T-integration

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T-integration is a numerical integration technique developed by Jon Michael Smith in the 1970s to facilitate command and control of space craft. Short for "tunable numerical integration", it uses a fixed step size and an iteration formula that depends on phase and gain parameters.

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.

T-Integration can be tuned to the problem it is being used to solve. T-Integration is based on information theory, not approximation theory. T-Integration has very simple frequency domain adjusting parameters: A phase adjusting parameter and a gain adjusting parameter. Interestingly, for open-loop problems, setting the gain 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) the simple first order T-Integrator out performs 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 simple matter of tuning the T-Integrator to the flight condition being simulated. In this sense the T-Integrator adapts to the problem it is trying to solve.

[edit] References

  • Eric W. Weisstein, T-Integration at MathWorld.
  • Smith, J. M. "Recent Developments in Numerical Integration", J. Dynam. Sys., Measurement and Control 96, Ser. G-1, No. 1, 61-70, Mar. 1974.
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