Linear Systems and Control: An Operator Perspective


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Multilinear functionals are generally referred to as tensors and provide a natural object of study in multi-dimensional signal and system analysis. We propose a decomposition of an arbitrary order N tensor by means of mutual orthogonal rank 1 tensors.

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In a sense, this decomposition generalizes the standard singular value decomposition of matrices to multi-variable data objects. It is shown how this decomposition can be used for approximating arbitrary tensors. The need for accurate modeling of physical phenomena often leads to large-scale dynamical systems that require long simulation times and large data storage.

For instance, one such example is provided by the discretization of partial differential equations over fine grids, which leads to large-scale systems of ordinary differential equations. In these settings, MOR seeks models of low dimension that accurately capture the input-output behavior of the large-scale system while requiring only a fraction of the large-scale simulation time and storage.

A powerful and versatile approach to MOR is provided by the Loewner framework for rational interpolation. This approach was introduced by Antoulas and Anderson in and a major advance was made by Mayo and Antoulas in It has since been successfully applied to two main categories of systems: a linear systems with multiple inputs and multiple outputs, and b linear parametric systems. It is the purpose of this talk to present the most recent extension of the Loewner framework to classes of non-linear differential-algebraic systems, namely bilinear and quadratic non-linear differential algebraic systems.

Using moment matching techniques, we compute families of parameterized reduced order models that achieve moment matching and preserve the block structure of the to-be-reduced model. In particular we adapt the theory to the particular case of the implementation of a specific control law. The result is a reduced order controller with a reduced number of blocks thatexhibit properties similar to the given control law.

Two types of Gramians are suggested, both satisfying generalized Lyapunov equations. The first is motivated by energy functionals, the second is taylored to yield an error bound for the truncated system.


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A new notion of symmetry for nonlinear systems was characterized recently. It plays an important role in linear systems theory and is expected to provide new insights to nonlinear systems. In this paper, we provide a novel framework of balanced realization for this class of systems and apply it for model order reduction preserving symmetry. We propose a 2-D model for these systems that incorporates the failure description.

Based on this model, we formulate the FDI problem in geometric language and state sufficient conditions for solvability of the problem. We also develop a FDI procedure based on an asymptotic observer of the state.


  • Arthur E. Frazho (Author of An Operator Perspective on Signals and Systems)?
  • An Operator Perspective on Signals and Systems | Arthur Frazho | Springer?
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  • These models satisfy strict algebraic conditions that can be directly fulfilled when the models are obtained by means of traditional modeling techniques by selecting the state variables in a "natural" way i. The situation is more critical when positive state-space models must be obtained by means of realization techniques from transfer functions since, in this case, the fulfillment of positivity conditions could call for the introduction of spurious dynamics and non minimal parameterizations.

    Linear and Non-Linear Systems (Solved Problems) - Part 1

    A possible alternative consists in using quasi-positive models; this paper discusses the pros and cons of these solutions. Differently from the classical techniques presented in the literature so far on this topic, which are based on the standard pole assignment algorithms and are therefore applicable only in the non-defective case, the method presented in this paper can be applied in the case of closed-loop eigenvalues with arbitrary multiplicity. We derive necessary and sufficient conditions for the losslessness and dissipativeness in terms of dissipation equality and a certain frequency domain inequality as a main result.

    In this talk I provide examples of situations where the two problems are not dual as well as a discussion of the parts of the two problems that are dual. More precisely we will overview the following subjects: - Two-level quantum system, Quantum harmonic oscillator, composite spin-spring system, Jaynes-Cummings model.

    Brussel Usevich, Konstantin Vrije Univ. Brussel Ishteva, Mariya Vrije Univ. This site is protected by copyright and trademark laws under US and International law. All rights reserved. This conference program is tentative and subject to change. Keywords: Networked Control Systems , Robust and H-Infinity Control , Linear Systems Abstract: This paper deals with robust synchronization of directed and undirected multi-agent networks with uncertain agent dynamics.

