Limit cycle analysis for electro‐hydraulic systems with friction

A systematic and straightforward methodology to analyze and quench the friction-induced limit cycle conditions in electro-hydraulic servovalve control systems with interval transport lag is developed in this paper. The nonlinearity is linearized by its corresponding describing function. The delay time in transmission line is considered. The stability-equation method accompanied with parameter plane method provides a useful tool for the establishment of necessary conditions to sustain a limit cycle in the constructed limit cycle surface. The portrayed surface characterizes clear relationship among limit cycle amplitude, frequency, transport delay, and the controller coefficients to be designed. The asymptotic stability region, the unstable region, and the limit cycle region are identified in the parameter plane. The variation of the transport lag which could accelerate the generation of limit cycles is investigated. A feasible stable region is characterized in the parameter plane to allow flexible choice of controller gains. The quenching of limit cycle is achieved by selecting controller gains from the asymptotic stability region. It is seen that the friction-induced limit cycles can be effectively quenched via the tuning of the controller gains.

In the machine tool industry and the semiconductor manufacture fields, the necessity of high precision and fast response is rapidly spreading. In general, it is well known that a positioning control system contains nonlinear friction force; it may cause a serious disadvantage on control performances such as tracking errors, limit cycles and undesired stick-slip motion at very low motion velocity. In order to achieve the high level control performance, the frictional effects have to be removed. This paper presents a novel nonlinear friction compensation approach based on the idea of P, PI/I-P+FF control with sliding mode compensator for a high precision stage using a synchronous piezoelectric device driver. P, PI/I-P+FF control consists of Proportional position control and Proportional-Integral/Integral-Proportional velocity control with velocity Feed-Forward compensator. Experimental results are given to show the effectiveness of our proposed control method compared to the conventional P, PI/I-P+FF control method. Especially, the positioning error of our proposed control was reduced by more than 2/3 compared to the case of the conventional control at a stepping motion.

The stability of a power reactor with multiple transport lags and nonlinear characteristics embodied in the controller is analyzed by parameter plane method. The analysis covers two reactor models: (a) A rather simple model, symbolically representing the features of a nonlinear controller, one transport lag and the reactor with temperature feedback, and (b) A more realistic model depicting a practical chemical fine-control reactor system characterized by multiple transport lags and mixing effect associated with the chemical fluid, the nonlinear characteristics of chemical injection valves, and the spatial distribution of poison inside the reactor, as well as the reactor temperature feedback. These analyses yield much useful information about the stability of both systems, such as the conditions of reactor stability and of the occurrence of limit cycles, as well as the aspects of discontinuous change of the limit cycle frequency. Use of this parameter plane method will greatly simplify evaluation ...

Nonlinear friction forces are a problem in classical Brownian motion for a long time [1]. They are described either by Langevin or by Fokker-Planck equations [2-4]. However, the rigorous generalized Langevin equation is linear, which points out that nonlinear friction forces possess a macroscopic hydrodynamic origin [1]. For this reason, they are not present in the modern quantum theory of open systems [5]. The scope of the present paper is to investigate the nonlinear friction effect on quantum mechanics. The analysis is based on a dissipative Madelung quantum hydrodynamics. Thus, the nonlinear friction is a result of nonlinear non-equilibrium thermodynamics in contrast to the traditional mechanical nonlinear friction forces. A quantum particle in vacuum is described by the Schrodinger equation

