Linear Time-Varying Dynamic Systems Optimization via Higher-Order Method: A Sub-Domain Approach

1999 ◽  
Vol 122 (1) ◽  
pp. 31-35 ◽  
Author(s):  
Xiaochun Xu ◽  
Sunil K. Agrawal
Keyword(s):  
Dynamic Systems ◽  
Linear Time ◽  
Transformation Matrix ◽  
Dynamic Equations ◽  
Time Varying ◽  
Nonsingular Transformation ◽  
Cost Functional ◽  
Optimality Equations ◽  
The Cost ◽  
Cost Functionals

A new procedure for optimization of linear time-varying dynamic systems has been proposed that uses transformations to embed the dynamic equations explicitly into the cost functional. This leads to elimination of Lagrange multipliers and characterization of the optimality equations by high-order differential equations in the same number of variables as number of control inputs. This procedure requires that the transformation matrix be nonsingular at all time within the domain. This paper extends this procedure to problems where a single nonsingular transformation matrix does not exist over the entire domain. In this paper, the time domain is partitioned into intervals such that a nonsingular transformation exists over each interval. The transformations are used to embed the dynamic equations into the cost functional. Variational analysis of the unconstrained cost functionals results in the optimality equations, which are solved efficiently by weighted residual methods. [S0739-3717(00)00601-2]

Diagonalization algorithms for linear time-varying dynamic systems

2000 ◽  
Vol 31 (8) ◽  
pp. 1053-1057 ◽  
Author(s):  
P. Van Der Kloet ◽  
F. L. Neerhoff
Keyword(s):  
Dynamic Systems ◽  
Linear Time ◽  
Time Varying

scholarly journals Parametric design to reduced-order functional observer for linear time-varying systems

2021 ◽  
pp. 002029402110211
Author(s):  
Da-Ke Gu ◽  
Li-Song Sun ◽  
Yin-Dong Liu
Keyword(s):  
Parametric Design ◽  
Linear Time ◽  
Design Flow ◽  
Observation Error ◽  
Time Varying ◽  
Nonsingular Transformation ◽  
Reduced Order ◽  
Functional Observer ◽  
Time Varying Systems ◽  
Generalized Sylvester Equation

This article studies the parametric design of reduced-order functional observer (ROFO) for linear time-varying (LTV) systems. Firstly, existence conditions of the ROFO are deduced based on the differentiable nonsingular transformation. Then, depending on the solution of the generalized Sylvester equation (GSE), a series of fully parameterized expressions of observer coefficient matrices are established, and a parametric design flow is given. Using this method, the observer can be constructed under the expected convergence speed of the observation error. Finally, two numerical examples are given to verify the correctness and effectiveness of this method and also the aircraft control problem.

Further Remarks on the Existence of Periodic Solutions of Linear Time Varying Periodic Fractional Order Systems

2017 ◽  
Author(s):  
XueFeng Zhang ◽  
YangQuan Chen
Keyword(s):  
Laplace Transform ◽  
Periodic Solutions ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Linear Time ◽  
Periodic Systems ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Order Linear ◽  
Existence Of Periodic Solutions

Existence of periodic solutions of fractional order dynamic systems is an important and difficult issue in fractional order systems field. In this paper, the non existence of completely periodic solutions and existence of partly periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic systems are discussed. A new property of Laplace transform of periodic function is derived. The non existences of completely periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic fractional order systems are presented by Laplace transform method and contradiction approach. The existence of partly periodic solutions of fractional order dynamic systems are proved by constructing numerical examples and considering Laplace transform property approaches. The examples and state figures are given to illustrate the effectiveness of conclusion presented.

A Lyapunov Function Approach to Energy Based Model Reduction

2001 ◽  
Author(s):  
Samuel Y. Chang ◽  
Christopher R. Carlson ◽  
J. Christian Gerdes
Keyword(s):  
Lyapunov Function ◽  
Model Reduction ◽  
Lyapunov Functions ◽  
Function Approach ◽  
Dynamic Equations ◽  
Unmodeled Dynamics ◽  
Time Varying ◽  
System Energy ◽  
Low Levels ◽  
The Cost

Abstract Model reduction based upon the idea of eliminating coordinates with low levels of associated power, energy or activity has been proposed by a number of researchers. None of these results, however, produce the sort of computable bounds on the neglected dynamics that would be useful in the design of controllers with guaranteed robustness properties. This paper outlines an approach to model reduction based upon Lyapunov functions that represent a modified version of the system energy of Lagrangian subsystems. The Lyapunov functions are used to bound the states of subsystems to be removed, enabling these states to be treated as time-varying perturbations in a simplified set of dynamic equations. In contrast to other results in energy-based model reduction, this approach provides bounds on the disturbances caused by the unmodeled dynamics though at the cost of the implementation ease associated with other methods.

