Inverse Dynamical Systems Research Articles

Mathematical model of the automatic control system in the problem of guaranteed accuracy

The formulation and solution of the mathematical problem of the synthesis of an automatic control system (ACS) of guaranteed accuracy under uncertain disturbances using the inverse dynamic system approach is considered. No restrictions are imposed on uncertain perturbations, and the perturbations themselves can be arbitrary. The procedure for forming the structure of the ACS with guaranteed accuracy, which ensures the preservation of the quality (stability and quality of the transient process) of the ACS during compensation of disturbances, and the features of this structure are shown. The proposed method of forming a corrective influence to compensate for the effects of disturbances as a function of the approximation of the ACS state variable to the limit of the permissible value. For the first time, it is shown that such a structure of ACS with guaranteed accuracy corresponds to a new mathematical structure - an algebraic differential equation (for a system with one degree of freedom) or a system of such equations (for a system with several degrees of freedom).

Recurrence properties for linear dynamical systems: An approach via invariant measures

We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector x0, for an adjoint operator T on a separable dual Banach space X, one can construct a T-invariant probability measure which contains x0 in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.

The Mean Sensitivity and Mean Equicontinuity in Uniform Spaces

Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

A note on stronger forms of sensitivity for inverse limit dynamical systems

In this paper we study stronger forms of sensitivity for inverse limit dynamical system which is induced from dynamical system on a compact metric space. We give the implication of stronger forms of sensitivity between inverse limit dynamical systems and original systems. More precisely, the inverse limit system is syndetically sensitive (resp. cofinitely sensitive, ergodically sensitive, multi-sensitive) if and only if original system is syndetically sensitive (resp. cofinitely sensitive, ergodically sensitive, multi-sensitive). Also, we prove that the inverse limit system is syndetically transitive if and only if original system is syndetically transitive.

Formula omitted]-mixing property and [formula omitted]-everywhere chaos of inverse limit dynamical systems

This paper investigates some chaotic properties via Furstenberg families generated by inverse limit dynamical systems. It is proved that the inverse limit dynamical system (lim⟵(X,f),σf) of a dynamical system (X,f) is ℱ-transitive (resp., ℱ-mixing, (ℱ1,ℱ2)-everywhere chaotic) if and only if the system (∩n=0∞fn(X),f|∩n=0∞fn(X)) is ℱ-transitive (resp., ℱ-mixing, (ℱ1,ℱ2)-everywhere chaotic), where ℱ, ℱ1 and ℱ2 are Furstenberg families.

Neural network assisted inverse dynamic guidance for terminally constrained entry flight.

This paper presents a neural network assisted entry guidance law that is designed by applying Bézier approximation. It is shown that a fully constrained approximation of a reference trajectory can be made by using the Bézier curve. Applying this approximation, an inverse dynamic system for an entry flight is solved to generate guidance command. The guidance solution thus gotten ensures terminal constraints for position, flight path, and azimuth angle. In order to ensure terminal velocity constraint, a prediction of the terminal velocity is required, based on which, the approximated Bézier curve is adjusted. An artificial neural network is used for this prediction of the terminal velocity. The method enables faster implementation in achieving fully constrained entry flight. Results from simulations indicate improved performance of the neural network assisted method. The scheme is expected to have prospect for further research on automated onboard control of terminal velocity for both reentry and terminal guidance laws.

Martelli’s Chaos in Inverse Limit Dynamical Systems and Hyperspace Dynamical Systems

In this paper we investigate Martelli’s chaos of inverse limit dynamical systems and hyperspace dynamical systems which are both induced from dynamical systems on a compact metric space. We give the implication of Martelli’s chaos among those systems. More precisely, we show that inverse limit dynamical system is Martelli’s chaos if and only if so is original system, and we prove that hyperspace dynamical system is Martelli’s chaos implies original system is Martelli’s chaos if the orbit of every single point set of original system is unstable in hyperspace dynamical system.

