Adaptive Boundary and Point Control of Linear Stochastic Distributed Parameter Systems
SIAM Journal on Control and Optimization
Robust adaptive control
Optimal design of adaptive tracking controllers for non-linear systems
Automatica (Journal of IFAC)
Model Reference Adaptive Control of Distributed Parameter Systems
SIAM Journal on Control and Optimization
Nonlinear and Adaptive Control Design
Nonlinear and Adaptive Control Design
Adaptive boundary control for unstable parabolic PDEs-Part II: Estimation-based designs
Automatica (Journal of IFAC)
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Approaching the problem of optimal adaptive control as "optimal control made adaptive," namely, as a certainty equivalence combination of linear quadratic optimal control and standard parameter estimation, fails on two counts: numerical (as it requires a solution to a Riccati equation at each time step) and conceptual (as the combination actually does not possess any optimality property). In this note we present a particular form of optimality achievable in Lyapunov-based adaptive control. State and control are subject to positive definite penalties, whereas the parameter estimation error is penalized through an exponential of its square, which means that no attempt is made to enforce the parameter convergence, but the estimation transients are penalized simultaneously with the state and control transients. The form of optimality we reveal here is different from our work in [Z. H. Li and M. Krstic, "Optimal design of adaptive tracking controllers for nonlinear systems," Automatica, vol. 33, pp. 1459-1473, 1997] where only the terminal value of the parameter error was penalized. We present our optimality concept on a PDE example--boundary control of a particular parabolic PDE with an unknown reaction coefficient. Two technical ideas are central to the developments in the note: a non-quadratic Lyapunov function and a normalization in the Lyapunov-based update law. The optimal adaptive control problem is fundamentally nonlinear and we explore this aspect through several examples that highlight the interplay between the non-quadratic cost and value functions.