Reduction of time-varying linear systems to controllable and observable form†

Necessary and sufficient conditions are given for the equivalence transformation of a time-varying linear system into controllable-observable, controllable-unobservable, uncontrollable-observable, and uncontrollable-unobservable sub-systems.     

Application of generalized Zubov's method to power system stability†

The generalized form of Zubov's partial differential equation (Szego 1962) is written using the dynamical equations of a power system. Then a Lyapunov function which satisfies the partial differential equation is obtained by using a transformation of state variables. The V function obtained is used to describe the region of stability. The application of the method to the power system stability problem is illustrated by considering a synchronous generator connected to an infinite bus, Three examples, using three different models for the synchronous machine, are given.     

Circle criterion and transient stability of power systems†

An increasing familiarity of power system engineers with frequency domain techniques has motivated the search for stability criteria for power systems that could be applied in the frequency domain. Following the advent of phase plane methods for second -order systems and the second method of Lyapunov, the Popov method gained popularity. This investigation attempts to apply an extension of Popov's method to determine the transient stability of a single machine infinite bus system under a large transient disturbance. Circle criterion provides simple rules for examining both the linear and non-linear portions of the power system in the frequency domain and for arriving at conditions to be imposed on these portions in order to ensure absolute stability. An upper and lower bound is established for the non-linear gain by treating the system's mathematical model as a set of non-linear algebraic equations and limiting values of gains are estimated by solving these equations. A numerical example is solved to illustrate the method of approach.     

L2-bounded stability for a stochastic system†

Both types of linear and non-linear systems with random coefficients are treated in this paper. Four theorems are proved by using the stochastic Liapunov functional, which give the sufficient condition in matrix form for exponential asymptotic stability in the largo with probability one and L₂-bounded stability with probability one.     

Hierarchical system identification of states and parameters in interconnected power systems†

This paper presents an application of hierarchical identification procedures of the previous paper to the identification of interconnected power system states and parameters from input—output observed data. A three-area interconnected power system model is used to demonstrate the decomposition of the original system based on its particular characteristics and the implementation of hierarchical algorithms for system identification. The adaptivity of these procedures to structural changes are also illustrated. Numerical results are obtained by conducting a digital simulation of the three-area system and using the hierarchical identification and coordination algorithms to estimate the states and unknown system parameters. Computational aspects of the hierarchical system identification solutions are discussed.     

Network models for co-state equations of linear and non-linear systems†

The paper presents a topological correspondence between a network and its adjoint which, in the final result, renders it possible to proceed to the adjoint network without going through the intermediate step of deriving the co-state equations. Algorithms are developed for arriving at the topology of adjoint networks for the terminal control problem of a linear network with a linear criterion as well as with a quadratic criterion. The results are also extended to non-linear networks. The topological interpretation of adjoint networks facilitates inclusion of the results into presently available programmes on computer-aided design, which, in fact, provides the primary motivation for the work.     

Adaptive estimation and stochastic control for uncertain models†

Ordinarily, in sequential state estimation problems, it is assumed that the parameters describing the system are completely known including the statistics of the noise terms. In many physical processes, however, the parameters, and even the mathematical model, may not be completely known. Hence, another class of problems may be considered in which the system model is in doubt. This type of problem is commonly referred to as ‘ Estimation-Under-Uncertainty ’. In this paper general formulations and solutions of state estimation and control problems are presented for a large class of linear stochastic systems with uncertain models, A general structure with continuous dynamical equations and discrete observations is adopted, since many physical situations are best modelled in this way.

The problem of estimation-under-uncertainty is formulated using the assumption that the model is one of a set of finite number of candidate models. Recursive estimation and identification algorithms are presented. A feedback control strategy is also developed for uncertain systems using the results derived for estimation-under-uncertainty,     

A continuous-time reduced-order filter for estimating the state vector of a linear stochastic system†

A design procedure for a, reduced-order filter is suggested and the differential equation obeyed by the estimate error covariance matrix is obtained. The problem of optimizing the filter within the design framework is partially solved.     

Estimation of states and parameters for a discrete-time system using quasilinearization†

An attempt is made here to demonstrate the usefulness of quasilinearization technique for the estimation of states and parameters of a discrete-time system. The estimation problem is viewed as a least-squared-errors problem which is then transformed into a Two Point Boundary Value (TPBV) problem by the use of the Discrete Maximum Principle. The TPBV problem is then solved by the quasilinearization technique.     

