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The effects of teaching a mathematics problem solving strategy on the problem solving ability of grade nine students

Smith, J. E.
Fonte: Brock University Publicador: Brock University
Tipo: Electronic Thesis or Dissertation
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Forty grade 9 students were selected from a small rural board in southern Ontario. The students were in two classes and were treated as two groups. The treatment group received instruction in the Logical Numerical Problem Solving Strategy every day for 37 minutes over a 6 week period. The control group received instruction in problem solving without this strategy over the same time period. Then the control group received the treat~ent and the treatment group received the instruction without the strategy. Quite a large variance was found in the problem solving ability of students in grade 9. It was also found that the growth of the problem solving ability achievement of students could be measured using growth strands based upon the results of the pilot study. The analysis of the results of the study using t-tests and a MANOVA demonstrated that the teaching of the strategy did not significaritly (at p s 0.05) increase the problem solving achievement of the students. However, there was an encouraging trend seen in the data.

The impact of integrated programming on student attitude and achievement in grade 9 academic mathematics and science

Cosentino, Cindy.
Fonte: Brock University Publicador: Brock University
Tipo: Electronic Thesis or Dissertation
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This research studioo the effect of integrated instruction in mathematics and~ science on student achievement in and attitude towards both mathematics and science. A group of grade 9 academic students received instruction in both science and mathematics in an integrated program specifically developed for the purposes of the research. This group was compared to a control group that had received science and mathematics instruction in a traditional, nonintegrated program. The findings showed that in all measures of attitude, there was no significant difference between the students who participated in the integrated science and mathematics program and those who participated in a traditional science and mathematics program. The findings also revealed that integration did improve achievement on some of the measures used. The performance on mathematics open-ended problem-solving tasks improved after participation in the integrated program, suggesting that the integrated students were better able to apply their understanding of mathematics in a real-life context. The performance on the final science exam was also improved for the integrated group. Improvement was not noted on the other measures, which included EQAO scores and laboratory practical tasks. These results raise the issue of the suitability of the instruments used to gauge both achievement and attitude. The accuracy and suitability of traditional measures of achievement are considered. It is argued that they should not necessarily be used as the measure of the value of integrated instruction in a science and mathematics classroom.

Advances in Pseudospectral Methods for Optimal Control

Fahroo, Fariba; Ross, I. Michael
Fonte: The American Institute of Aeronautics and Astronautics (AIAA) Publicador: The American Institute of Aeronautics and Astronautics (AIAA)
Tipo: Conference Paper
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The article of record as published may be located at http://arc.aiaa.org; Approved for public display, distribution unlimited; AIAA Guidance, Navigation and Control Conference and Exhibit ; Paper no. AIAA-2008-7309, Honolulu, Hawaii, 2008; Recently, the Legendre Pseudospectral (PS) method migrated from theory to fight application onboard the International Space Station for performing a finite-horizon, zero- propellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for infinite-horizon optimal control problems recently, the Legendre Pseudospectral (PS) method migrated from theory to flight application onboard the International Space Station for performing a finite-horizon, zero-propellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for in finite-horizon optimal control problems. Motivated by these technicalities, the concept of primal-dual weighted interpolation, introduced earlier by the authors, is used to articulate a united theory for all PS methods for optimal control. This theory illuminates the previously hidden fact of the unit weight function implicit in the Legendre PS method based on Legendre-Gauss-Lobatto points. The united framework also reveals why this Legendre PS method is the most appropriate method for solving finite-horizon optimal control problems with arbitrary boundary conditions. This conclusion is borne by a proper definition of orthogonality needed to generate convergent approximations in Hilbert spaces. Special boundary conditions are needed to ensure the convergence of the Legendre PS method based on the Legendre-Gauss-Radau (LGR) and the Legendre-Gauss (LG) points. These facts are illustrated by simple examples and counter examples which reveal when and why PS methods based on LGR and LG points fail. A new kind of consistency in the primal-dual weight functions allows us to generate dual maps (such as Hamiltonians...

