Control of Nonlinear Systems in Normal Form by Complementary Lyapunov Functions

2017 ◽  
Author(s):  
Andy Zelenak ◽  
Benito Fernández ◽  
Mitch Pryor
Keyword(s):  
Lyapunov Function ◽  
Normal Form ◽  
Lyapunov Functions ◽  
General Type ◽  
Order System ◽  
Time Varying ◽  
Control Lyapunov Function ◽  
New Approach ◽  
Complementary Function ◽  
Siso Systems

If a Lyapunov function is known, a dynamic system can be stabilized. However, the search for a Lyapunov function is often challenging. This paper takes a new approach to avoid such a search; it assumes a basic Control Lyapunov Function [CLF] then seeks to numerically diminish the value of the Lyapunov function. If a singularity arises during calculations with the default CLF, a complementary function is used. The complementary function eliminates a common cause of singularities with the default CLF. While many other algorithms from the literature use switched or time-varying CLF’s, the presented method is unique in that the CLF’s do not require prior calculation and the technique applies globally. The method is proven and demonstrated for SISO systems in normal form and then demonstrated on a higher-order system of a more general type.

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.

scholarly journals A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

2013 ◽  
Vol 58 (2) ◽  
pp. 290-303 ◽  
Author(s):  
Federico Bribiesca Argomedo ◽  
Christophe Prieur ◽  
Emmanuel Witrant ◽  
Sylvain Bremond
Keyword(s):  
Lyapunov Function ◽  
Diffusion Equation ◽  
Time Varying ◽  
Control Lyapunov Function ◽  
Strict Control

A Universal Feedback Controller for Discontinuous Dynamical Systems Using Nonsmooth Control Lyapunov Functions

2014 ◽  
Vol 137 (4) ◽  
Author(s):  
Teymur Sadikhov ◽  
Wassim M. Haddad
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Feedback Controller ◽  
Control Law ◽  
Control Lyapunov Function ◽  
Stabilizing Controller ◽  
Smooth Control ◽  
Constructive Feedback

The consideration of nonsmooth Lyapunov functions for proving stability of feedback discontinuous systems is an important extension to classical stability theory since there exist nonsmooth dynamical systems whose equilibria cannot be proved to be stable using standard continuously differentiable Lyapunov function theory. For dynamical systems with continuously differentiable flows, the concept of smooth control Lyapunov functions was developed by Artstein to show the existence of a feedback stabilizing controller. A constructive feedback control law based on a universal construction of smooth control Lyapunov functions was given by Sontag. Even though a stabilizing continuous feedback controller guarantees the existence of a smooth control Lyapunov function, many systems that possess smooth control Lyapunov functions do not necessarily admit a continuous stabilizing feedback controller. However, the existence of a control Lyapunov function allows for the design of a stabilizing feedback controller that admits Filippov and Krasovskii closed-loop system solutions. In this paper, we develop a constructive feedback control law for discontinuous dynamical systems based on the existence of a nonsmooth control Lyapunov function defined in the sense of generalized Clarke gradients and set-valued Lie derivatives.

Stabilization of Nonlinear Systems by Switched Lyapunov Function

2015 ◽  
Author(s):  
Andy Zelenak ◽  
Mitch Pryor
Keyword(s):  
Lyapunov Function ◽  
Coordinated Control ◽  
Order System ◽  
General Applicability ◽  
Control Effort ◽  
New Approach ◽  
Software Packages ◽  
Switched Lyapunov Function ◽  
Software And Hardware ◽  
Robotic Application

If a Lyapunov function is known, a dynamic system can be stabilized. However, computing a Lyapunov function is often challenging. This paper takes a new approach; it assumes a basic Lyapunov-like function then seeks to numerically diminish the Lyapunov value. If the control effort would have no effect at any iteration, the Lyapunov-like function is switched in an attempt to regain control. The method is tested on four simulated systems to give some perspective on its usefulness and limitations. A highly coupled 3rd order system demonstrates the approach’s general applicability and finally the coordinated control of 7 motors for a robotic application is considered. Details on the publicly available software packages for application agnostic software and hardware environments are also presented.

Finite-time control of uncertain time-varying systems in p-normal form

2020 ◽  
Author(s):  
Kanya Rattanamongkhonkun ◽  
Radom Pongvuthithum ◽  
Chulin Likasiri
Keyword(s):  
Normal Form ◽  
Finite Time ◽  
Control Law ◽  
Time Varying ◽  
Time Control ◽  
Special Cases ◽  
Time Varying Systems ◽  
A Chain ◽  
Uncertain Time ◽  
Power Integrator

Abstract This paper addresses a finite-time regulation problem for time-varying nonlinear systems in p-normal form. This class of time-varying systems includes a well-known lower-triangular system and a chain of power integrator systems as special cases. No growth condition on time-varying uncertainties is imposed. The control law can guarantee that all closed-loop trajectories are bounded and well defined. Furthermore, all states converge to zero in finite time.

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