Innovative Adaptive Control in Hypersonics: The MIT Rule Explained
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We inhabit a time where the significance of control engineering is more pronounced than ever. Over the past decade, advancements such as autonomous vehicles, the drastic reduction of space access costs through reusable rockets, and fully automated aircraft operations have become commonplace. These remarkable achievements owe their existence to contemporary control systems, which influence nearly every facet of our daily lives.
Among the diverse array of control methodologies, one particularly noteworthy approach is Adaptive Control, distinguished by its ability to self-learn. Much like the human brain, an adaptive controller possesses a form of plasticity, enabling it to adjust in real time based on both current and past experiences.
If you are eager to delve deeper into Adaptive Control, this article serves as the inaugural piece of a series designed to illuminate the foundational concepts and techniques within this fascinating domain of Control Theory. To set the stage, let’s embark on a journey back to the early 1960s, an era marked by the Space Race and the dawn of hypersonic manned flights, to explore one of the earliest adaptation laws in Adaptive Control Theory: the renowned “MIT rule.”
The MIT Rule: Learning through Gradient-Descent Optimization
The MIT rule was introduced by Osborn, Whitaker, and Kezer in a 1961 paper presented at the Institute of Aerospace Science (now AIAA), titled “New developments in the design of model reference adaptive control systems.” This rule offered a straightforward solution to the Model Reference Adaptive Control (MRAC) dilemma:
How can one adjust the controller’s parameters to align the output of a system (such as a vehicle) with that of a stable reference system, deemed optimal for its handling characteristics?
The authors proposed a method for updating the controller’s parameters ? by minimizing a parametric error function E(?) through a gradient descent adaptation strategy. Much like a marble rolling in a basin, the MIT rule acts like gravity, guiding the marble to the basin's lowest point, where its energy is minimized.
The parametric error function E(?) typically takes the form:
In this scenario, e(?) signifies a scalar model error, representing the disparity between the controlled vehicle's measured dynamics and those of the reference system.
Through the gradient-descent optimization, the adaptation law is derived by setting the time derivative of the control parameter (d**?**/dt) to be proportional to the negative gradient of the error function E(?), leading to:
Here, the proportional constant ? serves as a design parameter defining the adaptive gain.
While this equation encapsulates the original mathematical expression of the MIT rule, an alternate version exists that does not necessitate calculating the gradient, only requiring knowledge of its sign:
Despite its simplicity, the MIT rule signified a pivotal moment in adaptive control, yielding remarkable outcomes in numerous early MRAC applications. One of the most celebrated uses of the MIT rule can be traced back to the initial hypersonic research endeavors.
A Harmonious Union in Hypersonics: MRAC and the MIT Rule
During the exhilarating days of the Space Race, engineers from North American Aviation and Honeywell worked tirelessly to develop and test a new type of manned aircraft—the North American X-15. This engineering marvel emerged from NASA’s X-Plane program, developed in close collaboration with the US Air Force, and was crafted as a hypersonic research vehicle to investigate the conditions a spacecraft would encounter upon re-entering Earth’s atmosphere.
It was a time when the physics of hypersonics was still largely uncharted.
The X-15's first two prototypes, the X-15–1 and X-15–2, featured a flight control system with various fixed-gain settings selectable by the pilot based on the flight phase. Although the fixed-gain control system generally provided satisfactory handling characteristics in slow-changing flight conditions, pilots reported excessive cockpit workload during ballistic flights and re-entries due to the constant adjustments needed to manage the poor handling qualities in rapid flight transitions.
This degradation in handling quality manifested as temporary oscillations in the pitch and yaw axes during the re-entry phase, complicating the critical steering tasks for the pilot.
The challenges intensified during the transition between ballistic and atmospheric flight, around 125,000 feet, where the pilot had to manually switch control modes, leading to an increase in workload. The ballistic-control mode featured a separate flight control stick from the aerodynamic-control mode, utilizing reaction thrusters for maneuvering.
These shortcomings of the fixed-gain system paved the way for the advent of newer adaptive control technologies. In 1961, Honeywell seized the chance to test a novel adaptive flight control hardware in the X-15, originally designed for the X-20 Dyna-Soar prototype (the X-20 program and its MH-90 adaptive flight control system were ultimately canceled in December 1963).
This new adaptive controller, known as the MH-96 AFCS (Adaptive Flight Control System), was a rate-command, model-following controller employing a single scalar adaptive gain based on an MRAC architecture that utilized the MIT rule for its adaptation law.
