On Some Relations between Accretive, Positive, and Pseudocontractive Operators and Passivity Results in Hilbert Spaces and Nonlinear Dynamic Systems

This paper investigates some parallel relations between the operators (I-G) and G in Hilbert spaces in such a way that the pseudocontractivity, asymptotic pseudocontractivity, and asymptotic pseudocontractivity in the intermediate sense of one of them are equivalent to the accretivity, asymptotic accretivity, and asymptotic accretivity in the intermediate sense of the other operator. If the operators are self-adjoint then the obtained accretivity-type properties are also passivity-type properties. Such properties are very relevant in stability theory since they refer to global stability properties of passive feed-forward, in general, nonlinear, and time-varying controlled systems controlled via feedback by elements in a very general class of passive, in general, nonlinear, and time-varying controllers. These results allow the direct generalization of passivity results in controlled dynamic systems to wide classes of tandems of controlled systems and their controllers, described by G-operators, and their parallel interpretations as pseudocontractive properties of their counterpart (I-G)-operators. Some of the obtained results are also directly related to input-passivity, output-passivity, and hyperstability properties in controlled dynamic systems. Some illustrative examples are also given in the framework of dynamic systems described by extended square-integrable input and output signals.The author is very grateful to the Spanish Government and European Fund of Regional Development FEDER for Grant DPI2015-64766-R and to UPV/EHU for Grant PGC 17/33

Aspects and Some Results on Passivity and Positivity of Dynamic Systems

This paper is devoted to discuss certain aspects of passivity results in dynamic systems and the characterization of the regenerative systems counterparts. In particular, the various concepts of passivity as standard passivity, strict input passivity, strict output passivity and very strict passivity (i.e. joint strict input and output passivity) are given and related to the existence of a storage function and a dissipation function. Later on, the obtained results are related to external positivity of systems and positivity or strict positivity of the transfer matrices and transfer functions in the time invariant case. On the other hand, it is discussed how to achieve or how eventually to increase the passivity effects via linear feedback by the synthesis of the appropriate feed-forward or feedback controllers or, simply, by adding a positive parallel direct input-output matrix interconnection gain.This research is supported by the Spanish Government and by the European Fund of Regional Development FEDER through Grant DPI2015-64766-R and by UPV/EHU by Grant PGC 17/33

Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop

This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real transfer function in parallel with a non-linear and memory-free device. The properties of positivity and boundedness of the input–output energy are examined based on the “ad hoc” use of the Rayleigh energy theorem on the truncated relevant signals for finite time intervals. The cases of minimal and non-minimal state-space realizations of the linear part are characterized from a global asymptotic stability (asymptotic hyperstability) point of view. Some related extended results are obtained for the case when the linear part is both positive real and externally positive and for the case of incorporation of other linear components which are stable but not necessarily positive real.This research was funded by the Spanish Government and the European Commission, grant number RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and by the Basque Government, grant number IT1207-19. The APC was funded by grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE)

On the Characterization of Hankel and Toeplitz Operators Describing Switched Linear Dynamic Systems with Point Delays

This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressions are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.Ministerio de Educación (DPI2006-00714

On the External Positivity of SISO Linear Dynamic Systems under a Class of Nonzero and Possibly Negative Initial Conditions Eventually Subject to Incommensurate Point Internal and External Delays

The property of external positivity of dynamic systems is commonly defined as the non-negativity of the output for all time under zero initial conditions and any given non-negative input for all time. This paper investigates the extension of that property for a structured class of initial conditions of a single-input single-output (SISO) linear dynamic system which can include, in general, certain negative initial conditions. The above class of initial conditions is characterized analytically based on the structure of the transfer function. The basic study is performed in the delay-free case, but extensions are then given for systems subject to a finite number of internal and external, in general incommensurate, point delays and for the closed-loop dynamic systems which incorporate a feedback compensator. The formulation relies on calculating the output based on the impulse responses by considering the relation of the mentioned sets of structured initial conditions with the zero-state response which allows to keep the non-negativity of the zero-input response and that of the total response provided the non-negativity for all time of the zero-state response.This research was funded by Spanish Government and European Commission, grant number RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and by the Basque Government, grant numberIT1207-19-The APC was funded by Spanish Government and European Commission grant number RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE)

Total Stability Properties Based on Fixed Point Theory for a Class of Hybrid Dynamic Systems

Es reproducción del documento publicado en http://dx.doi.org/10.1155/2009/826438Robust stability results for nominally linear hybrid systems are obtained from total stability theorems for purely continuous-time and discrete-time systems by using the powerful tool of fixed point theory. The class of hybrid systems dealt consists, in general, of coupled continuous-time and digital systems subject to state perturbations whose nominal (i.e., unperturbed) parts are linear and, in general, time-varying. The obtained sufficient conditions on robust stability under a wide class of harmless perturbations are dependent on the values of the parameters defining the over-bounding functions of those perturbations. The weakness of the coupling dynamics in terms of norm among the analog and digital substates of the whole dynamic system guarantees the total stability provided that the corresponding uncoupled nominal subsystems are both exponentially stable. Fixed point stability theory is used for the proofs of stability. Ageneralization of that result is given for the case that sampling is not uniform. The boundedness of the state-trajectory solution at sampling instants guarantees the global boundedness of the solutions for all time. The existence of a fixed point for the sampled state-trajectory solution at sampling instants guarantees the existence of a fixed point of an extended auxiliary discrete system and the existence of a global asymptotic attractor of the solutions which is either a fixed point or a limit n globally stable asymptotic oscillation.Ministerio de Educación (Projecto DPI2006-00714); Gobierno Vasco (GIC07143-IT-269-07 y SAIOTEK S-PE08UN15

Hyperstability of Linear Feed-Forward Time-Invariant Systems Subject to Internal and External Point Delays and Impulsive Nonlinear Time-Varying Feedback Controls

This paper investigates the asymptotic hyperstability of a single-input–single-output closed-loop system whose controlled plant is time-invariant and possesses a strongly strictly positive real transfer function that is subject to internal and external point delays. There are, in general, two controls involved, namely, the internal one that stabilizes the system with linear state feedback independent of the delay sizes and the external one that belongs to an hyperstable class and satisfies a Popov’s-type time integral inequality. Such a class of hyperstable controllers under consideration combines, in general, a regular impulse-free part with an impulsive part.Basque Government, Grant IT1155-22

Some Fixed Point Properties of Self-Maps Constructed by Switched Sets of Primary Self-Maps on Normed Linear Spaces

This paper is devoted to the investigation of the existence of fixed points in a normed linear space X endowed with a norm ‖⋅‖ for self-maps f from T×X to X which are constructed from a given class of so-called primary self- maps being also from T×X to X. The construction of the self-maps of interest is performed via a so-called switching rule which is a piecewise-constant map from a set T to some finite subset of the positive integers or a sequence map which domain in some discrete subset of TMinisterio de Educación (DPI2009-07197)y Gobierno Vasco (GIC07143-IT-269-07

Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

Es reproducción del documento publicado en http://dx.doi.org/10.1155/2009/314581A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.Ministerio de Educación (DPI2006-00714; Gobierno Vasco (GIC07143-IT-269-07 y SAIOTEK S-PE08UN15