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 ren
der 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 $Dinmathbb{R}^n$.
Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tylers estimator [1] also called the Fixed-Point Estimator (FPE) [2], a shrinkage fixed-point estimator has been introduced
in [3]. First, this work extends the results of [3,4] by giving the general solution of the shrinkage fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $beta$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data.
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 c
hains. 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.
In this paper, we have studied the control problem of target-point based path following for car-type vehicles. This special path following task arises from the needs of vision based guidance systems, where a given target-point located ahead of the ve
hicle, in the visual range of the camera, must follow a specified path. A solution to this problem is developed through a non linear transformation of the path following problem into a reference trajectory tracking problem, by modeling the target point as a virtual vehicle. Bounded feedback laws must be then used on the real vehicles angular acceleration and the virtual vehicles velocity, to achieve stability. The resulting controller is globally asymptotically stable with respect and the proof is demonstrated using Lyapunov based arguments and a bootstrap argument. The effectiveness of this controller has been illustrated through simulations.
