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Quadratic Lyapunov Function (quadratic + lyapunov_function)

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Engineering100%

Selected Abstracts

Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 10 2002
Robert N. Shorten
In this paper, necessary and sufficient conditions are derived for the existence of a common quadra-tic Lyapunov function for a finite number of stable second order linear time-invariant systems. Copyright © 2002 John Wiley & Sons, Ltd. [source]

On the existence of common quadratic Lyapunov functions for second-order linear time-invariant discrete-time systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 10 2002
Mehmet Akar
Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function (CQLF) are given for two second-order linear time-invariant discrete-time systems. These conditions are later extended to an arbitrary number of systems. The conditions are readily verifiable both analytically and graphically. The paper also provides a constructive procedure for computing a CQLF when it exists. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Some new algebraic criteria for chaos synchronization of Chua's circuits by linear state error feedback control

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, Issue 3 2006
Xiaofeng Wu
Abstract The research on the sufficient criterion for chaos synchronization of the master,slave Chua's circuits by linear state error feedback control has received much attention and some synchronization criteria for special control matrix were proposed. In this paper, the above synchronization issue is investigated in the situation of general linear state error feedback controller with propagation delay of control signals from the master Chua's circuit. First of all, a master,slave synchronization scheme for Chua's circuits with propagation delay is given and the relevant error system is derived. Using a quadratic Lyapunov function and frequency domain method, three new algebraic synchronization criteria for the synchronization scheme with general control matrix are proven. They are applied to derive the synchronization criteria for simple control matrices. Some examples are given to show the sharpness of these new criteria compared with the known criteria. Copyright © 2006 John Wiley & Sons, Ltd. [source]

LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 1 2005
G. Chesi
Abstract The problem of estimating the domain of attraction (DA) of equilibria is considered for odd polynomial systems. Specifically, the computation of the optimal quadratic Lyapunov function (OQLF), i.e. the quadratic Lyapunov function (QLF) which maximizes the volume of the largest estimate of the DA (LEDA), is addressed. In order to tackle this double non-convex optimization problem, a relaxation approach based on homogeneous polynomial forms is proposed. The first contribution of the paper shows that a lower bound of the LEDA for a fixed QLF can be obtained via linear matrix inequalities (LMIs) based procedures. Also, condition for checking tightness of the lower bound are provided. The second contribution is a strategy for selecting a good starting point for the OQLF search, which is based on the volume maximization of the region where the time derivative of the QLFs is negative and is given in terms of LMIs. Several application examples are presented to illustrate the numerical behaviour of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd. [source]

A new absolute stability test for systems with state-dependent perturbations

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 14 2002
M. C. de Oliveira
Abstract In this paper, a new test for the absolute stability of nonlinear systems with state-dependent nonlinearities is developed. Scalar nonlinearities are assumed to lie in sectors. Using a Lur'e function as a Lyapunov function, a linear matrix inequalities (LMI) stability condition is derived. The new condition lets one go from a pure integral (Persidskii) to a pure quadratic Lyapunov function in an unified framework. Several results available in the literature are generated as particular cases of the new test. An example shows that the proposed condition can be much less conservative than available diagonal stability and passivity based methods, as the circle and Popov criteria. Tests for infinite as well as finite nonlinearity sectors can be easily generated, since the parameters of the nonlinearity sectors appear in the LMI condition in a very convenient way. This feature can also provide optimization of the absolute stability sector through convex programming techniques. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Linear, parameter-varying control and its application to a turbofan engine

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 9 2002
Gary J. BalasArticle first published online: 15 JUL 200
This paper describes application of parameter-dependent control design methods to a turbofan engine. Parameter-dependent systems are linear systems, whose state-space descriptions are known functions of time-varying parameters. The time variation of each of the parameters is not known in advance, but is assumed to be measurable in real-time. Three linear, parameter-varying (LPV) approaches to control design are discussed. The first method is based on linear fractional transformations which relies on the small gain theorem for bounds on performance and robustness. The other methods make use of either a single (SQLF) or parameter-dependent (PDQLF) quadratic Lyapunov function to bound the achievable level of performance. The latter two techniques are used to synthesize controllers for a high-performance turbofan engine. A LPV model of the turbofan engine is constructed from Jacobian linearizations at fixed power codes for control design. The control problem is formulated as a model matching problem in the ,, and LPV framework. The objective is decoupled command response of the closed-loop system to pressure and rotor speed requests. The performance of linear, ,, point designs are compared with the SQLF and PDQLF controllers. Nonlinear simulations indicate that the controller synthesized using the SQLF approach is slightly more conservative than the PDQLF controller. Nonlinear simulations with the SQLF and PDQLF controllers show very robust designs that achieve all desired performance objectives. Copyright © 2002 John Wiley & Sons, Ltd. [source]

On the existence of common quadratic Lyapunov functions for second-order linear time-invariant discrete-time systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 10 2002
Mehmet Akar
Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function (CQLF) are given for two second-order linear time-invariant discrete-time systems. These conditions are later extended to an arbitrary number of systems. The conditions are readily verifiable both analytically and graphically. The paper also provides a constructive procedure for computing a CQLF when it exists. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Performance analysis of reset control systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 11 2010
W. H. T. M. Aangenent
Abstract In this paper we present a general linear matrix inequality-based analysis method to determine the performance of a SISO reset control system in both the ,2 gain and ,2 sense. In particular, we derive convex optimization problems in terms of LMIs to compute an upperbound on the ,2 gain performance and the ,2 norm, using dissipativity theory with piecewise quadratic Lyapunov functions. The results are applicable to for all LTI plants and linear-based reset controllers, thereby generalizing the available results in the literature. Furthermore, we provide simple though convincing examples to illustrate the accuracy of our proposed ,2 gain and ,2 norm calculations and show that, for an input constrained ,2 problem, reset control can outperform a linear controller designed by a common nonlinear optimization method. Copyright © 2009 John Wiley & Sons, Ltd. [source]

LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 1 2005
G. Chesi
Abstract The problem of estimating the domain of attraction (DA) of equilibria is considered for odd polynomial systems. Specifically, the computation of the optimal quadratic Lyapunov function (OQLF), i.e. the quadratic Lyapunov function (QLF) which maximizes the volume of the largest estimate of the DA (LEDA), is addressed. In order to tackle this double non-convex optimization problem, a relaxation approach based on homogeneous polynomial forms is proposed. The first contribution of the paper shows that a lower bound of the LEDA for a fixed QLF can be obtained via linear matrix inequalities (LMIs) based procedures. Also, condition for checking tightness of the lower bound are provided. The second contribution is a strategy for selecting a good starting point for the OQLF search, which is based on the volume maximization of the region where the time derivative of the QLFs is negative and is given in terms of LMIs. Several application examples are presented to illustrate the numerical behaviour of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd. [source]

A team algorithm for robust stability analysis and control design of uncertain time-varying linear systems using piecewise quadratic Lyapunov functions

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 4 2001
H. L. S. Almeida
Abstract A team algorithm based on piecewise quadratic simultaneous Lyapunov functions for robust stability analysis and control design of uncertain time-varying linear systems is introduced. The objective is to use robust stability criteria that are less conservative than the usual quadratic stability criterion. The use of piecewise quadratic Lyapunov functions leads to a non-convex optimization problem, which is decomposed into a convex subproblem in a selected subset of decision variables, and a lower-dimensional non-convex subproblem in the remaining decision variables. A team algorithm that combines genetic algorithms (GA) for the non-convex subproblem and interior-point methods for the solution of linear matrix inequalities (LMI), which form the convex subproblem, is proposed. Numerical examples are given, showing the advantages of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd. [source]