The Cassini spacecraft dynamics telemetry during long Main Engine (ME) burns has indicated the presence of stable limit cycles between 0.03-0.05 Hz frequencies. These stable limit cycles cause the spacecraft to possess small but non-zero oscillating rates for extended periods of time. Even though these limit cycles have been observed in telemetry, they have not hindered the spacecraft performance to any degree of concern. The performance of the ME TVC system has been excellent throughout the prime and extended missions. However, it is important to understand these stable limit cycles from a controller design viewpoint. In this study, we hypothesize that the observed limit cycles in the spacecraft dynamics telemetry appear from a stable interaction between the unmodeled nonlinear elements and the linear ME TVC system. Many nonlinearities in the control system emerge from translating the linear engine gimbal actuator (EGA) motion into the rotational motion of the main engine. One such nonlinearity comes from the gear backlash in the EGA system, which is the focus of this paper. The limit cycle characteristics and behavior can be predicted by modeling this gear backlash nonlinear element via a describing function and studying the interaction of this describing function with the overall dynamics of the spacecraft. The linear ME TVC system and gear backlash nonlinearity responses are modeled analytically. The frequency, amplitude, and nature of the limit cycle are obtained from the Nyquist and describing function stability criteria. In addition, the ME TVC system along with the nonlinearity are modeled in Simulink ‡ . The simulation limit cycle response has similar characteristics as predicted analytically. The analytical and simulated limit cycle responses are compared to the flight telemetry for long burns such as the Saturn Orbit Insertion and Main Engine Orbit Trim Maneuvers. The limit cycle frequencies and peak-to-peak amplitudes of the X-axis spacecraft rate are 0.035 Hz/1.3 x 10 -4 rad/s, 0.058 Hz/2.0 x 10 -4 rad/s, and 0.054 Hz/1.0 x 10 -4 rad/s from telemetry, analytical, and simulated results, respectively. These limit cycle frequencies and amplitudes are on the same order of magnitude and exhibit stable oscillations. This work supports that the stable limit cycles occur due to the interaction between the unmodeled nonlinear elements and linear ME TVC system.

The vital purpose of a vehicle suspension system is to isolate the car body and hence passengers, from roadway unevenness disturbances. Implementation of passive suspension systems has continuously improved disconnection from disturbances through available deflection constraints to provide maximum isolation. In the majority of relevant reported research studies, a quarter car is modelled as moving vertically straight for both a viscous damper and a stiffness spring. The motivation for this study, reported here, is to extend the modelling to take account of the actual configuration of a test rig system. Accordingly, a new passive suspension system model is presented, which includes nonlinear lubricant friction forces that affect the linear support body bearings. The friction model established relies on dynamic system analysis and the fact of slipping body on lubricant bearings; this model captures most of the friction behaviours that have been observed experimentally. The suspension model is composed of a car body and wheel unit, and only vertical motion (bounce mode) is addressed. In addition, an active actuator is used to generate the system inputs as a road simulator. Therefore, a nonlinear hydraulic actuator, including the dynamic of servovalve and proportional–integral controller model, is established. This study is validated by experimental work, with simulations achieving C++compiler. As a result, a good agreement is obtained between the experimental and simulation results, that is, the passive suspension system with considered nonlinear friction and the nonlinear hydraulic actuator with servovalve equation models are entirely accurate and useful. The suggested proportional–integral controller successfully derives the hydraulic actuator to validate the control scheme. The ride comfort and handling response are close to that expected for the passive suspension system with road disturbances.

This paper presents the new vibration model of the horizontal washing machine (HWM) using a nonlinear geographic of the suspension system with nonlinear friction force - velocity (F-V) relationship of dampers obtained from experiment. Furthermore, using the dissipated energy function of dampers (Ed) criteria, some results of the effect of the spring stiffness on the dampers’ loading level are achieved. The simulation was done in Matlab/Simulink® environment. The results were then compared with experimental validation results to determine the suitability of the model. The conclusions can provide some available evidences for the design and improvement of nonlinear dampers.

