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A theory of system type for linear multivariable servomechanisms is developed. New characterizations of system type are formulated and two systematic methods for determining system types are derived. Both methods are algorithmic and easier to apply than the techniques given in earlier studies. The first applies to the case where the open-loop system is described by its transfer function matrix, the second where its time domain state-space representation is given. New results are also presented for characterizing and identifying the system type for certian composite systems from a knowledge of the individual subsystem types. (Author)

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We consider the failure of localized control in a nonlinear spatially extended system caused by extremely small amounts of noise. It is shown that this failure occurs as a result of a nonlinear instability. Nonlinear instabilities can occur in systems described by linearly stable but strongly nonnormal evolution operators. In spatially extended systems the nonnormality manifests itself in two different but complementary ways: transient amplification and spectral focusing of disturbances. We show that temporal and spatial aspects of the nonnormality and the type of nonlinearity are all crucially important to understanding and describing the mechanism of nonlinear instability. Presented results are expected to apply equally to other physical systems where strong nonnormality is due to the presence of mean flow rather than the action of control.

We consider the failure of localized control in a nonlinear spatially extended system caused by extremely small amounts of noise. It is shown that this failure occurs as a result of a nonlinear instability. Nonlinear instabilities can occur in systems described by linearly stable but strongly nonnormal evolution operators. In spatially extended systems the nonnormality manifests itself in two different but complementary ways: transient amplification and spectral focusing of disturbances. We show that temporal and spatial aspects of the nonnormality and the type of nonlinearity are all crucially important to understanding and describing the mechanism of nonlinear instability. Presented results are expected to apply equally to other physical systems where strong nonnormality is due to the presence of mean flow rather than the action of control.

The presentation centers on four classes of systems that can be treated as linear systems over a ring. These are: (1) discrete-time systems over a ring of scalars such as the integers; (2) continuous-time systems containing time delays; (3) large-scale discrete-time systems; and (4) time-varying discrete-time systems.

"Invariants of a System of Linear Partial Differential Equations, and the Theory of Congruences of Rays" is an article from American Journal of Mathematics, Volume 26 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370152

Brief summaries are given of time-varying linear system theory in both the conventional frequency-shift version and the novel frequency-shift version. The latter version of the theory is utilized to obtain the matched filter output for the bistatic echo from a moving, turning line target for a linear FM pulse in a non-dispersive medium.

This report summarizes linear time-varying systems theory and its application to active sonar signal processing when certain of the parameters are random variables. To keep the analysis within reasonable bounds the random parameters are described by their first- and second-order moments only. The report starts by outlining the theory and follows by describing two possible applications: one for the derivation of the scattering function of a moving, turning line target; the other in the use of pulse trains to measure scattering functions. (Author)