Asymptotic stability and boundedness of non-linear difference equations†

The problem of asymptotic stability and boundedness of non-linear difference equations is considered, In this paper we shall show some results which guarantee asymptotic stability for the trivial solution. Wo wish to present results which provide explicit regions of asymptotic stability. The results replace these words, 1 sufficiently small', by explicit estimates which are simple.

The concept of asymptotic stability to difference equations can be quite useful in the design of system response to arbitrary initial conditions.     

A criterion for the bounded-input bounded-output stability of a discrete-data system with a slope-restricted non-linearity†

This paper presents and proves a criterion for the bounded-input bounded-output (BIBO) stability of a class of discrete-data systems with a slope-restricted non-linearity. The proof is based on the principle of contraction mapping. A numerical example is given to illustrate the application of the criterion, and to discuss its implication in relation to some existing results.     

Non-linear semi-group theory and stability for infinite dimensional systems†

Non-linear semi-group theory and finite time stability are investigated for a class of non-linear partial differential equations of the parabolic type. A correlation between the finite time stability and stability with respect to an equivalent norm has been established for an unperturbed motion on a Banach space.     

A transformation for stability analysis of linear delay systems†

Sufficient conditions for stability and asymptotic stability on a half-open time interval are derived for linear time varying dynamic systems with delay. The derivation utilizes a linear, finite dimensional, time dependent transformation. Three examples show how the new results can bo used to determine stability properties of the original system from those of the transformed system.     

Domain of stability for a class of non-linear partial differential equations†

The domain of stability in the weak topology of a parabolic partial differential equation with negative feedbacks in two independent variables is determined explicitly in terms of the Weierstrass p-function. Locations of singularities are found for the equianharmonic case.     

Construction of regions of stability for non-linear systems containing time delays†

A method has been developed for constructing regions of stability for non-linear systems containing time delays. This method, based on modified Liapunov stability theorems, is applicable to higher-dimensional systems containing several time delays or time-varying delays. A reactor example is presented to illustrate the method.     

Global stability and performance of a simplified adaptive algorithm†

Most self-tuning and adaptive control algorithms usually use reference models, controllers, or identifiers of about the same order as the controlled plant. Since the dimension of the plants in the real world may be very large or unknown, implementation of adaptive control procedures may be difficult, or sometimes impossible. In this paper we prove global stability for a simple adaptive algorithm that can use low-order model reference and controllers, since no observers or identifiers are used in the adaptation process. The algorithm is basically fitted for systems that are denominated as ‘almost positive real’. It is shown that, at the price of bounded rather than vanishing output tracking errors, the simple algorithm can be applied in systems that can be stabilized via constant output feedback. These procedures are believed to reduce considerably the effort required for implementation of adaptive control in practical applications, especially in multivariable large-scale systems.     

A class of bootstrap estimators for linear system identification†

The paper deals with the identification of a process modelled by a stable, linear difference equation of known order, subject to additive output measurement noise, which is equally and independently distributed. It is shown that in the idealized situation, where the noise-free process output is assumed known, there is a set of three estimators which provide consistent estimates of the process parameters. Practical approximations to these estimators lead to on-line algorithms, in which the process parameters as well as the noise-free process outputs are estimated recursively in real time. These practical estimators are tested on simulated data obtained from time-invariant and slowly time-varying models. The performance of these estimators is compared with each other and also with some of the existing ‘bootstrap type’ estimators.     

A second-order method for state estimation of non-linear dynamical systems†

The paper is concerned with further elaboration of the concept of statistical approximation of a given nonlinear function and its application to the problem of state estimation of a non-linear noisy dynamical system from noise-corrupted observations. It is shown that under the assumption that the conditional probability density of the state variable is gaussian, it is possible to approximate a non-linear function of the state by a polynomial of arbitrary order. Using the second-order approximation, an algorithm is then developed for the problem of state estimation. Results obtained from this algorithm are compared with those obtained from a second-order algorithm based on Taylor's series expansion of the non-linear function.     

Euler-Lagrange conditions and estimation of states for a discrete-time linear system with time-delay†

In this paper, the Euler-Lagrange conditions for the extremization of a performance functional for a discrete time linear system with delay fire derived and this is then used for the estimation of states for the same system. The estimation problem is represented as a Two Point Boundary Value (TPBV) problem which is then solved numerically by using the steepest descent technique.     

Transient stability of a power system using a non-Luré-type Lyapunov function

A new viewpoint is offered on the construction of a non-Luré-type Lyapunov function for studying the dynamic behaviour of linear systems with a non-linear feedback of the ‘classical’ sector type. The approach makes it possible to show that a particular relation between the parameters defining the Lyapunov function must be satisfied in order to ensure an estimate of the attraction domain closest to the true stability boundary.