Control of Nonlinear Time-Periodic Systems Via Feedback Linearization

The problem of designing controllers for nonlinear time periodic systems is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (approximately) transformed into a linear control system. Then a controller can be designed using the well-known linear method to guarantee the stability of the system. We propose two approaches for the feedback linearization of the nonlinear time periodic system. The first 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. In this case the system equations can be 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. The second approach is a generalization of the classical exact feedback linearization method for autonomous systems but applicable to general time-periodic affine systems. By defining a time-dependent Lie operator, the input-output nonlinear time periodic problem is transformed into a linear autonomous problem for which control system can be designed easily. A sufficient condition under which the system is feedback linearizable is also given.

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

10.1115/1.2447190 ◽

2006 ◽

Vol 2 (2) ◽

pp. 124-131 ◽

Cited By ~ 6

Author(s):

Yandong Zhang ◽

S. C. Sinha

Keyword(s):

Nonlinear Systems ◽

Feedback Linearization ◽

Linear Time ◽

Sufficient Conditions ◽

Original System ◽

Computation Method ◽

Normal Form Theory ◽

Periodic Coefficients ◽

Quasi Linear System ◽

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.

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

10.1115/1.3124093 ◽

2009 ◽

Vol 4 (3) ◽

Cited By ~ 1

Author(s):

Yandong Zhang ◽

S. C. Sinha

Keyword(s):

Normal Form ◽

Design Problem ◽

Controller Design ◽

Design Method ◽

Observer Design ◽

Periodic Systems ◽

Normal Form Theory ◽

Periodic Coefficients ◽

System States ◽

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.

Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

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

10.1115/detc2007-34831 ◽

2007 ◽

Author(s):

Yandong Zhang ◽

S. C. Sinha

Keyword(s):

Normal Form ◽

Design Problem ◽

Controller Design ◽

Linear Time ◽

Design Method ◽

Observer Design ◽

Periodic Systems ◽

Normal Form Theory ◽

Periodic Coefficients ◽

For most complex dynamic systems, it is not possible to measure all system states in a direct fashion. 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 with time periodic coefficients is addressed. For linear time-periodic systems, it is shown that 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 the Chebyshev expansion technique. For the nonlinear time periodic systems, the observer design is investigated using the Poincare´ normal form technique. The local identity observer is designed by using a set of near identity coordinate transformations which can be constructed in the ascending order of nonlinearity. These observer design methods are implemented using a symbolic computational algorithm and several illustrative examples are given to show the effectiveness of the methods.

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

10.1115/1.4037797 ◽

2017 ◽

Vol 13 (2) ◽

Cited By ~ 5

Author(s):

Ashu Sharma ◽

S. C. Sinha

Keyword(s):

Parametric Excitation ◽

Periodic System ◽

Floquet Theory ◽

Picard Iteration ◽

Periodic Systems ◽

Transition Matrices ◽

Frequency Spectra ◽

Periodic Coefficients ◽

Stability Diagrams ◽

The Stability

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

Approximate Lyapunov-perron Transformations: Computation and Applications to Quasi-periodic Systems

10.1115/1.4050614 ◽

2021 ◽

Author(s):

Ashu Sharma

Keyword(s):

Analytical Solutions ◽

Periodic System ◽

Linear Part ◽

Closed Form Solution ◽

Hill Equation ◽

Periodic Systems ◽

Normal Form Theory ◽

Symbolic Form ◽

Stochastic Perturbations ◽

P Transformations

Abstract A new technique for the analysis of dynamical equations with quasi-periodic coefficients (so-called quasi-periodic systems) is presented. The technique utilizes Lyapunov-Perron (L-P) transformation to reduce the linear part of a quasi-periodic system into the time-invariant form. A general approach for the construction of L-P transformations in the approximate form is suggested. First, the linear part of a quasi-periodic system is replaced by a periodic system with a 'suitable' large principal period. Then, the state transition matrix of the periodic system is computed in the symbolic form using Floquet theory. Finally, Lyapunov-Floquet theorem is used to compute approximate L-P transformations. A two-frequency quasi-periodic system is studied and transformations are generated for stable, unstable and critical cases. The effectiveness of these transformations is demonstrated by investigating three distinct quasi-periodic systems. They are applied to a forced quasi-periodic Hill equation to generate analytical solutions. It is found that the closeness of the analytical solutions to the exact solutions depends on the principal period of the periodic system. A general approach to obtain the stability bounds on linear quasi-periodic systems with stochastic perturbations is also discussed. Finally, the usefulness of approximate L-P transformations to nonlinear quasi-periodic systems is presented by analyzing a quasi-periodic Hill equation with cubic nonlinearity using time-dependent normal form theory. The closed-form solution generated is found to be in good agreement with the exact solution.