Convergence of Pseudospectral Methods for constrained Nonlinear Optimal Control problems, IEEE (44th; December 12-14; Seville, Spain)

Kang, Wei; Gong, Qi; Ross, I. Michael
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
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The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15, 2005; We consider the optimal control of feedback linearizable dynamic systems subject to mixed state and control constraints. The optimal controller is allowed to be discontinuous including bang-bang control. Although the nonlinear system is assumed to be feedback linearizable, in general, the optimal control does not linearize the dynamics. The continuous optimal control problem is discretized using pseudospectral (PS) methods. We prove that the discretized problem is always feasible and that the optimal solution to the discretized, constrained problem converges to the possibly discontinuous optimal control of the continuous-time problem.

On the Pseudospectral Convector Mapping Theorem for Nonlinear Optimal Control, IEEE (45th; 2006)

Gong, Qi; Ross, I. Michael; Fahroo, Fariba; Kang, Wei
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
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The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; Proceedings of 45th IEEE Conference on Decision and Control Page(s): 2679 - 2686, Dec.13-15, 2006; In recent years, a large number of nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In an effort to better understand the PS approach to solving control problems, we present convergence results for problems with mixed state and control constraints. A set of sufficient conditions are proved under which the solution of the discretized optimal control problem converges to the continuous solution. Conditions for the convergence of the duals are described and illustrated. This leads to a clarification of covector mapping theorem and its connections to constraint qualifications.

Optimal Feedback Control Laws by Legendre Pseudospectral Approximations, ACC (2007; Arlington, VA)

Yan, Hui; Ross, I. Michael; Fahroo, Fariba
Fonte: The American Institute of Aeronautics and Astronautics (AIAA) Publicador: The American Institute of Aeronautics and Astronautics (AIAA)
Tipo: Conference Paper
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The article of record as published may be located at http://arc.aiaa.org; Approved for public display, distribution unlimited; Proceedings of the American Control Conference ; Arlington, VA June 25-27, 2001, pp. 2388-2393.; We develop state feedback control laws for linear time-varying systems with quadratic cost criteria by an indirect Legendre pseudospectral method. This method approximates the linear two-point boundary value problem to a system of algebraic equations by way of a differentiation matrix. The algebraic system is solved to generate discrete linear transformations between the states and controls at the Legendre-Gauss-Lobatto points. Since these linear transformations involve simple matrix operations, they can be computed rapidly and efficiently. Two methods are proposed: one that circumvents solving the differential Riccati equation by a discrete solution of the boundary value problem, and another that generates a predictor feedback law without the use of transition matrices. Thus our methods obviate the need for solving the time-intensive backward integration of the matrix Riccati differential equation or inverting ill-conditioned transition matrices. A numerical example illustrates the techniques and demonstrates the accuracy and efficiency of these controllers.

A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems

Gong, Qi; Kang, Wei; Ross, I. Michael
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
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The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; IEEE Transactions on Automatic Control, v. 51, no. 7 (2006 : July); We consider the optimal control of feedback linearizable dynamical systems subject to mixed state and control constraints. In general, a linearizing feedback control does not minimize the cost function. Such problems arise frequently in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. In this paper, we consider a pseudospectral (PS) method to compute optimal controls. We prove that a sequence of solutions to the PS-discretized constrained problem converges to the optimal solution of the continuous-time optimal control problem under mild and numerically verifiable conditions. The spectral coefficients of the state trajectories provide a practical method to verify the convergence of the computed solution. The proposed ideas are illustrated by several numerical examples.

