Energy-like Function Research Articles

Continuous finite-time regulation of Euler-Lagrange systems via energy shaping

In this paper, we provide conditions on the energy-like functions employed in the classical energy shaping control technique in order to solve the finite-time regulation problem of a class of Euler-Lagrange systems. Inherited from the energy shaping design, the obtained control schemes are described by the gradient of suitable artificial potential energy and energy dissipation functions. In order to achieve finite-time convergence at a desired constant position, these energy functions are provided with some homogeneity properties and designed to have an appropriate homogeneity degree. Using different energy function definitions, this paper provides different novel controllers as an example of the proposed control design methodology.

An adaptive tracking control method with swing suppression for 4-DOF tower crane systems

Abstract As an efficient transportation tool, a tower crane is widely used in construction sites. To improve the working efficiency, some automatic control methods have been proposed for tower crane systems, including some open loop methods, such as input shaping methods, trajectory planning methods, and so on. However, generally, tower cranes usually work in outdoor environment, which are sensitive to unavoidable external disturbances. Additionally, accurate system parameters are hard to obtain, which increases the control difficulty of the tower crane system. Considering these factors, in this paper, we propose an adaptive tracking control method, which achieves satisfactory tracking performance w.r.t. parameter uncertainties and external disturbances. Specifically, utilizing the passivity property, a shaped energy-like function is designed as a Lyapunov candidate, based on which, an adaptive tracking controller is proposed to deal with parameter uncertainties. Using Lyapunov stability analysis, together with LaSalle’s invariance principle, the closed-loop system is proven to be asymptotically stable. At last, a series of experimental tests are implemented to illustrate the satisfactory performance of the proposed method.

A new reconstruction method for a parabolic inverse source problem

ABSTRACTIn this paper, we focus on the detection of the shape and location of a discontinuous source term from the knowledge of boundary measurements. We propose a non-iterative reconstruction algorithm based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from an energy-like cost function. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. The efficiency of our algorithm is illustrated by some numerical simulations.

Aerial Co-Manipulation With Cables: The Role of Internal Force for Equilibria, Stability, and Passivity

This paper considers the study of cooperative manipulation of a cable-suspended load with two generic aerial robots and without explicit communication. The role of the internal force for the asymptotic stability of the beam-position/beam-attitude equilibria is analyzed in depth. Using a nonlinear Lyapunov-based approach, we prove that if a non-zero internal force is chosen then asymptotic stabilization of any desired beam-pose configuration can be achieved with a decentralized and communication-less master-slave admittance controller. If, conversely, a zero internal force is chosen, as done in the majority of the state-of-the-art algorithms, the attitude of the beam is not controllable without communication. Non-zero internal force can be interpreted then as a fundamental factor that enables the use of cables as implicit communication means between the two vehicles in replacement of the explicit ones. Furthermore, the formal proof of the system output-strictly passivity with respect to an energy-like storage function and a certain input-output pair is given. This proves the stability and the robustness of the method even during motion. The theoretical findings are validated through numerical simulations.

New results on the global asymptotic stability of certain nonlinear RLC circuits

This paper deals with the global asymptotic stability (GAS) of certain nonlinear RLC circuit systems using the direct Lyapunov method. For each system a suitable Lyapunov function or energy-like function is constructed and the direct Lyapunov method is applied to the related system. Then the invariant equilibrium point of each system that makes the system solution to the global asymptotic stable is determined. Some new explicit GAS conditions of certain nonlinear RLC circuit systems are derived by Lyapunov's direct method. The presented simulations are compatible with the new results. The results are given with proofs.

Kohn–Vogelius formulation and topological sensitivity analysis based method for solving geometric inverse problems

In this paper, we propose an alternative approach combining the advantages of the Kohn–Vogelius formulation and the topological sensitivity analysis method for solving geometric inverse problems. The Kohn–Vogelius formulation can rephrase the geometric inverse problem into a shape optimization one minimizing an energy-like function. The sensitivity analysis gives the leading term of the energy-like function variation with respect to the presence of a small geometry perturbation inside the computational domain. The obtained theoretical results lead to build a fast and accurate numerical reconstruction algorithm. The efficiency and accuracy of the proposed algorithm are illustrated by some numerical results.

