User:Jonmsmith/T-integration

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:

F_n = F_n−1 + G Δx (P f_n + (1−P) f_n−1).

Here f_n stands for f(x_n). The quantity F_n 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.

G and P can be any real numbers. They are not restricted to integer fractions as are the classical integrators. Obviously, this results in a doubly infinite set of non-classical first order integrators. Amazingly, out of this large set are unique pairs of P and G that is ideal for simulating continuous linear systems where the digital simulator root locus exactly matches the root locus of the analog linear system being simulated. Even more surprising is that a small set of these first order integrators can be made to used to match the Jacobian of the digital simulation to the Jacobian of the nonlinear system being simulated.

G and P can be selected empirically by matching digitally produced trajectory (numerically integrated trajectory) with a known real world analog 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. P and G can be selected empirically when getting a simulator certified by the FAA (or other certification organization) for flight training purposes. In these applications the simulation is tuned to satisfy the certifying organizations requirements for pilot flight handling evaluation and approval.

In other applications, various parts of a simulator can be developed by different organizations. When the pieces are brought together, each can be (usually is) out of phase with slightly different amplification, thus not working harmoniously. The simulation is slightly out of tune. The more the developers for a given simulation the more these problems occur. In these situations, the T-Integrator can be adjusted in gain and phase so that the simulator pieces are synchronized and can work in harmony. Tuning simulators is quite simple once you get the experience of adjusting phase and gain in the integrators in the simulation. After a while, like a piano tuner, it is an art that is easy to learn and anybody can do it, it just takes a bit of experience to tune a simulator and learn all the ways a simulation can get out of tune.

From a mathematical point of view, T-Integration is derived by block-diagramming the flow of information through a hybrid computer implementation of a continuous analog integrator. First: The integrand is converted to digital information by an analog-to-digital converter. The integrand is sampled and held for a sample period. The simplest analog to digital converter samples the integrand and the value of the integrand is held constant over the period of the sampler; simple enough. If the integrand is sinusoidal, and the period of the sampler is a seventh the period of the sinusoid, what results is a sequence of stair steps that roughly replicate the integrand. Second: careful inspection of the replica will show that it is shifted in time by a half sample period. To compensate for this delay requires a half sample period of "lead" compensation (digital "lead" filter). Third: careful inspection of the digital integrand shows that the average of the digital integrand is attenuated slightly compared with the continuous integrand. Thus, the digital integrand requires "gain" compensation so that the digital integrand will match the continuous integrand. Forth: It follows that if the digital integrand matches the continuous integrand, the area under both the digital, stair-step integrand function will be the same as the area under the continuous integrand. Now that means that the continuous integration of the digital integrand will exactly match continuous integration. Fifth: All that is required now is to sample the output of the analog integrator. What is between the integrand sampler and the integral sampler is the hybrid digital integrator. Now, to develop the numerical version of the hybrid integrator, take the Z-Transformation of the hybrid integrator from integrand sampler to integral sampler. Invert the Z-Transformation of the hybrid integrator into a difference equation and you have the T-Integrator.

If the hybrid integrator is used in a closed loop application, the feedback is delayed a full sample period (have to take the forward step before the feedback can be calculated). In this case, the compensation must be three halves of a sample period: A half to compensate for the analog-to digital converter delay and a full sample period to compensate for the loop closure delay. This process is derived carefully in the 1974 reference paper as cited below. The precise description of how the integrands produce the integrals is also shown in the paper on the discrete version of the mean value theorem of integral calculus also cited below. In simplest terms, the difference between the classical mean value theorem and the modern version is this: The classical mean value theorem only requires one parameter (the sampled value of the integrand) to make a rectangle exactly integrate a continuous integrand, while the modern version require two parameters (the phase compensation and the gain compensation)to make the stair-step function exactly integrate a continuous integrand.

What makes T-Integration different from classical numerical integration is the foundation for the derivation of T-Integration being information theory while the foundation for the derivation of classical numerical integrators is approximation theory. Phase and Gain controls are commonplace in information and control systems applications; not so with classical numerical integrators. The two theories were developed at different times for different reasons. Information theory, being the more modern of the two, is more suited to modern digital simulation, controls and information sciences applications. Further, the entire set of classically developed first order numerical integrators are found to be special cases of the more general T-Integrator for special case values of G and P.

Beyond the application of T-Integration for simulation is its potential for application in control systems. Since control systems can be designed using block diagrams, signal flow diagrams and digital programming diagrams the same principals apply to the development of the algorithms for controls as apply to the development of algorithms for simulation. For example, any flight control systems used in fighter aircraft or transport aircraft can employ integration as an integral part of control process. This can be implicit or explicit integration. In these cases the the T-Integrator can be embedded in the software design and used to tune the control system for various aircraft configurations and various flight phases. The G and P for automatic landing and for high Mach cruise are likely to be quite different. The G and P for combat configurations and flight regimes are also likely to be quite different. So it is that T-Integration has potential applications for control systems.

jonmsmith (talk) 01:48, 15 April 2008 (UTC)

References

   * Smith, J. M. "Recent Developments in Numerical Integration", J. Dynam. Sys., Measurement and Control 96, Ser. G-1, No. 1, 61-70, Mar. 1974.
   * Eric.W.Weisstein. "T-Integration Citation, CRC Concise Encyclopedia of Mathematics, Second Edition, pp.2986, 2003.
   * Smith, J. M. "Fast T-Integration." J. Mech. Eng. Sys. 1, 27-31, Jul. /Aug. 1990.
   * Smith, J. M. "Modern Numerical Integration Methods." In Mathematical Modeling and Digital Simulation, 2nd ed. New York: John Wiley, 1988.
   * 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.
   * Marc L. Sabin, "Bode Magnitude and Phase-Angle Characteristics of the Tunable Integrators," Vol. 6 Part 1, Proceedings of the Sixth Annual Pittsburgh Modeling and Simulation Conference, April 24-25, 1975.
   

• Weisstein, Eric W. "T-Integration". MathWorld.