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Results for “Lyapunov”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Lyapunov stability of smart inverters using linearized distflow approximation

Fast-acting smart inverters that utilize preset operating conditions to determine real and reactive power injection/consumption can create voltage instabilities (over-voltage, voltage oscillations and more) in an electrical distribution network if set-points are not properly configured. In this work, linear distribution power flow equations and droop-based Volt–Var and Volt–Watt control curves are used to analytically derive a stability criterion using Lyapunov analysis that includes the network operating condition. The methodology is generally applicable for control curves that can be represented as Lipschitz functions. The derived Lipschitz constants account for smart inverter hardware limitations for reactive power generation. A local policy is derived from the stability criterion that allows inverters to adapt their control curves by monitoring only local voltage, thus avoiding centralized control or information sharing with other inverters. The criterion is independent of the internal time-delays of smart inverters. Simulation results for inverters with and without the proposed stabilization technique demonstrate how smart inverters can mitigate voltage oscillations locally and mitigate real and reactive power flow disturbances at the substation under multiple scenarios. The study concludes with illustrations of how the control policy can dampen oscillations caused by solar intermittency and cyberattacks.

42 ENGINEERING↗
DOE OSTI2022

A Lyapunov-Based Generalized Dc-Side Controller Design for PV-Connected Systems

The objective of this paper is to realize a universal dc-side controller for photovoltaic (PV) systems where the control is agnostic to the downstream converter configuration. To achieve this, the downstream power converter and its controls are manipulated into an effective power control loop that is then cast into a generalized multi-loop design framework. On the dc side, a nonlinear small-signal model of the PV input is realized exclusively in terms of PV datasheet parameters (i.e., open-circuit voltage, short-circuit current, and maximum power point). Finally, a linear controller is used to modulate the dc-side PV system with the generic downstream power controller. A Lyapunov candidate is proposed to analyze the stability of the interconnected system and provide a streamlined approach for the controller design. The proposed design is validated on a 1k VA experimental setup that interfaces a PV module to the grid.

generalized PV controller↗
NASA NTRS1976

Extended Lyapunov stability criterion using a nonlinear algebraic relation with application to adaptive control

A new nonlinear stability criterion is developed by use of a class of Lyapunov functionals for model-reference adaptive systems (MRAS). Results are compared with traditional results, and a comparative design technique is used to illustrate its function in improving the transient response of an MRAS controller. For a particular system structure and class of input signals, the new stability criterion is shown to include traditional sufficiency stability conditions as a special case. An example is cited to illustrate the use of the nonlinear criterion and its definite advantages in helping improve the adaptive error transient response of a system. Analysis of results is effected by use of a linearization technique on the resulting adaptive equations.

Colburn, B. K.↗
NASA NTRS1979

Lyapunov stability and its application to systems of ordinary differential equations

An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem.

Kennedy, E. W.↗
NASA NTRS1987

Lyapunov exponents for infinite dimensional dynamical systems

Classically it was held that solutions to deterministic partial differential equations (i.e., ones with smooth coefficients and boundary data) could become random only through one mechanism, namely by the activation of more and more of the infinite number of degrees of freedom that are available to such a system. It is only recently that researchers have come to suspect that many infinite dimensional nonlinear systems may in fact possess finite dimensional chaotic attractors. Lyapunov exponents provide a tool for probing the nature of these attractors. This paper examines how these exponents might be measured for infinite dimensional systems.

Mhuiris, Nessan Mac Giolla↗
NASA NTRS1989

Numerical solution of large Lyapunov equations

A few methods are proposed for solving large Lyapunov equations that arise in control problems. The common case where the right hand side is a small rank matrix is considered. For the single input case, i.e., when the equation considered is of the form AX + XA(sup T) + bb(sup T) = 0, where b is a column vector, the existence of approximate solutions of the form X = VGV(sup T) where V is N x m and G is m x m, with m small is established. The first class of methods proposed is based on the use of numerical quadrature formulas, such as Gauss-Laguerre formulas, applied to the controllability Grammian. The second is based on a projection process of Galerkin type. Numerical experiments are presented to test the effectiveness of these methods for large problems.

Saad, Youcef↗
NASA NTRS1991

Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability

Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: (1) a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and (2) two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed.

