# Jason L. Speyer, David H. Jacobson's Primer on Optimal Control Theory (Advances in Design and PDF

By Jason L. Speyer, David H. Jacobson

ISBN-10: 0898716942

ISBN-13: 9780898716948

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Additional resources for Primer on Optimal Control Theory (Advances in Design and Control)

Sample text

49) Note that if φ is quadratic, the Newton–Raphson method converges to a minimum in one step. Accelerated Gradient Methods Since it is numerically ineﬃcient to compute φxx (xi ), this second partial derivative can be estimated by constructing n independent directions from a sequence of gradients, φTx (xi ), i = 1, 2, . . , n. For a quadratic function, this class of numerical optimization algorithms, called accelerated gradient methods, converges in n steps. The most common of these methods are the quasi-Newton methods, so called because as the estimate of φxx (xi ), called the Hessian, approaches the actual value, the method approaches the Newton–Raphson method.

U ψ =0 ψ Ty φ yT -P yT Lines of constant performance index. 7: Geometrical description of parameter optimization problem. 117) is a positive number chosen small so as not to violate the assumed linearity. 3. Minimization Subject to Constraints 43 (since P P T = P ). 119) where δf = fy δy = − fy P φTy = 0. The second-order constraint violation is then restored by going back to the constraint restoration step. 120) are met. Note that the constraint restoration and optimization steps can be combined given the assumed linearity.

Intuitively, the requirement is that the function φ takes on a minimum value on the tangent plane of the constraint. This is done by using the relation between δx and δu of δx = − fu (xo , uo ) δu. , fx δx + fu δu = 0. 83). 83) is veriﬁed as φˆuu (uo ) = 16a/b > 0, ensuring that φ is a locally constrained maximum at uo . 1 Let fi : Rn+m → R, i = 1, . . , n, be n continuously diﬀerentiable constraints and φ : Rn+m → R be the continuously diﬀerentiable performance index. 87) subject to fi (x, u) = ci , i = 1, .