Persistence of nonlinear hysteresis in fractional models of Josephson transmission lines

In this note, we depart from a model describing the transmission of electric currents in Josephson-junction chains, and provide a fractional generalization using Riesz discrete differential operators. The fractional model considered has generalized Hamiltonian- and energy-like functionals. The model and the energy functionals are fully discretized in order to investigate numerically the complex dynamics of the system when a sinusoidal perturbation at one end of the chain is imposed. As one of the most important results in this report, we establish the persistence of the nonlinear phenomena of supratransmission and infratransmission in Riesz fractional chains. Nonlinear hysteresis loops are obtained numerically for some values of the order of the fractional derivative, and numerical simulations of the propagation of monochromatic wave signals through the transmission line are presented using the nonlinear bistability of the system.

The Reaction Mass Biped: Geometric Mechanics and Control

Inverted Pendulum based reduced order models offer many valuable insights into the much harder problem of bipedal locomotion. While they help in understanding leg behavior during walking, they fail to capture the natural balancing ability of humans that stems from the variable rotational inertia on the torso. In an attempt to overcome this limitation, the proposed work introduces a Reaction Mass Biped (RMB). It is a generalization of the previously introduced Reaction Mass Pendulum (RMP), which is a multi-body inverted pendulum model with an extensible leg and a variable rotational inertia torso. The dynamical model for the RMB is hybrid in nature, with the roles of stance leg and swing leg switching after each cycle. It is derived using a variational mechanics approach, and is therefore coordinate-free. The RMB model has thirteen degrees of freedom, all of which are considered to be actuated. A set of desired state trajectories that can enable bipedal walking in straight and curved paths are generated. A control scheme is then designed for asymptotically tracking this set of trajectories with an almost global domain-of-attraction. Numerical simulation results confirm the stability of this tracking control scheme for different walking paths of the RMB. Additionally, a discrete dynamical model is also provided along-with an appropriate Geometric Variational Integrator (GVI). In contrast to non-variational integrators, GVIs can better preserve energy terms for conservative mechanical systems and stability properties (achieved through energy-like lyapunov functions) for actuated systems.

Nonlinear coupling control method for underactuated three-dimensional overhead crane systems under initial input constraints

In this paper, we present an enhanced coupling nonlinear control method for three-dimensional overhead crane systems under initial input constraints. The proposed control method can achieve superior control performance and strong robustness with respect to system parameter variations and external disturbances. Moreover, it guarantees ‘soft’ trolley start by introducing hyperbolic tangent functions into the controller. More precisely, we enhance the coupling behaviour between the trolley movement and the payload swing by fabricating two composite signals, based on which an energy-like storage function is established. Then, a nonlinear coupling control method under initial input constraints is derived directly. Lyapunov techniques and LaSalle’s invariance theorem are successfully adopted to find the asymptotic stability solution while satisfying the initial input constraints. Strict mathematical analysis of the control scheme with initial input constraints is provided as theoretical support for the superior performance of the controller. Simulation and experimental results are conducted to show the superior performance and strong robustness of the proposed control method.

A New Dynamic Performance Model of Motor Stalling and FIDVR for Smart Grid Monitoring/Planning

The stalling of highly concentrated constant torque induction motor loads due to system faults may result in fault induced delayed voltage recovery. This state can cause significantly depressed local voltage for several seconds after the fault is cleared and can also lead to widely cascaded system failure. Though there is extensive study conducted within the modeling of motor loads, the dynamic connection between the aggregated induction motor loads and the grid still needs further investigation. In this paper, a dynamic performance model is developed for motor stalling and over heat thermal tripping. Specifically, this dynamic model can be constructed with an energy-like Lyapunov function, and can be incorporated as part of power system dynamic cascading model. The simulation examples are carried out in an enhanced version of the IEEE 57 bus test system, as demonstration for feasibility. This model may be beneficial for smart grid monitoring and planning, as well as energy analysis for power system cascading failure.

