Limit Cycle Control Research Articles

Robust limit cycle control for finite-time generation of sustained oscillations in nonlinear systems with mixed dead-zone and saturation

This paper suggests a novel limit cycle control technique, based on the finite-time stability scheme, for producing sustained oscillations in a class of nonlinear complex systems. The considered systems are constrained to input dead-zone and saturation simultaneously and suffer from non-vanishing perturbations due to model uncertainties, and external disturbances. For this purpose, an innovative sliding manifold is proposed based on the topology of the wanted limit cycle which provides the conditions for developing the finite time stability concept for limit sets. With the aid of the finite-time command filter and the cooperation of the sliding mode control (SMC) method with the backstepping technique, a robust controller is constructed which guarantees the finite-time convergence of phase trajectories of the closed-loop system toward the wanted limit cycle. This leads to the creation of sustained oscillations, with the desirable amplitude and frequency, in the system's output. The stability analysis is proved based on the Lyapunov approach through the set stability concept. Finally, the effectiveness of the suggested methodology is confirmed by a variety of simulations.

Convolution/Volterra Reduced-Order Modeling for Nonlinear Aeroelastic Limit Cycle Oscillation Analysis and Control

A methodology for leveraging a combination of linear (through convolution integrals) and nonlinear (through Volterra series) reduced-order modeling (ROM) development was demonstrated on both a two- and three-degree-of-freedom (2-DOF and 3-DOF, respectively) aeroelastic system. The linear/nonlinear ROM approach was demonstrated against previous work in the field for a 2-DOF system and subsequently extended to a 3-DOF (flapped airfoil) system. Excellent agreement was demonstrated at limit cycle oscillation (LCO) onset (flutter) prediction between computational fluid dynamics, linear ROMs, and nonlinear ROMs. Although linear ROMs were sufficient to predict LCO onset, the nonlinear Volterra correction was required to estimate the amplitude of post-LCOs. Corrections that accounted for more coupling between degrees of freedom predicted the amplitudes more accurately. A study was undertaken to determine the minimal training set required to produce reasonable LCO amplitude estimates of the 3-DOF airfoil. A controller was also implemented in an effort to mitigate LCO onset and flutter divergence and to demonstrate the usefulness and power of leveraging the ROM for control design. A full-state feedback controller, tuned via a linear quadratic regulator (LQR), was shown to be effective in controlling the system below LCO onset. However, beyond LCO onset, this specific linear controller structure was observed to be ineffective in controlling the system. Future work will evaluate nonlinear control methods in which the ROM can be deployed as a part of the control system to update the LQR gains in a fully coupled approach.

A homeostasis criterion for limit cycle systems based on infinitesimal shape response curves.

Homeostasis occurs in a control system when a quantity remains approximately constant as a parameter, representing an external perturbation, varies over some range. Golubitsky and Stewart (J MathBiol 74(1-2):387-407, 2017) developed a notion of infinitesimal homeostasis for equilibrium systems using singularity theory. Rhythmic physiological systems (breathing, locomotion, feeding) maintain homeostasis through control of large-amplitude limit cycles rather than equilibrium points. Here we take an initial step to study (infinitesimal) homeostasis for limit-cycle systems in terms of the average of a quantity taken around the limit cycle. We apply the "infinitesimal shape response curve" (iSRC) introduced by Wang et al.(SIAM JApplDynSyst 82(7):1-43, 2021) to study infinitesimal homeostasis for limit-cycle systems in terms of the mean value of a quantity of interest, averaged around the limit cycle. Using the iSRC, which captures the linearized shape displacement of an oscillator upon a static perturbation, we provide a formula for the derivative of the averaged quantity with respect to the control parameter. Our expression allows one to identify homeostasis points for limit cycle systems in the averaging sense. We demonstrate in the Hodgkin-Huxley model and in a metabolic regulatory network model that the iSRC-based method provides an accurate representation of the sensitivity of averaged quantities.

