T-integration

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T-Integration is a numerical integration technique developed by Jon Michael Smith to facilitate simulating command and control of aircraft, space craft and similar dynamic systems. Short for "tunable numerical integration", it uses a fixed step size and an iteration formula that depends on phase and gain parameters 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 limit of the integral is denoted by x0 and Δx is the step size. T-integration is defined by the following recursive formula:

Fn = Fn−1 + G Δx (P fn + (1−P) fn−1), and F0 = 0.

Here fn stands for f(xn). The quantity Fn approximates

\int_{x_0}^{x_0+n\Delta x} 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,
  • P = 1/2: the trapezoid rule,
  • P = 1: the right-hand rectangle rule,
  • P = 3/2: the Adams’ Corrector rule,
  • P = Etc.

If G and/or P are other real numbers, then a set of new first order integrators is produced. Amazingly, out of this set is a unique pair of P and G that is ideal for simulating a given linear system. The simulator root locus exactly matches the root locus of the dynamic linear system being simulated. Even more surprising is that a small set of these first order integrators can match the Jacobian of a nonlinear system being simulated.

G and P can also be selected empirically by matching the numerically integrated trajectory with a known real world check case. This is particularly 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 combinations 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. This is particularly important when getting a simulator certified by the FAA (or other certification organization) for flight training purposes. You simply tune the simulator to satisfy the certifying organizations requirements for pilot flight handling evaluation and approval.

What makes T-Integration so different for classical numerical integration is that the foundation for the derivation of T-Integration is information theory while the foundation for the derivation of classical numerical integrators is approximation theory. The two theories were developed at different times. Information theory, being the more modern of the two, is more suited to modern digital simulation, controls and information sciences applications. Phase and Gain controls are commonplace in information and control systems applications; not so with classical numerical integrators.

T-Integration can be tuned to the problem it is being used to solve. For open-loop problems, setting the gain to 1 and varying the phase produces ALL classical first order numerical integrators and an infinity new integrators heretofore unknown. For closed loop applications the T-Integrator produces an infinity of non-classical integrators that produce exact numerical integration of linear systems and near exact integration of nonlinear systems.

For information systems applications (computer, control and communication and simulation) this first order T-Integrator out performed all numerical integrators based on classical approximation theory. Simulating aircraft motion for various aircraft configurations (gear up, gear down, flaps up, flaps down, engine out, stab-aug on, stab-aug off etc.) and dynamic conditions (high Mach, low Mach, take-off, landing etc.) becomes a matter of tuning the T-Integrator (finding the G and P) to the flight condition being simulated. In this sense the T-Integrator adapts to the problem it is trying to solve.

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