Pseudospectral Optimal Control for Military and Industrial Applications, IEEE (46th; 2007)

Gong, Qi; Kang, Wei; Bedrossian, Nazareth S.; Fahroo, Fariba; Sekhavat, Pooya; Bollino, Kevin; Lewis, Ryan
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
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The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; 46th IEEE Conference on Decision and Control ; Tutorial Session Paper no. 1739, Dec 2007; During the last decade, pseudospectral methods for optimal control, the focus of this tutorial session, have been rapidly developed as a powerful tool to enable new applications that were previously considered impossible due to the complicated nature of these problems. The purpose of this tutorial section is to introduce this advanced technology to a wider community of control system engineering. We bring in experts of pseudospectral methods from academia, industry, and military and DoD to present topics covering a large spectrum of pseudospectral methods, including the theoretical foundation, numerical techniques of pseudospectral optimal control, and military/industry applications.

Real-Time Computation of Neighboring Optimal Control Laws

Yan, Hui; Ross, I. Michael; Fahroo, Fariba
Fonte: The American Institute of Aeronautics and Astronautics (AIAA) Publicador: The American Institute of Aeronautics and Astronautics (AIAA)
Tipo: Conference Paper
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The article of record as published may be located at http://arc.aiaa.org; Approved for public display, distribution unlimited; Proceedings of AIAA/AAS - Astrodynamics Specialist Conference ; Paper no. AIAA-2002-4657, Monterey, California, Aug. 5-8, 2002; Feedback solutions to the neighboring optimal control problem are typically obtained by solving the matrix Riccati differential equation. In this paper, we propose a new approach based on solving linear algebraic equations in real-time. Our method is based on a Pseudospectral discretization of the linear time-varying boundary value problem that arises from an application of the Minimum Principle. We show how feedback control laws can be computed without any explicit integration, construction of transition matrices or solving the matrix Riccati differential equation. This facilitates a real-time implementation of the scheme and the design of predictive guidance and control laws. A numerical example of a low thrust orbit transfer problem shows the effectiveness of the method for neighboring optimal guidance.

Towards Real-Time Computation of Optimal Controls for Nonlinear Systems, AIAA (2002 ; Monterey, California)

Strizzi, John.; Ross, I. Michael; Fahroo, Fariba
Fonte: The American Institute of Aeronautics and Astronautics (AIAA) Publicador: The American Institute of Aeronautics and Astronautics (AIAA)
Tipo: Conference Paper
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The article of record as published may be located at http://arc.aiaa.org; Approved for public display, distribution unlimited; Proceedings of AIAA Guidance, Navigation, and Control Conference ; Paper no. AIAA-2002-4945, Monterey, California, Aug. 5-8, 2002; Computing optimal controls for nonlinear nonsmooth dynamical systems is an extremely challenging task. In recent years, we introduced a family of Pseudospectral methods for the accurate computation of optimal controls for highly nonlinear systems. While these methods are applicable for generic nonlinear systems, they can also be exploited for systems that exhibit certain differential-geometric properties. These techniques are encoded in the reusable software package, DIDO. In this paper, we harness the structured sparsity of Pseudospectral methods and show that a significant increase in computation speed can be achieved. This facilitates a rapid design of the outer-loop of a two-degree-of-freedom control system. We demonstrate the viability of our approach for several example problems involving smooth nonlinearities, path constraints and control saturation limits.

A Unified Computational Framework for Real-Time Optimal Control , IEEE (42nd; 2003; Maui, Hawaii)

Ross, I. Michael; Fahroo, Fariba
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
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The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; Proceedings of the 42nd IEEE ; Conference on Decision and Control ; Maui, Hawaii USA, December 2003; The dynamics of each agent of a multi-agent controlled dynamical system can be formulated in several possible ways: differential inclusion, flatness parameterization, higher-order inclusions and so on. A plethora of techniques have been proposed for each of these formulations but they are typically not portable across equivalent mathematical formulations. Further complications arise as a result of path constraints such as those imposed by obstacle avoidance or control saturation. In this paper, we present a unified computational framework based on pseudospectral methods to handle the optimal control of dynamical systems where the description of the governing equations or that of the path constraint is not a limitation. We illustrate our ideas by way of multiple formulations of a flexible link manipulator problem that includes a differentially flat formulation subject to control saturation. A comparison of our approach to a recent method reveals that we get an almost 30% improvement in the cost. Our results also show that equivalent mathematical formulations can yield varying run times leading to some surprising questions on flatness parameterization for real-time computation.

