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SIAM J. on Control and Optimization

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1964

Volume 2, Issue 3, pp. 267-449


An Analytic Theory of Modeling for a Class of Minimal-Energy Control Systems (Disturbance-Free Case)

W. J. Culver

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 267-294 (28 pages)

Online Publication Date: July 18, 2006

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A quantitative theory is developed for modeling a class of optimal control systems. A mathematical representation—a model system—is fit to an actual system solely on the basis of the respective optimal performances of the two systems, where performance is defined by a generalized quadratic criterion of the minimum energy, minimal endpoint-error type. The plant to be controlled is assumed to be linear time-varying (at least in the small), and the model is taken to be linear, but constant-coefficient.
Necessary and sufficient conditions are derived for achieving certain pertinent tasks of performance prediction and optimal control, wherein particular attention is paid to the accomplishment of these tasks by computer methods. It is found that the very structure of the plant representation may prohibit some model activities, e.g., if a certain inequality relation is not maintained between the respective dimensions of the state and control vectors.
Finally, the given performance index is used to partition the universe of linear systems into equivalence classes, and the conditions are presented for two systems to be performance-equivalent. These are shown to be the necessary and sufficient conditions for the optimal control laws of nonidentical systems to be, in fact, interchangeable in the large.

Minimum Effort Control Of Several Terminal Components

J. V. Breakwellt and F. Tung

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 295-316 (22 pages)

Online Publication Date: July 18, 2006

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The stochastic control problem of minimizing the total average velocity correction with several prescribed terminal variances in the presence of random injection and measurement errors is considered. It is shown that, for the case of linear feedback, this can be formulated as an optimization problem for an equivalent deterministic system whose states are the covariances of the predicted miss. However, the deterministic optimization problem is “degenerate” causing some difficulty in the computation of the feedback gain. It is shown that the optimum linear corrective strategy is, in general, discontinuous and consists of an initial period of no control, followed by a period of continuous control and finally a period of no control and possibly an impulse at the end. Equations are derived from which the variable feedback gain and the various time intervals can be computed. Two simple examples involving (1) the control of two terminal position components, and (2) the control of both the terminal position and the terminal velocity are considered in detail. Numerical results are given showing the comparison between this solution and that obtained by using the well known theory for the quadratic loss criterion. In particular, the computation includes, for the two position case, a gap in the information.

Some Mathematical Theory of the Penalty Method for Solving Optimum Control Problems

Kiyohisa Okamura

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 317-331 (15 pages) | Cited 3 times

Online Publication Date: July 18, 2006

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The penalty method is a powerful technique for solving the optimum control problems involving systems subject to holonomic side constraints. In the usual calculus of variations, the above problems are formulated in consideration of the Weierstrass-Erdmann corner conditions which add considerable complexity in practice. In the penalty method, however, the side constraints are eliminated by introducing a sequence of approximate formulations. Thus the Weierstrass-Erdmann corner conditions need not be checked.
When the penalty method is applied in the ordinary calculus the sequence of approximate formulations is proved to be equivalent to the original formulation in the limiting case. However, no mathematical rigor has been claimed when the penalty method is applied to the variational problems.
The author establishes, in this paper, some mathematical basis for the penalty method applied in the calculus of variations, particularly optimum control problems.

Optimal Control of Aperiodic Discrete-Time Systems

B. W. Jordan and E. Polak

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 332-346 (15 pages)

Online Publication Date: July 18, 2006

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Abstract Unavailable

Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering

W. M. Wonham

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 347-369 (23 pages)

Online Publication Date: July 18, 2006

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Abstract Unavailable

The Asymptotes of the Time Lag Root-Locus

Allan M. Krall

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 370-372 (3 pages)

Online Publication Date: July 18, 2006

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Abstract Unavailable

Time Optimal Control with Amplitude and Rate Limited Controls

W. W. Schmaedeke and D. L. Russell

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 373-395 (23 pages)

Online Publication Date: July 18, 2006

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Abstract Unavailable

Analysis of Linear Systems by Means of Laguerre Functions

James C. I. Dooge

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 396-408 (13 pages) | Cited 1 time

Online Publication Date: July 18, 2006

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The use of Laguerre functions is proposed for the analysis of heavily damped linear systems where the input is not subject to experimental control. An equation is derived which links the corresponding coefficients in the Laguerre function expansions of the input, the impulse response and the output of a linear system. This equation enables the third set of Laguerre coefficients to be calculated when the other two sets of coefficients are known. The connection between the Laguerre function expansion and the representation of the system response by a series of gamma distributions is noted and the latter series identified as defining an analog system composed entirely of branches of linear storage elements.

Penalty Functions and Bounded Phase Coordinate Control

D. L. Russell

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 409-422 (14 pages) | Cited 3 times

Online Publication Date: July 18, 2006

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This paper studies the use of two different kinds of penalty functions to obtain approximate and, in the limit, exact solutions to a class of bounded phase coordinate optimal control problems. The first type of penalty function assumes small values within the phase constraint and large values outside, while the second type is defined only within the phase constraints, assuming small values away from the constraint boundary but increasing to infinity as that boundary is approached.

Optimal-Thrust Trajectories in an Arbitrary Gravitational Field

Joseph G. Gurley

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 423-432 (10 pages)

Online Publication Date: July 18, 2006

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The problem of optimal-thrust trajectories is studied using a slight variation of the usual calculus of variations technique. The results include the usual first-order criteria for optimality, which are that the direction of thrust must be everywhere parallel to a solution $\psi $ of the adjoint differential equation, and that the magnitude of the thrust must be zero in regions where the magnitude of $\psi $ is less than a critical value, and equal to the maximum permissible value in regions where the magnitude of $\psi $ is greater than the critical value. Singular arcs, on which the magnitude of $\psi $ is continuously equal to the critical value, are shown to exist in the case of all except the simplest gravitational fields, and in some cases may form part of an optimal trajectory. A means of calculating the unique value of thrust required to sustain a singular arc is described, and a test for the optimality of such arcs is given. The test shows that a family of singular arcs discovered by D. F. Lawden is nonoptimal.

Minimax Control of Discrete Time Stochastic Systems

D. D. Sworder

J. Soc. Indus. and Appl. Math. Ser. A 2, pp. 433-449 (17 pages) | Cited 1 time

Online Publication Date: July 18, 2006

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Abstract Unavailable
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