Nonlinear Time Periodic Systems Research Articles

Stability Analysis of a Single-Phase Active Power Decoupling Converter Based on Discrete Harmonic State Space Model

Digitally controlled single-phase active power decoupling converter is a continuous-discrete hybrid system with nonlinear time-periodic (NTP) nature, thus the conventional linear time-invariant (LTI) model cannot accurately describe the system dynamic behavior. This paper presents a discrete harmonic state space (DHSS) modeling method to analyze the small-signal stability of a digitally controlled single-phase current source converter (SCSC) with an active power decoupling circuit (APDC). The proposed model retains the switching details, it has the potentiality to accurately describe the high-frequency dynamics. Moreover, the DHSS model not only transforms NTP system into LTI system, but also includes harmonic interaction dynamics. Furthermore, a LTI Toeplitz matrix is developed to analyze the oscillation phenomenon and system stability, and the stability boundaries of system parameters are obtained based on the eigenvalues loci. Finally, the physical experiment results validate the correctness of the theory analysis.

Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis

In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting, which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n-DOF generalization are used to exemplify these findings.

Revisited Harmonic Balance Trim Solution Method for Periodically-Forced Aerospace Vehicles

In this paper, a numerical method is proposed for determining the periodic state and control solutions of nonlinear time-periodic systems. Starting from an initial guess at the solution, the algorithm uses a harmonic balance technique to refine the solution through a gradient-based optimization approach. The algorithm introduces three major innovations when compared to previous techniques: it does not rely on state transition matrices, it simultaneously solves for the approximate higher-order linear time-invariant dynamics about the periodic solution, and it can be used to compute the harmonic control inputs that attenuate arbitrary state harmonics. Following a description of the algorithm, it is applied to three different aerospace vehicles with nonlinear time-periodic dynamics: a flapping-wing micro aerial vehicle, a helicopter, and a flapping-tail airplane. In all cases, the algorithm derives periodic solutions for the states and controls even when starting from very poor initial guesses. The algorithm has clear application in the development of advanced flight control laws that attenuate certain state harmonics and in the prediction of loads and vibrations.

Coupled Unsteady Aero-Flight Dynamics of Hovering Insects/Flapping Micro Air Vehicles

The main objective of this paper is to provide a rigorous coupling between body flight dynamics and flow dynamics (unsteady aerodynamics) in hovering of insects and flapping-wing micro air vehicles. A reduced-order unsteady aerodynamic model that takes into account the leading-edge vortex and rotational effects is integrated with a dynamic model to construct a flight dynamic model for flapping-wing micro air vehicles and insects near hover. Then, the averaging theorem is used to study the dynamic stability of such a nonlinear time-periodic system. Effects of unsteady aerodynamics on the trim/balance problem and stability of such an equilibrium are assessed. It is found that the flow dynamics induces asymmetric forward force and pitching moment, even though the flapping motion is symmetric. As such, balance of insects and flapping-wing micro air vehicles at hover cannot be achieved by purely symmetric flapping. Although unsteady aerodynamics considerably affects the balance/trim problem, its effect on stability is minor.

Dimensionless Approach to Multi-Parametric Stability Analysis of Nonlinear Time-Periodic Systems: Theory and Its Applications to Switching Converters

This paper proposes a dimensionless approach to analyze the multi-parametric stability behavior of switching converters, which can be characterized by a nonlinear time-periodic (NTP) system. The main objective is to analyze how multiple circuit parameters affect the stability patterns of the derived NTP system and to simplify the parametric complexity of such NTP system. In contrast to previous work, the proposed method focuses on the parametric resultant relationships of the NTP system in the sense of topological equivalence, and investigates its stability in terms of the homeomorphic NTP system. Firstly, an equivalent stability theory of NTP systems is proposed. Then, based on the equivalent theory, a normalized map is introduced and various interesting properties are derived so as to formulate the dimensionless approach. Moreover, the approximate solution of the NTP system in dimensionless parameter space is calculated by using the Galerkin method, and its stability pattern is identified with the help of eigenvalue analysis approach. Finally, a case study of one-cycle controlled Zeta PFC converter is discussed in detail to exemplify the application of the proposed method. These analytical results agree well with those ones obtained from experimental measurements.

