Direct Shooting Method Research Articles

Trajectory optimization involving sloshing media

This paper is concerned with the optimization of the transport motion of an open topped fluid filled container within a warehouse environment. In particular, optimal trajectories of the motion of the driver–container system in two-dimensional space will be investigated via numerical solutions of the model equations using sequential quadratic programming. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is the time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The systems of ordinary and partial differential equations, representing the dynamics of the models are solved numerically using a direct shooting method. The resulting non-linear programming problem is solved using sequential quadratic programming (SQP). Copyright © 2002 John Wiley & Sons, Ltd.

Numerical Computation of Optimal Feed Rates for a Fed-Batch Fermentation Model

In this paper, we consider a model for a fed-batch fermentation process which describes the biosynthesis of penicillin. First, we solve the problem numerically by using a direct shooting method. By discretization of the control variable, we transform the basic optimal control problem to a finite-dimensional nonlinear programming problem, which is solved numerically by a standard SQP method. Contrary to earlier investigations (Luus, 1993), we consider the problem as a free final time problem, thus obtaining an improved value of the penicillin output. The results indicate that the assumption of a continuous control which underlies the discretization scheme seems not to be valid. In a second step, we apply classical optimal control theory to the fed-batch fermentation problem. We derive a boundary-value problem (BVP) with switching conditions, which can be solved numerically by multiple shooting techniques. It turns out that this BVP is sensitive, which is due to the rigid behavior of the specific growth rate functions. By relaxation of the characteristic parameters, we obtain a simpler BVP, which can be solved by using the predicted control structure (Lim et al., 1986). Now, by path continuation methods, the parameters are changed up to the original values. Thus, we obtain a solution which satisfies all first-order and second-order necessary conditions of optimal control theory. The solution is similar to the one obtained by direct methods, but in addition it contains certain very small bang-bang subarcs of the control. Earlier results on the maximal output of penicillin are improved.

Flight envelopes for hypersonic aerorendezvous at constant altitude

Flight envelopes of a lifting space vehicle at constant-alti tude hypersonic coasting flight for the purpose of aerorendezvous with initial relative heading angles at 0,30, 60,..., and 180 deg are presented. Besides the final position, the final velocity vector (both magnitude and heading) also must be specified for aerorendezvous. Therefore, final conditions are very stringent and the coordinate-system rotation technique is used to ease the numerical computation and to save time. The two-point boundary-value problem resulting from the variational formulation is solved by using the direct shooting method. The flight envelopes are found to be functions of two angles: the initial position and the initial velocity heading of the space vehicle. Both are measured relative to the specified final velocity vector. The envelopes for 0 and 180 deg of initial relative heading angles are symmetric with respect to the longitudinal axis. There are two discontinuity points on each of the envelopes. The other flight envelopes are not symmetric, and each has one discontinuity point only.

Inverse Simulation for Nonlinear Systems Analysis

A heuristic analysis tool for nonlinear dynamical systems is described, which is based on the solution of a sequence of optimal control problems. These are solved by a direct shooting method, which uses information of system simulation to update the control parameters into a desired direction. It is possible to consider vector optimal control problems.

Optimal Control of a High Performance Wind Tunnel

Stringent accuracy and safety tolerances together with the demand to save operational costs require the optimal control of high performance wind tunnels. Based on the thermo-fluidmechanical balance principles a mathematical model of a cryogenic wind tunnel with lumped parameters is developed which is bilinear with time-delay in the mass flows and quadratic in the fan speed. Time optimal controls are discussed for this multivariable model depending on the variation of the state and control constraints and on the location of the control inputs. The computational procedure used is a direct shooting method with optimal discretization.

Optimal control of a high performance wind tunnel

Stringent accuracy and safety tolerances together with the demand to save operational costs require the optimal control of high performance wind tunnels. Based on the thermo-fluidmechanical balance principles a mathematical model of a cryogenic wind tunnel with lumped parameters is developed which is bilinear with time-delay in the mass flows and quadratic in the fan speed. Time optimal controls are discussed for this multivariable model depending on the variation of the state and control constraints and on the location of the control inputs. The computational procedure used is a direct shooting method with optimal discretization.

Direct shooting method, linearization, and nonlinear algebraic equations

The direct shooting method for the solution of boundary-value problems for ordinary, nonlinear differential equations is analyzed from the point of view of linearization. Some relations between this method and perturbation methods are established. The relations between the direct shooting algorithm and Whittaker's algorithm are also established, considering the problem from the point of view of solving nonlinear algebraic equations.

Direct shooting method for the solution of boundary-value problems

A new iterative method is developed to solve the boundary-value problems for ordinary nonlinear differential equations. The method requires that only the original system of differential equations is solved once in each iteration. The initial conditions for a new iteration are evaluated directly from the given boundary conditions and the initial and boundary conditions obtained in the previous iteration, thus avoiding the necessity of solving a system of algebraic equations. The convergence proofs of the method are given. Examples of the application of the method are presented and discussed.