56,084 research outputs found
On the formulation of a minimal uncertainty model for robust control with structured uncertainty
In the design and analysis of robust control systems for uncertain plants, representing the system transfer matrix in the form of what has come to be termed an M-delta model has become widely accepted and applied in the robust control literature. The M represents a transfer function matrix M(s) of the nominal closed loop system, and the delta represents an uncertainty matrix acting on M(s). The nominal closed loop system M(s) results from closing the feedback control system, K(s), around a nominal plant interconnection structure P(s). The uncertainty can arise from various sources, such as structured uncertainty from parameter variations or multiple unsaturated uncertainties from unmodeled dynamics and other neglected phenomena. In general, delta is a block diagonal matrix, but for real parameter variations delta is a diagonal matrix of real elements. Conceptually, the M-delta structure can always be formed for any linear interconnection of inputs, outputs, transfer functions, parameter variations, and perturbations. However, very little of the currently available literature addresses computational methods for obtaining this structure, and none of this literature addresses a general methodology for obtaining a minimal M-delta model for a wide class of uncertainty, where the term minimal refers to the dimension of the delta matrix. Since having a minimally dimensioned delta matrix would improve the efficiency of structured singular value (or multivariable stability margin) computations, a method of obtaining a minimal M-delta would be useful. Hence, a method of obtaining the interconnection system P(s) is required. A generalized procedure for obtaining a minimal P-delta structure for systems with real parameter variations is presented. Using this model, the minimal M-delta model can then be easily obtained by closing the feedback loop. The procedure involves representing the system in a cascade-form state-space realization, determining the minimal uncertainty matrix, delta, and constructing the state-space representation of P(s). Three examples are presented to illustrate the procedure
Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models
A unifying scheme based on an ancestor model is proposed for generating a
wide range of integrable discrete and continuum as well as inhomogeneous and
hybrid models. They include in particular discrete versions of sine-Gordon,
Landau-Lifshitz, nonlinear Schr\"odinger (NLS), derivative NLS equations,
Liouville model, (non-)relativistic Toda chain, Ablowitz-Ladik model etc. Our
scheme introduces the possibility of building a novel class of integrable
hybrid systems including multi-component models like massive Thirring, discrete
self trapping, two-mode derivative NLS by combining different descendant
models. We also construct inhomogeneous systems like Gaudin model including new
ones like variable mass sine-Gordon, variable coefficient NLS, Ablowitz-Ladik,
Toda chains etc. keeping their flows isospectral, as opposed to the standard
approach. All our models are generated from the same ancestor Lax operator (or
its q -> 1 limit) and satisfy the classical Yang-Baxter equation sharing the
same r-matrix. This reveals an inherent universality in these diverse systems,
which become explicit at their action-angle level.Comment: Latex, 20 pages, 2 figures, v3, final version to be published in J.
Math Phy
Group classification of heat conductivity equations with a nonlinear source
We suggest a systematic procedure for classifying partial differential
equations invariant with respect to low dimensional Lie algebras. This
procedure is a proper synthesis of the infinitesimal Lie's method, technique of
equivalence transformations and theory of classification of abstract low
dimensional Lie algebras. As an application, we consider the problem of
classifying heat conductivity equations in one variable with nonlinear
convection and source terms. We have derived a complete classification of
nonlinear equations of this type admitting nontrivial symmetry. It is shown
that there are three, seven, twenty eight and twelve inequivalent classes of
partial differential equations of the considered type that are invariant under
the one-, two-, three- and four-dimensional Lie algebras, correspondingly.
Furthermore, we prove that any partial differential equation belonging to the
class under study and admitting symmetry group of the dimension higher than
four is locally equivalent to a linear equation. This classification is
compared to existing group classifications of nonlinear heat conductivity
equations and one of the conclusions is that all of them can be obtained within
the framework of our approach. Furthermore, a number of new invariant equations
are constructed which have rich symmetry properties and, therefore, may be used
for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page
Partial differential equations from integrable vertex models
In this work we propose a mechanism for converting the spectral problem of
vertex models transfer matrices into the solution of certain linear partial
differential equations. This mechanism is illustrated for the
invariant six-vertex model and the resulting
partial differential equation is studied for particular values of the lattice
length.Comment: 19 pages. v2: affiliation and references updated, minor changes,
accepted for publication in J. Math. Phy
Scheduling Dimension Reduction of LPV Models -- A Deep Neural Network Approach
In this paper, the existing Scheduling Dimension Reduction (SDR) methods for
Linear Parameter-Varying (LPV) models are reviewed and a Deep Neural Network
(DNN) approach is developed that achieves higher model accuracy under
scheduling dimension reduction. The proposed DNN method and existing SDR
methods are compared on a two-link robotic manipulator, both in terms of model
accuracy and performance of controllers synthesized with the reduced models.
The methods compared include SDR for state-space models using Principal
Component Analysis (PCA), Kernel PCA (KPCA) and Autoencoders (AE). On the
robotic manipulator example, the DNN method achieves improved representation of
the matrix variations of the original LPV model in terms of the Frobenius norm
compared to the current methods. Moreover, when the resulting model is used to
accommodate synthesis, improved closed-loop performance is obtained compared to
the current methods.Comment: Accepted to American Control Conference (ACC) 2020, Denve
Unifying quantization for inhomogeneous integrable models
Integrable inhomogeneous versions of the models like NLS, Toda chain,
Ablowitz-Ladik model etc., though well known at the classical level, have never
been investigated for their possible quantum extensions. We propose a unifying
scheme for constructing and solving such quantum integrable inhomogeneous
models including a novel inhomogeneous sine-Gordon model, which avoids the
difficulty related to the customary non-isospectral flow by introducing the
inhomogeneities through some central elements of the underlying algebra.Comment: 12 pages, no figure, latex. Two new chapters on general inhom.
trigonometric models and inhom. SG model included. Accepted in Phys.Lett.
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