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘ invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘ reducibility conditions’. If these ‘ reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘ primary’ and ‘ secondary resonances’ present in the system. One can also recover ‘ resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

Observer Design for Nonlinear Systems With Time-Periodic Coefficients via Normal Form Theory

For most complex dynamic systems, it is not always possible to measure all system states by a direct measurement technique. Thus for dynamic characterization and controller design purposes, it is often necessary to design an observer in order to get an estimate of those states, which cannot be measured directly. In this work, the problem of designing state observers for free systems (linear as well as nonlinear) with time-periodic coefficients is addressed. It is shown that, for linear periodic systems, the observer design problem is the duality of the controller design problem. The state observer is constructed using a symbolic controller design method developed earlier using a Chebyshev expansion technique where the Floquet multipliers can be placed in the desired locations within the unit circle. For nonlinear time-periodic systems, an observer design methodology is developed using the Lyapunov–Floquet transformation and the Poincaré normal form technique. First, a set of time-periodic near identity coordinate transformations are applied to convert the nonlinear problem to a linear observer design problem. The conditions for existence of such invertible maps and their computations are discussed. Then the local identity observers are designed and implemented using a symbolic computational algorithm. Several illustrative examples are included to show the effectiveness of the proposed methods.

Development of a Feedback Linearization Technique for Parametrically Excited Nonlinear Systems via Normal Forms

The problem of designing controllers for nonlinear time periodic systems via feedback linearization is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (exactly or approximately) transformed into a linear time periodic control system. Then a controller can be designed to guarantee the stability of the system. Our approach is designed to achieve local control of nonlinear systems with periodic coefficients desired to be driven either to a periodic orbit or to a fixed point. The system equations are represented by a quasi-linear system containing nonlinear monomials with periodic coefficients. Using near identity transformations and normal form theory, the original close loop problem is approximately transformed into a linear time periodic system with unknown gains. Then by using a symbolic computation method, the Floquet multipliers are placed in the desired locations in order to determine the control gains. We also give the sufficient conditions under which the system is feedback linearizable up to the rth order.

Normal Forms for High Co-Dimension Bifurcations of Nonlinear Time-Periodic Systems with Nonsemisimple Eigenvalues

The normal forms for time-periodic nonlinear variational equations witharbitrary linear Jordan form undergoing bifurcations of highco-dimension are found. First, the equations are transformed via theLyapunov–Floquet (L–F) transformation into an equivalent form in whichthe linear matrix is constant with degenerate nonsemisimple lineareigenvalues while the nonlinear monomials have periodic coefficients. Byconsidering the resulting coupling of the bases of the near identitytransformation, the solvability condition for an arbitrary Jordan matrixis then derived. It is shown that time-independent and/or time-dependentnonlinear resonance terms remain in the normal form for various Jordanmatrices. Specifically, the normal forms for quadratic and cubicnonlinearities with the following linear Jordan forms are explicitlyderived: double zero eigenvalues (co-dimension two bifurcation), triplezero eigenvalues (co-dimension three bifurcation), and two repeatedpairs of purely imaginary eigenvalues (co-dimension two bifurcation). Acommutative system with cubic nonlinearities and a double inverted pendulum with a periodicfollower force are used as illustrative examples.

Normal forms and the structure of resonance sets in nonlinear time-periodic systems

The structure of time-dependent resonances arising in themethod of time-dependent normal forms (TDNF) for one andtwo-degrees-of-freedom nonlinear systems with time-periodic coefficientsis investigated. For this purpose, the Liapunov–Floquet (L–F)transformation is employed to transform the periodic variationalequations into an equivalent form in which the linear system matrix istime-invariant. Both quadratic and cubic nonlinearities are investigatedand the associated normal forms are presented. Also, higher-orderresonances for the single-degree-of-freedom case are discussed. It isdemonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2,...) solutions. The discussion is limited to the Hamiltonian case (which encompasses allpossible resonances for one-degree-of-freedom). Furthermore, it is alsoshown how a recent symbolic algorithm for computing stability andbifurcation boundaries for time-periodic systems may also be employed tocompute the time-dependent resonance sets of zero measure in theparameter space. Unlike classical asymptotic techniques, this method isfree from any small parameter restriction on the time-periodic term inthe computation of the resonance sets. Two illustrative examples (oneand two-degrees-of-freedom) are included.

Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincare map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.