Distributed H∞ state estimation for stochastic delayed 2-D systems with randomly varying nonlinearities over saturated sensor networks

In this paper, the distributed H∞ state estimation problem is investigated for the two-dimensional (2-D) time-delay systems. The target plant is characterized by the generalized Fornasini–Marchesini 2-D equations where both stochastic disturbances and randomly varying nonlinearities (RVNs) are considered. The sensor measurement outputs are subject to saturation restrictions due to the physical limitations of the sensors. Based on the available measurement outputs from each individual sensor and its neighboring sensors, the main purpose of this paper is to design distributed state estimators such that not only the states of the target plant are estimated but also the prescribed H∞ disturbance attenuation performance is guaranteed. By defining an energy-like function and utilizing the stochastic analysis as well as the inequality techniques, sufficient conditions are established under which the augmented estimation error system is globally asymptotically stable in the mean square and the prescribed H∞ performance index is satisfied. Furthermore, the explicit expressions of the individual estimators are also derived. Finally, numerical example is exploited to demonstrate the effectiveness of the results obtained in this paper.

Advances in computational Lyapunov analysis using sum-of-squares programming

The stability of an equilibrium point of a nonlinear dynamical system is typically determined using Lyapunov theory. This requires the construction of an energy-like function, termed a Lyapunov function, which satisfies certain positivity conditions. Unlike linear dynamical systems, there is no algorithmic method for constructing Lyapunov functions for general nonlinear systems. However, if the systems of interest evolve according to polynomial vector fields and the Lyapunov functions are constrained to be sum-of-squares polynomials then stability verification can be cast as a semidefinite (convex) optimization programme. In this paper we describe recent advances in sum-of-squares programming that facilitate advanced stability analysis and control design.

H∞ filtering for two-dimensional systems with mixed time delays, randomly occurring saturations and nonlinearities

This paper is concerned with the filtering problem for the two-dimensional (2-D) systems with mixed time delays which comprise both the discrete time-varying delays and the discrete distributed delays in states. Randomly occurring nonlinearities (RONs) and randomly occurring saturations (ROSs) are also considered, respectively, in the state-space model and the measurement output equation to reflect the nonlinear disturbance appearing in a random way and the limited communication capabilities of the sensors. The aim of this paper is focused on the design of a full-order filter such that the performance is guaranteed in the simultaneous presence of the mixed tim delays, RONs and ROSs. By utilizing the Kronecker product and the inequality technique, an energy-like function is developed to establish some sufficient criteria under which the filtering error system satisfies the prescribed disturbance attenuation performance. Furthermore, explicit expression of the filter parameters is also characterized. Numerical example provided in the end of the paper further demonstrates the effectiveness of the proposed design scheme.

Robust synchronization for 2-D discrete-time coupled dynamical networks.

In this paper, a new synchronization problem is addressed for an array of 2-D coupled dynamical networks. The class of systems under investigation is described by the 2-D nonlinear state space model which is oriented from the well-known Fornasini-Marchesini second model. For such a new 2-D complex network model, both the network dynamics and the couplings evolve in two independent directions. A new synchronization concept is put forward to account for the phenomenon that the propagations of all 2-D dynamical networks are synchronized in two directions with influence from the coupling strength. The purpose of the problem addressed is to first derive sufficient conditions ensuring the global synchronization and then extend the obtained results to more general cases where the system matrices contain either the norm-bounded or the polytopic parameter uncertainties. An energy-like quadratic function is developed, together with the intensive use of the Kronecker product, to establish the easy-to-verify conditions under which the addressed 2-D complex network model achieves global synchronization. Finally, a numerical example is given to illustrate the theoretical results and the effectiveness of the proposed synchronization scheme.

