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Register a new userStabilization of perturbed integrator chains using Lyapunov-Based Homogeneous Controllers
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In this paper, we present a Lyapunov-based homogeneous controller for the stabilization of a perturbed chain of integrators of arbitrary order $rgeq 1$. The proposed controller is based on homogeneous controller for stabilization of pure integrator chains. The advantages to control the homogeneity degree of the controller are also discussed. A bounded-controller with minimum amplitude of discontinuous control and a controller with fixed-time convergence are synthesized, using control of homogeneity degree, and their performances are shown in simulations. It is demonstrated that the homogeneous arbitrary HOSM controller cite{Levant2001} is a particular case of our controller.
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Consider the $n$-th integrator $dot x=J_nx+sigma(u)e_n$, where $xinmathbb{R}^n$, $uin mathbb{R}$, $J_n$ is the $n$-th Jordan block and $e_n=(0 cdots 0 1)^Tinmathbb{R}^n$. We provide easily implementable state feedback laws $u=k(x)$ which not only render the closed-loop system globally asymptotically stable but also are finite-gain $L_p$-stabilizing with arbitrarily small gain. These $L_p$-stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite-time stabilization of linear systems. We also provide additional $L_infty$-stabilization results for the case of both internal and external disturbances of the $n$-th integrator, namely for the perturbed system $dot x=J_nx+e_nsigma (k(x)+d)+D$ where $dinmathbb{R}$ and $Dinmathbb{R}^n$.
In this paper, we present Lyapunov-based adaptive controllers for the practical (or real) stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as Adaptive Higher Order Sliding Mode (AHOSM) controllers since they are designed for nonlinear SISO systems with bounded uncertainties such that the uncertainty bounds are unknown. Our main result states that, given any neighborhood N of the origin, we determine a controller insuring, for every uncertainty bounds, that every trajectory of the corresponding closed loop system enters N and eventually remains there. The effectiveness of these controllers is illustrated through simulations.
The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order.
In this paper, we present Lyapunov-based {color{black}time varying} controllers for {color{black}fast} stabilization of a perturbed chain of integrators with bounded uncertainties. We refer to such controllers as {color{black}time varying} higher order sliding mode controllers since they are designed for nonlinear Single-Input-Single-Output (SISO) systems with bounded uncertainties such that the uncertainty bounds are unknown. %{color{blue} OLD: Our main result states that, given any neighborhood $varepsilon$ of the origin, we determine a controller insuring, for every uncertainty bounds, that every trajectory of the corresponding closed loop system enters $varepsilon$ and eventually remains there. Furthermore, based on the homogeneity property, a new asymptotic accuracy, which depends on the size of $varepsilon$, is presented.} We provide a time varying control feedback law insuring verifying the following: there exists a family $(D(t))_{tgeq 0}$ of time varying open sets decreasing to the origin as $t$ tends to infinity, such that, for any unknown uncertainty bounds and trajectory $z(cdot)$ of the corresponding system, there exists a positive positve $t_z$ for which $z(t_z)in D(t_z)$ and $z(t)in D(t)$ for $tgeq t_z$. %enters convergence in finite time of all the trajectories to a time varying domain $D(t)$ shrinking to the origin and their maintenance there. Hence, since the function $eta(t)$ tends to zero, this leads the asymptotic convergence of all the trajectories to zero. The effectiveness of these controllers is illustrated through simulations.
Output-based controllers are known to be fragile with respect to model uncertainties. The standard $mathcal{H}_{infty}$-control theory provides a general approach to robust controller design based on the solution of the $mathcal{H}_{infty}$-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples.
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