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Register a new userPractical stabilization of perturbed integrator chains with unknown bounds
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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.
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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.
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 a generalization of the super-twisting algorithm for perturbed chains of integrators of arbitrary order. This Higher Order Super-Twisting (HOST) controller, which extends the approach of Moreno and als., is homegeneous with respect to a family of dilations and can be continuous. Its design is derived from a first result obtained for pure chains of integrators, the latter relying on a geometric condition introduced by the authors. The complete result is established using a homogeneous strict Lyapunov function which is explicitely constructed. The effectiveness of the controller is finally illustrated with simulations for a chain of integrator of order four, first pure then perturbed, where we compare the performances of two HOST controllers.
This paper studies an infinite horizon optimal control problem for discrete-time linear systems and quadratic criteria, both with random parameters which are independent and identically distributed with respect to time. A classical approach is to solve an algebraic Riccati equation that involves mathematical expectations and requires certain statistical information of the parameters. In this paper, we propose an online iterative algorithm in the spirit of Q-learning for the situation where only one random sample of parameters emerges at each time step. The first theorem proves the equivalence of three properties: the convergence of the learning sequence, the well-posedness of the control problem, and the solvability of the algebraic Riccati equation. The second theorem shows that the adaptive feedback control in terms of the learning sequence stabilizes the system as long as the control problem is well-posed. Numerical examples are presented to illustrate our results.
In this paper, we study a retailer price optimization problem which includes the practical constraints: maximum number of price changes and minimum amount of price change (if a change is recommended). We provide a closed-form formula for the Euclidean projection onto the feasible set defined by these two constraints, based on which a simple gradient projection algorithm is proposed to solve the price optimization problem. We study the convergence and solution quality of the proposed algorithm. We extend the base model to include upper/lower bounds on the individual product prices and solve it with some adjustments to the gradient projection algorithm. Numerical results are reported to demonstrate the performance of the proposed algorithm.
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