Instability of Thermally Induced Vibrations of Carbon Nanotubes via Nonlocal Elasticity

The dynamic stability problem of single-walled carbon nanotube embedded in a viscoelastic matrix under time-dependent temperature field is studied. Using nonlocal continuum mechanics parametric vibrations of the nano-scale Euler–Bernoulli beams are investigated. A theory of nonlocal elasticity with kernels of Helmholtz and bi-Helmholtz type is applied. The tube axial loading is caused by the Gaussian physically realizable temperature changes. The energy-like functionals are used in the instability analysis. Instability domains in the space of geometric and material parameters as well as temperature variance are presented.

The Fault Tolerable Control System Structure of SY-II Remote Operated Vehicle

In order to improve the Remote Operated Vehicle (ROV) into a reliable and maintainable one, this paper has presented a fault tolerance control system structure. Its surface platform includes fault diagnosis and tolerance modules. In fault diagnosis module, a second order sliding mode observer has been built on the basis of its dynamic model, so as to diagnose the thrusters’ conditions through the residue of command quantity. Coresponding with its thrusters’ position, a force allocation model has been established to realize fault tolerable control. Based on duality priciple, the energy-like function concerning ROV control is established. On the basis of the energy-like function, the recurrent strategy has been deduced for thrust allocation when some of thrusters are found failure. Simulations and experiments have been made to verify and analyse the control system structure. They have proved the SY-II control architecture is a reliable control system, it can detect the thruster fault precisely, realize fault tolerable control.

Statistical-Mechanics-Inspired Optimization of Sensor Field Configuration for Detection of Mobile Targets

This paper presents a statistical-mechanics-inspired procedure for optimization of the sensor field configuration to detect mobile targets. The key idea is to capture the low-dimensional behavior of the sensor field configurations across the Pareto front in a multiobjective scenario for optimal sensor deployment, where the nondominated points are concentrated within a small region of the large-dimensional decision space. The sensor distribution is constructed using location-dependent energy-like functions and intensive temperature-like parameters in the sense of statistical mechanics. This low-dimensional representation is shown to permit rapid optimization of the sensor field distribution on a high-fidelity simulation test bed of distributed sensor networks.

Topological sensitivity of energy cost functional for wave-based defect identification

This article is concerned with establishing the topological sensitivity (TS) against the nucleation of small trial inclusions of an energy-like cost function. The latter measures the discrepancy between two time-harmonic elastodynamic states (respectively defined, for cases where overdetermined boundary data is available for identification purposes, in terms of Dirichlet or Neumann boundary data for the same reference solid) as the strain energy of their difference. Such cost function constitutes a particular form of error in constitutive relation and may be used for e.g. defect identification. The TS is expressed in terms of four elastodynamic fields, namely the free and adjoint solutions for Dirichlet or Neumann data. A similar result is also given for the linear acoustic scalar case. A synthetic numerical example where the TS result is used for the qualitative identification of an inclusion is presented for a simple 2D acoustic configuration.

The shade avoidance syndrome: A non-Markovian stochastic growth model

Plants at high population density compete for light, showing a series of physiological responses known as the shade avoidance syndrome. These responses are controlled by the synthesis of the hormone auxin, which is regulated by two signals, an environmental one and an internal one. Considering that the auxin signal induces plant growth after a time lag, this work shows that plant growth can be modelled in terms of an energy-like function extremization, provided that the Markov property is not applied. The simulated height distributions are bimodal and right skewed, as in real community of plants. In the case of isolated plants, theoretical growth dynamics and speed correctly fit Arabidopsis thaliana experimental data reported in literature. Moreover, the growth dynamics of this model is shown to be consistent with the biomass production function of an independent model. These results suggest that memory effects play a non-negligible role in plant growth processes.

Some multistability properties of bidirectional associative memory recurrent neural networks with unsaturating piecewise linear transfer functions

Multistability is an important dynamical property in neural networks in order to enable certain applications where monostable networks could be computationally restrictive. This paper studies some multistability properties for a class of bidirectional associative memory recurrent neural networks with unsaturating piecewise linear transfer functions. Based on local inhibition, conditions for globally exponential attractivity are established. These conditions allow coexistence of stable and unstable equilibrium points. By constructing some energy-like functions, complete convergence is studied.