A control strategy for efficient slip ratio regulation of a pneumatic brake system for commercial vehicles

This paper presents a control strategy for efficient slip ratio regulation of a pneumatic brake system for commercial vehicles. A model-based estimator for brake pressure estimation has been developed. The braking torque applied to the wheel has been computed using the estimated brake pressure for the control of the wheel slip both in braking and traction situations. The vehicle velocity and wheel slip ratio estimation algorithms have been designed using only wheel speed sensors. The proposed slip regulation algorithm has also been successfully implemented for the antilock braking system (ABS) and traction control system (TCS). In ABS, the slip ratio and wheel acceleration are stabilized by a limit cycle control of the braking pressure. The TCS has been implemented by combining engine torque control and pneumatic brake pressure control. The brake controller is based on the valve switched control that incorporates the wheel dynamics and valve on/off characteristics. The ABS and TCS algorithms are integrated into the slip regulation algorithm to reduce the computation load of an Electrical Control Unit (ECU). Four-wheel independent slip monitoring and slip ratio control algorithms have been implemented on the ECU, and their performance has been investigated via both computer simulations and vehicle tests. Both results show that the proposed algorithms enhance the acceleration and braking performance without vehicle acceleration information. Moreover, the proposed split-mu strategy has improved the lateral stability during braking, and the acceleration performance during accelerating on the split-mu road. It has been shown via vehicle tests that, compared to the reference commercial algorithm, the braking distance was reduced by more than 4% on the split-mu and low-mu roads, and the acceleration performance was improved by 7.9% on the split-mu road.

Occasional stabilisation of limit cycle walking and control of chaos in the passive dynamics of the compass-gait biped model

Occasional stabilisation of limit cycle walking and control of chaos in the passive dynamics of the compass-gait biped model

Occasional stabilisation of limit cycle walking and control of chaos in the passive dynamics of the compass-gait biped model

It is known that the compass-gait biped model exhibits cyclic walking patterns and attractive complex phenomena, namely chaos and bifurcations. This work is concerned with the stabilisation of limit cycles and then the control of chaos. A new stabilisation process of the limit cycle walking is developed based on self-detection of the fixed point of the (un)stable limit cycle and on energy-shaping-based trajectory-tracking controller. The control process is applied in the beginning of the swing stage during a desired short time interval making hence the compass-gait biped robot to be completely passive on the remaining swing phase. We demonstrate that such occasional stabilisation of the limit cycles considerably increases the energetic efficiency of the bipedal locomotion. We show also that the proposed control method allows the compass-gait biped to walk efficiently and with a periodic gait down sloped surfaces of different angles.

Motor output and control input in flapping flight: a compact model of the deforming wing kinematics of manoeuvring hoverflies.

Insects are conventionally modelled as controlling flight by varying a few summary kinematic parameters that are defined on a per-wingbeat basis, such as the stroke amplitude, mean stroke angle and mean wing pitch angle. Nevertheless, as insects have tens of flight muscles and vary their kinematics continuously, the true dimension of their control input space is likely to be much higher. Here, we present a compact description of the deforming wing kinematics of 36 manoeuvring Eristalis hoverflies, applying functional principal components analysis to Fourier series fits of the wingtip position and wing twist measured over 26 541 wingbeats. This analysis offers a high degree of data reduction, in addition to insight into the natural kinematic couplings. We used statistical resampling techniques to verify that the principal components (PCs) were repeatable features of the data, and analysed their coefficient vectors to provide insight into the form of these natural couplings. Conceptually, the dominant PCs provide a natural set of control input variables that span the control input subspace utilized by this species, but they can also be thought of as output states of the flight motor. This functional description of the wing kinematics is appropriate to modelling insect flight as a form of limit cycle control.

Orbit-Attitude Coupled Tracking and Landing Control for an Asteroid

Recent research on asteroid exploration has shown tremendous potential to provide humankind detailed views of unexplored worlds in the inner solar system. A potential market of asteroid mining has aroused interest from commercial companies gradually. Asteroid exploration has the potential to offer the possibility to revolutionize the supply of many resources which are vital but exorbitant for human. A typical asteroid exploration mission is composed of remote asteroid detecting, approaching, hovering, landing and sampling, take-off and return, re-entry stage. Among these stages the tracking and landing control of the asteroid is crucial for the exploration. Rendezvous, approaching, fly-around and landing problem of an asteroid is investigated in this paper. The expected motion of the tracking spacecraft determined by the asteroid’ orbital and attitude status is presented firstly. According to the design of the OSIRIS-REx project, several constant thrusters are used for spacecraft to track the asteroid. Different control strategies like pulse control, saturated nonlinear control, limit cycle control and PWM based low-thrust control are developed to fulfil the asteroid’s tracking mission of different stages. Numerical simulations demonstrate the effectiveness of the tracking control.