Inverse limits and an implicitly defined difference equation from economics

The purpose of the paper is to introduce mathematicians to a cash-in-advance model from economics. We show that tools from inverse limits and dynamical systems developed in the last forty or so years are applicable to it.

A FOLIATION FOR UNKNOWN INPUTS RECONSTRUCTION

This paper deals with a new method to compute the left inverse for a class of dynamical systems. This method is based on a projection which decompose the state space into two transverse foliations: one is tangent to the unknown inputs and the other is transverse. Contrary to the traditional methods, our method enables us to write the inverse dynamic system under consideration. Within this work we highlight our propositions by examples.

Nonlinear inverse dynamic models of gas sensing systems based on chemical sensor arrays for quantitative measurements

Gas sensing systems based on low-cost chemical sensor arrays are gaining interest for the analysis of multicomponent gas mixtures. These sensors show different problems, e.g., nonlinearities and slow time-response, which can be partially solved by digital signal processing. Our approach is based on building a nonlinear inverse dynamic system. Results for different identification techniques, including artificial neural networks and Wiener series, are compared in terms of measurement accuracy.

CMAC Variable Structure Control of Three-Axis Simulation Turn-Table

The inverse dynamic system of the system controlled is mapped by CMAC and the principle of Variable structure control is applied to the design of CMAC network robust controller in this paper. The simulation results show the controller designed to have the ability of adaptive dynamic performances variations that are produced by the undeterministic system parameters and learning error of the CMAC network.

Nonlinear inversion-based output tracking

An inversion procedure is introduced for nonlinear systems which constructs a bounded input trajectory in the preimage of a desired output trajectory. In the case of minimum phase systems, the trajectory produced agrees with that generated by Hirschorn's inverse dynamic system; however, the preimage trajectory is noncausal (rather than unstable) in the nonminimum phase case. In addition, the analysis leads to a simple geometric connection between the unstable manifold of the system zero dynamics and noncausality in the nonminimum phase case. With the addition of stabilizing feedback to the preimage trajectory, asymptotically exact output tracking is achieved. Tracking is demonstrated with a numerical example and compared to the well-known Byrnes-Isidori regulator. Rather than solving a partial differential equation to construct a regulator, the inverse is calculated using a Picard-like interaction. When preactuation is not possible, noncausal inverse trajectories can be truncated resulting in the tracking-error transients found in other control schemes.

Inverse problems of radiative heat transfer in polydispersed media

We consider statements of inverse problems of radiative and coupled heat and mass-transfer in dispersed media. We review some of the recent papers and discuss the algorithm of the method of inverse dynamical systems.

Method of inverse dynamical systems for the reconstruction of internal sources and boundary conditions in heat transfer

A method for the inversion of linear dynamical systems is described; it can be used to investigate several inverse problems in the reconstruction of boundary conditions or internal sources in linear transfer equations.

Nearly periodic dynamical systems

The notion of near periodicity is shown to be somewhere between periodicity and almost periodicity. The relationship between near periodicity and various dynamical properties is established. It is shown that the set of non near periodic points cannot be totally disconnected. Nearly periodic semidynamical systems are studied via the associated inverse limit dynamical systems

The zero module and essential inverse systems

The theory of inverse dynamical systems has been and is continuing to be a keystone in the development of the theories of multivariable feedback control systems and of coding theory for reliable cornmunication. Advances in understanding of the role of plant inverses In control system design have brought about additional insights, for example, In the general areas of decoupled design and of realistic possibilities for closed-loop dynamical performance. Surprisingly enough, almost all the existing literature on inverse systems is cast in terms of matrices. Though the weD-known module theoretic approach to systems has been in place for a decade or more, this approach has not been fully exploited to bring out the foundations of a theory for inverse systems. This paper begins to lay such a foundation by developing a definition of zero module for a system. When inverse systems exist, their "pole modules" can be shown to contain the zero module in an appropriate algebraic sense. If the containment is tight, these inverses are called essential. Existence of and constructions for essential inverses are provided.