Response of Linear Time-Periodic Systems Subjected to Stochastic Excitations: A Chebyshev Polynomial Approach

1995 ◽  
Author(s):  
J. M. Spires ◽  
S. C. Sinha
Keyword(s):  
Chebyshev Polynomial ◽  
Original System ◽  
Invariant Form ◽  
Periodic Systems ◽  
Solution Technique ◽  
Mean Square ◽  
Stochastic Excitations ◽  
Mean Square Response ◽  
Time Invariant ◽  
Time Periodic

Abstract In many situations engineering systems modeled by a system of linear second order differential equations with periodic damping and stiffness matrices are subjected to stochastic excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it has been shown that the Liapunov-Floquet transformation matrix can be computed, and the original time-periodic system can be put into a time invariant form. In this paper, these techniques are applied in finding the transient mean square response and transient autocorrelation response of periodic systems subjected to stochastic forcing vectors. Two formulations are presented. In the first formulation, the mean square response of the original system is computed directly. In the second formulation, the original system is transformed to a time-invariant form. The autocorrelation response is found by determining the response of the time-invariant system. Both formulations utilize the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through Chebyshev polynomial expansion. Results for some time-invariant and periodic systems are included, as illustrative examples.

On The Response of Linear Time-Periodic Systems Subjected to Deterministic and Stochastic Excitations

1996 ◽  
Vol 2 (2) ◽  
pp. 219-249 ◽  
Author(s):  
J.M. Spires ◽  
S.C. Sinha
Keyword(s):  
Chebyshev Polynomial ◽  
Linear Time ◽  
Original System ◽  
Invariant Form ◽  
Periodic Systems ◽  
Solution Technique ◽  
Stiffness Matrices ◽  
Original Time ◽  
Time Invariant ◽  
Time Periodic

In many situations, engineering systems modeled by a set of linear, second-order differential equations, with periodic damping and stiffness matrices, are subjected to external excitations. It has been shown that the fundamental solution matrix for such systems can be efficiently computed using a Chebyshev polynomial series solution technique. Further, it is shown that the Liapunov-Floquet transformation matrix associated with the system can be computed, and the original time-periodic system can be put into a time- invariant form. In this paper, these techniques are applied in finding the transient response of periodic systems subjected to deterministic and stochastic forces. Two formulations are presented. In the first formulation, the response of the original system is computed directly. In the second formulation, first the original system is transformed to a time-invariant form, and then the response is found by determining the response of the time-invariant system. Both formulations use the convolution integral to form an expression for the response. This expression can be evaluated numerically, symbolically, or through a Chebyshev polynomial expansion technique. Results for some time-invariant and periodic systems are included, as illustrative examples.

scholarly journals Optimal Preview Control for Linear Discrete-Time Periodic Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Fucheng Liao ◽  
Mengyuan Sun ◽  
Usman
Keyword(s):  
Discrete Time ◽  
Original Problem ◽  
Control Method ◽  
Original System ◽  
Periodic Systems ◽  
Invariant System ◽  
Preview Control ◽  
Time Invariant ◽  
Time Periodic ◽  
Time Invariant System

In this paper, the optimal preview tracking control problem for a class of linear discrete-time periodic systems is investigated and the method to design the optimal preview controller for such systems is given. Initially, by fully considering the characteristic that the coefficient matrices are periodic functions, the system can be converted into a time-invariant system through lifting method. Then, the original problem is also transformed into the scenario of time-invariant system. Later on, the augmented system is constructed and the preview controller of the original system is obtained with the help of existing preview control method. The controller comprises integrator, state feedback, and preview feedforward. Finally, the simulation example shows the effectiveness of the proposed preview controller in improving the tracking performance of the close-loop system.

