Tracking and parameter identification for model reference adaptive control

Abstract

Summary We provide barrier Lyapunov functions for model reference adaptive control algorithms, allowing us to prove robustness in the input‐to‐state stability framework and to compute rates of exponential convergence of the tracking and parameter identification errors to zero. Our results ensure identification of all entries of the unknown weight and control effectiveness matrices. We provide easily checked sufficient conditions for our relaxed persistency of excitation conditions to hold. Our illustrative numerical example demonstrates the performance of the control methods.

Event-triggered control for linear time-varying systems using a positive systems approach

“…≤ e a(t−s) − 1 (B.10) when t ≥ s ≥ 0. Thus (B.9) implies that [8], for the use of this decomposition of fundamental solutions in adaptive control. The factoring (B.18) makes it possible to compute the fundamental solution values that are needed to check Assumption 3 when C = I.…”

“…In terms of the preceding matrices and constants and the matrices (8), and continuing our notation A = A 0 + A δ and B = B 0 + B δ , we are now ready to define our eventtriggered control. Our event-triggered feedback control and the corresponding event-triggering times t i are defined by…”

“…is satisfied, where x(0) = 0, B is from (8), and the constant ν > 0 satisfies the requirements from Section 3. The system (a)-(d) is an event-triggered dynamics that resets the control values at the times t i , using the following inductive argument, where the in (d) is used to express the fact that t i+1 is the infimum of all times t ≥ t i when (13) fails to hold when i is such that t i+1 < ∞.…”

Feedback stabilization and robustness analysis using bounds on fundamental matrices

“…This includes the special case where the sample times are t i = iν for all i ∈ Z 0 , in which case we can take η = ν and S = {(r, r + ν) ∈ R 2 : r ∈ [0, p 0 ]}. Although fundamental solutions for the known matrices M and M ± δ are used in the formula for the control and in (9), they can be computed from solving matrix valued differential equations, e.g., using the method from [10]. In fact, as noted in [10], we can compute Φ A (t, s) for any piecewise continuous locally bounded square matrix valued function A by writing Φ A (t, s) = α A (t)β A (s) where α A and β A are the unique solutions of the matrix differential equations…”

Improved performance of identification and adaptive control schemes using fractional operators

“…The domination case admits more robust designs, but, for the same reason, conservatism in the transient performance should be expected. Since adaptive schemes are justified in its capability to deal with uncertain settings, the design of high‐performance robust schemes becomes one of the main focus for adaptive researchers (e.g., the recent article 4 ).…”