Decentralized Adaptive Stabilization of Large-Scale Active-Passive Modular Systems

Adaptive control of unactuated dynamical systems through interconnections: Stability and performance guarantees

“…While we do not consider the actuated and unactuated system uncertainties to be time varying in the theoretical development of this article, we include this final condition on the restricted potential function such that our results can be readily extended for the practical case, in which the uncertainties are time varying. Second, as considered in References , a candidate restricted potential function that satisfies all the conditions stated in Definition has the form . Finally, as discussed in Reference , Definition is a generalized definition of the restricted potential function (Barrier Lyapunov function) used in References .…”

“…The proposed control is applied to stabilize the overall interconnected system in the presence of unknown physical interconnections as well as uncertainties in both the actuated and unactuated dynamics. The performance guarantees are enforced using a set-theoretic model reference adaptive control approach * such that the respective system error trajectories of the actuated and unactuated dynamics are restricted to stay inside user-defined *Note that a set-theoretic adaptive control architecture is utilized in the prior work of the authors; [35][36][37][38][39] however, they are not applicable as-they-are to the problem considered in this article (see Section 2). compact sets.…”

On Set-Theoretic Model Reference Adaptive Control of Uncertain Dynamical Systems Subject to Actuator Dynamics

“…whereW (t) Ŵ (t) − W (t), t ≥ 0, is the weight estimation error. Using the energy function V (e i ,W ) = φ( e i P ) + γ −1 i tr W TW , where D { e i (t) P : e i (t) P < } with P ∈ R n×n + being a solution of the Lyapunov equation in (12)…”

“…The first step of the proof is similar to the discussion given in the last paragraph of Section III. Specifically, consider the energy function given by V (e,W ) = φ( e P )+γ −1 tr W TW , where D { e(t) P : e(t) P < } with P ∈ R n×n + being a solution of the Lyapunov equation in (12) for R ∈ R n×n + . Using this energy function, one can calculateV e(t),W (t) ≤ − 1 2 αV (e(t),W (t)) + µ, where α λmin(R) λmax(P ) , d 2γ −1 iwẇ , and µ 1 2 αγ −1 iw 2 + d. In this calculation, we adopt similar steps as in the proof of Theorem 1 in [6]; thus, we refer interested readers to [6] for details.…”

“…To address this challenge, we generalize a recently developed set-theoretic model reference adaptive control architecture [6] (see also [7][8][9][10][11][12][13]), which has the capability to achieve "practical" performance guarantees, for uncertain dynamical systems subject to actuator dynamics using tools and methods from [4,5]. Specifically, we first show that the proposed set-theoretic model reference adaptive control architecture keeps the performance bounds between the uncertain dynamical system trajectories and the "modified" reference model trajectories within an a-priori, user-defined bound (unlike the results in [4,5]).…”