# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

2014 , Volume 4 , Issue 3

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2014, 4(3): 181-191 doi: 10.3934/naco.2014.4.181 +[Abstract](675) +[PDF](286.1KB)
Abstract:
In this paper, we provide a robust control approach for controlling the autonomous bicycle kinematics with the objective of stabilizing the bicycle steer $\delta$ and roll $\phi$ angles. The dynamical model is the so-called 'Whipples Bicycle Model', where the roll (lean) angle and the steer angle of the bicycle are the two outputs of the model and the torques across the roll and steer angle as the two control variables. Two control design methods are developed based on $H_\infty$ and $H_2$-norm optimization using dynamic output feedback. The ensuing results are compared with an adaptive control scheme. The autonomous bicycle was tested for varying velocities.
2014, 4(3): 193-207 doi: 10.3934/naco.2014.4.193 +[Abstract](560) +[PDF](355.0KB)
Abstract:
Recurrent neural networks (RNNs) have emerged as a promising tool in modeling nonlinear dynamical systems. The convergence is one of the most important issues of concern among the dynamical properties for the RNNs in practical applications. The reason is that the viability of many applications of RNNs depends on their convergence properties. We study in this paper the convergence properties of the weighted state space search algorithm (WSSSA) -- a derivative-free and non-random learning algorithm which searches the neighborhood of the target trajectory in the state space instead of the parameter space. Because there is no computation of partial derivatives involved, the WSSSA has a couple of salient features such as simple, fast and cost effective. In this study we provide a necessary and sufficient condition that required for the convergence of the WSSSA. Restrictions are offered that may help assure convergence of the of the WSSSA to the desired solution. The asymptotic rate of convergence is also analyzed. Our study gives insights into the problem and provides useful information for the actual design of the RNNs. A numerical example is given to support the theoretical analysis and to demonstrate that it is applicable to many applications.
2014, 4(3): 209-225 doi: 10.3934/naco.2014.4.209 +[Abstract](555) +[PDF](548.6KB)
Abstract:
In this paper we propose two two-step methods for image zooming using duality strategies. In the first method, instead of smoothing the normal vector directly as did in the first step of the classical LOT model, we reconstruct the unit normal vector by means of Chambolle's dual formulation. Then, we adopt the split Bregman iteration to obtain the zoomed image in the second step. The second method is based on the TV-Stokes model. By smoothing the tangential vector and imposing the divergence free condition, we propose an image zooming method based on the TV-Stokes model using the dual formulation. Furthermore, we give the convergence analysis of the proposed algorithms. Numerical experiments show the efficiency of the proposed methods.
2014, 4(3): 227-239 doi: 10.3934/naco.2014.4.227 +[Abstract](492) +[PDF](1800.6KB)
Abstract:
Schilders' factorization can be used as a basis for preconditioning indefinite linear systems which arise in many problems like least-squares, saddle-point and electronic circuit simulations. Here we consider its application to resistor network modeling. In that case the sparsity of the matrix blocks in Schilders' factorization depends on the sparsity of the inverse of a permuted incidence matrix. We introduce three different possible permutations and determine which permutation leads to the sparsest inverse of the incidence matrix. Permutation techniques are based on types of sub-digraphs of the network of an incidence matrix.
2014, 4(3): 241-267 doi: 10.3934/naco.2014.4.241 +[Abstract](866) +[PDF](707.4KB)
Abstract:
Microgrids are smaller, self-contained electricity grids featuring distributed generation (e.g., solar photovoltaic panels, wind turbines, biomass), energy storage technologies, and power system control devices that enable self-coordinated operations. Microgrids can be seen as a key technology for greater integration of renewable energy resources. However, the uncertain nature in power generated by these resources poses challenges to its integration into the electric grid. In this paper, we present a demand-side management stochastic optimization model to operate an isolated microgrid under uncertain power generation and demand.
2014, 4(3): 269-285 doi: 10.3934/naco.2014.4.269 +[Abstract](553) +[PDF](772.1KB)
Abstract:
The timetabling problem is to find a schedule of activities in space/time that satisfies a prescribed set of operational and resource constraints and which maximizes an objective function that reflects the value of the schedule. Constructing an effective timetable is always a challenging task for any scheduler. Most literature research focuses on specific applications and the resulting models are not easily applied to problems other than those for which they were designed for. In this paper, we construct a general model for university course timetabling. Our model incorporates a total of 17 different types of requirements identified in the literature as well as three new constraint types that we think should be part of the restrictions in a general university based timetabling model. An integer programming (IP) model is presented which incorporates restrictions that need to be satisfied and requests that are included in the objective function. We implement and test our models using the AIMMS mathematical software package. Computational results on a number of case studies are favorable and demonstrate the value of our approach.