A new Lagrangian formulation with complex currents is developed and yields a direct and simple method for modeling three-phase permanent-magnet and induction machines. The Lagrangian is the sum a mechanical one and of a magnetic one. This magnetic Lagrangian is expressed in terms of rotor angle, complex stator and rotor currents. A complexification procedure widely used in quantum electrodynamic is applied here in order to derive the Euler-Lagrange equations with complex stator and rotor currents. Such complexification process avoids the usual separation into real and imaginary parts and simplifies notably the calculations. Via simple modifications of such magnetic Lagrangians we derive new dynamical models describing permanent-magnet machines with both saturation and saliency, and induction machines with both magnetic saturation and space harmonics. For each model we also provide its Hamiltonian thus its magnetic energy. This energy is also expressed with complex currents and can be directly used in Lyapunov and/or passivity based control. Further, we briefly investigate the observability of this class of Euler-Lagrange models, in the so-called sensorless case when the measured output is the stator current and the load torque is constant but unknown. For all the dynamical models obtained via such variational principles, we prove that their linear tangent systems are unobservable around a one-dimensional family of steady-states attached to the same constant stator voltage and current. This negative result explains why sensorless control of three-phase electrical machines around zero stator frequency remains yet a difficult control problem.
The leader-following consensus problem for multiple Euler-Lagrange systems was studied recently by the adaptive distributed observer approach under the assumptions that the leader system is neurally stable and the communication network is jointly connected and undirected. In this paper, we will study the same problem without assuming that the leader system is neutrally stable, and the communication network is undirected. The effectiveness of this new result will be illustrated by an example.
The study of fuzzy fractional variational problems in terms of a fractional Liouville-Caputo derivative is introduced. Necessary optimality conditions for problems of the fuzzy fractional calculus of variations with free end-points are proved, as well as transversality conditions.
This paper derives sufficient conditions for bounded distributed connectivity-preserving coordination of Euler-Lagrange systems with only position measurements and with system uncertainties, respectively. The paper proposes two strategies that suitably scale conventional gradient-based controls to account for the actuation bounds and to reserve sufficient actuation for damping injection. For output feedback control of networked systems with only position measurements, the paper incorporates a first-order filter to estimate velocities and to inject damping for stability. For networks of uncerta…
This paper considers the fixed-time control problem of a multi-agent system composed of a class of Euler-Lagrange dynamics with parametric uncertainty and a dynamic leader under a directed communication network. A distributed fixed-time observer is first proposed to estimate the desired trajectory and then a fixed-time controller is constructed by transforming uncertain Euler-Lagrange systems into second-order systems and utilizing the backstepping design procedure. The overall design guarantees that the synchronization errors converge to zero in a prescribed time independent of initial condit…
The problem of minimizing an integral functional of a vector-valued Lagrangian on a set of admissible arcs with given endpoints is considered. The problem is tackled by embedding it into a set-optimization problem such that the image space becomes a complete lattice. This procedure allows to define the concepts of minimizer and of infimizer, as two completely different notions. An infimizer is proved to contain minimizers or at least minimizing sequences for the linearly scalarized problems, with the scalarizing parameters being elements in the dual of the ordering cone. In this way set-valued…
The set of closed (or holonomic) measures provides a useful setting for studying optimization problems because it contains all curves, while also enjoying good compactness and convexity properties. We study the way to do variational calculus on the set of closed measures. Our main result is a full description of the distributions that arise as the derivatives of variations of such closed measures. We give examples of how this can be used to extract information about the critical closed measures. The condition of criticality with respect to variations leads, in certain circumstances, to the Eul…
