Relative Stability Via the Direct Method of Lyapunov

1964 ◽  
Vol 86 (1) ◽  
pp. 87-90 ◽  
Author(s):  
W. G. Vogt
Keyword(s):  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Control Theory ◽  
Linear Approximation ◽  
Relative Stability ◽  
Direct Method ◽  
Positive Definite ◽  
Error Criterion ◽  
Definition Of ◽  
Classical Control

A mathematically rigorous concept of relative stability based on the v-functions of the direct method of Lyapunov is introduced. Two systems of the type representable by x˙ = f(x) are considered, where under the proper restrictions on f(x), a Lyapunov function, v(x) is uniquely determined by a positive definite error criterion r(x) and the equation v˙(x) = −r(x). The definition of the relative stability proposed, eventually leads to conditions on the linear approximation systems which are sufficient to assure the relative stability of the nonlinear systems. This leads to conditions on the eigenvalues of the linear approximation system which are necessary but not sufficient for relative stability. Additional conditions on the choice of the error criteria are needed. The present definition permits the gap between concepts of stability in classical control theory and that due to the direct method of Lyapunov to be at least partially bridged.

Complex lag synchronization of two identical chaotic complex nonlinear systems

Open Physics ◽  
2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Emad Mahmoud ◽  
Kholod Abualnaja
Keyword(s):  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Chaotic Attractors ◽  
State Variables ◽  
Lag Synchronization ◽  
Phase Errors ◽  
New Type ◽  
Complex Nonlinear Systems ◽  
Definition Of ◽  
Made In

AbstractMuch progress has been made in the research of synchronization for chaotic real or complex nonlinear systems. In this paper we introduce a new type of synchronization which can be studied only for chaotic complex nonlinear systems. This type of synchronization may be called complex lag synchronization (CLS). A definition of CLS is introduced and investigated for two identical chaotic complex nonlinear systems. Based on Lyapunov function a scheme is designed to achieve CLS of chaotic attractors of these systems. The effectiveness of the obtained results is illustrated by a simulation example. Numerical results are plotted to show state variables, modulus errors and phase errors of these chaotic attractors after synchronization to prove that CLS is achieved.

scholarly journals Stability Analysis of Fractional-Order Nonlinear Systems with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yu Wang ◽  
Tianzeng Li
Keyword(s):  
Stability Analysis ◽  
Nonlinear Systems ◽  
Fractional Order ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Delay System ◽  
Fractional Order Systems ◽  
Lyapunov Direct Method ◽  
Definition Of ◽  
Fractional Order Nonlinear System

Stability analysis of fractional-order nonlinear systems with delay is studied. We propose the definition of Mittag-Leffler stability of time-delay system and introduce the fractional Lyapunov direct method by using properties of Mittag-Leffler function and Laplace transform. Then some new sufficient conditions ensuring asymptotical stability of fractional-order nonlinear system with delay are proposed firstly. And the application of Riemann-Liouville fractional-order systems is extended by the fractional comparison principle and the Caputo fractional-order systems. Numerical simulations of an example demonstrate the universality and the effectiveness of the proposed method.

Adaptive control based on Barrier Lyapunov function for a class of full-state constrained stochastic nonlinear systems with dead-zone and unmodeled dynamics

2021 ◽  
pp. 014233122098563
Author(s):  
Fei Shen ◽  
Xinjun Wang ◽  
Xinghui Yin
Keyword(s):  
Adaptive Control ◽  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Dead Zone ◽  
Small Neighborhood ◽  
Unmodeled Dynamics ◽  
Stochastic Nonlinear Systems ◽  
Dynamic Surface ◽  
Barrier Lyapunov Function ◽  
Full State