Mathematical modeling and optimal control of battlefield information flow

Phillips, Donovan D.
Fonte: Monterey, California. Naval Postgraduate School Publicador: Monterey, California. Naval Postgraduate School
Tipo: Tese de Doutorado Formato: xiv, 145 p.
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The U.S. Army's Future Force requires information dominance to succeed, yet finds itself with an ever-increasing gap between its capacity to collect information and its information processing capacity--with little understanding of how to efficiently utilize scarce processing resources. In this investigation, a model is proposed to adequately represent the flow of information within a command and control context toward the end of optimally controlling this flow. The model is conjectured to be NP-hard in general. Closed-form optimal solutions are derived for special cases of the model, while other cases are shown to be NP-hard. A case of the model is shown to equate to a special case of the quadratic assignment problem not previously known to have a closed-form solution, and such a solution is derived. Upper and lower bounds are derived for more general cases of the model, and heuristic strategies are proposed and tested in discrete event simulation. Strong empirical evidence is produced to demonstrate the effectiveness and robustness of one heuristic.; TRADOC Analysis Center; US Army (USA) author.

Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, ACA (2004; Boston, Massachusetts)

Ross, I. Michael; Fahroo, Fariba
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
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The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; Proceeding of the 2004 American Control Conference Boston, Massachusetts ; vol. 2, page(s):1610-1615, June 30-July 2, 2004; We consider a fairly general class of state-constrained nonlinear hybrid optimal control problems that are based on coordinatizing Sussmann's model. An event set generalizes the notion of a guard set, reset map, endpoint set as well as the switching set. We present a pseudospectral (PS) knotting method that discretizes the continuous-time variables of the problem. The discrete event conditions are imposed over the PS knots leading to a large, sparse, mixed-variable programming (MVP) problem. The Karush-Kuhn-Tucker conditions for the MVP are transformed in a manner that makes them closely resemble the discretized necessary conditions obtained from the hybrid minimum principle. A set of closure conditions are introduced to facilitate commuting the operations of dualization and discretization. An immediate consequence of this is a hybrid covector mapping theorem that provides an order-preserving transformation of the Lagrange multipliers associated with the discretized problem to the discretized covectors associated with the hybrid optimal control problem.

Convergence of Pseudospectral Methods for a Class of Discontinuous Optimal Control

Kang, Wei; Qi Gong; Ross, I. Michael
Fonte: Escola de Pós-Graduação Naval Publicador: Escola de Pós-Graduação Naval
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Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on; The article of record may be found at http://dx.doi.org/10.1109/CDC.2005.1582587; We consider the optimal control of feedback linearizable dynamic systems subject to mixed state and control constraints. The optimal controller is allowed to be discontinuous including bang-bang control. Although the nonlinear system is assumed to be feedback linearizable, in general, the optimal control does not linearize the dynamics. The continuous optimal control problem is discretized using pseudospectral (PS) methods. We prove that the discretized problem is always feasible and that the optimal solution to the discretized, constrained problem converges to the possibly discontinuous optimal control of the continuous-time problem.

Analysis and Control of Period-Doubling Bifurcation in Buck Converters Using Harmonic Balance

Fang, Chung-Chieh; Abed, Eyad H.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/10/2012 Português
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56.38442%
Period doubling bifurcation in buck converters is studied by using the harmonic balance method. A simple dynamic model of a buck converter in continuous conduction mode under voltage mode or current mode control is derived. This model consists of the feedback connection of a linear system and a nonlinear one. An exact harmonic balance analysis is used to obtain a necessary and sufficient condition for a period doubling bifurcation to occur. If such a bifurcation occurs, the analysis also provides information on its exact location. Using the condition for bifurcation, a feedforward control is designed to eliminate the period doubling bifurcation. This results in a wider range of allowed source voltage, and also in improved line regulation.; Comment: Published in the International Journal of Latin American Applied Research, 31(3), pp. 149-156, Jul. 2001, Special theme issue: Bifurcation Control: Methodologies and Applications, In Honor of the 65th Birthday of Professor Leon O. Chua

