Henryk Górecki

A Generalization of the Orlando Formula — Symbolic Manipulation Approach

In this paper some new results on polynomials whose roots are linear combinations (with fixed integer coefficients) of the roots of other polynomial are presented. This research is motivated by a new method for calculation of a generalized integral criterion with the integrand function being an arbitrary polynomial of the transient error.

Analytic solution of transcendental equations

Analytic solution of transcendental equations A decomposition technique of the solution of an n-th order linear differential equation into a set of solutions of 2-nd order linear differential equations is presented.

Algebraic Condition for Decomposition of Large-Scale Linear Dynamic Systems

Algebraic Condition for Decomposition of Large-Scale Linear Dynamic Systems The paper concerns the problem of decomposition of a large-scale linear dynamic system into two subsystems. An equivalent problem is to split the characteristic polynomial of the original system into two polynomials of lower degrees. Conditions are found concerning the coefficients of the original polynomial which must be fulfilled for its factorization. It is proved that knowledge of only one of the symmetric functions of those polynomials of lower degrees is sufficient for factorization of the characteristic polynomial of the original large-scale system. An algorithm for finding all the coefficients of the decomposed polynomials and a general condition of factorization are given. Examples of splitting the polynomials of the fifth and sixth degrees are discussed in detail.

Extrema Subject to Equality Constraints

The previous chapter dealt with the necessary and sufficient conditions for the existence of extrema of functions in the interiors of their domains without any additional constrains.

Elements of Variational Calculus

The oldest variational problem considered by man was the choice of the shortest path connecting two points. The obvious solution was the segment of the straight line.

Linear Quadratic Problems [1]

The linear quadratic problem plays an important role in the theory and applications of automatic control. Many theoretical results have led to simple and effective and efficient control algorithms which at first found application in the military and space control and then have been employed for to process control, power systems, biological systems and economy, to just name a few.

Elements of Multicriteria Optimization [3, 16, 17]

Practical optimization problems often require taking into account more than one criterion. They are expressed in the form of different performance indices which often cannot be reduced to one overall criterion. Moreover, the criteria often contradict one another. In such situations it makes sense to look for an optimal trade-off.

Parametric Optimization of Continuous Linear Dynamic Systems

Parametric optimization includes a range of problems in which the mathematical model of a considered system is in principle determined, and what remains to be done is to determine the optimum values of certain free parameters which are used in the equations which describe the system.

Fundamental Properties and Requirements of Control Systems

One of the fundamental properties that is required of a control system is that the state and output response to a bounded control input acting in the time interval ((t_0, infty )) be also bounded. This property is called the stability in the Laplace sense. Yet there is another definition of stability which is more often used. Namely, it is the stability in the Lyapunov sense.

Unconstrained Extrema of Functions

In this chapter we will consider a problem which has a crucial importance, and will often be a point of departure for many other problems to be considered in the next chapters, that is, the finding of a maximum or minimum (extremum, in general) of a real function.