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Optimality in Optimal Control Problems
Published in Simant Ranjan Upreti, Optimal Control for Chemical Engineers, 2016
Now Equation (3.20) is certainly satisfied if the coefficient of δy is zero, or equivalently λ·=−(Fy+λGy),0≤t≤tf The above equation is known as the Euler Lagrange equation in honor of Swiss mathematician Leonard Euler (1707–1783) and French mathematician Joseph Luis de Lagrange (1736–1813). The Euler-Lagrange equation is also called the adjoint or costate equation since it defines the adjoint or costate variable λ.
Optimal control of brakes and steering for autonomous collision avoidance using modified Hamiltonian algorithm
Published in Vehicle System Dynamics, 2019
Yangyan Gao, Timothy Gordon, Mathias Lidberg
We now re-state the optimisation problem: to minimise lateral off-tracking into the adjacent lane () subject to the constraint of avoiding collision at : where is the time at point . However, between and , the optimal control is simply , and hence . Further, for this type of problem, the control will remain at its limits [13]. Hence the optimisation can be restricted to phase 1 and the optimality condition – Equation (5) is simplified to the minimisation of a terminal cost at with θ as the control variable: This optimal control problem is solved in the Appendix, where the optimal control can be obtained by solving the coupled state and costate equations as a two-point boundary value problem. The optimal control law is found to be in the form of a ‘bilinear tangent law’ [13]: Constants are determined, though not fully explicitly, in terms of the problem parameters. Here we solve the optimisation problem numerically and use the analytical solution as a check on accuracy. We apply the MATLAB built-in function bvp4c [14] for solving the two-point boundary-value problem given in the Appendix. Figure 2 shows the numerical solution (solid line) and is compared to analytical solution obtained from Equation (7) (indicated with stars). Figure 3 gives further validation of the solution, as the Hamiltonian function should equate to zero (see Appendix).