    Given a network with identical nominal dynamics, we allow uncertainty in the form of coprime factor perturbations of the transfer matrix of the agent dynamics. These perturbations are assumed to be stable and have H-infinity norm that is bounded by an a priori given desired tolerance. We derive state space equations for dynamic observer based protocols that achieve robust synchronization in the presence of such uncertainty. We show that this robust synchronization of the network by the dynamic protocol is equivalent to robust stabilization of a single linear system by all controllers from a given set of feedback controllers.

    The synchronizing protocols are expressed in terms of real symmetric solutions to algebraic Riccati equations related to the nominal agent dynamics, and contain a weighting factor depending on the eigenvalues of the graph Laplacian. We obtain an achievable interval, i. For undirected networks, the supremum of this interval is proportional to the square root of the quotient of the smallest and the largest eigenvalue of the graph Laplacian.

    Keywords: Systems on Graphs , Nonlinear Systems and Control , Robust and H-Infinity Control Abstract: In this paper, we deal with robust synchronization problems for uncertain dynamical networks of identical Lur'e systems diffusively interconnected by means of measurement outputs. In contrast to stabilization of one single Lur'e system with a passive static nonlinearity in the negative feedback loop, in our paper the feedback nonlinearities are assumed to be incrementally passive. We assume that the interconnection topologies among these Lur'e agents are undirected and connected throughout this paper.

    A distributed dynamical protocol is proposed. We establish sufficient conditions for the existence of such protocol that robustly synchronizes the Lur'e dynamical network. The protocol parameter matrices are computed in terms of the system matrices of the individual agent, but also the second smallest and the largest eigenvalues of the Laplacian matrix associated with the interconnection topology. Keywords: Large Scale Systems , Linear Systems Abstract: This note proposes a unified approach to analyse linear time-invariant consensus problems via the use of integral quadratic constraints IQCs without recourse to loop transformations, which may cloud the inherent structural properties of the multi-agent networked systems.

    The main technical hindrance to using IQCs lies in the presence of the marginally stable integral action in consensus setups. It is shown that by working with conditions defined on modified signal spaces of interests and exploiting the graph structure underlying the connections between the dynamic systems, IQC methods can be applied directly to consensus analysis. A decentralised and scalable condition for consensus is proposed in this setting, which generalises some of the existing results in the literature.

    Keywords: Systems on Graphs , Networked Control Systems , Linear Systems Abstract: The paper addresses the synchronization problem for a network of identical, linear time-invariant state-space models. The notion of synchronizability is investigated and a set of sufficient and necessary conditions relating synchronizability to the dynamical properties of the subunits are provided. The paper also extends recent results about synchronization of passive linear systems by proving that networks of linear, detectable, passive systems can be synchronized by any possibly directed connected interconnection topology.

    The theory is illustrated with several examples. Keywords: Networked Control Systems , Large Scale Systems , Systems on Graphs Abstract: This paper is concerned with state synchronization of linear agents subject to input saturation over a fixed undirected communication graph. We first derive a sufficient condition for achieving the synchronization via relative state feedback control law for any initial condition.

    Based on this analysis result, we present a linear matrix inequality LMI condition for designing the synchronizing state feedback gain.


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    The present LMI condition is scalable as long as we can calculate the eigenvalues of the Laplacian of the communication graph, and is readily solved by an existing convex programming algorithm. The proposed controller takes under consideration a functional decoupling control strategy realized using a geometric approach and the inversability property of the DC-drives with which the Robotino is equipped. For a given control structure the functional controllability is proven for motion trajectories of class C3, continuous functions with third derivative also continuous.

    Horizon, Vertical and Angular motions are considered and once the decoupling between these motions is obtained, a Model Predictive Control MPC strategy is used in combination with an inverse drive model. Simulation results using real data of Robotino are shown. Keywords: Nonlinear Systems and Control , Algebraic Systems Theory Abstract: Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, the feedback interconnected system is in the same class with a generating series that can be computed explicitly via the antipode of a certain Faa di Bruno type Hopf algebra.

