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From the Ray Prouty Archives: Parameter Identification

By Ray Prouty | January 12, 2018

Since about 1970, a new term has entered the vocabulary of those who analyze helicopter stability and control. The term is “parameter identification,” or as some people say, "system identification." What does it mean?

Starting simple

Let’s start with a straightforward example. Suppose you have an object supported on a spring with a damper in parallel. If you pull the objects down, let go and then record its displacement, you will obtain a time history of an oscillation.

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If you show this time history to someone with a little experience in the dynamics of systems, he will be able to evaluate — or identify — two parameters of the system:

  • The ratio of the spring constant to the weight, and
  • The ratio of the damping constant to the weight. He does this by working with the measured period of the oscillation and the rate of decay of the envelope that encloses it. If he knows the object’s weight, he can tell you the spring rate, “k,” in pounds per inch. He can also tell you the damping constant, “c,” in pounds per inch per second.

Add complication

An airplane or a helicopter in flight obeys the same dynamic laws as the spring-mass-damper system. But the analysis is more complicated because the aircraft moves in six degrees of freedom rather than just one. It can move horizontally, vertically and sideward, as well as rotating in pitch, roll and yaw.

Starting from some trim condition, changes in the speed along (or the angular rate around) any of the degrees of freedom.

These changes would all seem to be due to the equivalent of dampers. But by looking at the system slightly differently, we can relate some of the changes to effective spring effects. Examples are the stability in forward flight of pitch provided by the horizontal stabilizer and the directional stability provided by a tail rotor or vertical stabilizer.

In the language of the aircraft analyst, all the change parameters are known as “stability derivatives,” and they are used in a set of “linear” equations of motion.

For an aircraft in free flight with six degrees of freedom, it is theoretically possible to have 62 (36) stability derivatives. How so? Because for each motion along or around one of the three axes, a force or moment may be generated in not only the axis being directly affected, but in all of the other five degrees of freedom as well. (In some rather sophisticated analyses, the rotor coning, longitudinal flapping and lateral flapping are considered separate degrees of freedom, thus raising the theoretical number of stability derivatives to 81.)

In addition, there is another set of derivatives. They represent the forces and moments produced when the pilot moves any one of his controls: collective, longitudinal cyclic, lateral cyclic and tail rotor pedals. This adds another possible 24 derivatives to our collection of 36 — to give 60.

Fortunately, many of the derivatives either physically do not exist or are small enough to be ignored. This leaves only about half that are really important to the analysis of flying qualities. The values of most of these stability derivatives depend on speed. In hover, therefore, even more derivatives drop out and the ones that remain have quite different values than they do in forward flight.

Estimated values of all of these stability derivatives can be obtained even before the aircraft flies. They are obtained using combinations of analysis, wind-tunnel results and judicious guess.

Obtaining flight-test data

However, the aircraft in flight may not behave in accordance with these initial estimates. Thus, analysts are motivated to use flight-test results to check and possibly modify the values of the stability derivatives. This is important for such things as making good simulators, providing a better analysis for correcting flying-quality problems or even developing a sophisticated autopilot that will automatically reconfigure itself for optimum performance in any flight condition.

Flight testing is done with a helicopter that is instrumented to record accelerations, rates and attitudes. The aircraft is first trimmed out in the flight condition to be investigated. To excite, or trigger, the various modes of response, all four pilot control inputs may be used: longitudinal cyclic, lateral cyclic, collective and pedals. Figure 64-3 shows several types of control inputs used.

Step, pulse or doublet

A step control input is good for a short-time response. But for longer time histories, a pulse or doublet that represents flying through a gust must be used. The special sequence of sharp-edged pulses known as the “3211” will adequately excite both the short-period modes — which damp out quickly — and the long-period modes that might take 20 or more seconds to develop. Figure 64-4 shows how the pitch rate of an Aerospatiale (now Eurocopter) Puma at 80 kts responds to this kind of input in the longitudinal system.

To work in the “frequency domain” instead of the “time domain,” one carries out a frequency sweep using sinusoidal inputs at increasing frequency.

Since it is the helicopter’s basic characteristics that are desired, these tests should be done with the stability and control augmentation system (SCAS) turned off. That is, unless the helicopter is so unstable that adequate time histories cannot be obtained without SCAS.

Flight-test data can contain “noise” because the sensors pick up vibration, or the data is being taken when the air is not absolutely calm. Therefore, the data may have to be “smoothed” before analysis. A special filter to do this is known as “Kalman Filter.” Figure 64-5 shows the result of treating on test’s time history with a filter.

Extracting derivatives

There are several ways to obtain the derivatives from the flight-test time histories. Each starts with a set of six equations of motion with first estimates for the 60 derivatives. The methods for extracting the derivatives may be thought of as sophisticated trial-and-error systems that attempt to curve-fit calculated results to the test data by modifying each of the derivatives.

Without going into the mathematics, we can at least name some of the techniques: Ordinary Least Squares, Weighted Least Squares, Recursive Least Squares, Deterministic Least Squares, Maximum Likelihood, Statistical Linearized Filter and Extended Kalman Filter.

Many techniques require several passes through the data. Each pass produces better estimates. An enormous amount of calculation is required, so these techniques only become practical when high-speed computers were introduced.

Several helicopters have been subjected to these techniques. Here’s one discovery: The values of the final stability derivatives that make simulation satisfactorily agree with flight test data can be more than 50% different from the original theoretical estimates.

This is most striking in those derivatives that are important in the Dutch roll mode (a combination of yaw and roll). These derivatives govern directional stability, dihedral effect, yaw damping and roll damping.

It is evident that some of these are strongly affected by the main rotor’s wake as it impinges on the tail surfaces in ways that are difficult to predict. This difficulty is what makes the parameter-identification process so valuable as the development of a helicopter matures beyond its first flight.

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