Stability Of Periodic Systems Research Articles

Parametric Stability Analysis of Active Magnetic Bearing Supported Rotor System With a Novel Control Law Subject to Periodic Base Motion

Active magnetic bearings (AMBs) are being continuously explored for industrial applications mainly because of their friction-free operation. However, if a rotor-shaft-AMB system is used in applications such as turbo engine of an aircraft or in the propeller shaft of a ship, it would be subject to parametric excitation because of the moving base of the system. Parametric excitation is known to cause instability in systems. This paper addresses the issue of such flexible rotor-AMB system and performs the parametric stability analysis. A novel control law based on the constitutive relationship of a viscoelastic material called the four-element control law is developed. Because of the presence of periodic base motion terms in the system state matrix, the system can be classified as a linear periodic system. Thus, the stability analysis is carried out using Floquet–Liapunov theory on stability of periodic system, for which an approximate state transition matrix is first found and stability of the system is then decided based on the absolute value of eigenvalue. Three different base motions (roll, pitch, and yaw) were considered for evaluating the performance of the proposed control law. Simulation results reveal that the proposed control law has better stability characteristics than the proportional-integral-derivative (PID) control law.

Stabilization of periodic systems with aftereffect by finite-dimensional approximations

We study the stabilization problem for a linear periodic system of differential equations with aftereffect. Approximating systems are described by differential equations with finite-dimensional Volterra operators. We construct admissible controls in the class of piecewise continuous functions by the feedback principle. We establish a connection between the approximating stabilization problem and that of the optimal stabilization for an autonomous linear system of difference equations.

Assessing Asymptotic Stability of Linear Continuous Time-Varying Systems by Computing the Envelope of all Trajectories

In this note, necessary and sufficient numerical conditions for asymptotic stability and for uniform asymptotic stability of linear continuous time-varying systems are derived. For a given set of initial conditions, a tube containing all the trajectories of the system is constructed in the state space. At each instant of time, there exists an initial condition inside the set such that the resulting trajectory attains the border of the tube. Based on the above formulation, necessary and sufficient conditions for asymptotic stability and for uniform asymptotic stability are expressed through the solution of a linear differential Lyapunov equation. The conditions can deal with the stability of periodic systems as well. One of the main characteristics of the proposed necessary and sufficient conditions is that the only assumption on the dynamical matrix of the time-varying system is continuity. Examples from the literature illustrate the superiority of the proposed conditions when compared to other methods.

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.

Stabilization of Periodic Systems: Overview and Advances

In this paper we consider the problem of stability and the problem of stabilization via state-feedback of periodic linear systems in discrete-time. First, the case of known parameter systems is considered. Then, both norm-bounded and polytopic uncertain systems are dealt with. The basic tools for the analysis are the robust Lyapunov paradigm and the periodic Linear Matrix Inequality technique.

Dynamic Stability of Vehicle Systems via Poincaré Maps

SUMMARY Computer simulations of a driven vehicle are performed to assess and quantify the automobile stability. The analytical methodology is based on PoincarS maps and Floquet theory, which are commonly used to investigate the stability of periodic systems. The approach accommodates the study of the complex dynamics of vehicle locomotion under various configurations of the system. The proposed method does not require any knowledge regarding the inner structure of the vehicle model. Kinematic data is simply collected at the specific instants to conduct the analysis. It is found that this method has the potential for real time warning and control of vehicle instabilities.

Analytical stability analysis of periodic systems by Poincaré mappings with application to rotorcraft dynamics

Apoint mappinganalysis is employed to investigate the stability of periodic systems. The method is applied to simplified rotorcraft models. The proposed approach is based on a procedure to obtain an analytical expression for the period-to-period mapping description of system's dynamics, and its dependence on system's parameters. Analytical stability and bifurcation conditions are then determined and expressed as functional relations between important system parameters. The method is applied to investigate the parametric stability of flapping motion of a rotor and the ground resonance problem encountered in rotorcraft dynamics. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Stability analysis of higher dimensional systems, such as the ground resonance problems, by perturbation methods is not straightforward, and is usually very cumbersome.

Normal form and stability of periodic systems with internal resonance: PMM vol. 40, n≗ 3, 1976, pp. 431–438

We examine the problem of the Liapunov stability of the zero solution of a multidimensional nonlinear differential equation system with periodic coefficients and holomorphic right-hand sides. We investigate the critical case when the characteristic equation of the linearized system has only complex-conjugate roots equal to unity in modulus and when definite integral relations (internal resonance) exist between the characteristic indices and the frequency of the unperturbed motion. For any type of resonance we give a normal form of the system by reducing the original problem to the problem of stability under internal resonance for autonomous systems, considered earlier in [1] ; this enables us to obtain necessary and sufficient conditions for the stability of a model system, i.e. For the most important cases of odd-order resonance. We show that in the majority of cases the instability of the model system implies the instability of the complete system. We show that an even-order resonance can lead to the asymptotic stability of the system. For a second-order system we give necessary and sufficient conditions for asymptotic stability with respect to the first nonlinear term in the normal form. We indicate the extension of the results obtained to the stability of Hamiltonian systems.