrete-time and continuous-time models do not mean they are mathematically equivalent. They simply mean that the models are constructed according to similar assumptions and thus they may have similar properties. In fact, analogous models often share many identical mathematical properties, yet there are certain fundamental differences between them. For example, one- or two-dimensional discrete-time iterative maps can show chaotic behaviors, but their continuous-time counterparts never show chaos. We will discuss this issue in more detail later.

Nonetheless, knowing these analogous relationships between discrete-time and continuous-time models is helpful in several ways. First, they can provide convenient pathways when you develop your own mathematical models. Some natural phenomena may be conceived more easily as discrete-time, stepwise processes, while others may be better conceived as continuous-time, smooth processes. You can start building your model in either way, and when needed, convert the model from discrete-time to continuous-time or vice versa. Second, Equation \ref{6.20} offers a simple method to numerically simulate continuous-time models. While this method is rather crude and prone to accumulating numerical errors, the meaning of the formula is quite straight forward, and the implementation of simulation is very easy, so we will use this method in the following section. Third, the relationship between the coefficient matrices given in Equation \ref{6.25} is very helpful for understanding mathematical differences of stability criteria between discrete-time and continuous-time models. This will be detailed in the next chapter.

Exercise 6.3

Consider the dynamics of a system made of three parts, A, B, and C. Each takes a real-valued state whose range is [−1,1]. The system behaves according to the following state transitions:

  • A adopts B’s current state as its next state.
  • B adopts C’s current state as its next state.
  • C adopts the average of the current states of A and B as its next state.

First, create a discrete-time model of this system, and then convert it into a continuous-time model using Equation \ref{6.25}.