Bibliography: p. 417-423

Psychotherapy can be defined as: (1) a process between at least two people one of whom is a client/patient and the other is a therapist/counselor; (2) an environment, where psychological phenomena can be observed, changed, or modified and where adaptive experiences and thoughts can be facilitated; (3) a longitudinal commitment for both client and therapist (Garfield, 1990). The number of clients or therapists in a psychotherapy can differ depending on the therapeutic setting (e.g., individual and group therapy). The medical model of psychotherapy could be overly reductionist when trying to capture the process of change (Wampold & Imel, 2015). Psychotherapeutic intervention can be modeled by a set of variables interlinked by mathematical equations of varying complexity. The medical model of psychotherapy assumes linear mechanistic relationships between intervention and outcome variables (e.g., a linear dose-effect relationship). While we can describe an intervention or experiment with only two variables connected by a simple function (e.g., cognitive restructuring predicts increased well-being in patients undergoing cognitive behavioral therapy), more general models require increasing complexity (Sivakumar, 2004). It is for example necessary to investigate how, for which population specifically, and under which conditions a specific intervention works (McGill et al., 2020). Simpler, narrower models can be then seen as an approximation of the more general ones, either by omitting less impactful variables, or by simplifying (e.g., linearizing) the relationships. However, from the psychotherapy process perspective, the interaction between client and therapist can be considered a complex dynamic system evolving in time as psychotherapy proceeds forward (Gelo & Salvatore, 2016). Therefore, modeling psychotherapy process as a complex dynamic system has a higher potential to bring insight into the nuanced nature of the dyadic (or systemic) interaction among the participants and its connection to outcome than traditional linear approaches (Petticrew, 2015). Unfortunately, classic self-reported data are usually not suitable for more advanced methods of complexity modeling. Therefore, psychotherapeutic phenomena are often investigated using data and methods from closely related scientific disciplines such as computational neurosciences. Change in psychotherapy is often nonlinear and context-specific (Guastello et al., 2008). These complex models often lead to sets of equations that do not have analytical solutions and require use of advanced mathematical tools to be workable (Strogatz, 2018). Important group of the so-called dynamic models concerns with the development of variables with time, and are typically described by differential or difference equations. For instance, complex psychotherapeutic change processes include time-series or chains of events emerging after the therapeutic intervention. Nonlinearity of these equations coupled with increasing complexity might produce irregular and disorderly behavior, where outcome of a process varies greatly with initial conditions - chaotic behavior (Schiepek et al., 2017). This might limit the predictive value of the model. Mathematical properties of such models (not only in psychotherapy) are studied in dynamic systems theory and its related fields (such as chaos theory). Nowadays, complexity science in psychotherapy is an already well-established field of study. However, a review aggregating all the literature and summarizing what mathematical methods and models are used is still missing. We aim to provide such review in this study.

The linear programming (LP) is one of the most popular necessary optimization tool used for data analytics as well as in various scientific fields. However, the current state-of-art algorithms suffer from scalability issues when processing Big Data. For example, the commercial optimization software IBM CPLEX cannot handle an LP with more than hundreds of thousands variables or constraints. Existing algorithms are fundamentally hard to scale because they are inevitably too complex to parallelize. To address the issue, we study the possibility of using the Belief Propagation (BP) algorithm as an LP solver. BP has shown remarkable performances on various machine learning tasks and it naturally lends itself to fast parallel implementations. Despite this, very little work has been done in this area. In particular, while it is generally believed that BP implicitly solves an optimization problem, it is not well understood under what conditions the solution to a BP converges to that of a corresponding LP formulation. Our efforts consist of two main parts. First, we perform a theoretic study and establish the conditions in which BP can solve LP [1,2]. Although there has been several works studying the relation between BP and LP for certain instances, our work provides a generic condition unifying all prior works for generic LP. Second, utilizing our theoretical results, we develop a practical BP-based parallel algorithms for solving generic LPs, and it shows 71x speed up while sacrificing only 0.1 accuracy compared to the state-of-art exact algorithm. As a result of the study, the PIs have published two conference papers and two follow-up journal papers are under submission. We refer the readers to our published work for details.

The linear programming (LP) is one of the most popular necessary optimization tool used for data analytics as well as in various scientific fields. However, the current state-of-art algorithms suffer from scalability issues when processing Big Data. For example, the commercial optimization software IBM CPLEX cannot handle an LP with more than hundreds of thousands variables or constraints. Existing algorithms are fundamentally hard to scale because they are inevitably too complex to parallelize. To address the issue, we study the possibility of using the Belief Propagation (BP) algorithm as an LP solver. BP has shown remarkable performances on various machine learning tasks and it naturally lends itself to fast parallel implementations. Despite this, very little work has been done in this area. In particular, while it is generally believed that BP implicitly solves an optimization problem, it is not well understood under what conditions the solution to a BP converges to that of a corresponding LP formulation. Our efforts consist of two main parts. First, we perform a theoretic study and establish the conditions in which BP can solve LP [1,2]. Although there has been several works studying the relation between BP and LP for certain instances, our work provides a generic condition unifying all prior works for generic LP. Second, utilizing our theoretical results, we develop a practical BP-based parallel algorithms for solving generic LPs, and it shows 71x speed up while sacrificing only 0.1 accuracy compared to the state-of-art exact algorithm. As a result of the study, the PIs have published two conference papers and two follow-up journal papers are under submission. We refer the readers to our published work for details.