Volume 6: 13th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

10.1115/detc2017-68041 ◽

2017 ◽

Author(s):

Ashu Sharma ◽

Subhash C. Sinha

Keyword(s):

Parametric Excitation ◽

Periodic System ◽

Floquet Theory ◽

Periodic Systems ◽

Frequency Spectra ◽

Periodic Coefficients ◽

Approximate Approach ◽

Varying Coefficients ◽

P Transformation ◽

The Stability

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC) ◽

Unification of Poincaré and Floquet Theory for Time Periodic Systems

10.1115/detc2020-22524 ◽

2020 ◽

Author(s):

Susheelkumar C. Subramanian ◽

Sangram Redkar

Keyword(s):

Closed Form ◽

Periodic System ◽

Floquet Theory ◽

Linear Time ◽

Mathieu Equation ◽

Transformation Matrix ◽

Closed Form Expression ◽

Periodic Systems ◽

Form Expression ◽

Abstract As per Floquet theory, a transformation matrix (Lyapunov Floquet transformation matrix) converts a linear time periodic system to a linear time-invariant one. Though a closed form expression for such a matrix was missing in the literature, this method has been widely used for studying the dynamical stability of a time periodic system. In this paper, the authors have derived a closed form expression for the Lyapunov Floquet (L-F) transformation matrix analytically using intuitive state augmentation, Modal Transformation and Normal Forms techniques. The results are tested and validated with the numerical methods on a Mathieu equation with and without damping. This approach could be applied to any linear time periodic systems.

A Simple Approach for the Computation of Lyapunov-Floquet Transformations for the Mathieu Equation

10.1115/detc2021-71028 ◽

2021 ◽

Author(s):

Ashu Sharma

Keyword(s):

Periodic System ◽

Linear Time ◽

Mathieu Equation ◽

Adjoint System ◽

Simple Approach ◽

Periodic Systems ◽

Linear Ordinary Differential Equations ◽

Constant Coefficients ◽

Time Invariant ◽

Abstract Lyapunov-Floquet (L-F) transformations reduce linear ordinary differential equations with time-periodic coefficients (so-called linear time-periodic systems) to equations with constant coefficients. The present work proposes a simple approach to construct L-F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential term and a periodic term. Using this Floquet form of a solution, the ordinary differential equation corresponding to a linear time-periodic system reduces to an eigenvalue problem. Next, eigenanalysis is performed to obtain the general solution and subsequently, the state transition matrix of the time-periodic system is constructed. Then, the Lyapunov-Floquet theorem is used to compute L-F transformation. The inverse of L-F transformation is determined by defining the adjoint system to the time-periodic system. Mathieu equation is investigated in this work and L-F transformations and their inverse are generated for stable and unstable cases. These transformations are very useful in the design of controllers using time-invariant methods and in the bifurcation studies of nonlinear time-periodic systems.

Bifurcation Control of Nonlinear Systems With Time-Periodic Coefficients

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.1636194 ◽

2003 ◽

Vol 125 (4) ◽

pp. 541-548 ◽

Cited By ~ 9

Author(s):

Alexandra Da´vid ◽

S. C. Sinha

Keyword(s):

Bifurcation Control ◽

Nonlinear Feedback ◽

Periodic Load ◽

Normal Form Theory ◽

Center Manifold Reduction ◽

Periodic Coefficients ◽

Codimension One ◽

Form Theory ◽

Time Invariant ◽

In this study, a method for the nonlinear bifurcation control of systems with periodic coefficients is presented. The aim of bifurcation control is to stabilize post bifurcation limit sets or modify other nonlinear characteristics such as stability, amplitude or rate of growth by employing purely nonlinear feedback controllers. The method is based on an application of the Lyapunov-Floquet transformation that converts periodic systems into equivalent forms with time-invariant linear parts. Then, through applications of time-periodic center manifold reduction and time-dependent normal form theory completely time-invariant nonlinear equations are obtained for codimension one bifurcations. The appropriate control gains are chosen in the time-invariant domain and transformed back to the original variables. The control strategy is illustrated through the examples of a parametrically excited simple pendulum undergoing symmetry-breaking bifurcation and a double inverted pendulum subjected to a periodic load in the case of a secondary Hopf bifurcation.

Computation of Lyapunov–Perron transformation for linear quasi-periodic systems

Journal of Vibration and Control ◽

10.1177/1077546321993568 ◽

2021 ◽

pp. 107754632199356 ◽

Cited By ~ 1

Author(s):

Susheelkumar C Subramanian ◽

Peter MB Waswa ◽

Sangram Redkar

Keyword(s):

Closed Form ◽

Periodic System ◽

Linear Time ◽

Closed Form Expression ◽

Periodic Systems ◽

Form Expression ◽

State Augmentation ◽

Time Invariant ◽

Time Periodic ◽

Time Invariant System

The transformation of a linear time periodic system to a time-invariant system is achieved using the Floquet theory. In this work, the authors attempt to extend the same toward the quasi-periodic systems, using a Lyapunov–Perron transformation. Though a technique to obtain the closed-form expression for the Lyapunov–Perron transformation matrix is missing in the literature, the application of unification of multiple theories would aid in identifying such a transformation. In this work, the authors demonstrate a methodology to obtain the closed-form expression for the Lyapunov–Perron transformation analytically for the case of a commutative quasi-periodic system. In addition, for the case of a noncommutative quasi-periodic system, an intuitive state augmentation and normal form techniques are used to reduce the system to a time-invariant form and obtain Lyapunov–Perron transformation. The results are compared with the numerical techniques for validation.