confident and consistent partial learning of recursive functions abstract partial learning is a criterion where the learner infinitely often outputs one correct conjecture while every other hypothesis is issued only finitely often this paper addresses two variants of partial learning in the setting of inductive inference of functions first confident partial learning requires that the learner also on those functions which it does not learn singles out exactly one hypothesis which is output infinitely often second essentially class consistent partial learning is partial learning with the additional constraint that on the functions to be learnt almost all hypotheses issued are consistent with all the data seen so far the results of the present work are that confident partial learning is more general than explanatory learning incomparable with behaviourally correct learning and closed under union essentially class consistent partial learning is more general than behaviourally correct learning and incomparable with confident partial learning furthermore it is investigated which oracles permit to learn all recursive functions under these criteria for confident partial learning some non high oracles are omniscient for essentially class consistent partial learning all pa complete and all oracles of hyperimmune turing degree are omniscient

rectifiable oscillations of self adjoint and damped linear differential equations of second order abstract asymptotic and oscillatory behaviours near x 0 of all solutions y y x of self adjoint linear differential equation p p q p y q y 0 on 0 t will be studied where p p x and q q x satisfy the so called hartman wintner type condition we show that the oscillatory behaviour near x 0 of p p q is characterised by the nonintegrability of q p on 0 t moreover under this condition we show that the rectifiable resp unrectifiable oscillations near x 0 of p p q are characterised by the integrability resp nonintegrability of q p 3 4 on 0 t next some invariant properties of rectifiable oscillations in respect to the liouville transformation are proved also sturm s comparison type theorem for the rectifiable oscillations is stated furthermore previous results are used to establish such kind of oscillations for damped linear second order differential equation y g x y f x y 0 and especially the bessel type damped linear differential equations are considered finally some open questions are posed for the further study on this subject

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

controllability of generalized linear time invariant systems this note deals with the problem of c controllability of generalized systems a new criterion for determining c controllability of generalized systems is established with controllability established it is shown that it is always possible to transfer the system from the initial position to any target by a control vector valued function whose elements are polynomials of bounded degree the polynomial coefficients are obtained as a solution to a set of linear equations

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

decoupling by restricted static state feedback the general case in this paper we tackle the general block decoupling problem for linear constant dynamical systems c a b with m inputs and p outputs with restricted static state feedback in other words with control laws of type u gfx gv we give a necessary and sufficient condition of existence for such laws which generalizes the one previously given in 2 for the simple case k p leq m where k denotes the number of blocks to be decoupled

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

algorithms to verify generic causality and controllability of descriptor systems graph theoretic algorithms are developed for determining generic causality and controllability of structured descriptor systems sds s for the case of nonsingular e a completely graph theoretic characterization of generic controllability of sds s is developed the case of singular e is partially resolved with graph theoretic characterization of necessary conditions and sufficient conditions for generic causality and controllability developed separately although not stated explicitly these results can be easily translated into a graph theoretic algorithm to test for generic observability of sds s

decoupling of square singular systems via proportional state feedback necessary and sufficient conditions for the decoupling of a solvable square singular system ex t ax t bu t with output y t dx t through an admissible control law of the form u t kx t hr t where h is a square nonsingular matrix it has been shown that for a given singular system that satisfies these conditions a propagational state feedback exists for which the system s transfer function is a diagonal nonsingular and proper rational matrix the proofs of the main results are constructive and provide a procedure for computing an appropriate proportional state feedback

eigenvalue placement for generalized linear systems this paper deals with some aspects of eigenvalue placement by state feedback for generalized linear systems described by e ax bu where e is a singular map it is shown that controllability of the infinite eigenvalues of the pencil se a is equivalent to the existence of a state feedback map which assigns those eigenvalues to pre specified complex numbers a procedure for the assignment of all eigenvalues to distinct complex numbers is also discussed

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

a geometric approach to eigenstructure assignment for singular systems moore s well known result on the feedback assignment of eigenstructure in linear state space systems 5 was generalized to descriptor systems in 9 in this note we make some useful extensions to fill out the theory to begin with we do not assume controllability of the plant it is shown that the maximum number of finite closed loop eigenvalues is equal to dim el where l is the supremal a e r b invariant subspace in order to relate the possible closed loop structure to the open loop plant structure we define a set of indexes for descriptor systems which are in many respects similar to the controllability indexes of state space systems in terms of these controllability indexes we give a systematic approach to selecting the closed loop eigenvectors a major contribution of this note is the fact that it is geometric in nature and hence avoids any decomposition of the system matrices into a special form e g weierstrass form 1 the geometric setting introduced here should constitute a basis for further research in generalizing the well known results in geometric system theory disturbance decoupling input output decoupling etc to singular systems

decoupling in the design and synthesis of singular systems abstract necessary and sufficient conditions for the decoupling of singular systems using p d proportional derivative feedback are determined given a system which satisfies these conditions the class of all feedback matrices which decouple the system is characterized this class is used to determine the number of closed loop pole zeros which can be specified for the decoupled system and to develop a synthesis technique for the realization of desired closed loop pole zero configurations

exact pole assignment with regularity by output feedback in descriptor systems part 2 in this paper the first of two we examine the problem of modal control of linear time invariant multivariable descriptor systems in particular for systems satisfying a certain inequality we will give necessary and sufficient conditions for the existence of an output feedback matrix possibly complex which exactly assigns an arbitrary set of r the number of independent semi state variables of the system distinct complex numbers to a generalized eigenspectrum of the closed loop system while ensuring that the resulting system has a unique smooth solution for each admissible initial condition furthermore it is shown that if the set of proposed poles contains only real elements the assignment can be carried out using real feedback next we present conditions that are necessary and sufficient for pole assignability by real feedback of a self conjugate set of complex numbers provided that the system satisfies a further inequality

an approach for pole assignment in singular systems pole assignment in a singular system edx dt ax bu is discussed it is shown that the problem of assigning the roots of det se a bf by applying a proportional feedback u fx r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system an immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems the approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system

output feedback decoupling of generalized state space systems abstract the input output decoupling problem of generalized state space systems via pure proportional output feedback is studied here for the first time wherein the following four major issues are resolved the necessary and sufficient conditions for the problem to have a solution the general analytical expressions of the decoupling controller matrices the general analytical expressions of the diagonal elements of the decoupled closed loop system and the structural properties of the closed loop system