Class Of Second-order Dynamical Systems Research Articles

The Impact of Damping in Second-Order Dynamical Systems with Applications to Power Grid Stability

We consider a broad class of second-order dynamical systems and study the impact of damping as a system parameter on the stability, hyperbolicity, and bifurcation in such systems. We prove a monotonic effect of damping on the hyperbolicity of the equilibrium points of the corresponding first-order system. This provides a rigorous formulation and theoretical justification for the intuitive notion that damping increases stability. To establish this result, we prove a matrix perturbation result for complex symmetric matrices with positive semidefinite perturbations to their imaginary parts, which may be of independent interest. Furthermore, we establish necessary and sufficient conditions for the breakdown of hyperbolicity of the first-order system under damping variations in terms of observability of a pair of matrices relating damping, inertia, and Jacobian matrices, and propose sufficient conditions for Hopf bifurcation resulting from such hyperbolicity breakdown. The developed theory has significant applications in the stability of electric power systems, which are one of the most complex and important engineering systems. In particular, we characterize the impact of damping on the hyperbolicity of the swing equation model which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations.

Hinf Tracking for a Class of Second-Order Dynamical Systems

The problem of tracking for second-order dynamical systems subject to disturbance input is considered by using a complete parametric design approach. The goal of this problem is to design a controller such that the output of the controlled system robustly asymptotically tracks the output of the reference model, and the transfer function from the exogenous disturbance to the tracking error meets a prescribed norm upper bound constraint. Based on the complete parametric solution to a class of generalized second-order Sylvester matrix equations, complete parameterizations for all the controller gain matrices are established in terms of two set of freedom parameters. Also, based on these parametric gain matrices, the prescribed norm upper bound constraint is transformed into an equivalent constraint condition which restricts the choice of the freedom parameters. An example is utilized to show the effect of the proposed approach.

Eigenstructure assignment in a class of second-order dynamic systems

Eigenstructure assignment using the proportional-plus-derivative feedback controller in a class of second-order dynamic system is investigated. Simple, general, complete parametric expressions for both the closed-loop eigenvector matrix and the feedback gains are established based on two simple Smith form reductions. The approach utilizes directly the original system data and involves manipulations only on n-dimensional matrices. Furthermore, it reveals all the degrees of freedom which can be further utilized to achieve additional system specifications. An example shows the effect of the proposed approach.