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Exciting multi-DOF systems by feedback resonance
Abstract
The mechanism of entrainment to natural oscillations in a class of (bio)mechanical systems described by linear models is investigated. Two new nonlinear control strategies are proposed to achieve global convergence to a prescribed resonance mode of oscillation within a finite time. The effectiveness of the proposed methods for resonance entrainment is demonstrated by examples of computer simulation for linear and nonlinear systems.
... However, the fundamental physical principles and mechanisms underlying such activity have not been described. In this brief comment, we attempt to fill this gap and to describe how knowledge of such mechanisms enables us to design optimal and energy-efficient control systems [5,6,7]. ...
... With the accumulated knowledge regarding the mechanism of CPG functioning, it has now become possible to develop energyefficient control systems for mechanical oscillatory systems many degrees of freedom [7]. Such systems will exhibit robustness and the capability to adapt to various operating conditions. ...
... The CPG is mimicked by the control systems, regulating the complex periodic movements of many robotic devices, which when coupled to any mechanical structure can induce self-excited periodic motion. A more simplified version of the controller has been designed recently to self-oscillate the system at the particular natural mode of interest [33]. ...
... However, the present control scheme is a general in nature that encompasses the above two control configurations as special cases. The mode of oscillation generated by the proposed control is simple as compared to that studied in [33] to excite multi-DOF mechanical systems at resonant frequencies. A decentralized control is proposed in [40] is able to excite a multi-DOF system at all its resonant frequencies by providing fully actuated configuration, giving the actuators a phase lag of 90 0 and constant gains in a wide range of frequencies. ...
Recently, scientists have utilized self-excited oscillations in many mechanical and micromechanical applications and accordingly, several experimental and theoretical research activities have been undertaken with the objective of artificially inducing self-excited oscillation in mechanical systems. This paper considers some theoretical aspects of generation of self-excited oscillation in a two degrees-of-freedom under-actuated spring-mass-damper system by centralized nonlinear feedback. The control force depends on the response of both masses. The proposed control law is slightly general in nature as it entails both collocated and non-collocated control as special cases. The response of the nonlinear feedback control is studied with the help of the method of averaging. The present study shows that by appropriately selecting the control gains it is possible to excite the desired mode(s) of oscillation irrespective of initial conditions. Even it is possible to excite dual mode oscillation having the characteristics of quasiperiodic oscillations. The theoretical analysis of the dynamics of the system with the control is validated by simulation results, performed in MATLAB Simulink.
... The organization of swinging using feedback is more difficult in multidimensional nonlinear systems. We note that feedback resonance in such systems is studied in some works, for example, in [Efimov, Fradkov, and Iwasaki, 2013]. Serious difficulties arise here because these mechanical systems have several oscillatory degrees of freedom, i.e., several natural frequencies and modes. ...
This article is devoted to the study of controlled movements of spatial double pendulum with non-parallel cylindrical joints axes. The collinear control is used to swinging of the system by feedback. The most important property of collinear control is the ability of increasing system oscillations only on one oscillation mode. A modification of the collinear control law with variable gain depending on the energy level is investigated. It allows to control the system motions more flexible than in the case of constant gain. As a result, it is possible to observe a smooth transition from a linear oscillation mode to a nonlinear one with a gradual output to a steady oscillation motion with a given energy level. The obtained results are clearly illustrated by graph dependencies that demonstrate the swinging of the system on one oscillation mode from small to finite amplitudes.
... The case of 1DOF system models (pendulum or vibroactuator) is well studied [Andrievskii et al., 2001;Fradkov, 1999], as well as the case of synchronization for two or more 1DOF systems [Blekhman et al., 2002]. However for multi-DOF systems, like multirotor vibration actuator a number of problems need further study [Efimov et al., 2013]. ...
In this paper the control of oscillations in the two-rotor vibration unit is studied. It is assumed that the velocity of the oscillation of the platform cannot be accurately measured. The time-varying observer is proposed to restore it. In order to guarantee stability of the frequency and amplitude of oscillations of the vibrating parts of a two-rotor vibration unit special control algorithms based on speed-gradient methodology. Simulation results confirm stability of the synchronous rotation modes of the unbalanced rotors of the vibration unit.
... The control systems regulating the complex periodic movements of many robotic devices mimic the CPG so that, when coupled to any mechanical structure, self-excited periodic motion can be induced. Recently, a simplified version of the controller has been designed to make the system self-oscillate in a particular natural mode of interest (Efimov et al., 2013). ...
