An official website of the United States government
A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc.
more »
« less
Reusable Entry Vehicle Trajectory Optimization Using Multiple-Domain Radau Collocation
The trajectory optimization of a reusable launch vehicle entry is studied. The objective is to maximize the crossrange during atmospheric entry subject to a constraint on the stagnation point heating rate. The problem is solved by partitioning the domain of the independent variable into multiple subdomains such that each subdomain consists of a segment where the heating rate constraint is either active or inactive. Additional necessary conditions for optimality are enforced in segments where the heating rate constraint is active. A multiple-domain Legendre-Gauss-Radau direct collocation method is then used to solve the partitioned problem. Key features of the stagnation point heating rate constraint are presented, and it is observed that the aforementioned approach is capable of solving the problem under consideration more accurately than traditional direct collocation methods.
more »
« less
- Award ID(s):
- 2031213
- PAR ID:
- 10481407
- Publisher / Repository:
- American Institute of Aeronautics and Astronautics
- Date Published:
- ISBN:
- 978-1-62410-699-6
- Format(s):
- Medium: X
- Location:
- National Harbor, MD & Online
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A modified form of Legendre-Gauss orthogonal direct collocation is developed for solving optimal control problems whose solutions are nonsmooth due to control discon- tinuities. This new method adds switch time variables, control variables, and collocation conditions at both endpoints of a mesh interval, whereas these new variables and collocation con- ditions are not included in standard Legendre-Gauss orthogonal collocation. The modified Legendre-Gauss collocation method alters the search space of the resulting nonlinear programming problem and optimizes the switch point of the control solution. The transformed adjoint system of the modified Legendre- Gauss collocation method is then derived and shown to satisfy the necessary conditions for optimality. Finally, an example is provided where the optimal control is bang-bang and contains multiple switches. This method is shown to be capable of solving complex optimal control problems with nonsmooth solutions.more » « less
-
A new method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method developed in this paper employs a modified form of the Legendre-Gauss-Radau (LGR) orthogonal direct collocation method in which an additional variable and two additional constraints are included at the end of a mesh interval. The additional variable is the switch time where a discontinuity occurs. The two additional constraints are a collocation condition on each differential equation that is a function of control along with a control constraint at the endpoint of the mesh interval that defines the location of the nonsmoothness. These additional constraints modify the search space of the NLP in a manner such that an accurate approximation to the location of the nonsmoothness is obtained. An example with a nonsmooth solution is used throughout the paper to illustrate the improvement of the method over the standard Legendre-Gauss-Radau collocation method.more » « less
-
null (Ed.)A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre– Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh inter- val. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Fur- thermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.more » « less
-
null (Ed.)A general-purpose C++ software program called CGPOPS is described for solving multiple-phase optimal control problems using adaptive direct orthogonal collocation methods. The software employs a Legendre-Gauss-Radau direct orthogonal collocation method to transcribe the continuous optimal control problem into a large sparse nonlinear programming problem (NLP). A class of hp mesh refinement methods are implemented that determine the number of mesh intervals and the degree of the approximating polynomial within each mesh interval to achieve a specified accuracy tolerance. The software is interfaced with the open source Newton NLP solver IPOPT. All derivatives required by the NLP solver are computed via central finite differencing, bicomplex-step derivative approximations, hyper-dual derivative approximations, or automatic differentiation. The key components of the software are described in detail, and the utility of the software is demonstrated on five optimal control problems of varying complexity. The software described in this article provides researchers a transitional platform to solve a wide variety of complex constrained optimal control problems.more » « less
