Method of Conjugate Gradients for Optimal Control Problems with State Variable Constraints PDF Download
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Author: T. S. Fong
Publisher:
ISBN:
Category :
Languages : en
Pages : 94
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Book Description
A review of the computational method of conjugate gradients for linear and nonlinear operator equations is given with emphasis in applying this technique to state variable constraint control problems. The first and second Frechet derivatives of the performance functional are derived. The search directions generated in the iteration process for the optimal control are locally conjugate with respect to the second Frechet derivative. The convergence is along the expanding sequence of sets, the itersection of the linear spaces spanned by the search directions and the set of admissible controls. The computational technique is applied to two state variable constraint problems, in one of which a penalty function is employed to convert the constraint problem to an unconstrained one in addition to the approach considering the constraints directly. For this same problem the method of steepest descent also is studied, and comparison of the results obtained is made and discussed. (author).
Author: T. S. Fong
Publisher:
ISBN:
Category :
Languages : en
Pages : 94
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Book Description
A review of the computational method of conjugate gradients for linear and nonlinear operator equations is given with emphasis in applying this technique to state variable constraint control problems. The first and second Frechet derivatives of the performance functional are derived. The search directions generated in the iteration process for the optimal control are locally conjugate with respect to the second Frechet derivative. The convergence is along the expanding sequence of sets, the itersection of the linear spaces spanned by the search directions and the set of admissible controls. The computational technique is applied to two state variable constraint problems, in one of which a penalty function is employed to convert the constraint problem to an unconstrained one in addition to the approach considering the constraints directly. For this same problem the method of steepest descent also is studied, and comparison of the results obtained is made and discussed. (author).
Author: Thomas Shu Fong
Publisher:
ISBN:
Category : Mathematical optimization
Languages : en
Pages : 176
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Book Description
Author: John Kendall Willoughby
Publisher:
ISBN:
Category : Adaptive control systems
Languages : en
Pages : 128
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Book Description
The method of conjugate gradients (CG) has been shown to be a rapidly converging and efficient means of solving unconstrained optimal control problems. This dissertation presents some theoretical and computational characteristics of three modifications to the CG algorithm which make it applicable to control problems with terminal state variable constraints. The penalty function method and the projection method have been used to adapt ordinary gradient methods to constrained problems. It is concluded here that the penalty function technique is no more or less advantageous with the CG method than with other gradient techniques. The projection method is shown to be theoretically less compatible with the CG algorithm than with other gradient methods. However, a stepsize adjustment policy is suggested that preserves the rapid convergence that is characteristic of the CG method. It is also shown that nonlinear instead of linear terminal constraints cause no additional theoretical of computational difficulty. A third adaptation of the CG method is given which is original to this study. The method, called the modified conjugate gradient method (MCG), is applied to constrained problems by using constant Lagrange multipliers which converge to their optimal values as the iteration proceeds. A unique feature of the MCG method is that each control iterate produced by the method causes the constraints to be satisfied exactly. Furthermore, the technique is equally applicable to nonlinear and linear terminal state constraints. (Author).
Author: D. H. Jacobson
Publisher:
ISBN:
Category :
Languages : en
Pages : 29
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Book Description
A slack variable is used to transform an optimal control problem with a scalar control and a scalar inequality constraint on the state variables into an unconstrained problem of higher dimension. It is shown that, for a p-th order constraint, the p-th time derivative of the slack variable becomes the new control variable. The usual Pontryagin Principle of Lagrange multiplier rule gives necessary conditions of optimality. There are no discontinuities in the adjoint variables. A feature of the transformed problem is that any nominal control function produces a feasible trajectory. The optimal trajectory of the transformed problem exhibits singular arcs which correspond, in the original constrained problem, to arcs which lie along the constraint boundary; this suggests a duality between singular and state-constrained problems, which should be explored. Generalizations of the approach to cases where the constraint and the control are vectors of equal dimension, as well as to problems involving multiple constraints and a single control variable, are considered. Owing to the appearance of singular arcs in the solution of the transformed problem, a direct application of second-order or second-variation algorithms is not possible. However, gradient or conjugate gradient methods are applicable and computations, using the conjugate gradient method, are presented to illustrate the usefulness of the transformation technique. (Author).
Author: Kees C. P. Machielsen
Publisher:
ISBN:
Category : Boundary value problems
Languages : en
Pages : 232
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Book Description
Author: C. T. Leondes
Publisher: Elsevier
ISBN: 1483194590
Category : Technology & Engineering
Languages : en
Pages : 276
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Book Description
Advances in Control Systems: Theory and Applications, Volume 8 provides information pertinent to significant progress in the field of control and systems theory and applications. This book focuses on applications to large-scale systems. Organized into seven chapters, this volume begins with an overview of an effective algorithm for dynamic system organization with state variable constraints. This text then explores a number of effective techniques for the analysis and syntheses of final value control systems. Other chapters consider some significant problems associated with the practical application of Kalman Filter techniques. This book discusses as well the most significant and fundamental work on the international scene in the development of effective algorithms for dynamic system optimization. The final chapter deals with the application of modern control methods of complex industrial process control problems. This book is a valuable resource for mathematicians, control system engineers, physical scientists, economists, econometricians, and research workers.
Author: Radoslaw Pytlak
Publisher: Springer
ISBN: 3540486623
Category : Science
Languages : en
Pages : 224
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Book Description
While optimality conditions for optimal control problems with state constraints have been extensively investigated in the literature the results pertaining to numerical methods are relatively scarce. This book fills the gap by providing a family of new methods. Among others, a novel convergence analysis of optimal control algorithms is introduced. The analysis refers to the topology of relaxed controls only to a limited degree and makes little use of Lagrange multipliers corresponding to state constraints. This approach enables the author to provide global convergence analysis of first order and superlinearly convergent second order methods. Further, the implementation aspects of the methods developed in the book are presented and discussed. The results concerning ordinary differential equations are then extended to control problems described by differential-algebraic equations in a comprehensive way for the first time in the literature.
Author:
Publisher:
ISBN:
Category : Aeronautics
Languages : en
Pages : 1278
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Book Description
Author: Ivan Borges Oliveira
Publisher:
ISBN:
Category :
Languages : en
Pages : 147
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Book Description
(Cont.) Standard logarithmic barrier functions and Newton methods are employed to address the hard constraints on control variables of the type Umin
Author: G. R. Hennig
Publisher:
ISBN:
Category :
Languages : en
Pages : 61
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Book Description
The paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, a state variable inequality constraint, and terminal constraints. The problem is to find the state x(t), the control u(t), and the parameter pi so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. (Modified author abstract).