Limit Cycle Synchronization of Nonlinear Systems with Matched and Unmatched Uncertainties Based on Finite-Time Disturbance Observer

This paper focuses on the limit cycle control of uncertain nonlinear systems in the presence of both matched and unmatched uncertainties. For this purpose, first, a virtual nonlinear system is constructed which has the desired limit cycle in its phase trajectories. The Lyapunov stability theorem for positive limit sets is utilized to confirm the stability of the created limit cycle in the virtual system. Next, by using a modified nonsingular terminal sliding mode control method, a robust controller is designed to synchronize the trajectories of the actual nonlinear system with the corresponding virtual one in a finite time. It is assumed that the actual system suffers from both matched and unmatched uncertainties and/or unknown external disturbances. A nonlinear terminal sliding surface is designed with the aim of a finite-time disturbance observer to tackle these unknown terms. The finite-time Lyapunov stability theorem is utilized to confirm the stability and robustness of the designed control law. Through simulation results, the finite-time stabilization of the synchronization errors and appropriate performance of the proposed control law are verified.

Robust limit cycle control in a class of nonlinear discrete-time systems

ABSTRACTThis paper studies generation of robust periodic solutions in a class of nonlinear discrete-time system. The sustained oscillations, with the desired frequency and amplitude, are achieved through the creation of the appropriate elliptic limit cycle in the phase plane of the uncertain closed-loop discrete-time system. In the first step, the nominal control law is designed to enforce the trajectories of the nominal closed-loop system to converge to the desired limit cycle. Next, considering uncertain terms, an additional robustifying term is designed. This term is added to the nominal controller to sustain the desirable stable oscillations in the presence of uncertain terms. The resulted robust controller brings the trajectories of the uncertain closed-loop discrete-time system to a boundary layer (with adjustable width) around the desired limit cycle. Moreover, the domain of attraction of the limit cycle and also the ultimate boundary layer around it are calculated via the Lyapunov analysis. Additionally, in order to verify the applicability of the proposed method, it is implemented on the discretised model of a spring–damper system. Computer simulations confirm the theoretical results in generating robust stable oscillations.

Passive Energy-based Control via Energy Tanks and Release Valve for Limit Cycle and Compliance Control

Abstract The problem of generating periodic solutions for a fully actuated robot is considered in this work. The result is achieved by extending the concept of limit cycle control via energy regulation, in order to guarantee passivity of the closed-loop system. Such a feature is particularly relevant in case of physical interaction with the environment, since a non-passive system could cause problems in terms of stability, robustness and safety. As energy-based limit cycle control can be used also for regulation control, the paper extends as well recent results in passivation of projection-based null space compliance control to the case of adjustable maximum allowed activity in the system.

Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties

ABSTRACTIn this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.

Energy Based Limit Cycle Control of Elastically Actuated Robots

A new control law for elastic joint robots that allows to regulate an energy function of the system to a desired value is presented in this technical note. Being able to either remove energy from the system or inject into it, oscillations can be both damped out and induced. The proposed nonlinear dynamic state feedback controller forces the system to evolve on a submanifold of the configuration space. The reduced dynamics of the system and of the controller itself are similar to a single elastic joint, for which an asymptotically stable limit cycle is obtained regulating an energy function to a positive desired value. When the desired value of the energy function is chosen to be zero, then the asymptotically stable limit cycle reduces to an asymptotically stable equilibrium point. In this case the oscillations are damped out and the desired task-space configuration is reached. The design of the controller extensively uses the concept of conditional stability, so that the limit cycle can be designed for a lower dimensional dynamical system, although it will result to be a limit cycle for the whole system.

Generating 1-DOF limit cycle walking at target walking speed by feed-forward and feedback limit cycle control

We propose the feed-forward and feedback (FaF) control systems for generating the limit cycle walking at a target walking speed by the combined rimless wheel (CRW) model. The proposed FaF control systems can calculate a control input constantly based on the mathematical analysis of the current and target walking states. As a result, first, the limit cycle walking at the target walking speed is generated by numerical simulations when the walker is driven by the constant FaF control system. Second, for controlling the convergence speed, we extend the FaF control to the two-period stepwise control systems. The limit cycle walking at the target walking speed is still generated, and the convergence speed is controlled by the settling time parameter. Finally, the real-time walking state updating is considered in the FaF control systems to handle the disturbances. In this case, we find that for generating a target walking speed, a precise mathematical model is not necessary for generating the target state, but can make the control input stable and efficient.

