The Challenge of Limit-Cycles

The success of the new electronic control system concepts was based on the use of electrical signals from sensors (primarily rate gyros and accel­erometers) that could be fed into the flight control system to control air­craft motion. As these electronic elements began to play a larger role, a different dynamic phenomenon came into play. "Limit-cycles” are a com­mon characteristic of nearly all mechanical-electrical closed-loop systems and are related to the total gain of the feedback loop. For an aircraft flight control system, total loop gain is the product of two variables: (1) the mag­nitude of the aerodynamic effectiveness of the control surface for creating rotational motion (aerodynamic gain) and (2) the magnitude of the artifi­cially created control surface command to the control surface (electrical gain). When the aerodynamic gain is low, such as at very low airspeeds, the electrical gain will be correspondingly high to command large sur­face deflections and rapid aircraft response. Conversely, when the aero­dynamic gain is high, such as at high airspeed, low electrical gains and small surface deflections are needed for rapid airplane response.

These systems all have small dead bands, lags, and rate limits (non­linearities) inherent in their final, real-world construction. When the

total feedback gain is increased, the closed-loop system will eventually exhibit a small oscillation (limit-cycle) within this nonlinear region. The resultant total loop gain, which causes a continuous, undamped limit – cycle to begin, represents the practical upper limit for the system gain since a further increase in gain will cause the system to become unstable and diverge rapidly, a condition which could result in structural failure of the system. Typically the limit-cycle frequency for an aircraft control system is between two and four cycles per second.

Notice that the limit-cycle characteristics, or boundaries, are depen­dent upon an accurate knowledge of control surface effectiveness. Ground tests for limit-cycle boundaries were first devised by NASA Dryden Flight Research Center (DFRC) for the X-15 program and were accomplished by using a portable analog computer, positioned next to the airplane, to gen­erate the predicted aerodynamic control effectiveness portion of the feed­back path.[683] The control system rate gyro on the airplane was bypassed, and the analog computer was used to generate the predicted aircraft response that would have been generated had the airplane been actually flying. This equivalent rate gyro output was then inserted into the control system. The total loop gain was then gradually increased at the analog computer until a sustained limit-cycle was observed at the control surface. Small stick raps were used to introduce a disturbance in the closed-loop system in order to observe the damping characteristics. Once the limit-cycle total loop gain boundaries were determined, the predicted aerodynamic gains for various flight conditions were used to establish electrical gain limits over the flight envelope. These ground tests became routine at NASA Dryden and at the Air Force Flight Test Center (AFFTC) for all new aircraft.[684] For subsequent production aircraft, the resulting gain schedules were programmed within the flight control system computer. Real-time, direct measurements of air­speed, altitude, Mach number, and angle of attack were used to access and adjust the electrical gain schedules while in flight to provide the highest safe feedback gain while avoiding limit-cycle boundaries.

Although the limit-cycle ground tests described above had been per­formed, the NASA-Northrop HL-10 lifting body encountered limit-cycle

oscillations on its maiden flight. After launch from the NB-52, the telem­etry data showed a large limit-cycle oscillation of the elevons. The oscil­lations were large enough that the pilot could feel the aircraft motion in the cockpit. NASA pilot Bruce Peterson manually lowered the pitch gain, which reduced the severity of the limit-cycle. Additional aerodynamic problems were present during the short flight requiring that the final landing approach be performed at a higher-than-normal airspeed. This caused the limit-cycle oscillations to begin again, and the pitch gain was reduced even further by Peterson, who then capped his already impres­sive performance by landing the craft safely at well over 300 mph. NASA engineer Weneth Painter insisted the flight be thoroughly analyzed before the test team made another flight attempt, and subsequent analysis by Robert Kempel and a team of engineers concluded that the wind tunnel predictions of elevon control effectiveness were considerably lower than the effectiveness experienced in flight.[685] This resulted in a higher aero­dynamic gain than expected in the total loop feedback path and required a reassessment of the maximum electrical gain that could be tolerated.[686]