Active nonlinear saturation-based control for suppressing the free vibration of a self-excited plant

A nonlinear saturation-based control strategy for the suppression of the free vibration of a self-excited plant is presented. The self-excitation to have the classical form of that of the van der Pol oscillator is considered. The control technique is implemented by coupling the active absorber with the plant via a specific set of quadratic nonlinearities. The perturbation method of multiple scales is employed to find the first-order approximate solutions to the governing equations. Then a stability analysis is conducted for the response of the system and the performance of the control strategy is investigated. A parametric investigation is carried out to see the effects of changing the damping ratio of the absorber, and the value of the feedback gain on the responses of the plant and the absorber. Finally, the perturbation solutions are verified by numerical integration of the governing differential equations. It is demonstrated that the saturation-based control method is effective in reducing the vibration response of the self-excited plant when the absorber’s frequency is exactly tuned to one-half the natural frequency of the plant.     

Controlling chaotic orbits in mechanical systems with impacts

We stabilize desired unstable periodic orbits, embedded in the chaotic invariant sets of mechanical systems with impacts, by applying a small and precise perturbation on an available control parameter. To obtain such perturbation numerically, we introduce a transcendental map (impact map) for the dynamical variables computed just after the impacts. To show how to implement the method, we apply it to an impact oscillator and to an impact-pair system.     

Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.     

A Multi-Layered Potential Field Method for Multi-Agent Navigation,Searching and Tracking

Collaborating multi-agent systems can handle complex tasks with several or changing mission objectives. We developed a potential field method that allows various information layers to influence the control over a group of vehicles. The gradient of the potential field is the driving force for local action, whereas the global waypoint is determined by the minimum of the agent's potential field. The driving force to the global waypoint is a virtual spring-mass-damper system that pulls the agent towards its waypoint, restricted by the local gradient of the agent's potential field. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A tuning criterion for the inertial tuned damper. Design using phasors in the Argand–Gauss plane

A new approach to the design of a dynamic damper for a monomass oscillator is presented; the design procedure is then applied to control a multimodal oscillator. This new dynamics emerged from an analysis by means of phasors (rotating vectors in the Argand–Gauss plane) which revealed the phase relations between the damper and main oscillator. In particular this work introduces a geometric formalism, based on the use of phasors in the complex plane, for the sizing of inertial dampers applied to multimodal structural oscillators. Their damping effect depends on the fact that the response of the secondary oscillator (the damper) delays the response of the primary mass by 90°, so that the elastic force transmitted by the damper becomes a viscous force on the controlled oscillator. When such condition occurs we say that the damper is “tuned” to the main oscillator; the damping induced by the damper serves to limit the displacement of main oscillator. Our geometrical approach provides a method whose language is close to that of structural mechanics, thus paving the way to the professionals for: (i) sizing the damper parameters and (ii) evaluating the stability to the damped system and its performance limits. The aim of the development is that of exploring the use of dampers to control the response of buildings under horizontal seismic and aerodynamic loads.     

Robust stabilization control design for multivariable nonlinear uncertain systems

The paper develops a new control technique for multivariablenonlinear systems in the presence of uncertainties and externaldisturbances. The proposed design method does not require thatthe uncertainties should satisfy matching conditions; nor doesit require that the nominal system should be stable or prestabilized.The robust-control strategy is established using concepts fromvariable-structure theory and is based on Lyapunov stabilitytheory. The control possesses a quite simple structure whichis related to the given uncertainty bounds.     

Exponential stability and partial averaging

The exponential stability of singularly perturbed time-varying systems is investigated. It turns out that, under natural conditions, exponential stability of an averaged system is equivalent to exponential stability of the perturbed system for small perturbation parameters. Explicit estimates for both, the approximation of single trajectories and the order of the exponential decay, are obtained. The method of proof does not require smoothness of the averaged system.     

Numerical optimization of tuned mass absorbers attached to strongly nonlinear Duffing oscillator

We investigate the dynamics of the vertically forced Duffing oscillator with suspended tuned mass absorber. Three different types of tuned mass absorbers are taken into consideration, i.e., classical single pendulum, dual pendulum and pendulum-spring. We numerically adjust parameters of absorbers to obtain the best damping properties with the lowest mass of attached system. The modification of classical case (single pendulum) gives the decrease of Duffing system amplitude. We present strategy of parameters tuning which can be easily applied in a large class of systems.     

Optimal control of an elastic string with viscous damping

We study bilinear optimal control of a wave equation with one spatial dimension. The problem describes oscillations of an elastic string with viscous damping, and the damping coefficient is taken as the control. The objective functional involves driving the state solution close to a desired profile and incurring a cost on the control. The optimal control is characrerized in terms of an optimality system.     

Controlling bifurcation and chaos of a plastic impact oscillator

A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.     

Control of time-periodic systems via symbolic computation with application to chaos control

A general method for the control of linear time-periodic systems employing symbolic computation of Floquet transition matrix is considered in this work. It is shown that this method is applicable to chaos control. Nonlinear chaotic systems can be driven to a desired periodic motion by designing a combination of a feedforward controller and a feedback controller. The design of the feedback controller is achieved through the symbolic computation of fundamental solution matrix of linear time-periodic systems in terms of unknown control gains. Then, the Floquet transition matrix (state transition matrix evaluated at the end of the principal period) can determine the stability of the system owing to classical techniques such as pole placement, Routh–Hurwitz criteria, etc. Thus it is possible to place the Floquet multipliers in the desired locations to determine the control gains. This method can be applied to systems without small parameters. The Duffing’s oscillator, the Rössler system and the nonautonomous parametrically forced Lorenz equations are chosen as illustrative examples to demonstrate the application.     

