Dynamic mathematical model with example

A dynamic mathematical model represents a system with equations in which time is a variable. The state of the system at any time depends on its earlier states. It is written as differential equations for continuous systems or difference equations for discrete-time systems, and is solved analytically or numerically (simulation).

General form with state vector x(t), input u(t):

dx/dt = f(x, u, t)

Characteristics

• Contains derivatives (or time-step differences) of the variables.
• Needs initial conditions, e.g. x(0) = x0.
• Output depends on both the input and the history of the system.
• Can be linear or nonlinear, deterministic or stochastic.

Example 1: population growth

If the growth rate is proportional to the population P, dP/dt = k P, P(0) = P0. The solution is P(t) = P0 e^(kt). For k = 0.02 per year, P0 = 10,000, after 10 years P = 10,000 e^0.2 = 12,214.

Example 2: mass-spring-damper

M x'' + D x' + K x = F(t). This second-order equation models vibration of a vehicle suspension. Setting x1 = x, x2 = x' gives the state form x1' = x2, x2' = (F - D x2 - K x1)/M.

Example 3: cooling of a body

dT/dt = -k (T - Ta). T falls exponentially towards ambient temperature Ta.

Example 4: discrete form

Bank balance: B(n+1) = B(n) (1 + r) + d(n) (difference equation, r = interest rate, d = deposit).

Solution methods

• Analytical (integration) when the equation is simple.
• Numerical integration (Euler: x(t+h) = x(t) + h f(x,t), Runge-Kutta) on a digital computer.
• Analog computers, which solve the equations with integrators and summers.