Haddad, Wassim M.↗
NASA NTRS1992

A generalized Lyapunov theory for robust root clustering of linear state space models with real parameter uncertainty

The problem of analyzing and designing controllers for linear systems subject to real parameter uncertainty is considered. An elegant, unified theory for robust eigenvalue placement is presented for a class of D-regions defined by algebraic inequalities by extending the nominal matrix root clustering theory of Gutman and Jury (1981) to linear uncertain time systems. The author presents explicit conditions for matrix root clustering for different D-regions and establishes the relationship between the eigenvalue migration range and the parameter range. The bounds are all obtained by one-shot computation in the matrix domain and do not need any frequency sweeping or parameter gridding. The method uses the generalized Lyapunov theory for getting the bounds.

Yedavalli, R. K.↗
NASA NTRS1998

Sufficient Conditions for the Stability of a Lur'e System With Two Nonzero Inputs Based on Lyapunovs Direct Method

The Lur'e problem is a famous problem in nonlinear control theory. The analysis of this system is usually restricted to Lur'e systems in which the external input is zero. One form of this problem involves the stability analysis of the Lur'e system with two nonzero inputs. The stability analysis of this type of system is traditionally performed based on L stability theory. The objective of this paper is to present sufficient conditions for the stability of this system based on Lyapunov's direct method. This is a conceptually simpler approach to the existing analysis based on L stability theory and it provides some additional information about the stability of the system.

Adetona, Olawale↗
NASA NTRS2004

Intelligent, Robust Control of Deteriorated Turbofan Engines via Linear Parameter Varying Quadratic Lyapunov Function Design

A method for accommodating engine deterioration via a scheduled Linear Parameter Varying Quadratic Lyapunov Function (LPVQLF)-Based controller is presented. The LPVQLF design methodology provides a means for developing unconditionally stable, robust control of Linear Parameter Varying (LPV) systems. The controller is scheduled on the Engine Deterioration Index, a function of estimated parameters that relate to engine health, and is computed using a multilayer feedforward neural network. Acceptable thrust response and tight control of exhaust gas temperature (EGT) is accomplished by adjusting the performance weights on these parameters for different levels of engine degradation. Nonlinear simulations demonstrate that the controller achieves specified performance objectives while being robust to engine deterioration as well as engine-to-engine variations.

Turso, James A.↗
NASA NTRS2004

Low-thrust Orbit Transfers Using Candidate Lyapunov Functions with a Mechanism for Coasting

We consider low-thrust orbit transfers around a central body, where specified changes are sought in orbit elements except true anomaly. The desired changes in the remaining five elements can be arbitrarily large. Candidate Lyapunov functions are created based on analytic expressions for maximum rates of change of the orbit elements and the desired changes in the elements. These functions may be thought of as proximity quotients because they provide R measure of the proximity to the target orbit. The direction of thrust needed for steepest descent to the target orbit is also available analytically. The thrust is shutoff if the effectivity of the thrust at the current location on the osculating orbit is below some threshhold value. Thus, the equations of motion can be numerically integrated to obtain quickly and simply a transfer to the target orbit. A series of transfers can be easily computed to assess the trade-off between propellant mass and flight time. Preliminary comparisons to optimal solutions show that the method, while sub-optimal, performs well.

Candidate Lyapunoc functions↗
NASA NTRS2013

Lyapunov Orbits in the Jupiter System Using Electrodynamic Tethers

Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. The effect of tether forces on periodic orbits in Jupiter-satellite systems are investigated. A perturbation force is added to the restricted three-body problem model and a series of simplifications allows development of a conservative system that retains the Jacobi integral. Expressions are developed to find modified locations of equilibrium positions. Modified families of Lyapunov orbits are generated as functions of tether size and Jacobi integral. Zero velocity curves and stability analyses are used to evaluate the dynamical properties of tether-modified orbits.

Cassini↗
NASA NTRS2024

Shape Servoing of Deformable Objects Using Model Estimation and Barrier Lyapunov Function

An adaptive shape servoing control method is presented in this article to manipulate a deformable object into a desired shape in 3-D. A finite-point-based representation of the deformable object is used and the deformation Jacobian matrix is approximated using Fourier series basis functions. The unknown parameters of the deformation Jacobian are learned by using the velocity applied to a control point on the object and corresponding change of positions of the points describing the entire object. An integral concurrent learning (ICL)-based parameter update law is designed along with a constrained controller to satisfy the state constraints on the motion of the control point using Barrier Lyapunov function analysis. ICL-based parameter update law uses data history of velocity and corresponding positions of the points along with their current values. An efficient algorithm to update the history stack using singular value maximization is proposed based on the structure of the regressor matrix. Simulations using a physical simulator and experiments using a robot platform are performed to validate the performance of the proposed controller on two different deformable objects.

Vrithik Raj Guthikonda↗