Optimal damping ratio for linear second-order systems

In this paper, it is shown that the optimal damping ratio for linear second-order systems that results in minimum-time no-overshoot response to step inputs is of bang-bang type. The optimal damping ratio is zero at the outset and is switched to some maximum value at an appropriate instant of time. The switching time is shown to be a function of the maximum damping ratio and the system natural frequency. Furthermore, it is shown that the larger the maximum damping ratio is, the shorter it takes for the system to reach the desired set point. Finally, it is shown that, if the optimal damping ratio is switched as a function of the system state, then the minimum-time no-overshoot criterion is satisfied, irrespective of the magnitude of the uncertainty in the value of the system natural frequency.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.     

On linear vibrational systems with one dimensional damping II

We are interested in the quadratic eigenvalue problem of damped oscillations where the damping matrix has dimension one. This describes systems with one point damper. A generic example is a linearn-mass oscillator fixed on one end and damped on the other end. We prove that in this case the system parameters (mass and spring constants) are uniquely (up to a multiplicative constant) determined by any given set of the eigenvalues in the left half plane. We also design an effective construction of the system parameters from the spectral data. We next propose an efficient method for solving the Ljapunov equation generated by arbitrary stiffness and mass matrices and a one dimensional damping matrix. The method is particularly efficient if the Ljapunov equation has to be solved many times where only the damping dyadic is varied. In particular, the method finds an optimal position of a damper in some 60n ³ operations. We apply this method to our generic example and show, at least numerically, that the damping is optimal (in the sense that the solution of a corresponding Ljapunov equation has a minimal trace) if all eigenvalues are brought together. We include some perturbation results concerning the damping factor as the varying parameter. The results are hoped to be of some help in studying damping matrices of the rank much smaller than the dimension of the problem.     

Voltage amplification transfer function analysis of the operational amplifier in the negative capacitance circuits for vibration control with piezoceramics

Piezo elements due to their ability of converting mechanical energy into electrical energy and vice versa can be found in numerous mechanical vibration damping and absorbing applications. A desired effect may be customized by an external impedance shunt branch connected to the plates of the piezo element. The negative capacitance connected in serial with the passive shunt significantly improves the damping and absorbing performance of such systems. Negative capacitance circuit is built of an impedance converter realized by the operational amplifier. Since the amplifiers are not the ideal elements, the performance of the proposed systems is limited. This is due to the maximum voltages and currents that the operational amplifiers are able to generate. This effect causes instabilities and limits the operational area of the impedance converter. In the paper, the amplification transfer function of the non-ideal operational amplifier in the negative impedance converter is studied, and the necessary modification with the additional passive elements is proposed. The influence of the certain imperfections in the design, and its improvements are shown on the system consisting of the 1DOF mechanical oscillator, and a shunted piezo element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust intelligent sliding model control using recurrent cerebellar model articulation controller for uncertain nonlinear chaotic systems

In this paper, a robust intelligent sliding model control (RISMC) scheme using an adaptive recurrent cerebellar model articulation controller (RCMAC) is developed for a class of uncertain nonlinear chaotic systems. This RISMC system offers a design approach to drive the state trajectory to track a desired trajectory, and it is comprised of an adaptive RCMAC and a robust controller. The adaptive RCMAC is used to mimic an ideal sliding mode control (SMC) due to unknown system dynamics, and a robust controller is designed to recover the residual approximation error for guaranteeing the stable characteristic. Moreover, the Taylor linearization technique is employed to derive the linearized model of the RCMAC. The all adaptation laws of the RISMC system are derived based on the Lyapunov stability analysis and projection algorithm, so that the stability of the system can be guaranteed. Finally, the proposed RISMC system is applied to control a Van der Pol oscillator, a Genesio chaotic system and a Chua’s chaotic circuit. The effectiveness of the proposed control scheme is verified by some simulation results with unknown system dynamics and existence of external disturbance. In addition, the advantages of the proposed RISMC are indicated in comparison with a SMC system.     

Recovering unknown model parameters contained in a class of time-variant chaotic dynamical systems

This work deals mainly with the problem of recovering all unknown parameters for a class of time-variant chaotic dynamical systems from given time sequence. Based on synchronization between a chaotic sender system and an additional receiver system, a procedure, which combines a linear feedback technique with updated feedback gain and an adapted control strategy associated with the law of estimated parameters, is developed to dynamically determine the values of unknown parameters contained in the sender system. To promote widespread applications, the structure of the receiver system can be independent of that of the sender system. The effectiveness of this procedure is guaranteed by the periodic version of the classical LaSalle invariance principle of differential equations. Illustrations are presented for a harmonically excited Duffing oscillator and a four dimensional chaotic oscillator. The numerical results reveal the present procedure not only can precisely recover unknown model parameters, but also can rapidly response to sudden changes in unknown parameters. In addition, it has great robustness against the disturbance of noise.     

Stabilizing chaotic system on periodic orbits using multi-interval and modern optimal control strategies

In this paper, optimal approaches for controlling chaos is studied. The unstable periodic orbits (UPOs) of chaotic system are selected as desired trajectories, which the optimal control strategy should keep the system states on it. Classical gradient-based optimal control methods as well as modern optimization algorithm Particle Swarm Optimization (PSO) are utilized to force the chaotic system to follow the desired UPOs. For better performance, gradient-based is applied in multi-intervals and the results are promising. The Duffing system is selected for examining the proposed approaches. Multi-interval gradient-based approach can put the states on UPOs very fast and keep tracking UPOs with negligible control effort. The maximum control in PSO method is also low. However, due to its inherent random behavior, its control signal is oscillatory.