This paper investigates the problem of adaptive control based on Barrier Lyapunov function for a class of full-state constrained stochastic nonlinear systems with dead-zone and unmodeled dynamics. To stabilize such a system, a dynamic signal is introduced to dominate unmodeled dynamics and an assistant signal is constructed to compensate for the effect of the dead zone. Dynamic surface control is used to solve the “complexity explosion” problem in traditional backstepping design. Two cases of symmetric and asymmetric Barrier Lyapunov functions are discussed respectively in this paper. The proposed Barrier Lyapunov function based on backstepping method can ensure that the output tracking error converges in the small neighborhood of the origin. This control scheme can ensure that semi-globally uniformly ultimately boundedness of all signals in the closed-loop system. Two simulation cases are proposed to verify the effectiveness of the theoretical method.

scholarly journals Local modal participation analysis of nonlinear systems using Poincaré linearization

2019 ◽  
Vol 99 (1) ◽  
pp. 803-811 ◽  
Author(s):  
Boumediene Hamzi ◽  
Eyad H. Abed
Keyword(s):  
Nonlinear Systems ◽  
Autonomous Systems ◽  
Analytical Formula ◽  
Linear Case ◽  
Local Mode ◽  
Initial Condition ◽  
Initial State ◽  
State Participation ◽  
Symmetry Assumption ◽  
Definition Of

AbstractThe paper studies an extension to nonlinear systems of a recently proposed approach to the definition of modal participation factors. A definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. While the definition is general, the resulting measures depend on the assumed uncertainty law governing the system initial condition, as in the linear case. The work follows Hashlamoun et al. (IEEE Trans Autom Control 54(7):1439–1449 2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure over an uncertain set of system initial state. Poincaré linearization is used to replace the nonlinear system with a locally equivalent linear system. It is found that under a symmetry assumption on the distribution of the initial state, the tractable calculation and analytical formula for mode-in-state participation factors found for the linear case persists to the nonlinear setting. This paper is dedicated to the memory of Professor Ali H. Nayfeh.

Control Theory and Decision Making in Organisations A Reconnaissance

1970 ◽  
Vol 3 (3) ◽  
pp. T46-T48 ◽  
Author(s):  
G. L. Mallen
Keyword(s):  
Control Theory ◽  
Control Systems ◽  
Decision Process ◽  
Transfer Functions ◽  
Social Systems ◽  
Simulation Methods ◽  
Industrial Dynamics ◽  
Simulation Approach ◽  
Decision Systems ◽  
Classical Control

Differences between the domains of application of classical control theory and applied cybernetics are examined. It is suggested that a unifying concept for the understanding of both simple mechanical control systems and complex social systems is that of the decision process. Simple decision systems are equated to those for which transfer functions can be specified. Complex systems demand a simulation approach. No prescriptive organisational control theory based on simulation methods yet exists but one is required and is seen to be emerging from such diverse fields as artificial intelligence and Industrial Dynamics.

Robust stability and memory state feedback control for a class of switched nonlinear systems with multiple time-varying delays

2021 ◽  
pp. 107754632098598
Author(s):  
Marwen Kermani ◽  
Anis Sakly
Keyword(s):  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Feedback Stabilization ◽  
State Feedback Control ◽  
Multiple Time ◽  
Time Varying ◽  
Switched Nonlinear Systems ◽  
Common Lyapunov Function ◽  
Suggested Approach ◽  
Aggregation Techniques

This study is concerned with the stability analysis and the feedback stabilization problems for a class of uncertain switched nonlinear systems with multiple time-varying delays. Unusually, more general time delays, which depend on the subsystem number, are considered. In this regard, by constructing a novel common Lyapunov function, using the aggregation techniques and the Borne and Gentina criterion, new algebraic stability and feedback stabilization conditions under arbitrary switching are derived. The proposed results are explicit and obtained without searching a common Lyapunov function through the linear matrix inequalities approach, considered a difficult matter in this case. At last, two numerical simulation examples are shown to prove the practical utility of the suggested approach.

scholarly journals Low Model Analysis and Synchronous Simulation of the Wave Mechanics

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wenyuan Duan ◽  
Heyuan Wang ◽  
Meng Kan
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Chaotic System ◽  
Invariant Set ◽  
Positive Definite ◽  
Chaotic Attractors ◽  
Wave Mechanics ◽  
Wave Dynamics ◽  
Simulation Results ◽  
Positive Invariant Set

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

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