T-Integration

T-integration, which stands for "tunable numerical integration," is a fast, accurate, and numerically stable numerical integration formula given by

where  is the integral,  is the integrand,  and  are "phase" and "gain" tuning parameters,  refers to the number of the iteration being evaluated, and  is the integration step size.

The method was developed during the Apollo era to figure out how to simulate the digitally controlled Apollo command module during rendezvous and lunar landing operations. It was needed because none of the classical numerical integrators worked when trying to simulate the digital flight control systems maneuvering a space craft to a lunar landing.

For , varying  from 0 to 2 gives many classical first-order integrators:

1.  and : Euler integrator,

2.  and : trapezoidal rule,

3.  and : Rectangular rule,

4.  and : Adams' method.

REFERENCES:

Fowler, M. "A New Numerical Method for Simulation." Simulation 6, 90-92, Feb. 1976.

Smith, J. M. "Recent Developments in Numerical Integration." J. Dynam. Sys., Measurement and Control 96, Ser. G-1, No. 1, 61-70, Mar. 1974.

Smith, J. M. "Zero-Order T-Integration and Its Relation to the Mean Value Theorem." In Proceedings of the Sixth Annual Pittsburgh Modeling and Simulation Conference, Part 1, April 24-25, 1975.

Smith, J. M. "Modern Numerical Integration Methods." In Mathematical Modeling and Digital Simulation, 2nd ed. New York: John Wiley, 1988.

Smith, J. M. "Fast T-Integration." J. Mech. Eng. Sys. 1, 27-31, Jul./Aug. 1990.

Smith, J. M. "Jon Michael Smith on T-Integration: Trade Secrets in Numerical Analysis." http://members.aol.com/jsmith46ws/ni1.htm.