A method is presented for direct trajectory optimization and costate estimation using global collocation at Legendre-Gauss-Radau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the terminal point. The resulting integration matrix is nonsingular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finite-horizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem (NLP) to the costates of the optimal control problem. Finally, * M.S. Student, Dept. it is shown that a previously developed Radau collocation method, which is restricted to infinite-horizon problems, is subsumed by the method presented in this paper. The results of this paper show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions to general finite-horizon optimal control problems.
A method is presented for direct trajectory optimization and costate estimation using global collocation at Legendre-Gauss-Radau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integral from the initial point to the interior LGR points and the terminal point. The resulting integration matrix is nonsin-gular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finite-horizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the non-linear programming problem (NLP) to the costates of the optimal control problem. Finally, * M.S. Student, it is shown that a previously developed Radau collocation method, which is restricted to infinite-horizon problems, is subsumed by the method presented in this paper. The results of this paper show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions to general finite-horizon optimal control problems.
Local anesthetics are used very often in medicine and dentistry. They have few adverse effects, but the increased use of these drugs has resulted in a higher incidence of local and systemic anesthetic toxicity (LAST). From the initial symptoms to the deleterious effects on cardiac and the central nervous system, LAST is an important consequence of which we should be aware. LAST is known since the introduction and use of local anesthetics; it was originally associated with seizures and respiratory failure. However, in the 1970s, side effects on the heart were also identified, as the fatal cardiac toxicity associated with bupivacaine was discovered in healthy patients. Prevention and safe administration of regional anesthesia remains primary factors in the avoidance of the toxicity of these drugs. When a patient has LAST, treatment should be started immediately to reduce seizures. If there is cardiac arrest, follow ACLS guidelines. Intravenous lipids improve cardiac conduction, contractility and coronary perfusion by removing liposoluble local anesthetic from cardiac tissue.
Herein, we describe the development of a new strategy for the synthesis of unsaturated oligoesters via sequential metal- and reagent-free insertion of vinyl sulfoxonium ylides into the O–H bond of carboxylic acid. Like two directional coupling of amino acids (N- to C-terminal and C- to N-terminal) in peptide synthesis, the present approach offers a strategy in both directions to synthesize oligoesters. The sequential addition of the vinyl sulfoxonium ylide to the carboxylic acids (acid iteration sequence) in one direction and the sequential addition of the carboxylic acids to the vinyl sulfoxonium ylide (ylide iteration sequence) in another direction yield (Z)-configured unsaturated oligoesters. To perform this iteration, we have developed a highly regioselective insertion of vinyl sulfoxonium ylide into the X–H (X = O, N, C, S, halogen) bond of acids, thiols, phenols, amines, indoles, and halogen acids under metal-free reaction conditions. The insertion reaction is applied to a broad range of substrates (>50 examples, up to 99% yield) and eight iterative sequences. Mechanistic studies suggest that the rate-limiting step depends on the type of X–H insertion.
A method is presented for costate estimation in nonlinear optimal control problems using multiple-interval collocation at Legendre-Gauss or Legendre-Gauss-Radau points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuous-time optimal control problem are given. When the optimal costate is continuous, the transformed adjoint systems of the nonlinear programming problems are discrete representations of the continuous-time first-order optimality conditions. If, however, the optimal costate is discontinuous, then the transformed adjoint systems are not discrete representations of the continuous-time firstorder optimality conditions. In the case where the costate is discontinuous, the accuracy of the costate approximation depends on the locations of the mesh points. In particular, the accuracy of the costate approximation is found to be significantly higher when mesh points are located at discontinuities in the costate. Two numerical examples are studied and demonstrate the effectiveness of using the multiple-interval collocation approach for estimating costate in continuous-time nonlinear optimal control problems. = initial time ut = control on time domain t 2 t 0 ; t f u = control on time domain 2 1; 1 U j = control approximation at time point j w j = jth Legendre-Gauss or Legendre-Gauss-Radau quadrature weight X = state approximation on time domain 2 1; 1 xt = state on time domain t 2 t 0 ; t f x = state on time domain 2 1; 1 X j = state approximation at time point j = Lagrange multiplier associated with discretized inequality path constraint = Lagrange multiplier associated with inequality path constraint ij = Kronecker delta function = accuracy tolerance j = Lagrange multiplier of discretized dynamic constraint at j = continuous costate on the time domain 2 1; 1 j = costate approximation at time point j = Lagrange multiplier associated with quadrature constraint = jump in the costate at mesh point = transformed time domain 1; 1 = Mayer cost = boundary condition function = Lagrange multiplier associated with the discretized boundary condition = Lagrange multiplier associated with the boundary condition
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