Control of Dynamic Systems With Nonlinearities and Time Varying Parameters

1993 ◽  
Author(s):  
Dirk Söffker ◽  
Peter C. Müller
Keyword(s):  
Dynamic Systems ◽  
Disturbance Rejection ◽  
Linear Time ◽  
Time Varying ◽  
Varying Parameters ◽  
Linear Time Invariant ◽  
Time Variance ◽  
Disturbance Rejection Control ◽  
Time Varying Parameters ◽  
Time Invariant

Abstract The well-known theory of disturbance rejection control and the experience of using a generalized technique with universal fault model for building observers and regulators for the estimation and compensation of disturbances and unmodeled or uncertain effects as well, could be used for controlling dynamic systems with time varying parameters and nonlinearities. Based on a linear time-invariant model the effects of non-linearities and unmodeled dynamics are estimated by an extended observer scheme. Using this information these dynamic effects will be compensated by the developed compensation scheme. Here also different compensation techniques of disturbance rejection control are discussed, compared, and modified. The simulation example of an inverted flexible pendulum shows the efficiency of the method controlling an unstable mechanical system without exact knowledge of structure and parameters of nonlinearity and time-variance.

SOME ANALYTIC CALCULATIONS OF THE CHARACTERISTIC EXPONENTS

2007 ◽  
Vol 17 (10) ◽  
pp. 3675-3678 ◽  
Author(s):  
P. VAN DER KLOET ◽  
F. L. NEERHOFF ◽  
N. H. WANING
Keyword(s):  
Dynamic Systems ◽  
Linear Time ◽  
Equilibrium Points ◽  
Analytic Solutions ◽  
Nonlinear Dynamic Systems ◽  
Time Varying ◽  
Variational Equations ◽  
Linear Time Invariant ◽  
Characteristic Exponents ◽  
Time Invariant

As is well known, the variational equations of nonlinear dynamic systems are linear time-varying (LTV) by nature. In the modal solutions for these LTV equations, the earlier introduced dynamic eigenvalues play a key role. They are closely related to the Lyapunov- and Floquet-exponents of the corresponding nonlinear systems. In this contribution, we present some simple examples for which analytic solutions exist. It is also demonstrated by example how the classical linear time-invariant (LTI) solutions are related to the equilibrium points of the general LTV solutions.

scholarly journals On Nonnegative Solutions of Fractionalq-Linear Time-Varying Dynamic Systems with Delayed Dynamics

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Dynamic Systems ◽  
Fractional Derivatives ◽  
Linear Time ◽  
Asymptotic Properties ◽  
Time Varying ◽  
Caputo Fractional Derivatives ◽  
Nonnegative Solutions ◽  
Time Varying Systems ◽  
Fractional Differential ◽  
The Stability

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based onq-calculus and Caputo fractional derivatives on any order.

scholarly journals Global Stability of Polytopic Linear Time-Varying Dynamic Systems under Time-Varying Point Delays and Impulsive Controls

2010 ◽  
Vol 2010 ◽  
pp. 1-33 ◽  
Author(s):  
M. de la Sen
Keyword(s):  
Dynamic System ◽  
Linear Systems ◽  
Dynamic Systems ◽  
Linear Time ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Impulsive Controls ◽  
The Stability ◽  
Time Invariant ◽  
Point Delays

This paper investigates the stability properties of a class of dynamic linear systems possessing several linear time-invariant parameterizations (or configurations) which conform a linear time-varying polytopic dynamic system with a finite number of time-varying time-differentiable point delays. The parameterizations may be timevarying and with bounded discontinuities and they can be subject to mixed regular plus impulsive controls within a sequence of time instants of zero measure. The polytopic parameterization for the dynamics associated with each delay is specific, so that(q+1)polytopic parameterizations are considered for a system withqdelays being also subject to delay-free dynamics. The considered general dynamic system includes, as particular cases, a wide class of switched linear systems whose individual parameterizations are timeinvariant which are governed by a switching rule. However, the dynamic system under consideration is viewed as much more general since it is time-varying with timevarying delays and the bounded discontinuous changes of active parameterizations are generated by impulsive controls in the dynamics and, at the same time, there is not a prescribed set of candidate potential parameterizations.

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