Many devices and processes utilize self-excited oscillation to enhance performance. Recently, much research work has been devoted to the induction of self-excited oscillation in mechanical systems by nonlinear feedback. The present paper investigates the efficacy of a displacement feedback technique in generating self-excited oscillation at the desired mode(s) in a multiple degrees-of-freedom mechanical system. The controller couples the system with a bank of second-order filters and generates the required control force as a nonlinear function of the filter output. The describing function method theoretically explores the dynamics of the system with the control law. The control cost of the controller is studied for the proper choice of the filter parameters. The analytical results are substantiated by the numerical simulation results. The present study reveals that the proposed control laws, if used in an appropriate way, can generate self-excited oscillation in the system at the desired mode(s).
... The control systems regulating the complex periodic movements of many robotic devices mimic the CPG that when coupled to any mechanical structure can induce self-excited periodic motion. Recently, a simplified version of the controller has been designed to make the system self-oscillate in a particular natural mode of interest [36]. ...
The article proposes an acceleration feedback based technique for exciting modal self-oscillation in a class of multi degrees-of-freedom mechanical systems. The controller comprises a bank of second-order filters and the control law is formulated as the nonlinear function of the filter output. A design methodology is developed to excite self-oscillation in any desired mode or combination of modes (mixed-mode oscillation). The choice of control parameters takes into account the control cost and robustness of the controller. The effects of structural damping on the system performance are also studied. Analytical results are confirmed by numerical simulations. An adaptive control is proposed to maintain the oscillation amplitude at the desired level.
... The control systems regulating the complex periodic movements of many robotic devices mimic the CPG as nonlinear oscillators that, when coupled to any mechanical structure, can induce selfexcited periodic motion. Recently, a more simplified version of the controller has been designed to make the system self-oscillate in a particular natural mode of interest (Efimov et al., 2013). ...
The paper presents a new nonlinear control design methodology for inducing modal, self-excited oscillation in a class of multi degrees-of-freedom mechanical systems. The system is assumed to be fully actuated and the controller operates in a centralized fashion based on the direct nonlinear velocity feedback. The linear part of the controller is designed to assign negative modal damping in the mode to be excited and positive modal damping in other modes. The nonlinear modal interaction is investigated using averaging analysis and different excitation regimes are delineated in the control parameter space. The nonlinear part of the controller is optimized to minimize the control cost. Numerical simulations of the control system are performed to substantiate the analytical results and validate the accuracy of the design. The control method is also shown to have some degree of robustness against parametric variations over the nominal values.
Periodic orbits often describe desired state trajectories of dynamical systems in various engineering applications. Stability analysis of periodic solutions lays a foundation for control design to achieve convergence to a prescribed orbit. Here we consider a class of perturbed nonlinear systems with fast and slow dynamics and develop a novel averaging method for analyzing the local exponential orbital stability of a periodic solution. A framework is then proposed for feedback control design to stabilize a natural oscillation of an uncertain nonlinear system using a synchronous adaptive oscillator. The idea is applied to linear mechanical systems and a design theory is established. In particular, we propose a controller based on the Andronov-Hopf oscillator with additional adaptation mechanisms for estimating the unknown natural frequency and damping parameters. We prove that, with sufficiently slow adaptation, the estimated parameters locally converge to their true values and entrainment to the natural oscillation is achieved as part of an orbitally stable limit cycle. Numerical examples demonstrate that adaptation and convergence can in fact be fast.
Harmonic oscillators are widespread in industrial applications. One of their key properties is the large amplification of excitation signals at their eigenfrequency - broadly known as resonance. This paper investigates how to excite harmonic oscillators such that resonance is reached in minimum time. We use Pontryagin's Minimum Principle to derive time-optimal control laws to reach resonance for both the undamped and damped harmonic oscillator. For the former our derivations even lead to a time-optimal feedback law given through a switching curve. For application purposes we compare this feedback law with a much simpler sliding-mode based control law for controlling a simplified model of an industrial vibratory device. In order to smoothly apply both control laws, additional state and parameter estimators are presented to address the time-varying nature of the application.
We consider a class of multibody robotic systems inspired by dynamics of animal locomotion, such as swimming and crawling. Distinctive properties of such systems are that the stiffness matrix is asymmetric due to skewed restoring force from the environment, and the damping matrix is a scalar multiple of the inertia matrix when the body is flat like those of fish. Extending the standard notion to this class, we define the natural oscillation as a free response under the damping compensation to yield marginal stability. The natural oscillation of the body provides a basic gait for locomotion. We propose a class of simple nonlinear feedback controllers to achieve entrainment to the natural oscillation of the body, resulting in a prescribed average velocity. As an example, a link chain system with symmetric mechanical structure is considered, and its natural oscillation is shown to exhibit traveling waves appropriate for undulatory locomotion. Controllers are designed under various conditions to achieve prescribed locomotion speeds by natural oscillations. In particular, it is shown how design parameters can be chosen to increase the rate of convergence, and how the locomotion speed can be set by adjusting the natural frequency through body stiffness compensation.
