The MIT Rule: Adapting through Gradient-Descent Optimization

The MIT Rule was first introduced by Osborn, Whitaker, and Kezer in a paper presented at the Institute of Aerospace Science (now known as AIAA) in 1961 titled “New Developments in the Design of Model Reference Adaptive Control Systems.” It offered a straightforward approach to the Model Reference Adaptive Control (MRAC) challenge:

> How can we adjust the controller’s parameters to ensure that the response of the vehicle aligns closely with that of a stable reference system, regarded as optimal for handling qualities?

The authors proposed updating the controller's parameters ? by minimizing a parametric error function E(?) through a gradient descent adaptation law. The MIT Rule can be likened to gravity acting on a marble in a basin, guiding it to rest at the lowest point, where energy is minimized. This encapsulates the fundamental principle of the MIT Rule.

The parametric error function E(?) typically takes the form:

Here, e(?) is defined as a scalar model error, representing the discrepancy between the actual dynamics of the controlled vehicle and those of the reference system.

By employing the gradient-descent optimization technique, the adaptation law is derived by making the time derivative of the control parameter (d?/dt) proportional to the negative gradient of the error function E(?), leading to:

Here, the proportional constant ? acts as a design parameter that determines the adaptive gain.

While this equation represents the original mathematical formulation of the MIT Rule, an alternative version exists that does not require gradient computation—only the sign.

Though the MIT Rule and its variations may seem simple, they marked a significant milestone in the history of adaptive control, yielding impressive results in many early MRAC applications.

One notable implementation of the MIT Rule occurred during the pioneering hypersonic research programs.

A Perfect Pairing for Hypersonics: MRAC and the MIT Rule

During the exhilarating era of the Space Race, engineers from North American Aviation and Honeywell raced against time to create a new type of manned aircraft capable of exploring hypersonic flight: the North American X-15.

This engineering marvel emerged as part of NASA’s X-Plane program, developed in close partnership with the US Air Force. It served as a hypersonic research vehicle, aimed at understanding the flight conditions a spacecraft would encounter during re-entry to Earth.

> It was a time when the physics of hypersonics had yet to be fully understood.

The initial X-15 prototypes, X-15–1 and X-15–2, featured a flight control system with various fixed-gain settings that pilots could select based on the flight phase. However, despite providing adequate handling qualities in slower, time-varying flight conditions, pilots reported excessive workload during ballistic flights and reentries.

The challenges stemmed from the constant adjustments pilots had to make in response to the inadequate handling qualities offered by the fixed-gain controller during rapidly changing flight conditions.

Handling quality issues led to transient oscillations in the pitch and yaw axes during the re-entry phase, complicating control during this critical flight period.

These challenges intensified the pilot's workload, especially during transitions between ballistic and atmospheric flight.

The X-15's ballistic-control mode operated independently from the aerodynamic-control mode, utilizing reaction thrusters for pitch, yaw, and roll momentum.

The deficiencies of the fixed-gain control system opened the door for innovative adaptive control technologies. Around 1961, Honeywell seized the chance to test a new adaptive flight control hardware in the X-15, originally intended for the canceled X-20 Dyna-Soar prototype.

This new adaptive controller, designated the MH-96 AFCS (Adaptive Flight Control System), employed a rate command model-following structure featuring a scalar adaptive gain, based on an MRAC architecture utilizing the MIT Rule for its adaptation law.

The third prototype, X-15–3, was the only one equipped with the MH-96 adaptive control system, marking it as one of the first aircraft in aviation history to incorporate an adaptive controller.

The pitch axis controller architecture mirrored that shown in Figure 4, with the reference system defined by a first-order low-pass filter having a time constant (?) of 0.5 seconds.

The goal of this architecture was to maintain the controller's adaptive gain (represented as Kq in Figure 9) at the highest feasible level without causing unacceptable high-frequency instabilities.

In this manner, the adaptive controller would rapidly align the vehicle dynamics with the reference model.

The upper limit of the adaptive gain, defining the delicate boundary between stability and instability, was an unknown parameter reliant on flight conditions and actuator dynamics.

The MH-96 adaptive control system aimed to operate at the edge of instability, where the handling qualities of the X-15 closely matched those of the reference model.

Understanding this concept is crucial to grasping how the MIT Rule was applied in the MH-96 controller.

Operating near the edge of instability, the adaptive demand path of the MH-96 controller exhibited limit cycle oscillations with frequencies close to the natural frequency of the servo-actuator loop (90rad/s for the elevons and 70rad/s for the rudder).

With this knowledge, Honeywell's control engineers identified the need to select an error function E(K) of the form:

Where T represents a threshold between acceptable and unacceptable oscillation amplitudes, K is the adaptive gain, and ||u(K)|| indicates the amplitude of the adaptive demand component centered around the natural frequency of the servo-actuator loop.

By applying the MIT Rule to minimize this error, engineers derived the following adaptation law for the K gain (with the actual adaptation law including additional rate limits and saturation):

Considering:

The final formulation of the MIT Rule becomes:

If oscillation amplitudes fall below the threshold T, the adaptive gain K increases, and vice versa.

The adaptive gain K was also instrumental in the MH-96 adaptive system for automatic blending between aerodynamic and ballistic control modes.

With lower values of K, the ballistic-control mode would automatically disengage, while exceeding a switch-on threshold indicated a decrease in aerodynamic control power, re-engaging the reaction control thrusters and facilitating a seamless control transition during ascent and reentry.

Additionally, maintaining a constant reference system throughout the flight envelope allowed the MH-96 to deliver consistent performance and good handling qualities across all control axes.

With this adaptation strategy, the X-15–3 equipped with the MH-96 adaptive flight control system outperformed its fixed-gain predecessors (X-15–1 and X-15–2).

A compelling summary of pilot feedback regarding the MH-96 adaptive system is found in the following 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 upcoming installment 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 control engineering, I would assert that “a Simulink model is worth a thousand equations.”

With the relatively straightforward architecture of the MH-96 control system, you'll find that implementing it in a Simulink model allows for a tangible demonstration of 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.

[6] NASA Technical Report. Experience With the X-15 Adaptive Flight Control System.

Rodney Rodríguez Robles is an aerospace engineer, cyclist, blogger, and advocate for cutting-edge technology, living the dream in the aerospace industry he aspired to as a child. He writes about coding, the history of aeronautics, control engineering, rocket science, and the technology enhancing everyday life.

Connect with me on LinkedIn and Twitter; I would love to hear from you!