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

Abstract

In applications where the bandwidth of the actuator dynamics is not sufficiently high, the stability of model reference adaptive controllers can be degraded drastically as it is well known. To this end, one of the effective approaches for tackling the challenge of limited actuator bandwidth is the hedging method in which using a "modified" reference model, the adaptation becomes "excluded" from the actuator dynamics. With this approach, however, the performance bounds between the uncertain dynamical system trajectories and the "ideal" reference model trajectories can be conservative and depend on the bounds on the system uncertainties; therefore, no "practical" performance guarantees exist. To address this challenge, we generalize a recently developed set-theoretic model reference adaptive control architecture, which has the capability to achieve "practical" (i.e., user-defined) performance guarantees, for uncertain dynamical systems subject to actuator dynamics. Specifically, we first show that the proposed architecture keeps the performance bounds between the uncertain dynamical system trajectories and the "modified" reference model trajectories within an a-priori, user-defined bound. We next show that the error bounds between the "ideal" reference model trajectories and the uncertain dynamical system trajectories is characterized by this user-defined bound as well as the actuator bandwidth limit, and hence, is "computable" using a given set of adaptive control design parameters. Finally, as a byproduct, our illustrative numerical example shows that the time rate of change of the actual control signal (i.e., the output of the actuator dynamics) becomes less in magnitude as compared with the the set-theoretic model reference adaptive control case without actuator dynamics.

Adaptive Set-Theoretic Emulator Reference Architecture (ASTERA): Control of Uncertain Dynamical Systems with Performance Guarantees and Smooth Transients

“…The contribution of this paper was an adaptive set-theoretic emulator reference architecture (ASTERA). The proposed architecture was a generalization of both set-theoretic model reference adaptive control methods [18][19][20][21][22][23][24] and state emulator-based or state emulator-like standard adaptive control methods [10][11][12][13][14][15][16][17] to simultaneously achieve user-defined performance guarantees and smooth transients. Specifically, the state emulator adopted in the ASTERA altered the trajectories of a given reference model for the purpose of achieving smooth transients -a property that does not necessarily exist in standard set-theoretic model reference adaptive control methods.…”

“…In fact, the common denominator of the contributions in [4][5][6][7][8][9][10][11][12][13][14][15][16][17] is that their respective worst-case system error bounds are conservative and not user-defined, since they are generally calculated using classical Lyapunov theory arguments that gives sufficient stability conditions only. To address this problem from a standard model reference adaptive control point, set-theoretic model reference adaptive control methods, which are predicated on a set-theoretic controller construction using generalized restricted potential functions (generalized barrier Lyapunov functions), are proposed in [18][19][20][21][22][23][24] (we refer to, for example, [ Section 1,19] for details on other adaptive control methods with strict performance guarantees such as [25][26][27][28][29][30][31][32]). In particular, the key feature of [18][19][20][21][22][23][24] is to guarantee the weighted Euclidean norm between the uncertain dynamical system state and the given reference model state to be less than a-priori, user-defined scalar worst-case bound, where this implies that they achieve strict performance guarantees.…”

“…To address this problem from a standard model reference adaptive control point, set-theoretic model reference adaptive control methods, which are predicated on a set-theoretic controller construction using generalized restricted potential functions (generalized barrier Lyapunov functions), are proposed in [18][19][20][21][22][23][24] (we refer to, for example, [ Section 1,19] for details on other adaptive control methods with strict performance guarantees such as [25][26][27][28][29][30][31][32]). In particular, the key feature of [18][19][20][21][22][23][24] is to guarantee the weighted Euclidean norm between the uncertain dynamical system state and the given reference model state to be less than a-priori, user-defined scalar worst-case bound, where this implies that they achieve strict performance guarantees. However, unlike the contributions in [4][5][6][7][8][9][10][11][12][13][14][15][16][17], these approaches can suffer from high-frequency oscillations since their system error dependent learning rate can be high for specific time instants to enforce the aforementioned user-defined scalar worst-case bound.…”

Human-in-the-Loop Systems with Inner and Outer Feedback Control Loops: Adaptation, Stability Conditions, and Performance Constraints

“…whereŴ (t) ∈ R (s+n+ny)×m is the estimate of W (t). Following the set-theoretic model reference adaptive control architecture presented in [9] (see also [10][11][12][13][14][15]), let the update law for (13) be given bẏ…”

“…The generalizations of the set-theoretic model reference adaptive control architecture to the unstructured system uncertainties, actuator failures, actuator dynamics were then studied in [12][13][14][15]. Within the scope of this paper, we use this new architecture in [9] for enforcing a user-defined performance constraint on the norm of the system error trajectories, where we explicitly show how the selection of this constraint affects the overall physical system.…”