Many devices and processes utilize self-excited oscillations either as the working principle or as the performance enhancer. The present paper investigates a nonlinear velocity feedback control method for generating artificial self-excited oscillations in a class of two degrees-of-freedom mechanical systems. The paper derives the conditions of existence and stability of the modal oscillations and numerically corroborates this. It also proposes a methodology to design the control system for inducing natural oscillations in one of the desired modes. Other than modal oscillations, the system can also be designed to oscillate in a non-modal state with the desired frequency and amplitude-ratio within a specific range (depending on the system). Numerical simulations confirm the analytical results. Further analysis shows that for each frequency of excitation, the control cost is minimum when the system operates at an optimal amplitude-ratio.
The problem of numerical evaluation of the excitability index for oscillatory systems is considered. It is shown that the speed-gradient excitation provides exact solution to the maximum energy problem for a second order linear oscillator over the infinite time interval. Upper and lower bounds for the total energy of the system in the steady-state oscillation mode and the excitability index are estimated. Exact value of the accessible system energy for the harmonic excitation case is found.
A variant of super-twisting differentiator is proposed. Lyapunov function is designed and an estimate on finite time of derivatives estimation is given. The differentiator is equipped with hybrid adaptation algorithm that ensures global differentiation ability independently on amplitude of the differentiated signal and measurement noise.
The issue of spatially distributed systems regulation is studied on the example of lattices of linear oscillators. The problem of natural wave stabilization is formulated and solved applying passivity-based approach. By natural wave stabilization we understand the desired energy and phase assignment for oscillators (the waves frequencies can be chosen from the lattice natural spectrum). The proposed controls are applied to 1D and 2D lattices confirming efficiency of the proposed solution.
. A measurable function x : J ae R! X (X a metric space) is said to be C-meagre if C ae X is non-empty and, for every closed set K ae X with K " C = ;, x Gamma1 (K) has finite Lebesgue measure. This concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations, extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems. 1 Introduction By way of motivation, consider the initial-value problem x = f(x), x(0) = 2 R N , with f : R N ! R N locally Lipschitz, and denote its unique solution by x : t 7! OE(t; ) defined on its maximal interval of existence I = [0; ). Let g : R N ! R be continuous. Assume that x is a bounded solution of the initial-value problem (in which case, I = [0; 1)), then g ffi x is uniformly continuous. If, in addition, g ffi x 2 L 1 , then we may conclude (by an elementary observation frequently referred to as Barbalat's lemma [5] (see, also, [18, x21,...
The problem of oscillations stabilization for the case of unknown natural frequency of vibration machine is considered. An algorithm of the problem solution for the second order linear model of the machine with uncertain parameters, external disturbances and partial noisy measurements is proposed.
Foundations of Theory of Differential Equations with Discontinuous Right-Hand Sides Auxiliary Algebraic Statements on Solutions of Matrix Inequalities of a Special Type Dichotomy and Stability of Nonlinear Systems with Multiple Equilibria Stability of Equilibria Sets of Pendulum-Like Systems.
Because the theoretical part of the book is based on the calculus of variations, the exposition is very transparent and requires mostly a trivial mathematical background. In the case of open-loop optimal control, this leads to Pontryagin's Minimum Principle and, in the case of closed-loop optimal control, to the Hamilton-Jacobi-Bellman theory which exploits the principle of optimality. Many optimal control problems are solved completely in the body of the text. Furthermore, all of the exercise problems which appear at the ends of the chapters are sketched in the appendix. The book also covers some material that is not usually found in optimal control text books, namely, optimal control problems with non-scalar-valued performance criteria (with applications to optimal filtering) and Lukes' method of approximatively-optimal control design. Furthermore, a short introduction to differential game theory is given. This leads to the Nash-Pontryagin Minimax Principle and to the Hamilton-Jacobi-Nash theory. The reason for including this topic lies in the important connection between the differential game theory and the H-control theory for the design of robust controllers. © Springer-Verlag Berlin Heidelberg 2007. All rights are reserved.
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
The possibilities of studying nonlinear behavior of physical systems by small feedback action are discussed. Analytical bounds of possible system energy change by feedback are established. It is shown that for a 1-DOF nonlinear oscillator, the change of energy by feedback can reach the limit achievable for a linear oscillator by a harmonic (non-feedback) action. The results are applied to different physical problems: evaluating the amplitude of action leading to escape from a potential well; stabilizing unstable modes of a nonlinear oscillator (pendulum); using feedback testing signals in spectroscopy. These and related studies are united by similarity of their goals (examination possibilities and limitations for changing a system behavior by feedback) and by unified methodology borrowed from cybernetics (control science). They, therefore, constitute a part of physics which can be called cybernetical physics.
