Igor B. Yadykin

On properties of gramians of continuous control systems

A new approach to the study of controllability and observabihty by means of analysis of the controhabihty and observabihty matrices and gramians was proposed. It rehes on the lemma of expansion of the matrix integrals proved by expanding the resolvents of the matrices Am and A and the properties of convolution of the matrix function in the complex domain. Analysis of the structural properties of gramians demonstrated that the coefficients of the characteristic equations and the Faddeev matrices in the decomposition of the resolvent of the system dynamics matrix play the leading part in their formation. An example illustrating the properties of gramians was given.

Gramians Method of Steady-State Stability Analysis for Large Electrical Power Systems

Abstract This paper provides a new approach to computing infinite and finite time Gramians and cross-Gramians relying on the usage of the Laplacian transformation of matrix exponential time functions and expansion of the product of these functions. The expansions are bilinear and quadratic forms of sequences of the Faddeev matrices generated by resolvents of original matrices. Matrix identities are obtained for bilinear and quadratic forms of these Faddeev sequences. Asymptotic expansions of controllability and observability Gramians of dynamic systems at the proximity of stability limit are found. Illustrative examples are given to demonstrate the perspective of using Gramians for the small-signal stability analysis of the electric power systems.

On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Consideration was given to the methods for solution of the differential and algebraic Lyapunov and Sylvester equations in the time and frequency domains. Their solutions are represented as various finite and infinite grammians. The proposed approach to calculation of the grammians lies in expanding them as the sums of the matrix bilinear or quadratic forms generated with the use of the Faddeev matrices and representing each the solution of the linear matrix algebraic equation corresponding to an individual matrix eigenvalue. A lemma was proved representing explicitly the finite and infinite grammians as the matrix exponents depending on the combined spectrum of the original matrices. This result is generalized to the cases where the spectrum of one matrix contains an eigenvalue of the multiplicity two. Examples illustrating calculation of the finite and infinite grammians were discussed.

On Expansion of Gramians and Cross-Gramians of Continuous Control Systems over Spectra of System Dynamics Matrices

Abstract The paper suggests a new approach to computing infinite and finite time Gramians and Cross-Gramians relying on the use of the Laplace transformation to matrix exponential time functions and expansion of the product of these functions. The expansions are bilinear and quadratic forms for sequences of Faddeevs matrices generated by resolvents of original matrices. Matrix identities are obtained for bilinear and quadratic forms of these Faddeevs sequences. Asymptotic expansions are found for expansion of controllability and observability Gramians of dynamic systems under control modes at time preceding the system arriving to the stability boundary. An illustrative example demonstrates a technique of computing finite time controllability Gramians.

On Gramians Properties of Continuous Control Systems

Abstract A new approach to the controllability and observability properties, based on controllability and observability matrices links, is considered. The approach is based on matrices A, A m integrals expansion Lemma and properties of the matrices function convolution in complex plane. Coefficients of characteristic equations and Faddeevs matrices in the resolvent expansion of the system dynamic matrix play a dominant role in matrices integrals expansion forming. The gramians expansion on dynamic system matrix spectrum provide a new notion for the solutions of the Lyapunov and Sylvester equations. The results are illustrated by the example for two-zones furnace control.

Characterization of power systems near their stability boundary by Lyapunov direct method

Abstract We propose a new method for the small-signal stability analysis of power systems based on the spectral decomposition of a square H 2 norm of the transfer function. Compared with the dynamics of H 2 and H ∞ norms of the transfer functions, the analysis of the behavior of individual eigen-components allows the earlier identification of the pre-fault condition occurrence. Since each eigen-component is associated with a particular eigenvector, the potential sources of instability can easily be localized and tracked in real time. An important class of systems operating under the pre-fault conditions near the boundary of stability is considered. We demonstrate that in such cases several ill-stable modes can increase the system energy up to a critical level much earlier due to their synergetic effect. In particular, an ill-stable low-frequency mode can act as a catalyst increasing the energy in the system. An illustrative test for the stability analysis of a real small power grid at Russky Island is provided.

Intelligent control of power generation states

Abstract The topicality of the problem of power generating facility identification is justified against the background of European and Russian power grids interconnection. Intelligent control techniques for power generation states is presented. A technology for estimating the dynamics of power grids generating facilities participation in overall primary frequency regulation in contingencies is developed based on frequency and generating capacity time series. Process identification algorithms based on virtual models design using process data archives and knowledge bases are discussed. Associative search methods are used for identification algorithms development.

Multi-agent Approach to Design of Multimodal Intelligent Immune System for Smart Grid

Abstract The design of intelligent control system for active adaptive smart grid is considered. To avoid the loss of stability of electric energy systems, and to detect inner and outer threats, the multi-agent based concept of design of universal intelligent immune system is suggested.

Optimal tuning of PID controllers for MIMO bilinear plants

An optimal method is proposed for tuning PID controllers for bilinear MIMO plants in the state-space description when the input signals are piece-wise functions of time. The design method for the parameters of the controller is based on optimizing the proximity of the transient responses in the open-loop system and its implicit reference model. The parameters of the bilinear MIMO plant are estimated via parametric identification algorithms and then used in the tuning algorithms. Optimal tuning algorithms are designed for continuous-time PID controllers for bilinear systems. A numerical example of tuning the parameters of a two-channel controller for a bilinear plant is presented. The synthesized controller is shown to demonstrate good performance over a wide range of coordinate-parametric inputs.

H2-optimal tuning algorithms for fixed structure controllers

An H2-method of optimal tuning is proposed for a fixed order controller. The SISO plant model is considered in state space. The H2-method of tuning parameter design is based on the minimization of a transient process closeness criterion for appropriate open-loop and closed-loop control systems and their reference models. The controller tuning algorithms use the plant parameter estimations obtained during the plant parameter identification. The analytical expressions are obtained for the square of H2-norm of a stable dynamic system. The following theorem is proven: the minimum necessary conditions for the functionals of transfer function H2-norm of open-loop and closed-loop systems are the same as the minimum necessary conditions for the Frobenius norm of the controller parameter tuning polynomial.