A Martingale Approach and Time-Consistent Sampling-based Algorithms for Risk Management in Stochastic Optimal Control

Huynh, Vu Anh; Kogan, Leonid; Frazzoli, Emilio
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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In this paper, we consider a class of stochastic optimal control problems with risk constraints that are expressed as bounded probabilities of failure for particular initial states. We present here a martingale approach that diffuses a risk constraint into a martingale to construct time-consistent control policies. The martingale stands for the level of risk tolerance over time. By augmenting the system dynamics with the controlled martingale, the original risk-constrained problem is transformed into a stochastic target problem. We extend the incremental Markov Decision Process (iMDP) algorithm to approximate arbitrarily well an optimal feedback policy of the original problem by sampling in the augmented state space and computing proper boundary conditions for the reformulated problem. We show that the algorithm is both probabilistically sound and asymptotically optimal. The performance of the proposed algorithm is demonstrated on motion planning and control problems subject to bounded probability of collision in uncertain cluttered environments.

Formation Control with Triangulated Laman Graphs

Chen, Xudong; Belabbas, M. -A.; Basar, Tamer
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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Formation control deals with the design of decentralized control laws that stabilize agents at prescribed distances from each other. We call any configuration that satisfies the inter-agent distance conditions a target configuration. It is well known that when the distance conditions are defined via a rigid graph, there is a finite number of target configurations modulo rotations and translations. We can thus recast the objective of formation control as stabilizing one or many of the target configurations. A major issue is that such control laws will also have equilibria corresponding to configurations which do not meet the desired inter-agent distance conditions; we refer to these as undesired equilibria. The undesired equilibria become problematic if they are also stable. Designing decentralized control laws whose stable equilibria are all target configurations in the case of a general rigid graph is still an open problem. We propose here a partial solution to this problem by exhibiting a class of rigid graphs and control laws for which all stable equilibria are target configurations.

A stochastic density matrix approach to approximation of probability distributions and its application to nonlinear systems

Vladimirov, Igor G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonormal basis functions with positive semi-definite Hermitian matrices of unit trace. Such matrices are called stochastic density matrices in order to reflect an analogy with the quantum mechanical density matrices. The SDM approximation of a PDF satisfies the normalization condition and is nonnegative everywhere in contrast to the truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic structure, such as the Hermite polynomial and Fourier bases, the SDM approximation can be chosen so as to satisfy given moment specifications and can be optimized using a quadratic proximity criterion. We apply the SDM approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion processes governed by nonlinear stochastic differential equations. This leads to an ordinary differential equation for the SDM dynamics of the approximating PDF. As an example, we consider the Smoluchowski SDE on a multidimensional torus.; Comment: 12 pages, 3 figures. A brief version of this paper will appear in the proceedings of the IEEE Multi-Conference on Systems and Control, 21-23 September 2015, Sydney, Australia

Algebraic background for numerical methods, control theory and renormalization

Manchon, Dominique
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/01/2015 Português
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We review some important algebraic structures which appear in a priori remote areas of Mathematics, such as control theory, numerical methods for solving differential equations, and renormalization in Quantum Field Theory. Starting with connected Hopf algebras we will also introduce augmented operads, and devote a substantial part to pre-Lie algebras. Other related algebraic structures (Rota-Baxter and dendriform algebras, NAP algebras) will be also mentioned.; Comment: Proceedings of the COmbinatorics and COntrol conference, Benasque, Spring 2010 (to appear)

On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

Debrabant, Kristian; Lang, Jens
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.