The X-15–3, the third prototype, was the sole aircraft equipped with the MH-96 adaptive control system, making it one of the first in aviation history to feature such technology. The controller's architecture for the pitch axis mirrored that illustrated in the previous figures, with a reference system defined by a first-order low-pass filter with a time constant of 0.5 seconds.
The primary aim of this architecture was to maintain the controller's adaptive gain (denoted by Kq) at the highest possible level without inducing unacceptable high-frequency instabilities, allowing for quick alignment of the vehicle dynamics with the reference model.
The upper limit of the adaptive gain, which delineates the fine line between stability and instability, is an unknown parameter dependent on flight conditions and actuator dynamics.
Operating on the edge of instability was indeed the objective of the MH-96 adaptive control system, as this state optimized the handling qualities of the X-15 to align closely with those of the reference model.
At this critical juncture, the adaptive demand path of the MH-96 controller would exhibit limit cycle oscillations near the natural frequency of the servo-actuator loop (90 rad/s for the elevons and 70 rad/s for the rudder). Recognizing this phenomenon, Honeywell’s control engineers determined that the error function E(K) should be shaped as follows:
Here, T serves as a threshold distinguishing acceptable from unacceptable oscillation amplitudes, K is the adaptive gain, and ||**u(K)**|| indicates the amplitude of the adaptive demand component around a frequency bandwidth centered on the natural frequency of the servo-actuator loop.
Applying the MIT rule to minimize this error yields the adaptation law for the K gain (the actual adaptation law incorporates additional rate limits and saturation):
This results in the following equation:
Notably, when the oscillation amplitude is below the threshold T, the adaptation gain K increases, and vice versa. The adaptive gain K also played a role in the MH-96 adaptive system for seamlessly blending between the aerodynamic and ballistic control modes.
For low values of K, the ballistic-control mode would automatically disengage, but once K exceeded a predefined threshold (indicating a reduction in aerodynamic control power), the reaction control thrusters re-engaged, facilitating a nearly seamless control transition during ascent and re-entry.
As the reference system remained constant throughout the flight envelope, the MH-96 demonstrated consistent response and commendable handling qualities across all control axes. Thanks to this adaptation law, the X-15–3 and its MH-96 adaptive flight control system achieved superior performance compared to its fixed-gain predecessors (X-15–1 and X-15–2).
A succinct summary of the pilots' feedback on the MH-96 adaptive system can be found in this excerpt from a NASA report:
“The true superiority of the X-15 AFCS was that it unburdened the pilot. The airplane was stable at any dynamic pressure and at any angle of attack. The AFCS inspired confidence and allowed the pilot to spend time cross-checking flight instruments, checking subsystems, and 'sightseeing.'” — Pilot observations from Experience with the X-15 Adaptive Flight Control System report.
What Lies Ahead?
In the subsequent chapter of this series, we will explore how to construct a simplified Simulink model of the X-15’s MH-96 AFCS.
As the saying goes, a picture is worth a thousand words, but in the realm of control engineering, I contend that “a Simulink model is worth a thousand equations.”
Given the relatively straightforward architecture of the MH-96 control system, you will discover the ease of implementing it in a Simulink model, providing an insightful look at the MIT rule's performance in a practical context.
See you in the next chapter!
References
[1] Iven M.Y. Mareels, Brian D.O. Anderson, Robert R. Bitmead, Marc Bodson, Shankar S. Sastry, Revisiting the Mit Rule for Adaptive Control, IFAC Proceedings Volumes, Volume 20, Issue 2, 1987, Pages 161–166, ISSN 1474–6670
[2] NASA Armstrong Fact Sheet: X-15 Hypersonic Research Program
[3] Orr J.S., Statler I.C., Barshi I. (2015) The X-15 3–65 Accident: An Aircraft Systems and Flight Control Perspective. In: Sgobba T., Rongier I. (eds) Space Safety is No Accident. Springer, Cham.
[4] NASA Technical Report: The X-20 Flight Control System Development
[5] Dydek, Zachary, Anuradha Annaswamy, and Eugene Lavretsky. “Adaptive Control and the NASA X-15–3 Flight Revisited.” IEEE Control Systems Magazine 30.3 (2010): 32–48. Web.© 2010 IEEE.
[6] NASA Technical Report. Experience With the X-15 Adaptive Flight Control System.
Rodney Rodríguez Robles is an aerospace engineer, cyclist, blogger, and cutting-edge technology advocate, realizing a dream in the aerospace sector he envisioned as a child. He discusses coding, aeronautics history, control engineering, rocket science, and technologies that enhance daily life.
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