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Two efficient SVD/Krylov algorithms for model order reduction of large scale systems
No full textWe present two efficient algorithms to produce a reduced order model of a time-invariant linear dynamical system by approximate balanced truncation.
Attention is focused on the use of the structure and the iterative construction via Krylov subspaces of both controllability and observability matrices to compute low-rank approximations of the Gramians or the Hankel operator. This allows us to take advantage of any sparsity in the system matrices and indeed the cost of our two algorithms is only linear in the system dimension. Both algorithms efficiently produce good low-rank approximations (in the least square sense) of the Cholesky factor of each Gramian and the Hankel operator. The second algorithm works directly on the Hankel operator, and it has the advantage that it is independent of the chosen realization. Moreover it is also an approximate Hankel norm method. The two reduced order models produced by our methods are guaranteed to be stable and balanced. We study the convergence of our iterative algorithms and the properties of the fixed point iteration. We also discuss the stopping criteria and the choice of the reduced order
Model Order Reduction or How to make everything as simple as possible but not simpler
No full textLarge complex mathematical models are regularly used for simulation and prediction. However, in control design it is common practice to work with as simple models as possible, because they are easier to analyse and evaluate.
There is a strong need for methods and tools that can take a complex model and deduce simple models for various purposes such as control design. A simple but good model captures much knowledge. It points out the basic properties
and can give good insight about the process.
For simple linear time-invariant models there is a well-established theory and commercially available tools for design of controllers with given speci�cations. Real experiments or simulations using more complex models are then used to verify that the designed controller really works well. For nonlinear models the methods are much less developed. It is simple to derive a linearization on symbolic form from a nonlinear model. It is much more dif�cult to give explicit expressions for stationary operating points since these calculations involve nonlinear equation systems.
The main idea in model reduction is that a high-dimensional state vector is actually belonging to a low-dimensional subspace [1, 2, 4]. Provided that the low-rank subspace is known, the original model can be projected on it to
obtain a required low-dimensional approximation [3]. The goal of every model reduction method is to �nd such a low-dimensional subspace.
In this talk I will introduce model reduction and I will overview some of the most used methods
CONTROL OF A COMPLIANT HUMANOID ROBOT IN DOUBLE SUPPORT PHASE: A GEOMETRIC APPROACH.
No full textEnhancing energy e�ciency of bipedal walking is an important research problem that has been approached by design of recently developed compliant bipedal robots such as CoMan. While compliance leads to energy e�ciency, it also complicates the walking control system due to further under-actuated degrees of freedom (DoF) associated with the compliant actuators. This problem becomes more challenging as the constrained motion of the robot in double support is considered. In this paper this problem is approached from a multi-variable geometric control aspect to systematically account for the compliant actuators dynamics and constrained motion of the robot in double support phase using a detailed electro-mechanical model of CoMan. It is shown that the formulation of constraint subspace is non-trivial in the case of non-rigid robots. A step-wise numerical algorithm is provided and the e�ectiveness of the
proposed method is illustrated via simulation, using a ten DoF model of CoMan
Attractors near grazing-sliding bifurcations
No full textIn this paper we prove, for the first time, that multistability can occur in 3-dimensional Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexsist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three
orbit with two sliding segments may coexist
Multiple attractors in grazing-sliding bifurcations in an explicit example of a Filippov type flow
No full textWe present a constructed explicit example of a three-dimensional Filippov type flow where we show the birth of multiple attractors in grazing-sliding bifurcations. To the best of our knowledge, it is the first such an example of a Filippov type flow where grazing-sliding bifurcation is shown to trigger birth of multiple attractors, reported in the literature. Three qualitatively different scenarios are shown; namely, birth of period-two and period-three stable orbits with one sliding segment, chaotic attractor coexisting with stable period-three orbit characterised by a segment of sliding motion, and a coexistence of a period-three sliding orbit with two sliding segments and a
limit cycle with no sliding segments. Our work reveals an important feature of the normal form map used to construct the Filippov flow that would produce the desired dynamics. Namely, due the fact that the normal form that we use is valid only locally around the grazing-sliding bifurcations, the scale of the variation of the bifurcation parameter past the grazing-sliding had to be carefully chosen to see the dynamics predicted by the map. In other words, sufficiently small neighbourhood where the normal form is valid, in the context of nonsmooth bifurcations, seems to mean different order of magnitude in the range of the bifurcation parameter variation than in the context
of smooth bifurcations
Involutions in the Automorphism Groups of Small Sporadic Simple Groups
Get PDFFor each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators
Weighted projective spaces and iterated Thom spaces.
Get PDFFor any (n+1)-dimensional weight vector chi of positive
integers, the weighted projective space P(chi) is a projective toric variety, and has orbifold singularities in every case other than CP^n. We study the algebraic topology of P(chi), paying particular attention to its localisation at individual primes p. We identify certain p-primary weight vectors pi for which P(pi) is homeomorphic to an iterated Thom space over S^2, and discuss how any P(chi) may be reconstructed from its p-primary factors. We express Kawasaki's computations of the integral cohomology ring H^*(P(\chi);Z) in terms of iterated Thom isomorphisms, and recover Al Amrani's extension to complex K-theory. Our methods generalise to arbitrary complex oriented
cohomology algebras E^*(P(chi)) and their dual homology
coalgebras E_*(P(\chi)), as we demonstrate for complex cobordism theory, which is the universal example. In particular, we describe a fundamental class in Omega_{2n}^U(P(\chi)), which may be interpreted as a resolution of singularities
An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems
No full textWe develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials.
It incorporates scaling of
the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable
conditioning and backward stability properties,
and a preprocessing step that
reveals and deflates the zero and infinite eigenvalues contributed by singular
leading and trailing matrix coefficients.
The algorithm is backward stable for quadratics that are not too heavily damped.
Numerical experiments show that
our MATLAB implementation of the algorithm,
quadeig, outperforms the MATLAB
function polyeig in terms of both stability and efficiency
Backward stability of iterations for computing the polar decomposition
Get PDFAmong the many iterations available for computing the polar decomposition
the most practically useful are the scaled Newton iteration and
the recently proposed dynamically weighted Halley iteration.
Effective ways to scale these and other iterations are known,
but their numerical stability is much less well understood.
In this work we show that a general iteration
for computing the unitary polar factor
is backward stable under two conditions.
The first condition requires that the iteration is implemented in a mixed
backward--forward stable manner
and the second requires that the mapping
does not significantly decrease the size of any singular value relative to the
largest singular value.
Using this result we show that the
dynamically weighted Halley iteration
is backward stable when it is
implemented using Householder QR factorization
with column pivoting and either row pivoting or row sorting.
We also prove the backward stability of the scaled Newton iteration under
the assumption that matrix inverses are computed in a mixed
backward-forward stable fashion;
our proof is much shorter than a previous one of
Kielbasinski and Zietak.
We also use our analysis to explain the instability of
the inverse Newton iteration and to show that the Newton-Schulz iteration
is only conditionally stable.
This work shows that by carefully blending perturbation analysis with
rounding error analysis it is possible to produce a general result that can
prove the backward stability or
predict or explain the instability (as the case may be)
of a wide range of practically interesting iterations for the polar decomposition
A framework for the development of implicit solvers for incompressible flow problems
Get PDFThis survey paper reviews some recent developments in the design of
robust solution methods for the Navier-Stokes equations
modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses
a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system. Numerical experiments are presented that illustrate the effectiveness of our generic approach. It is further shown that the basic solution strategy can be readily extended to more complicated models, including unsteady flow problems with coupled physics and steady flow problems that are nondeterministic in the sense that they have uncertain input data