Comparison of Poincaré Normal Forms and Floquet Theory for Analysis of Linear Time Periodic Systems

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Susheelkumar C. Subramanian ◽  
Sangram Redkar
Keyword(s):  
Floquet Theory ◽  
Normal Forms ◽  
Linear Time ◽  
Mathieu Equation ◽  
Temporal Variations ◽  
Periodic Systems ◽  
Comparative Results ◽  
Time Invariant ◽  
Identity Transformations ◽  
Time Periodic

Abstract In this work, the authors draw comparisons between the Floquet theory and Normal Forms technique and apply them towards the investigation of stability bounds for linear time periodic systems. Though the Normal Forms technique has been predominantly used for the analysis of nonlinear equations, in this work, the authors utilize it to transform a linear time periodic system to a time-invariant system, similar to the Lyapunov–Floquet (L–F) transformation. The authors employ an intuitive state augmentation technique, modal transformation, and near identity transformations to facilitate the application of time-independent Normal Forms. This method provides a closed form analytical expression for the state transition matrix (STM). Additionally, stability analysis is performed on the transformed system and the comparative results of dynamical characteristics and temporal variations of a simple linear Mathieu equation are also presented in this work.

scholarly journals On Fault Detection in Linear Discrete-Time, Periodic, and Sampled-Data Systems

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
P. Zhang ◽  
S. X. Ding
Keyword(s):  
Fault Detection ◽  
Linear Time ◽  
Periodic Systems ◽  
Data Systems ◽  
Sampled Data ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Problem Formulations ◽  
Time Invariant ◽  
Time Periodic

This paper gives a review of some standard fault-detection (FD) problem formulations in discrete linear time-invariant systems and the available solutions. Based on it, recent development of FD in periodic systems and sampled-data systems is reviewed and presented. The focus in this paper is on the robustness and sensitivity issues in designing model-based FD systems.

Some Ideas on the Local Control of Nonlinear Systems With Time-Periodic Coefficients

1999 ◽  
Author(s):  
Alexandra Dávid ◽  
S. C. Sinha
Keyword(s):  
Normal Form ◽  
Local Control ◽  
State Feedback ◽  
Feedback Controller ◽  
Periodic Systems ◽  
State Feedback Controller ◽  
Linearized System ◽  
Linear State ◽  
Time Invariant ◽  
Time Periodic

Abstract In this paper ideas on local control of linear and nonlinear time-periodic systems are presented. Our first goal is to stabilize the system far away from bifurcation points. In this case, the classical linear state feedback stabilization based on pole placement is generalized such that it is applicable to time-periodic systems. The linear state-feedback controller design involves computation of the fundamental solution matrix of the system in a symbolic form as function of the control parameters. Next, we focus on the bifurcation control of time-periodic systems. When the linearized system is in a critical case of stability (i.e. when it has Floquet multipliers on the unit circle of the complex plane) and the critical modes are uncontrollable in the linear sense, then a purely nonlinear state-feedback controller is designed to stabilize the equilibrium at the bifurcation point and ensure the stability of the bifurcated nontrivial solution. When the linearized system is linearly controllable, then it is shown that an appropriately chosen linear state-feedback control can also modify the nonlinear features of the bifurcations, such as stability or size of the limit cycles or quasi-periodic limit sets. The control techniques are based on a series of transformations that convert the system into a time-invariant form. First, the Lyapunov-Floquet transformation is used to make the linear part of the periodic system time-invariant. Then, time-periodic center manifold reduction and time-dependent normal form theory are applied to obtain the simplest nonlinear form of a system undergoing bifurcation. For most codimension one bifurcations the normal form is completely time-invariant and therefore, it is a rather simple task to choose the appropriate control gains. These ideas are illustrated by an example of a parametrically excited simple pendulum undergoing symmetry breaking bifurcation.

Cramer–Rao Bound Development for Linear Time Periodic Systems

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
Chris S. Schulz ◽  
Donald L. Kunz ◽  
Norman M. Wereley
Keyword(s):  
Linear Time ◽  
System Model ◽  
Periodic Systems ◽  
Accurate Representation ◽  
Linear Time Invariant ◽  
Lti System ◽  
Identification Techniques ◽  
Cramer Rao Bound ◽  
Time Invariant ◽  
Time Periodic

System identification techniques are often used to determine the parameters required to define a model of a linear time invariant (LTI) system. The Cramer–Rao bound can be used to validate those parameters in order to ensure that the system model is an accurate representation of the system. Unfortunately, the Cramer–Rao bound is only valid for LTI systems and is not valid for linear time periodic (LTP) systems such as a helicopter rotor in forward flight. This paper describes an extension of the Cramer–Rao bound to LTP systems and demonstrates the methodology for a simple LTP system.

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

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 ◽  
Time Periodic

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.

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