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Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms
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.
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.
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.
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
Transient response from Liapunov stability equation
Transient response from Liapunov stability equation
Liapunov function for analyzing stability of nonlinear equilibrium solutions in hydrodynamics
Liapunov stability theory generalized and applied to class of partial differential equations
Semigroups, groups and Liapunov stability of partial differential equation, obtaining operator differential equations in Hilbert spaces
Applications of group theory and Liapunov stability to partial differential equations
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.
Investigating Lyapunov stability properties of solutions of Volterra integrodifferential equations
Asymptotic stability for solving differential equations according to Liapunov
An extension of model reference adaptive control is presented that accommodates use of a time-varying reference model. Specifically, the reference model is taken to be a time-varying convex combination of two linear, time-invariant models. The design is intended to act as a way to smoothly transition between two different reference models without resorting to a scheduled switch. It also provides the ability to use an interpolated reference model when the plant is operating between design points. The time variation of the combination must satisfy some requirements to ensure stability but is otherwise user choice. Subject to these requirements, bounded tracking error behavior is demonstrated via Lyapunov stability analysis for the single-input, single-output, output feedback case. Tracking error convergence is asymptotic when time variation ceases. The proposed design is demonstrated in simulation of a numerical model.
The positive limit sets of the solutions of a contingent differential equation are shown to possess an invariance property. In this connection the 'invariance principle' in the theory of Lyapunov stability is extended to systems with unknown, bounded, time-varying parameters, and thus to a large and important class of nonautonomous systems. Asymptotic stability criteria are obtained and applied to guaranteed cost control problems.
This paper describes two attitude control laws suitable for atmospheric flight vehicles with a steady angular momentum bias in the vehicle yaw axis. This bias is assumed to be provided by an internal flywheel, and is introduced to enhance roll and pitch stiffness. The first control law is based on Lyapunov stability theory, and stability proofs are given. The second control law, which assumes that the angular momentum bias is large, is based on a classical PID control. It is shown that the large yaw-axis bias requires that the PI feedback component on the roll and pitch angle errors be cross-fed. Both control laws are applied to a vehicle simulation in the presence of disturbances for several values of yaw-axis angular momentum bias. It is seen that both control laws provide a significant improvement in attitude performance when the bias is sufficiently large, but the nonlinear control law is also able to provide improved performance for a small value of bias. This is important because the smaller bias corresponds to a smaller requirement for mass to be dedicated to the flywheel.