The Cassini spacecraft dynamics telemetry during long main engine burns has indicated the presence of stable limit cycles between 0.03 and 0.05 Hz frequencies. Even though these limit cycles have been observed in telemetry, they have not hindered the spacecraft performance to any degree of concern. The performance of the main engine thrust vector control system has been excellent throughout the prime and extended missions. However, it is important to understand these stable limit cycles from a controller design viewpoint. It is proposed that the observed limit cycles in the dynamics telemetry appear from a stable interaction between the unmodeled nonlinear elements and the linear main engine thrust vector control system. This paper focuses on one such nonlinearity that emerges from the gear backlash in the engine gimbal actuator system. The gear backlash nonlinearity is modeled via a describing function, and the interaction between the nonlinear element model and the overall dynamics of the spacecraft is studied. With this interaction, the stable limit cycle is reproduced analytically and in simulation. The results are compared with the actual telemetry from long burns like Saturn orbit insertion and main engine orbit trim maneuvers. The predicted stable limit cycle frequency and amplitude show good agreement within 6% error between the analytical, simulation, and telemetry results. This work claims that the stable limit cycles occur due to the stable interaction between the unmodeled nonlinear elements and linear main engine thrust vector control system.

In this paper, a largeorder system is reduced by using the self adaptive bat algorithm (SABA) to a reduced-order approximation.The numerator coefficients of a desired reduced-order system are optimized based on integral square error minimization as an objective function pertaining to a unit-step as input.The denominator of the reduced order model is solved by stability equation method.The efficacy of the proposed method is tested with SMIB test systems to get a corresponding reduced-order system.The results are satisfactory in terms of minimum error with the proposed method as compared to Routh approximation and Stability equation based reduced models.

Limit cycle analysis of a hydroelectric system: A new approach

Limit cycles in self-excited multi-degree-of-freedom systems

This study examines the stability properties and robustness of nonlinear general type 2 TSK fuzzy control systems (NGT2 TSKs). Due to their higher design flexibility and nonlinear functions applied in their rule consequences, NGT2 TSK fuzzy controllers (NGT2 TSK-Cs) outperform other types of fuzzy controllers in the presence of complex nonlinearities and high degrees of uncertainty. With the help of describing function and parameter plane approaches, the limit cycles and asymptotically stable regions are determined to analyze the stability. Control systems with high unacceptable vibration levels, such as limit cycles, could become unstable as a result. Therefore, particle swarm optimization (PSO) is used to identify the minimum limit cycle region to enhance stability. To assess the robustness of closed-loop systems during uncertainty, minimum gain and phase margins are calculated. Finally, simulation examples are presented to verify the validity of the suggested methodology.

A systematic limit cycle suppression method is presented to predict and suppress the sustained limit cycles persisting in a class of uncertain control systems containing inherent nonlinearity. The aim is to determine the feasible controller parameter sets in the parameter plane for the entire uncertain system. A family of constant limit cycle amplitude loci is plotted. Successful application to pitch orientation control system with parametric uncertainties and nonlinearity is demonstrated. System output response is further investigated.

This paper deals with a new method for model order reduction of linear continuous time interval system. This new method is based on the Kharitonov’s theorem, the Stability equation method and the error minimization by Differential Evolution. The reduced order interval model is determined by using Kharitonov’s polynomials, which make use of the Kharitonov’s theorem and general form of the stability equation method for denominator, while the numerator is obtained by minimizing the integral square error between the transient responses of original and reduced order models using Differential Evolution algorithm. This method generates stable reduced order interval system if the original higher order system is stable and retains the steady-state value. The proposed method is illustrated with the help of typical numerical example considered from the literature.

Oscillations caused by solid friction are treated experimentally in a rotational system. A ring-like body is connected to a spring and can oscillate coaxially with a rotating shaft under the application of spring force and nonlinear friction force between sliding contact surfaces. Pre-driving of the rotating shaft just after the cleaning of sliding surfaces has a great influence on the characteristic curve betwen kinetic friction moment and peripheral velocity of the shaft. Pre-driving produces a change in configuration of the characteristic curve, therefore the corresponding amplitude-velocity curve and the period-velocity curve obtained experimentally show different appearances from those without pre-driving. These curves are qualitatively similar closely to theoretical ones derived from Lienard's construction.