A plain approach for center manifold reduction of nonlinear systems with external periodic excitations

2019 ◽  
Vol 26 (11-12) ◽  
pp. 929-940 ◽  
Author(s):  
Peter MB Waswa ◽  
Sangram Redkar ◽  
Susheelkumar C Subramanian
Keyword(s):  
Nonlinear Systems ◽  
Center Manifold ◽  
Order Reduction ◽  
Order System ◽  
State Behavior ◽  
Center Manifold Reduction ◽  
Parametrically Excited ◽  
First Case ◽  
External Frequency ◽  
System States

We present a relatively plain, direct and intuitive methodology that accomplishes model order reduction on the invariant center manifold of large-dimensional nonlinear systems subjected to external periodic excitations. This methodology is applicable to parametrically excited (periodic coefficients) and constant coefficient nonlinear systems. Our approach is empowered by intuitive augmentation of system states and is further liberated from typical special strategies such as the need for a ‘book-keeping’ parameter, a detuning parameter and minimal excitation. Consequently, it is relatively straightforward to implement and applicable to a broad range of nonlinear systems with external periodic excitations. Two illustrative application cases are investigated, and their subsequent results validate the accuracy of our approach. The first case considers a parametrically excited system with a single external periodic frequency excitation. A mass-spring-damper system with constant coefficients and two heterogeneous external frequency excitations is considered in the second example. In both cases, the long-term steady-state behavior analysis shows exact or cogent agreement between the original and the reduced-order system.

scholarly journals Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

2005 ◽  
Vol 41 (1-3) ◽  
pp. 237-273 ◽  
Author(s):  
S. C. SINHA ◽  
SANGRAM REDKAR ◽  
VENKATESH DESHMUKH ◽  
ERIC A. BUTCHER
Keyword(s):  
Nonlinear Systems ◽  
Order Reduction ◽  
Parametrically Excited

scholarly journals Continuous Prescribed-Time Sliding Mode Control with A Prescribed-Time ESO for Second-Order Nonlinear Systems with Mismatched Disturbance

2021 ◽  
Author(s):  
Xiaozhe Ju ◽  
Feng Wang ◽  
Yingzi Guan ◽  
Shihao Xu
Keyword(s):  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Sliding Mode ◽  
Fixed Time ◽  
Second Order ◽  
Order System ◽  
Practical Applications ◽  
Mode Control ◽  
System States ◽  
Mismatched Disturbances

Abstract This paper aims to settle the continuous prescribed-time stabilization problem of second-order nonlinear systems with mismatched disturbances. A continuous prescribed-time sliding mode control (CPTSMC) method with a prescribed-time extended state observer (PTESO) is proposed. The PTESO can precisely estimate the unknown states and disturbances, with its upper bound for the settling time (UBST) prescribed by only one parameter more tightly than existing finite-time or fixed-time ESOs. Furthermore, as a common concern for ESOs, the peaking value problem is well addressed. Then, a novel prescribed-time convergent form with little conservatism and simple tuning procedures is designed, and the internal mechanism in acquiring higher transient performance is explicitly researched. By using the estimated states and disturbances, the CPTSMC makes system states converge in a chattering-alleviated manner following the novel prescribed-time form. In addition to proving that the UBST of the whole system is tightly prescribed by only one design parameter, we show the continuity of the CPTSMC and the boundedness of all system signals, which are vital for practical applications. Ultimately, numerical simulations on the second-order system and a DC motor servo verify the efficiency of the proposed control system.

Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems

2005 ◽  
Vol 128 (4) ◽  
pp. 458-468 ◽  
Author(s):  
Venkatesh Deshmukh ◽  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Order Reduction ◽  
Second Order ◽  
Degree Of Freedom ◽  
Structural Systems ◽  
Parametrically Excited ◽  
Reduced Models ◽  
Linear And Nonlinear ◽  
Time Invariant ◽  
Time Periodic

Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.

Control Hopf Bifurcations Based on Center Manifold

2011 ◽  
Vol 204-210 ◽  
pp. 99-105
Author(s):  
Geng Jun Zhu ◽  
Yun Shi Xiao ◽  
Xiu Shan Cai
Keyword(s):  
Nonlinear Systems ◽  
Center Manifold ◽  
Order System ◽  
Hopf Bifurcations ◽  
Stabilization Problem

The stabilization problem of nonlinear systems with Hopf bifurcations is studied in this paper. The link between a feedback law and a center manifold is obtained. By choosing a feedback law that stabilizes the dynamics of a center manifold, then it can stabilize the full order system. The simulation shows the effectiveness of the method.

scholarly journals Reduction principle at work

2021 ◽  
Vol 8 (1) ◽  
pp. 46-74
Author(s):  
Christian Pötzsche ◽  
Evamaria Russ
Keyword(s):  
Center Manifold ◽  
Reduction Principle ◽  
Critical Stability ◽  
Center Manifold Reduction ◽  
Infinite Dimensional ◽  
Linearized Stability ◽  
Dichotomy Spectrum ◽  
Nonautonomous Case ◽  
Necessary And Sufficient ◽  
Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

scholarly journals Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jinbin Wang ◽  
Rui Zhang ◽  
Lifenq Ma
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Two Dimensional ◽  
Main Attention ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Physical Phenomena ◽  
Manifold Reduction ◽  
Theoretical Results

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

A new model order reduction method for the design of compensator by using moment matching algorithm

2019 ◽  
Vol 42 (3) ◽  
pp. 472-484 ◽  
Author(s):  
Arvind Kumar Prajapati ◽  
Rajendra Prasad
Keyword(s):  
Model Reduction ◽  
Lower Order ◽  
Large Scale ◽  
Order Reduction ◽  
Order System ◽  
Moment Matching ◽  
Matching Algorithm ◽  
New Model ◽  
Factor Division Algorithm ◽  
Lower Order System

The aim of this paper is the construction of a new model reduction technique for large scale stable linear dynamic systems. It is principally focused on the dominant modes and time moments retention. This reduction implicates the translation of the overall important features confined in the large scale complete order model into the lower order system, allowing the computation of approximant denominator by using generalized pole clustering method. The approximant numerator is obtained by means of the factor division algorithm. As a result, a lower order system is obtained. To demonstrate its effectiveness, to highlight some fundamental of its features, and to accomplish its accuracy, a comparative study is done. Two standard numerical examples are taken, where approximant model computed by the proposed method is compared with the reduced order models computed from the recently proposed methods as well as well-known model reduction schemes. The paper is also emphasized on the design of compensator by using moment matching algorithm with the help of the reduced model. The design of compensator is validated and illustrated with the help of a standard numerical example taken from the literature.

Hopf Bifurcation for Semilinear FDEs in General Banach Spaces

2020 ◽  
Vol 30 (09) ◽  
pp. 2050130 ◽  
Author(s):  
Shangzhi Li ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Banach Spaces ◽  
Center Manifold ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Delay Effect ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Equivariant Hopf Bifurcation ◽  
Manifold Reduction

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction.

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