A NEW DIRECT ADAPTIVE REGULATOR WITH ROBUSTNESS ANALYSIS OF SYSTEMS IN BRUNOVSKY FORM

2010 ◽  
Vol 20 (04) ◽  
pp. 319-339 ◽  
Author(s):  
DIMITRIOS THEODORIDIS ◽  
YIANNIS BOUTALIS ◽  
MANOLIS CHRISTODOULOU
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Robustness Analysis ◽  
Modeling Error ◽  
Nonlinear Dynamical ◽  
Neuro Fuzzy ◽  
Fuzzy Dynamical System ◽  
Error Terms ◽  
System States ◽  
The Stability ◽  
Adaptive Regulator

The direct adaptive regulation of unknown nonlinear dynamical systems in Brunovsky form with modeling error effects, is considered in this paper. Since the plant is considered unknown, we propose its approximation by a special form of a Brunovsky type neuro–fuzzy dynamical system (NFDS) assuming also the existence of disturbance expressed as modeling error terms depending on both input and system states plus a not-necessarily-known constant value. The development is combined with a sensitivity analysis of the closed loop and provides a comprehensive and rigorous analysis of the stability properties. The existence and boundness of the control signal is always assured by introducing a novel method of parameter hopping and incorporating it in weight updating laws. Simulations illustrate the potency of the method and its applicability is tested on well known benchmarks, as well as in a bioreactor application. It is shown that the proposed approach is superior to the case of simple recurrent high order neural networks (HONN's).

Neuro – Fuzzy Control Schemes Based on High Order Neural Network Function Approximators

2010 ◽  
pp. 450-483 ◽  
Author(s):  
Dimitris C. Theodoridis ◽  
M. A. Christodoulou ◽  
Yiannis S. Boutalis
Keyword(s):  
Nonlinear Dynamical Systems ◽  
High Order ◽  
Nonlinear Dynamical ◽  
Network Function ◽  
Neuro Fuzzy ◽  
Control Scheme ◽  
Fuzzy Dynamical System ◽  
Novel Method ◽  
Square Systems ◽  
System States

The indirect or direct adaptive regulation of unknown nonlinear dynamical systems is considered in this chapter. Since the plant is considered unknown, we first propose its approximation by a special form of a fuzzy dynamical system (FDS) and in the sequel the fuzzy rules are approximated by appropriate high order neural networks (HONN’s). The system is regulated to zero adaptively by providing weight updating laws for the involved HONN’s, which guarantee that both the identification error and the system states reach zero exponentially fast. At the same time, all signals in the closed loop are kept bounded. The existence of the control signal is always assured by introducing a novel method of parameter hopping, which is incorporated in the weight updating laws. The indirect control scheme is developed for square systems (number of inputs equal to the number of states) as well as for systems in Brunovsky canonical form. The direct control scheme is developed for systems in square form. Simulations illustrate the potency of the method and comparisons with conventional approaches on benchmarking systems are given.

Stability Theory via Vector Lyapunov Functions

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Stability Theory ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Comparison System ◽  
Nonlinear Dynamical ◽  
Vector Lyapunov Functions ◽  
Lasalle Theorem ◽  
System States ◽  
The Stability

This chapter describes a fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions. It first introduces the notation and definitions before developing stability theorems via vector Lyapunov functions for continuous-time and discrete-time nonlinear dynamical systems. It then extends the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. It also presents a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii–LaSalle theorem. In the analysis of large-scale nonlinear interconnected dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem.

Is There a Relation Between Synchronization Stability and Bifurcation Type?

2020 ◽  
Vol 30 (08) ◽  
pp. 2050123
Author(s):  
Zahra Faghani ◽  
Zhen Wang ◽  
Fatemeh Parastesh ◽  
Sajad Jafari ◽  
Matjaž Perc
Keyword(s):  
Complex Networks ◽  
Nonlinear Dynamical Systems ◽  
General Relation ◽  
Dynamical Behavior ◽  
Stability Function ◽  
Nonlinear Dynamical ◽  
Practical Applications ◽  
Stability Functions ◽  
Stability And Bifurcation ◽  
The Stability

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

Adaptive Fuzzy Sliding Mode Control for a Class of Nonlinear Dynamical Systems

2011 ◽  
Vol 71-78 ◽  
pp. 4309-4312 ◽  
Author(s):  
Wen Da Zheng ◽  
Gang Liu ◽  
Jie Yang ◽  
Hong Qing Hou ◽  
Ming Hao Wang
Keyword(s):  
Sliding Mode Control ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Adaptive Controller ◽  
Adaptive Sliding Mode Control ◽  
Nonlinear Dynamical ◽  
Mode Control ◽  
Fuzzy Sliding Mode ◽  
The Stability ◽  
Theory Of Lyapunov Stability

This paper presents a FBFN-based (Fuzzy Basis Function Networks) adaptive sliding mode control algorithm for nonlinear dynamic systems. Firstly, we designed an perfect control law according to the nominal plant. However, there always exists discrepancy between nominal and actual mode, and the FBFN was applied to approximate the uncertainty. After that, the adaptive law was designed to update the parameters of FBFN to alleviate the approximating errors. Based on the theory of Lyapunov stability, the stability of the adaptive controller was given with a sufficient condition. Simulation example was also given to illustrate the effectiveness of the method.