On the Theory of the Solution of a System of Simultaneous Non-Linear Partial Differential Equations of the First Order. Nanson, E Proceedings of the Royal Society of London (1854-1905). 1875-01-01. 24:337–344

The invariants of input-output systems under the action of the feedback group was examined. The approach used the theory of Lie groups and concepts of modern differential geometry, and illustrated how the latter provides a basis for the discussion of the analytic structure of systems. Finite dimensional linear systems in a single independent variable are considered. Lessons of more general situations (e.g., distributed parameter and multidimensional systems) which are increasingly encountered as technology advances are presented.

 Berkeley EE221a Linear System Theory Lecture Notes -- Fa17

 Berkeley EE221a Linear System Theory Lecture Notes -- Fa17

 Berkeley EE221a Linear System Theory Lecture Notes -- Fa17

Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.

Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.

Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.

Historically, radar returns from the sea surface have generally been modeled one of two ways--either (a) a quasi-deterministic fashion usingelectromagnetic scattering theory combined with a random realization of physical sea surface based on insights from hydrodynamics or (b) usinga primarily statistical formulation of the amplitude envelope using assumptions about the distribution of scatterers within a resolution cell basedon vague physical insights. However, these models suffer from excessive computational and/or conceptual complexity or have highly restrictiveregimes of applicability. In this work, we present an alternative characterization of sea clutter returns utilizing statistical linear time-varyingsystem theory in an attempt to provide a more general model that works in a wider variety of circumstances while retaining a competitively lowcomputational burden. It is hoped that the compactness of this mathematical representation will facilitate more rapid development of effectiveclutter mitigation techniques in the future.

On the Theory of the Solution of a System of Simultaneous Non-Linear Partial Differential Equations of the First Order. Nanson, E Proceedings of the Royal Society of London (1854-1905). 1874-01-01. 23:510–510

Frequency domain analogies, polynomial matrices, and linear systems

Frequency domain analogies, polynomial matrices, and linear systems

Conical shell theory and piston theory aerodynamics are used to study the aeroelastic stability of the thermal protection system (TPS) on the NASA Hypersonic Inflatable Aerodynamic Decelerator (HIAD). Structural models of the TPS consist of single or multiple orthotropic conical shell systems resting on several circumferential linear elastic supports. The shells in each model may have pinned (simply-supported) or elastically-supported edges. The Lagrangian is formulated in terms of the generalized coordinates for all displacements and the Rayleigh-Ritz method is used to derive the equations of motion. The natural modes of vibration and aeroelastic stability boundaries are found by calculating the eigenvalues and eigenvectors of a large coefficient matrix. When the in-flight configuration of the TPS is approximated as a single shell without elastic supports, asymmetric flutter in many circumferential waves is observed. When the elastic supports are included, the shell flutters symmetrically in zero circumferential waves. Structural damping is found to be important in this case. Aeroelastic models that consider the individual TPS layers as separate shells tend to flutter asymmetrically at high dynamic pressures relative to the single shell models. Several parameter studies also examine the effects of tension, orthotropicity, and elastic support stiffness.

We establish from first principles a perturbative framework that allows us to compute reaction rates for processes taking place in nonequilibrium $O (N)$ linear-sigma systems in broken phase. The system of our concern is quasiuniform system near equilibrium or nonequilibrium quasistationary system. We employ the closed-time-path formalism and use the so-called gradient approximation. No further approximation is introduced. In the course of construction of the framework, we obtain the gap equation that determines the effective masses of $\pi$ and of $\sigma$, and the generalized Boltzmann equation that describes the evolution of the number-density functions of $\pi$ and of $\sigma$.

19 p. 28 cm

19 p. 28 cm

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