Adaptive Control Theory : Guaranteed Robustness with Fast Adaptation

Controllers for linear systems can be analyzed for their performance and robustness in the frequency domain by using Bode plots, the Nyquist plot, the Nichols chart, and other tools. In the frequency domain, transient response can be analyzed in terms of bandwidth and resonant frequency properties, steadystate behavior is analyzed in terms of low-frequency gain, and robustness is analyzed in terms of gain, phase, and time-delay margins. Feedback control transfer functions can be selected to trade off performance and robustness. Frequency-domain analysis is highly developed in many application areas, including aircraft control [1], where there are strict performance requirements in terms of speed of response and overshoot, robustness requirements in the presence of varying or unknown parameters and disturbances, and requirements to limit control bandwidth due to the presence of vibratory system modes. In aircraft control, the reliability of linear systems design has led to gain scheduling for flight controllers, which allows the design of controllers with guaranteed performance and robustness in prescribed regions of the flight envelope. Gain scheduling requires linear models, which are obtained using extensive and expensive wind tunnel testing. Research in adaptive flight control started in the 1950s with a thrust to develop autopilots for supersonic aircraft with large flight envelopes while reducing the amount of gain-scheduling of linear controllers. Adaptive controllers were developed to cope with unknown dynamics, timevarying parameters, and unknown disturbances. There was intense early interest but also several failures of adaptive controllers [2]. Early developments in adaptive control were not sufficiently supported by theoretical analysis. Subsequent applications of Lyapunov theory and other techniques allowed rigorous analysis of adaptive controllers, resulting in modifications in parameter tuning laws, enhancements such as projection methods and robustifying terms, and guarantees on the ultimate boundedness of estimation errors and tracking errors. Tuning of adaptive controllers remains difficult, with parameters often being tuned by trial-and-error or simulation. Transient response and control signals do not have guaranteed bounds, and it is difficult to obtain robustness guarantees. The result is a continued reliance on gain-scheduled controllers. In the 1980s the aerospace industry was transformed by the advent of fly-by-wire digital avionics systems. These systems improved the ability to control aircraft using digital controllers. Yet new configurations of aircraft are aerodynamically unstable, have more significant nonlinearities, and are required to have larger flight envelopes. Particularly difficult are the supermaneuverability requirements, which demand flight at high angles of attack, where linear approximation models are not valid. Military and aging commercial aircraft applications require the design of control systems that can recover from structural or actuator failures. These developments generated a renewed interest in adaptive control, with adaptive controllers often used to augment gain-scheduled controllers. This combination affords some possibility of combining proven linear system design methods with adaptive control solutions. Model reference adaptive controllers (MRAC) provide an approach to combining linear control design methods with adaptive controllers, since linear design techniques can be used to design the reference models so that they conform to MILSPEC requirements [3] and provide performance improvement when tracking errors are small. However, bounds on the transient behavior of control signals cannot be given, and the designed controls are often I EEE Control Systems Magazine welcomes suggestions for books to be reviewed in this column. Please contact either Michael Polis or Zongli Lin, associate editors for book reviews.