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The Beauty of Simple Adaptive Control --
Old and New results in Stability Analysis of Nonlinear Systems

Abstract: Simple Adaptive Control (SAC) techniques have initially been conceived because of the need for adaptive control methods in large-scale systems, where the use of models and controllers of the order of the plant was naturally excluded. While initially SAC was considered (even by its own developers) to be just a modest version of the standard MRAC, the workshop shows that it was only before appropriate mathematical tools of analysis (that could reveal its real potential yet were missing at the time) had been developed. Further developments showed that they can easily be applied to such applications as missiles, planes, satellites, fine motion control, etc. The talk explains how various drawbacks related to the classical Model Reference Adaptive Control (MRAC) methodology have been addressed and eliminated. To this end, it will be shown that the conditions needed for robust stability have been significantly mitigated. Recent developments in nonlinear systems stability analysis tools lead to clear proofs of SAC stability in real-world application and realistic environments. Previously unstable “counterexamples” to MRAC are revisited and shown to be just simple, successful, and stable applications of SAC. Realistic examples from various domains of flight control, guidance and aerospace are also 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 to illustrates show the simplicity of implementation of SAC as an Add-On to classical control design towards improving performance beyond anything that could be obtained otherwise.

To this end, although Lyapunov stability theory is the customary basis of any modern stability analysis, its direct application requires fitting to the system a Positive Definite function whose derivative ”along all trajectories of the system” is Negative Definite. If this is the case, it is easy to see that both the Lyapunov function and its derivative ultimately vanish and therefore, one can reach the desired conclusion of asymptotic stability. However, because in most non-trivial problems the derivative is at most Negative Semi-Definite, various extensions to the basic Lyapunov stability theory have been sought. Because early extensions of Lyapunov stability theory were only covering autonomous systems of the form x = f (x) , various alternatives were sought for nonautonomous systems of the form x = f (x,t) . In particular, a successful alternative has been provided by Barbalat Lemma that under some conditions allowed concluding that the Lyapunov derivative ultimately vanishes and therefore, as many examples illustrated, asymptotic stability conclusions could be drawn. Texts on Nonlinear Control gave various formulations intending to mitigate prior conditions that would facilitate application of Barbalat Lemma to stability analysis. However, Barbalat Lemma that deals with function theory and not necessarily with theory of systems stability imposes conditions of uniform continuity of functions and even continuity of derivatives that again could limit its applicability. Besides, even when applicable, it only ends with partial results. As even recent publications illustrate, although extensions of LaSalle's Invariance Principle to nonautonomous systems have been available at least since 1976, they have remained surprisingly unknown for large circles of 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 prospective users with respect to its usefulness. It is hoped that the review of LaSalle’s Invariance Principle along with a new extended Invariance Principles and the comparative parallel presentation of various alternatives to stability analysis may help showing the extreme

• Review of need for adaptation
• Life is with uncertainty: need various gains for various situation, either change in nominal parameters or various operational conditions
• Nonstationary gains: right gains at right times. Good idea, yet… must be careful.
• Example: The danger of “safe” fixed control in nonstationary environments
• Model Reference Adaptive Control (MRAC)
• First Adaptive Control ideas: ingenious and failure
• Basic MRAC: First rigorous proofs, problematic SPR conditions
• “Classical” MRAC: Stability in ideal situations , problems otherwise:
• Unmodeled dynamics
• Persistent excitation
• Bursting, etc.
• Simple Adaptive Control (SAC)
• From MRAC towards Simple Adaptive Control (SAC)
• Can Adaptive Control use and extend ideas from Optimization to the world of uncertainty?
• The Simple Adaptive Control (SAC)
• Gentle introduction of Stability conditions in Engineering terms
• Minimum Phase and conditions on CB
• Continuous mitigation of stability conditions
• Claim and Proof: Same conditions that allow LTI design also guarantee stability with SAC. Advantage: the guarantee of stability allows reaching superior performance.
• Discrete-time systems
• Nonstationary and Nonlinear systems
• Robustness with Noise
• Brief review of stability analysis for nonlinear systems
• Lyapunov stability approach: examples and limitations
• Simple example and LTI example
• First extensions: Krasovskii and LaSalle, Only Autonomous systems
• Further extensions: Barbalat and (the real and ignored) LaSalle, Nonautonomous systems
• A new Invariance Principle, safe tool for proofs of stability
• Example problems
• Application of SAC Approach
• Unstable MRAC “counterexamples” become simple examples for SAC
• Explicit design case: Combining Classical LTI design and SAC guarantee stability and lead to superior performance for Non-minimum Phase UAV.
• Missile
• Flexible structures: new developments relax the need for collocation
• Example Problems
• Advanced issues (time permitting): gain convergence, participants proposed examples

Workshop length and schedule: The workshop takes a full day and is splitted into two parts with a break in between.