Let $M$ be a smooth connected and complete manifold of dimension $n$, and $Δ$ be a smooth nonholonomic distribution of rank $m\leq n$ on $M$. We prove that, if there exists a smooth Riemannian metric on $Δ$ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $Δ$ on $M$. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.
This paper focuses on the problem of constructing time-varying feedback laws that asymptotically stabilize a given part of the state variables for nonlinear control-affine systems. It is assumed that the class of systems under consideration satisfies nonlinear controllability conditions with respect to the stabilizable variables. Under these assumptions, a time-periodic feedback control is constructed explicitly by using the inversion of the matrix composed of the control vector fields and their Lie brackets. The proposed control design scheme is applied to solving the leader-following problem…
The problem of tracking an arbitrary curve in the state space is considered for underactuated driftless control-affine systems. This problem is formulated as the stabilization of a time-varying family of sets associated with a neighborhood of the reference curve. An explicit control design scheme is proposed for the class of controllable systems whose degree of nonholonomy is equal to 1. It is shown that the trajectories of the closed-loop system converge exponentially to any given neighborhood of the reference curve provided that the solutions are defined in the sense of sampling. This conver…
This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures ex…
A class of nonlinear control-affine systems with bounded time-varying drift is considered. It is assumed that the control vector fields together with their iterated Lie brackets satisfy Hormander's condition in a neighborhood of the origin. Then the problem of exponential stabilization is treated by exploiting periodic time-varying feedback controls. An explicit parametrization of such controllers is proposed under a suitable non-resonance assumption. It is shown that these controllers ensure the exponential stability of the closed-loop system provided that the period is small enough. The prop…
Implementation of Time-Varying Controllers for a Nonholonomic Mobile Robot: Experimental Studies
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；University of Klagenfurt（克拉根福大学）；National Acad. of Sciences of Ukraine（乌克兰国家科学院）
We develop a predictor-feedback control design for multi-input nonlinear systems with distinct input delays, of arbitrary length, in each individual input channel. Due to the fact that different input signals reach the plant at different time instants, the key design challenge, which we resolve, is the construction of the predictors of the plant's state over distinct prediction horizons such that the corresponding input delays are compensated. Global asymptotic stability of the closed-loop system is established by utilizing arguments based on Lyapunov functionals or estimates on solutions. We …
This article deals with optimizing problems classified by the kinds of restrictions as required in differential geometry and in mechanics: holonomic and nonholonomic. The central issue relates to dual nonholonomic programs (what they mean and how they are solved?) when the nonholonomic constraints are given by Pfaff equations. The original results are surprising and include aspects derived from the Vranceanu theory of nonholonomic manifolds, from the geometric distributions theory and from Darboux's theorem on canonical coordinates in which we can express a Pfaff form. On these ways we get als…
In this paper we use an affine connection formulation to study an optimal control problem for a class of nonholonomic, under-actuated mechanical systems. In particular, we aim at minimizing the norm-squared of the control input to move the system from an initial to a terminal state. We consider systems evolving on general manifolds. The class of nonholonomic systems we study in this paper includes, in particular, wheeled-type vehicles, which are important for many robotic locomotion systems. The two special aspects of this optimal control problem are the nonholonomic constraints and under-actu…
The paper considers a motion control problem for kinematic models of nonholonomic wheeled systems. The class of maneuverable wheeled systems is defined consisting of systems that can follow any sufficiently smooth non-stop trajectory on the plane. A sufficient condition for maneuverability is obtained. The design of control law that stabilizes motion along the desired trajectory on the plane is performed in two steps. On the first step the trajectory on the configuration manifold of the system and the input function are constructed that ensure the exact reproduction of the desired trajectory o…
This paper studies (single-time and multitime) optimal control problems on a nonholonomic manifold (described either by the kernel of a Gibbs-Pfaff form or by the span of appropriate vector fields). For both descriptions we analyse: infinitesimal deformations and adjointness, single-time optimal control problems, multitime optimal control problem of maximizing a multiple integral functional, multitime optimal control problem of maximizing a curvilinear integral functional, Curvilinear functionals depending on curves, optimization of mechanical work on Riemannian manifolds. Also we prove that a…
We study the tracking of a trajectory for a nonholonomic system by recasting the problem as a constrained optimal control problem. The cost function is chosen to minimize the error in positions and velocities between the trajectory of a nonholonomic system and the desired reference trajectory, both evolving on the distribution which defines the nonholonomic constraints. The problem is studied from a geometric framework. Optimality conditions are determined by the Pontryagin Maximum Principle and also from a variational point of view, which allows the construction of geometric integrators. Exam…
We consider the four-dimensional nonholonomic distribution defined by the 4-potential of the electromagnetic field on the manifold. This distribution has a metric tensor with the Lorentzian signature $(+,-,-,-)$, therefore, the causal structure appears as in the general relativity theory. By means of the Pontryagin's maximum principle we proved that the equations of the horizontal geodesics for this distribution are the same as the equations of motion of a charged particle in the general relativity theory. This is a Kaluza -- Klein problem of classical and quantum physics solved by methods of …
In this paper, we describe a constrained Lagrangian and Hamiltonian formalism for the optimal control of nonholonomic mechanical systems. In particular, we aim to minimize a cost functional, given initial and final conditions where the controlled dynamics is given by nonholonomic mechanical system. In our paper, the controlled equations are derived using a basis of vector fields adapted to the nonholonomic distribution and the Riemannian metric determined by the kinetic energy. Given a cost function, the optimal control problem is understood as a constrained problem or equivalently, under some…
In this paper, the problem of non-fragile finite-time stabilization for linear discrete mean-field stochastic systems is studied. The uncertain characteristics in control parameters are assumed to be random satisfying the Bernoulli distribution. A new approach called the ``state transition matrix method" is introduced and some necessary and sufficient conditions are derived to solve the underlying stabilization problem. The Lyapunov theorem based on the state transition matrix also makes a contribution to the discrete finite-time control theory. One practical example is provided to validate th…
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 …
Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard conjecture states that that set should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of …
University of Surrey（萨里大学）；Dalian University of Technology（大连理工大学）
The ability to direct a Probabilistic Boolean Network (PBN) to a desired state is important to applications such as targeted therapeutics in cancer biology. Reinforcement Learning (RL) has been proposed as a framework that solves a discrete-time optimal control problem cast as a Markov Decision Process. We focus on an integrative framework powered by a model-free deep RL method that can address different flavours of the control problem (e.g., with or without control inputs; attractor state or a subset of the state space as the target domain). The method is agnostic to the distribution of proba…