This paper concerns optimal control problems for a class of sweeping processes governed by discontinuous unbounded differential inclusions that are described via normal cone mappings to controlled moving sets. Largely motivated by applications to hysteresis, we consider a general setting where moving sets are given as inverse images of closed subsets of finite-dimensional spaces under nonlinear differentiable mappings dependent on both state and control variables. Developing the method of discrete approximations and employing generalized differential tools of first-order and second-order varia…
The derivation of second-order ordinary differential equations (ODEs) as continuous-time limits of optimization algorithms has been shown to be an effective tool for the analysis of these algorithms. Additionally, discretizing generalizations of these ODEs can lead to new families of optimization methods. We study discretizations of an Euler-Lagrange equation which generate a large class of accelerated methods whose convergence rate is $O(\frac{1}{t^p})$ in continuous-time, where parameter $p$ is the order of the optimization method. Specifically, we address the question asking why a naive exp…
On Composite Adaptive Continuous Finite-Time Control of a class of Euler-Lagrange systems
University of Guadalajara（瓜达拉哈拉大学）；Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学）；Laboratoire des signaux et systèmes (L2S), CNRS（法国国家科学研究中心信号与系统实验室）
Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann-Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag-Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results per…
In this paper, we study the leader-following consensus problem of multiple Euler-Lagrange systems subject to an uncertain leader system. We first establish an adaptive distributed observer for a neutrally stable linear leader system whose system matrix is not known exactly. Under standard assumptions, this adaptive distributed observer can estimate and pass the leader's state to each follower through the communication network of the system without knowing the leader's system matrix exactly. Under the additional assumption that the leader's state is persistently exciting, this adaptive distribu…
This paper introduces a class of time-varying controllers for Euler-Lagrange systems such that the convergence occurs at an arbitrary finite time, independently of initial conditions, and free of chattering. The proposed controller is based on a mapping technique and is designed in two steps: First, a conventional (obstacle avoidance) asymptotically stable controller is specified for the nominal system; then, by a simple substitution, a prescribed-time (obstacle avoidance) controller is achievable for the perturbed system. It is proved that the proposed scheme is uniformly prescribed-time stab…
This paper investigates the impact of bounded actuation on the connectivity-preserving consensus of two classes of multi-agent systems, with kinematic agents and with Euler- Lagrange agents. The investigation establishes that: (1) there exists a class of gradient-based controls which drive kinematic multi-agent systems to connectivity-preserving consensus even if they saturate; (2) actuator saturation restricts the initial states from which Euler-Lagrange multi-agent systems can be synchronized while preserving their local connectivity; (3) Euler-Lagrange multi-agent systems with unbounded act…
The paper studies optimal control problem described by higher order evolution differential inclusions (DFIs) with endpoint and state constraints. In the term of Euler-Lagrange type inclusion is derived sufficient condition of optimality for higher order DFIs. It is shown that the adjoint inclusion for the first order DFIs, defined in terms of locally adjoint mapping, coincides with the classical Euler-Lagrange inclusion. Then a duality theorem is proved, which shows that Euler-Lagrange inclusions are "duality relations" for both problems. At the end of the paper duality problems for third orde…
A family of optimal control problems for a single and two coupled spinning particles in the Euler-Lagrange formalism is discussed. A characteristic of such problems is that the equations controlling the system are implicit and a reduction procedure to deal with them must be carried on. The reduction of the implicit control equations arising in these problems will be discussed in the slightly more general setting of implicit equations defined by invariant one-forms on Lie groups. As an instance, the first order differential equations describing the extremal solutions of an optimal control probl…
In order to solve fractional variational problems, there exist two theorems of necessary conditions: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that involves only Caputo derivatives. In this article, we make a comparison solving a particular fractional variational problem with both methods to obtain some conclusions about which method gives the optimal solution.
Small-Gain Theorem Based Distributed Prescribed-Time Convex Optimization For Networked Euler-Lagrange Systems
School of Artificial Intelligence and Automation（人工智能与自动化学院）；Key Laboratory of Imaging Processing and Intelligence Control（图像处理与智能控制重点实验室）；Graduate school of advanced science and engineering（先进科学与工程研究生学校）
