scholarly journals Error Bound for Non-Zero Initial Condition Using the Singular Perturbation Approximation Method

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 232 ◽  
Author(s):  
Adnan Daraghmeh ◽  
Naji Qatanani
Keyword(s):  
Singular Perturbation ◽  
Error Bound ◽  
Initial Conditions ◽  
A Priori ◽  
Approximation Error ◽  
Linear Control Systems ◽  
Zero Initial Condition ◽  
Reduction Techniques ◽  
Balanced Model ◽  
Perturbation Approximation

In this article, we present up to date results on the balanced model reduction techniques for linear control systems, in particular the singular perturbation approximation. One of the most important features of this method is it allows for an a priori L 2 and H ∞ bounds for the approximation error. This method has been successfully applied for systems with homogeneous initial conditions, however, the main focus in this work is to derive an L 2 error bound for singular perturbation approximation for system with inhomogeneous initial conditions, extending the work by Antoulas et al. The theoretical results are validated numerically.

Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation

2019 ◽  
Vol 353 ◽  
pp. 295-307
Author(s):  
Adnan Daraghmeh ◽  
Carsten Hartmann ◽  
Naji Qatanani
Keyword(s):  
Singular Perturbation ◽  
Model Reduction ◽  
Linear Systems ◽  
Initial Conditions ◽  
Balanced Model ◽  
Perturbation Approximation

scholarly journals Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 85
Author(s):  
Andreas Rauh ◽  
Julia Kersten
Keyword(s):  
Linear Systems ◽  
Initial Conditions ◽  
A Priori ◽  
Real Life ◽  
Reachability Analysis ◽  
Interval Methods ◽  
Space Representation ◽  
Uncertain Parameters ◽  
Time Varying ◽  
Similarity Transformations

Continuous-time linear systems with uncertain parameters are widely used for modeling real-life processes. The uncertain parameters, contained in the system and input matrices, can be constant or time-varying. In the latter case, they may represent state dependencies of these matrices. Assuming bounded uncertainties, interval methods become applicable for a verified reachability analysis, for feasibility analysis of feedback controllers, or for the design of robust set-valued state estimators. The evaluation of these system models becomes computationally efficient after a transformation into a cooperative state-space representation, where the dynamics satisfy certain monotonicity properties with respect to the initial conditions. To obtain such representations, similarity transformations are required which are not trivial to find for sufficiently wide a-priori bounds of the uncertain parameters. This paper deals with the derivation and algorithmic comparison of two different transformation techniques for which their applicability to processes with constant and time-varying parameters has to be distinguished. An interval-based reachability analysis of the states of a simple electric step-down converter concludes this paper.

Large deviations in linear control systems with nonzero initial conditions

2015 ◽  
Vol 76 (6) ◽  
pp. 957-976 ◽  
Author(s):  
B. T. Polyak ◽  
A. A. Tremba ◽  
M. V. Khlebnikov ◽  
P. S. Shcherbakov ◽  
G. V. Smirnov
Keyword(s):  
Large Deviations ◽  
Control Systems ◽  
Initial Conditions ◽  
Linear Control ◽  
Linear Control Systems

scholarly journals An integral equation formulation of the N-body dielectric spheres problem. Part 1: Numerical analysis

2020 ◽  
Author(s):  
Muhammad Hassan ◽  
Benjamin Stamm
Keyword(s):  
Integral Equation ◽  
Numerical Analysis ◽  
Error Analysis ◽  
Convergence Rates ◽  
A Priori ◽  
Approximation Error ◽  
Spherical Particles ◽  
Integral Equation Formulation ◽  
Dielectric Spheres ◽  
The Stability

In this article, we analyse an integral equation of the second kind that represents the solution of N interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants- and thus the approximation error- with respect to the number N of involved dielectric spheres. We develop a new a priori error analysis that demonstrates N-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of N.

A numerical method for the approximation of reachable sets of linear control systems

2017 ◽  
Vol 36 (2) ◽  
pp. 423-441 ◽  
Author(s):  
Lizhen Shao ◽  
Fangyuan Zhao ◽  
Guangda Hu
Keyword(s):  
Numerical Method ◽  
Control Systems ◽  
Optimization Problems ◽  
Approximation Error ◽  
Outer Approximation ◽  
Reachable Sets ◽  
Linear Control ◽  
Superior Performance ◽  
Linear Control Systems ◽  
Inner Approximation

Abstract In this article, a numerical method for the approximation of reachable sets of linear control systems is discussed. First a continuous system is transformed into a discrete one with Runge–Kutta methods. Then based on Benson’s outer approximation algorithm for solving multiobjective optimization problems, we propose a variant of Benson’s algorithm to sandwich the reachable set of the discrete system with an inner approximation and an outer approximation. By specifying an approximation error, the quality of the approximations measured in Hausdorff distance can be directly controlled. Furthermore, we use an illustrative example to demonstrate the working of the algorithm. Finally, computational experiments illustrate the superior performance of our proposed algorithm compared to a recent algorithm in the literature.

