Title:: Optimal control with aerospace applications URL: Zotero Link: [Longuski et al_2014_Optimal control with aerospace applications.pdf](zotero://select/library/items/44YGTV5E) we adjoin terminal constraints to the cost functional through the use of additional Lagrange multipliers. We refer to this approach as the “adjoined method” and note that it has become a sort of gold standard in the literature since the revised printing of Bryson and Ho’s Applied Optimal Control in 1975” Yellow Highlight [Page 6](zotero://open-pdf/library/items/44YGTV5E?page=6&annotation=XTRZE6IV) In parameter optimization” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=3QENPRY3) we minimize a function of a finite number of parameters” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=WI5RGE8F) Optimal control” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=S39UMDIY) seeks x(t), an n-vector, that minimizes something called a functional” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=JMFMQR5N) Equation (1.2) is a necessary condition” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=9A7WS7IQ) We note that f can be maximized by minimizing −f.” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=2844AM3G) ∂2f ∂xi∂xj (evaluated at x∗), is a positive-definite matrix, which provides a sufficient condition for a local minimum” Yellow Highlight [Page 22](zotero://open-pdf/library/items/44YGTV5E?page=22&annotation=RJQJGR3H) We note that we can write the “variation”: δJ = ∂f ∂x1 δx1 + ··· + ∂f ∂xn δxn (1.6) for an infinitesimal, arbitrary displacement δx and conclude that at a stationary point (i.e., Eq. (1.2)): δJ = 0” Yellow Highlight [Page 24](zotero://open-pdf/library/items/44YGTV5E?page=24&annotation=KJDBBIDP) Determine the rectangle of maximum perimeter that can be inscribed in a unit circle.” Yellow Highlight [Page 25](zotero://open-pdf/library/items/44YGTV5E?page=25&annotation=X8Y2SS2E) Given a circular field, what is the biggest rectangle that we can make within it? J = f (x, y)=−4(x + y)” Yellow Highlight [Page 25](zotero://open-pdf/library/items/44YGTV5E?page=25&annotation=L4KRX48A) In the Hohmann transfer, only two impulses are applied. Can the total cost ( V) be reduced with more impulses?” Yellow Highlight [Page 31](zotero://open-pdf/library/items/44YGTV5E?page=31&annotation=WLSA36FN) In 1959 three articles” Yellow Highlight [Page 31](zotero://open-pdf/library/items/44YGTV5E?page=31&annotation=M8PWB6GW) showed that if the radius ratio rf/ro is sufficiently large, other transfers exist which require less propellant than the Hohmann transfer” Yellow Highlight [Page 31](zotero://open-pdf/library/items/44YGTV5E?page=31&annotation=8JGJQQEX) These other transfers are based on the three-impulse, bi-elliptic transfer” Yellow Highlight [Page 31](zotero://open-pdf/library/items/44YGTV5E?page=31&annotation=6FBTJH6S) The problem is to find the path, y = y(x), to minimize the time given by Eq. (2.18).” Yellow Highlight [Page 44](zotero://open-pdf/library/items/44YGTV5E?page=44&annotation=8LX24BMS) The general problem is to find y(x) that makes J stationary (such that small changes in y(x) make no change in J).” Yellow Highlight [Page 45](zotero://open-pdf/library/items/44YGTV5E?page=45&annotation=6GRESRLF) It was discovered that y(x) must satisfy the Euler-Lagrange equation” Yellow Highlight [Page 45](zotero://open-pdf/library/items/44YGTV5E?page=45&annotation=UATHGVQ2) The problem of launching a satellite into orbit is closely related to the brachistochrone problem.” Yellow Highlight [Page 45](zotero://open-pdf/library/items/44YGTV5E?page=45&annotation=AT7P4NM6) scalar J is called a functional because it maps functions [the path x(t) and control u(t)] into a single number” Yellow Highlight [Page 46](zotero://open-pdf/library/items/44YGTV5E?page=46&annotation=8PZJVHXR) typical examples for J are the propellant used to launch a spacecraft into orbit or the time for a particle to travel between points” Yellow Highlight [Page 46](zotero://open-pdf/library/items/44YGTV5E?page=46&annotation=94TQDYTR) (t, ε) represents a family of curves that are near the optimal path” Yellow Highlight [Page 59](zotero://open-pdf/library/items/44YGTV5E?page=59&annotation=UDWDX6FB) When ε is set to zero, the curve is the optimal path” Yellow Highlight [Page 59](zotero://open-pdf/library/items/44YGTV5E?page=59&annotation=LBXL9FMH) In our search for the optimal path we will find that derivatives with respect to parameters (such as ε), as well as derivatives with respect to time, must be taken” Yellow Highlight [Page 59](zotero://open-pdf/library/items/44YGTV5E?page=59&annotation=YBTUR929) first variation of a function x(t, ε)” Blue Highlight [Page 59](zotero://open-pdf/library/items/44YGTV5E?page=59&annotation=67DXW4B6) ∂x(t, ε) ∂ε ε=0” Yellow Highlight [Page 59](zotero://open-pdf/library/items/44YGTV5E?page=59&annotation=B252MG9G) η(t)” Yellow Highlight [Page 59](zotero://open-pdf/library/items/44YGTV5E?page=59&annotation=Q6KLEC95) the Euler-Lagrange theorem” Yellow Highlight [Page 61](zotero://open-pdf/library/items/44YGTV5E?page=61&annotation=BC32QMVI) provides the necessary conditions for trajectory optimization” Yellow Highlight [Page 61](zotero://open-pdf/library/items/44YGTV5E?page=61&annotation=VW7XCU77) The Euler-Lagrange equation is simpler to derive than the theorem because the Euler-Lagrange equation only solves for a path” Yellow Highlight [Page 61](zotero://open-pdf/library/items/44YGTV5E?page=61&annotation=AKWHDJT6) the Weierstrass condition appears to be identical to Pontryagin’s Minimum Principle. The main difference is that the former requires the control to be a continuous, unbounded function of time while the latter is far more general in allowing the control to be a “measurable function” (which includes piecewise continuous, bounded functions of time). We are interested in applications in which the control may be a piecewise continuous function (e.g., bang-bang control in which a thruster is turned on and off)” Yellow Highlight [Page 114](zotero://open-pdf/library/items/44YGTV5E?page=114&annotation=6RAUTQTH) %% Import Date: 2023-06-11T15:06:19.922+01:00 %%