Inducing sustained oscillations in a class of nonlinear discrete time systems

Inducing sustained oscillations in a class of nonlinear discrete time systems is studied in this paper. The novelty of this paper is based on the proposed approach in generating stable oscillations according to limit cycle control. The limit cycle control is not formulated for nonlinear discrete time systems of any order and this paper concentrates on this issue. Considering the stable limit cycle as a positive limit set for the dynamical system, a nonlinear control law is designed to create the considered limit cycle in the phase trajectories of the closed-loop nonlinear discrete time system to achieve oscillations with the desirable amplitude and frequency. For this purpose, firstly, the limit cycle control is proposed for second-order nonlinear discrete time systems. The stability analysis of the generated limit cycle is done via a suitable Lyapunov function. Also, the domain of attraction of the created limit cycle is calculated. The proposed method is then extended for nonlinear discrete time systems of any order via the backstepping technique. Finally, computer simulations are performed for a practical example to demonstrate the ability of the designed controller in generating stable oscillations.

Limit Cycle Control Using Energy Function Regulation With Friction Compensation

In this letter, we consider the problem of generating periodic solutions for fully actuated robots with unknown disturbances, which can be modeled using a regressor matrix. We extend our previous work on limit cycle control based on energy function regulation for the case when disturbance torques are acting on the system, e.g., torques due to friction. Since the controller is designed in two independent steps, the compensation of the friction cannot be carried on with standard techniques and it will be split in two steps as well. In the first one, we reduce the dimension of the dynamical system and use a sliding mode approach for friction compensation, and in the second, we produce the desired limit cycle and use an adaptive approach for friction compensation. Crucial for the analysis is the concept of conditional attractiveness with semidefinite Lyapunov functions, that we formulate in letter to show the attractiveness of a closed orbit of the whole system, even if it is designed assuming a reduced dynamics. Finally, we validate our approach with experiments on a humanoid robot.

Control of limit cycles in buck converters

Purpose – The purpose of this paper is to investigate limit cycles in digitally Proportional, Integral and Derivative (PID) controlled buck regulators. Filtering is examined as a means of removing the limit cycles in digitally controlled buck regulators. Design/methodology/approach – The paper explains why limit cycles occur in a digitally PID controlled buck converter. It then proceeds to propose two filters for their elimination. Results indicate the effectiveness of each of the filters. Findings – The paper gives a mathematical analysis of the occurrence of limit cycles in digitally controlled PID buck regulators. It finds that notch and comb filters are effective for the purpose of eliminating limit cycles in buck regulators. Originality/value – The paper employs a model of the buck regulator inclusive of the inductor loss – this was not done to date for this type of work. The paper analyses PID control. This was not done in the manner given. The paper addresses filtering as a means of removing limit cycles. It examines the effect of changing the digital controller parameters on the requirements of the filters.

Switching based limit cycle control for compliantly actuated second-order systems

This paper derives a stability statement for a novel, switching based limit cycle control. The stability proof is based on multiple Lyapunov functions and a new interpretation of contraction analysis. By showing that the dissipated energy on the cycle increases with increasing velocity, while the injected energy is constant, the emergence of an attractive limit cycle is shown. The approach applies for general, nonlinear, and compliantly actuated second-order systems, with positive definite plant parameters and non-aperiodic solutions. An analysis of the controller parameters reveals, that for the majority of parameters, global attractiveness of the limit cycle can be guaranteed.

Limit Cycle Generation for Multi-Modal and 2-Dimensional Piecewise Affine Control Systems

This paper considers a limit cycle control problem of a multi-modal and 2-dimensional piecewise affine control system. Limit cycle control means a controller design method to generate a limit cycle for given piecewise affine control systems. First, we deal with a limit cycle synthesis problem and derive a new solution of the problem. In addition, theoretical analysis on the rotational direction and the period of a limit cycle is shown. Next, the limit cycle control problem for piecewise affine control system is formulated. Then, we obtain matching conditions such that the piecewise affine control system with the state feedback law corresponds to the reference system which generates a desired limit cycle. Finally, in order to indicate the effectiveness of the new method, a numerical simulation is illustrated. paper. The outline of this paper is as follows. We first consider a limit cycle synthesis problem and derive its new solution in Section II. Some theoretical properties are also shown. Then, in Section III, a formulation of limit cycle control problem is presented, and necessary and sufficient conditions for the problem, which are called matching conditions, are derived. Finally, a numerical simulation is shown in order to confirm the effectiveness of the new method in Section IV.

A describing function approach to the design of robust limit-cycle controllers

The design of robust limit-cycle controllers is introduced for autonomous systems with separable SISO nonlinearities. The objective is to design a controller to secure specified robust oscillation amplitude and frequency. The method consists of quasi-linearization of the nonlinear element via a Describing Function (DF) approach and then shaping the loop to reach desired limit-cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot. An example is given to illustrate the robustness of the controlled system to uncertainties in the linear subsystem model.