User:Jonmsmith/T-integration

T-integration is a numerical integration technique developed by Jon Michael Smith to facilitate digital computer controlling and simulating aircraft, space craft and similar computer controlled dynamic systems. Short for "Tunable Numerical Integration", it is a fixed-step iteration formula whose integrand can be adjusted in phase and amplitude. T-integrators all have phase and gain adjustable parameters that are similar to phase and gain adjustable parameters in modern aircraft autopilots. The simplest version of the T-integrator algorithm is as follows:

Let f(x) denote the integrand and P and G the phase and gain parameters. Furthermore, the left-hand side of the range of integration is denoted by x0 and Δx is the step size. T-integration is defined by the following recursive formula:
 * $$F(x_0) = 0$$
 * $$F(x_n) = F(x_{n-1}) + G (\Delta x) (Pf(x_n)+(1-P)f(x_{n-1})) \quad n \ge 1$$

where
 * $$x_n = x_0 + n \, \Delta x$$

The function F approximates the integral of f i.e.
 * $$F(x_n) \approx \int_{x_0}^{x_n} f(x) \,\mathrm{d}x. $$

If G = 1, then the method reduces to the following well known numerical integration techniques for the given values of P:
 * P = 0: the left-hand rectangle rule known as Euler's method
 * P = 1/2: the trapezoid rule,
 * P = 1: the right-hand rectangle rule,
 * P = 3/2: the Adams–Bashfourth corrector rule, (see Linear multistep method)

If G and/or P are other real numbers, then a different set of first order integrators is produced. T-integration can be tuned to the problem it is being used to solve by empirically selecting values of G and P to match the numerically integrated trajectory with a known real world check case. This is useful when simulating aircraft motion for various aircraft configurations. For example, G and P can be selected to match the real motion of the aircraft with the landing gear up, gear down, flaps up, flaps down, high Mach, low Mach, right engine out, left engine out and combination's of these and other aircraft configurations. In these applications, G and P are changed depending on the landing gear handle position, the flap handle position, the throttle position etc.