Hans Blomberg

Polynomial operators in non-linear systems theory

The concept of generalized polynomial operators is introduced and applied to the theory of non-linear systems. Several properties similar to those previously derived for systems described e.g. by Volterra functional series are studied. The main attention is given to inverting of the operators in question. A local inverse is constructed and the region where it is valid is determined. The construction is applied to solving of certain types of non-linear differential equations and to seeking sufficient conditions for BIBO stability of the corresponding systems.

Structured system models Part 1. Controllability and observability indices

The controllability indices for multivariable linear systems are ordinarily defined by selecting linearly independent columns of the controllability matrix. The usual two selection schemes result in Hermite indices and Kronecker indices. To express the information on uncontrollable modes, ‘augmented’ indices are introduced. The observability indices are treated in an analogous way. The relationships between the indices and a polynomial matrix representation of the system are studied. The augmented observability Hermite (Kronecker) indices are found to coincide with the degrees of the diagonal entries of the polynomial matrix having suitable diagonal blocks of upper triangular (canonical row proper) form. The relationships for controllability indices are analogous. As an application the augmented indices are utilized to construct observable or controllable state-space representations from non-observable or non-controllable representations.

FOUNDATIONS OF THE POLYNOMIAL THEORY FOR LINEAR SYSTEMS

Abstract This paper briefly presents the fundamental facts of the polynomial systems theory as applied by Rosenbrock, Wolovich and others. The theory utilizes an operator algebra which is a natural generalization of the usual operational calculus based on the Laplace and Z-transforms. It is shown how this operator algebra can be used in an efficient way for solving a number of typical problems relating to the analysis and synthesis of linear time-invariant feedback systems and estimators.

Structured system models Part 3. An application to feedback compensator design

The controllability and observability indices are studied and applied to the feedback compensator design. The compensator design method uses polynomial matrices as system models. As the main result, a new algorithm is introduced for the construction of a first candidate for the feedback compensator. A new algorithm is also given for constructing a state-space model from polynomial matrix models. Such a realization is needed if there is originally only a polynomial matrix model for the system.

Some New Aspects on Descriptor form Representation of Systems

Abstract Properties of the descriptor form representation of linear time-invariant multivariable systems have been studied using polynomial system methodology. A new normal form for the input-output equivalence on descriptor form representations has been presented. This normal form makes the complicated and partly misleading controllability and observability considerations presented in the literature clear and easy to carry out.

Controllability and Observability Indices of Structured Systems; an Application to Feedback Compensator Design

Abstract The controllability indices for multivariable linear systems are defined by selecting linearly independent columns of the controllability matrix. The two selection schemes yield Hermite indices and Kronecker indices. To express the information on uncontrollable modes “augmented” indicca arc introduced. For structured systems defined to have their parameters either fixed at zero or independent and free there exist generic indices, i.e. indices which are actual indices for “almost all” free parameter values. The generic indices can be determined using graph—theoretic methods. Observability indices are defined and studied in an analogous way. The relationships between the indices and a polynomial matrix representation of the system are given and applied to improve a feedback compensator design method based on the polynomial matrix representation. New methods for removing uncontrollable and unobservable modes are also given.

A CAD PACKAGE FOR MULTIVARIABLE CONTROL SYSTEMS PRESENTED BY POLYNOMIAL MATRICES

Abstract The basic theory for analysis and synthesis of differential systems using polynomial matrix representation is considered. Concepts like controllability, observability, stability etc. are defined and methods based on elementary row operations for checking them are given. As an example of synthesis methods an algorithm for synthesis of observers is given. A similar algorithm for synthesis of feedback compensators has been presented elsewhere. Computer aided design of multivariable control systems is performed interactively using a program which enables manipulation of polynomial matrices, and a simulation program that uses polynomial matrices as models of differential systems. The program for manipulation of polynomial matrices performs row operations and other transformations, calculates determinants and their roots etc. Synthesis problems can also be considered by studying different regulator compositions, constructing estimators and feedback compensators. The simulation program communicates with the manipulation program by means of polynomial matrices stored in files. Simulation results are presented by graphic displays or by separate plotter and design decisions can be based on these results. The use of the CAD package is demonstrated by considering an observer synthesis problem.

A CAD Package for Multivariable Control Systems Presented by Polynomial Matrices

Abstract The basic theory for analysis and synthesis of differential systems using polynomial matrix representation is considered. Concepts like controllability, observability, stability etc. are defined and methods based on elementary row operations for checking them are given. As an example of synthesis methods an algorithm for synthesis of observers is given. A similar algorithm for synthesis of feedback compensators has been presented elsewhere. Computer aided design of multivariable control systems is performed interactively using a program which enables manipulation of polynomial matrices, and a simulation program that uses polynomial matrices as models of differential systems. The program for manipulation of polynomial matrices performs row operations and other transformations, calculates determinants and their roots etc. Synthesis problems can also be considered by studying different regulator compositions, constructing estimators and feedback compensators. The simulation program communicates with the manipulation program by means of polynomial matrices stored in files. Simulation results are presented by graphic displays or by separate plotter and design decisions can be based on these results. The use of the CAD package is demonstrated by considering an observer synthesis problem.

On the Feedback Control of Multivariable Linear Systems

Abstract The paper deals with the feedback control of systems governed by ordinary linear time-invariant differential equations. The problem is discussed in terms of a simple module theory for a class of linear systems. A procedure for synthetizing the feedback loop is suggested. The procedure is based on the basic theory for polynomial matrices and related facts. The use of the procedure is illustrated by means of simple examples. Several possible candidates for the feedback loop are synthetized and discussed. The sensivity of the stability of the feedback control system with respect to small variations in the various parameter values is considered.

On the On-Line Coordination Under Uncertainty of Hierarchically Controlled Dynamical Systems Containing Buffer Storages

The problem of on-line coordination of a hierarchically controlled dynamical system under uncertainty by using the interaction balance principle is discussed. For the sake of simplicity, a two-level system consisting of two subsystems interconnected by means of buffer storages is considered. The uncertainty leads to a coordination method based on a “balance in the mean” condition. It turns out that the weakness of the coupling between the subsystems is an important concept. In the case considered the buffer storages contribute to the weakness of this coupling to a noticeable degree.