The Road to Chaos by Time-Asymmetric Hebbian Learning in Recurrent Neural Networks

2007 ◽  
Vol 19 (1) ◽  
pp. 80-110 ◽  
Author(s):  
Colin Molter ◽  
Utku Salihoglu ◽  
Hugues Bersini
Keyword(s):  
Supervised Learning ◽  
Hebbian Learning ◽  
Learning Algorithm ◽  
Initial Conditions ◽  
A Priori ◽  
Computational Cost ◽  
Chaotic Regime ◽  
Dynamical Regime ◽  
New Form ◽  
The Impact

This letter aims at studying the impact of iterative Hebbian learning algorithms on the recurrent neural network's underlying dynamics. First, an iterative supervised learning algorithm is discussed. An essential improvement of this algorithm consists of indexing the attractor information items by means of external stimuli rather than by using only initial conditions, as Hopfield originally proposed. Modifying the stimuli mainly results in a change of the entire internal dynamics, leading to an enlargement of the set of attractors and potential memory bags. The impact of the learning on the network's dynamics is the following: the more information to be stored as limit cycle attractors of the neural network, the more chaos prevails as the background dynamical regime of the network. In fact, the background chaos spreads widely and adopts a very unstructured shape similar to white noise. Next, we introduce a new form of supervised learning that is more plausible from a biological point of view: the network has to learn to react to an external stimulus by cycling through a sequence that is no longer specified a priori. Based on its spontaneous dynamics, the network decides “on its own” the dynamical patterns to be associated with the stimuli. Compared with classical supervised learning, huge enhancements in storing capacity and computational cost have been observed. Moreover, this new form of supervised learning, by being more “respectful” of the network intrinsic dynamics, maintains much more structure in the obtained chaos. It is still possible to observe the traces of the learned attractors in the chaotic regime. This complex but still very informative regime is referred to as “frustrated chaos.”

scholarly journals Better Baltic Sea wave forecasts: improving resolution or introducing ensembles?

Ocean Science ◽  
2018 ◽  
Vol 14 (6) ◽  
pp. 1435-1447
Author(s):  
Torben Schmith ◽  
Jacob Woge Nielsen ◽  
Till Andreas Soya Rasmussen ◽  
Henrik Feddersen
Keyword(s):  
Baltic Sea ◽  
Wave Height ◽  
Spectral Resolution ◽  
Significant Wave Height ◽  
Initial Conditions ◽  
A Priori ◽  
The Baltic Sea ◽  
Significant Wave ◽  
Operational Forecasts ◽  
The Baltic

Abstract. The performance of short-range operational forecasts of significant wave height (SWH) in the Baltic Sea is evaluated. Forecasts produced by a base configuration are intercompared with forecasts from two improved configurations: one with improved horizontal and spectral resolution and one with ensembles representing uncertainties in the physics of the forcing wind field and the initial conditions of this field. Both of the improved forecast classes represent an almost equal increase in computational costs. Therefore, the intercomparison addresses the question of whether more computer resources would be more favorably spent on enhancing the spatial and spectral resolution or, alternatively, on introducing ensembles. The intercomparison is based on comparisons with hourly observations of significant wave height from seven observation sites in the Baltic Sea during the 3-year period from 2015 to 2017. We conclude that for most wave measurement sites, the introduction of ensembles enhances the overall performance of the forecasts, whereas increasing the horizontal and spectral resolution does not. These sites represent offshore conditions, in that they are well exposed from all directions, are a large distance from the nearest coast and in deep water. Therefore, there is the a priori expectation that a detailed shoreline and bathymetry will not have any impact. Only at one site do we find that increasing the horizontal and spectral resolution significantly improves the forecasts. This site is situated in nearshore conditions, close to land and a nearby island, and is therefore shielded from many directions. Consequently, this study concludes that to improve wave forecasts in offshore areas, ensembles should be introduced. For near shore areas, in comparison, the study suggests that additional computational resources should be used to increase the resolution.

Sign in / Sign up

Export Citation Format

Share Document