    This defines the feedback product of two generating series. Existing methods to compute this antipode are based on matrix inversion and are known to be inefficient. This paper has two objectives. The first is to use some recent advances in the area to produce a completely recursive algorithm for computing the antipode of the Hopf algebra for the SISO feedback group. The second objective is to provide a Mathematica implementation of the algorithm and evaluate its performance against the existing method using matrix inversion. Extension of the results to nonlinear systems has run into many problems.

    Consequently impulsive nonlinear systems are usually described by the effect of impulses, thus giving rise to jumps in the state space. However, it is not clear a priori how these jumps are to be related to the effective impulsive inputs the causes to the system. We derive such results for bilinear systems of arbitrary order, and discuss further extensions to other classes of systems.

    The method is justifiable by taking an approach touching on non-standard analysis. It is based on first principles considering singular functions as sequences of regular functions, pretty much as they are thought in an undergraduate signal course, and leads to a definition of insensible times and functions. The latter provide the fine structure extension.

    Keywords: Nonlinear Systems and Control , Hybrid Systems , Transportation Systems Abstract: In this paper, we propose a new observer design for nonlinear systems that can be linearized using a change of coordinates and a singular time re-scaling. Our observer is a switched system and the observer error dynamics are described, after time re-scaling, by a switched linear system that is uniformly exponentially stable.

    We also give necessary and sufficient conditions for linearizability via a change of coordinates and a singular time re-scaling. Our methods are illustrated by an example coming from the ABS literature. Keywords: Optimal Control , Biological Systems , Robotics Abstract: This paper solves the optimization problem for a simplified one-dimensional worm model when the friction force depends on the direction of the motion.

    Read Linear Systems and Control: An Operator Perspective (Chapman & Hall/CRC Pure and Applied

    The motion of the worm is controlled by the actuator force f t which is assumed to be piecewise continuous and always generates the same force in the opposite directions. The paper derives the necessary condition for the force which maximizes the average velocity or minimizes the power over a unit distance. The maximum excursion of the worm body and the force are bounded.

    A simulation is given at the end of the paper. Foster , B. His answer was YES, with reservations. Foster created a family of thitherto unknown networks for the purpose and gave explicit formulas for the capacitors and inductors of those networks to realize the desired impedance. When and only when the computed component values turned out to be positive, the impedance was realizable. Keywords: Mathematical Theory of Networks and Circuits , Linear Systems Abstract: Since the data in the Foster-Ladenheim catalog are more than seventy-five 75 years old see first lecture , some of you may regard talking about it as scientific archeology.

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    But no, our forefathers, the old boys, surprisingly, knew much more about things than we thought they did. Actually, the Ladenheim catalog is a rich source of information, some on the surface, some to be made precise. A rediscovery of some tricks of classical invariant theory. Cultural obstacles between engineering and mathematics, but also sloppy teaching of mathematics to engineering students in the Boston area in the early twentieth century.

    References

    Keywords: Mathematical Theory of Networks and Circuits , Mechanical Systems Abstract: The motivation provided by mechanical network synthesis to make a fresh attack on certain questions in circuit synthesis will be briefly recalled. The classical early work on RLC synthesis, beginning with the works of Foster and Cauer and culminating in the Bott-Duffin construction, will be explained in a tutorial manner.

    The proof in T. Hughes and M. There is a close connection between the Cauchy index of a real-rational function and many classical algebraic results relating to pairs of polynomial functions [2]. Using this connection, it is possible to derive algebraic constraints on circuit impedance functions relating to the precise numbers of inductors and capacitors in that circuit.

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    Linear Systems and Control: An Operator Perspective Linear Systems and Control: An Operator Perspective
    Linear Systems and Control: An Operator Perspective Linear Systems and Control: An Operator Perspective
    Linear Systems and Control: An Operator Perspective Linear Systems and Control: An Operator Perspective
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