The possibilities of studying nonlinear physical systems by feedback action are discussed. The feedback resonance phenomenon in nonlinear oscillators and the Speed–Gradient based method of creating feedback resonance are described. Examples of feedback resonance in 1-DOF system (controlled pendulum) and in 2-DOF system (two coupled pendulums) are studied by computer simulation.
The existing results on stabilization of invariant sets for nonlinear systems based on speed–gradient method and the notion of V-detectability are overviewed and extended. Applications to control of oscillations in pendulum, cart–pendulum and spherical pendulum systems are presented. Copyright © 2001 John Wiley & Sons, Ltd.
The problem of oscillations stabilization with unknown natural frequency of vibration machine is considered. An algorithm of the problem solution for the second order linear model of the machine with uncertain parameters, external disturbances and partial noisy measurements is proposed.
Mechanical systems can often be controlled efficiently by exploiting a resonance. An optimal trajectory minimizing an energy cost function is found at (or near) a natural mode of oscillation. Motivated by this fact, we consider the natural entrainment problem: the design of nonlinear feedback controllers for linear mechanical systems to achieve a prescribed mode of natural oscillation for the closed-loop system. We adopt a set of distributed central pattern generators (CPGs) as the basic control architecture, inspired by biological observations. The method of multivariable harmonic balance (MHB) is employed to characterize the condition, approximately, for the closed-loop system to have a natural oscillation as its trajectory. Necessary and sufficient conditions for satisfaction of the MHB equation are derived in the forms useful for control design. It is shown that the essential design freedom can be captured by two parameters, and the design parameter plane can be partitioned into regions, in each of which approximate entrainment to one of the natural modes, with an error bound, is predicted by the MHB analysis. Control mechanisms underlying natural entrainment, as well as limitations and extensions of our results, are discussed.
Preferred stride frequency (PSF) of human walking has been shown to be predictable as the resonant frequency of a force-drive harmonic oscillator (FDHO). The purpose of this study was to determine whether walking at the PSF and FDHO leads to minimal metabolic and mechanical costs. Subjects walked on a level treadmill at the PSF, FDHO, and frequencies above and below. Effects of stride length (SL) and speed (S) were assessed by two conditions, one in which SL was constant and the other in which S was constant. The predictability of PSF from resonance was replicated. Walking at the PSF and FDHO frequencies resulted in metabolic costs which were not significantly different (P greater than 0.05). A U-shaped oxygen consumption curve was observed with the minimum at the PSF and FDHO conditions when S was constant. A two-component curve in which a breakpoint was observed was found in the SL constant condition. A significant increase in metabolic cost was observed above the PSF/FDHO (P less than 0.01). Internal work (power) values were not significantly different between walking frequencies for the S constant condition. In the SL constant condition, internal work values showed linear increases as frequency increased. It was concluded that PSF of walking arises from the interface of the resonance properties of the limbs as oscillators and the tendency of biological systems to self-optimize.
Timing of the repetitive movements that constitute any rhythmic behavior is regulated by intrinsic properties of the central
nervous system rather than by sensory feedback from moving parts of the body. Evidence of this permits resolution of the long-standing
controversy over the neural basis of rhythmic behavior and aids in the identification of this mechanism as a general principle
of neural organization applicable to all animals with central nervous systems.
The neuronal circuit controlling the rhythmic movements in animal locomotion is called the central pattern generator (CPG). The biological control mechanism appears to exploit mechanical resonance to achieve efficient locomotion. The objective of this paper is to reveal the fundamental mechanism underlying entrainment of CPGs to resonance through sensory feedback. To uncover the essential principle, we consider the simplest setting where a pendulum is driven by the reciprocal inhibition oscillator. Existence and properties of stable oscillations are examined by the harmonic balance method, which enables approximate but insightful analysis. In particular, analytical conditions are obtained under which harmonic balance predicts existence of an oscillation at a frequency near the resonance frequency. Our result reveals that the resonance entrainment can be maintained robustly against parameter perturbations through two distinct mechanisms: negative integral feedback and positive rate feedback.
The problem of global (local) stabilizability of an invariant set
of passive nonlinear controlled system is considered. A new notion of
detectability of the zero value set of a smooth storage function is
introduced. It is shown that this property plays the fundamental role in
the solution of invariant set stabilization problems and in defining the
global (local) stabilizing regulator. Sufficient conditions, implying
that passive nonlinear system has a detectable zero value set for a
smooth storage function, are proposed
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Computation of the excitability index for linear oscillators Feedback resonance in single and coupled 1-DOF oscillators
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Predicting the minimal energy costs of human walking. Medicine and Science in Sport and Exercise. v23 i4
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Hybrid adaptive resonance control using adaptive observers for vibration machines
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