Bifurcation Control of Ro¨ssler System

2003 ◽  
Author(s):  
Z. Q. Wu ◽  
P. Yu
Keyword(s):  
Feedback Control ◽  
Nonlinear Dynamical Systems ◽  
Equilibrium Points ◽  
New Method ◽  
Control Law ◽  
Equilibrium Solutions ◽  
Nonlinear Dynamical ◽  
The Stability ◽  
Parameter Values ◽  
Feasible System

In this paper, a new method is proposed for controlling bifurcations of nonlinear dynamical systems. This approach employs the idea used in deriving the transition variety sets of bifurcations with constraints to find the stability region of equilibrium points in parameter space. With this method, one can design, via a feedback control, appropriate parameter values to delay either static, or dynamic or both bifurcations as one wishes. The approach is applied to consider controlling bifurcations of the Ro¨ssler system. The uncontrolled Ro¨ssler has two equilibrium solutions, one of which exhibits static bifurcation while the other has Hopf bifurcation. When a feedback control based on the new method is applied, one can delay the bifurcations and even change the type of bifurcations. An optimal control law is obtained to stabilize the Ro¨ssler system using all feasible system parameter values. Numerical simulations are used to verify the analytical results.

scholarly journals State Observation for Lipschitz Nonlinear Dynamical Systems Based on Lyapunov Functions and Functionals

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1424 ◽  
Author(s):  
Angelo Alessandri ◽  
Patrizia Bagnerini ◽  
Roberto Cianci
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Nonlinear Dynamical Systems ◽  
Estimation Error ◽  
Stability Conditions ◽  
Nonlinear Dynamical ◽  
State Observation ◽  
State Observers ◽  
The Stability

State observers for systems having Lipschitz nonlinearities are considered for what concerns the stability of the estimation error by means of a decomposition of the dynamics of the error into the cascade of two systems. First, conditions are established in order to guarantee the asymptotic stability of the estimation error in a noise-free setting. Second, under the effect of system and measurement disturbances regarded as unknown inputs affecting the dynamics of the error, the proposed observers provide an estimation error that is input-to-state stable with respect to these disturbances. Lyapunov functions and functionals are adopted to prove such results. Third, simulations are shown to confirm the theoretical achievements and the effectiveness of the stability conditions we have established.

ON THE BIRTH OF MULTIPLE LIMIT CYCLES IN NONLINEAR SYSTEMS

1996 ◽  
Vol 06 (12b) ◽  
pp. 2587-2603 ◽  
Author(s):  
JORGE L. MOIOLA ◽  
GUANRONG CHEN
Keyword(s):  
Limit Cycles ◽  
Nonlinear Dynamical Systems ◽  
Approximation Technique ◽  
Hopf Bifurcations ◽  
Systems Methodology ◽  
Nonlinear Dynamical ◽  
Parameter Perturbations ◽  
Feedback Control Systems ◽  
Stability Indexes ◽  
The Stability

Degenerate (or singular) Hopf bifurcations of a certain type determine the appearance of multiple limit cycles under system parameter perturbations. In the study of these degenerate Hopf bifurcations, computational formulas for the stability indexes (i.e., curvature coefficients) are essential. However, such formulas are very difficult to derive, and so are usually computed by different approximation methods. Inspired by the feedback control systems methodology and the harmonic balance approximation technique, higher-order approximate formulas for such curvature coefficients are derived in this paper in the frequency domain setting. The results obtained are then applied to a study of nonlinear dynamical systems within the region of one periodic solution, bypassing a direct investigation of the multiple limit cycles and some tedious discussion of the complex multiplicity issue. Finally, we will show that several types of stability bifurcations can be controlled based on the results obtained in this paper.

INDIRECT ADAPTIVE CONTROL OF UNKNOWN MULTI VARIABLE NONLINEAR SYSTEMS WITH PARAMETRIC AND DYNAMIC UNCERTAINTIES USING A NEW NEURO-FUZZY SYSTEM DESCRIPTION

2010 ◽  
Vol 20 (02) ◽  
pp. 129-148 ◽  
Author(s):  
DIMITRIOS THEODORIDIS ◽  
YIANNIS BOUTALIS ◽  
MANOLIS CHRISTODOULOU
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
High Order ◽  
Control Signal ◽  
Parameter Uncertainties ◽  
Fuzzy Partitioning ◽  
Neuro Fuzzy ◽  
Square Systems ◽  
System States ◽  
Dynamic Uncertainties

The indirect adaptive regulation of unknown nonlinear dynamical systems with multiple inputs and states (MIMS) under the presence of dynamic and parameter uncertainties, is considered in this paper. The method is based on a new neuro-fuzzy dynamical systems description, which uses the fuzzy partitioning of an underlying fuzzy systems outputs and high order neural networks (HONN's) associated with the centers of these partitions. Every high order neural network approximates a group of fuzzy rules associated with each center. The indirect regulation is achieved by first identifying the system around the current operation point, and then using its parameters to device the control law. Weight updating laws for the involved HONN's are provided, which guarantee that, under the presence of both parameter and dynamic uncertainties, both the identification error and the system states reach zero, while keeping all signals in the closed loop bounded. The control signal is constructed to be valid for both square and non square systems by using a pseudoinverse, in Moore-Penrose sense. The existence of the control signal is always assured by employing a novel method of parameter hopping instead of the conventional projection method. The applicability is tested on well known benchmarks.

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