Simple Adaptive Control and New Results in Stability Analysis
Itzhak Barkana, BARKANA Consulting
Haim Weiss, RAFAEL Advanced Defence Systems
Ilan Rusnak, RAFAEL Advanced Defence Systems
Simple Adaptive Control (SAC) techniques have been conceived for large-scale systems. While initially considered to be just a modest version of the standard Model Reference Adaptive Control (MRAC), it was only before appropriate mathematical tools of analysis had been developed. Further developments showed that they can easily be applied to such applications as robots, planes, missiles, satellites, fine motion control, etc. Drawbacks related to classical MRAC have been addressed and eliminated and conditions needed for robust stability have been mitigated. Recent developments in nonlinear systems stability analysis tools lead to clear proofs of SAC stability in realistic environments. Realistic examples are used to show that indeed SAC is the Stable Direct MRAC methodology. A non-minimum-phase and unstable UAV will be used as a detailed case-study. Application to real hardware will be demonstrated.
The basic Lyapunov stability theory requires fitting a Positive Definite function to the system whose derivative is Negative Definite. Because early extensions of Lyapunov stability theory were only covering autonomous systems, various alternatives were sought for nonautonomous systems. Yet they impose conditions of uniform continuity that again could limit its applicability. Besides, even when applicable, it only ends with partial results. Although extensions of LaSalle’s Invariance Principle to nonautonomous systems have been available at least since 1976, they have remained surprisingly unknown for the nonlinear control community. Moreover, even if assumably known, misinterpretations of its larger mathematical scope (that covers much more than mere asymptotic stability) may have misled the users with respect to its usefulness. The review of LaSalle’s Invariance Principle together with presentation of various alternatives to stability analysis may help showing the extreme efficiency of the new Theorems of Stability to nonlinear systems stability analysis.