Numerical computation of neighboring optimum feedback control schemes in real-time

A modification of the theory of neighboring extremals is presented which leads to a new formulation of a linear boundary value problem for the perturbation of the state and adjoint variables around a reference trajectory. On the basis of the multiple shooting algorithm, a numerical method for stable and efficient computation of perturbation feedback schemes is developed. This method is then applied to guidance problems in astronautics. Using as much stored a priori information about the precalculated flight path as possible, the only computational work to be done on the board computer for the computation of a regenerated optimal control program is a single integration of the state differential equations and the solution of a few small systems of linear equations. The amount of computation is small enough to be carried through on modern board computers for real-time. Nevertheless, the controllability region is large enough to compensate realistic flight disturbances, so that optimality is preserved.

[1]  J. V. Breakwell,et al.  On the conjugate point condition for the control problem , 1965 .

[2]  William B. Gragg,et al.  On Extrapolation Algorithms for Ordinary Initial Value Problems , 1965 .

[3]  T. E. Hull.  NUMERICAL SOLUTIONS OF INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS , 1975 .

[4]  H. Kelley.  Guidance theory and extremal fields , 1962 .

[5]  E. Fehlberg.  Classical Fifth-, Sixth-, Seventh-, and Eighth-Order Runge-Kutta Formulas with Stepsize Control , 1968 .

[6]  A. Bryson,et al.  Optimization and Control of Nonlinear Systems Using the Second Variation , 1963 .

[7]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[8]  A. Bryson,et al.  A SUCCESSIVE SWEEP METHOD FOR SOLVING OPTIMAL PROGRAMMING PROBLEMS , 1965 .

[9]  A. Bryson,et al.  Second-Order Optimality Conditions for Variable End Time Terminal Control Problems , 1973 .

[10]  J. Stoer,et al.  Numerical treatment of ordinary differential equations by extrapolation methods , 1966 .

[11]  W. F. Powers,et al.  A method for comparing trajectories in optimum linear perturbation guidance schemes. , 1968 .

[12]  L. J. Wood.  Perturbation guidance for minimum time flight paths of spacecraft. , 1972 .

[13]  William F. Powers,et al.  Techniques for improved convergence in neighboring optimum guidance , 1969 .

[14]  A. Sedgwick,et al.  An effective variable-order variable-step adams method. , 1973 .

[15]  A. E. Bryson,et al.  A neighboring optimum feedback control scheme based on estimated time-to-go with application to re-entry flight paths , 1968 .

[16]  Imsong Lee,et al.  Optimal Trajectory, Guidance, and Conjugate Points , 1965, Inf. Control..

[17]  W. H. Enright,et al.  Test Results on Initial Value Methods for Non-Stiff Ordinary Differential Equations , 1976 .

[18]  P. Dyer,et al.  Optimization of control systems with discontinuities and terminal constraints , 1969 .

[19]  E. Fehlberg,et al.  Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems , 1969 .

[20]  H. Kelley.  An optimal guidance approximation theory , 1964 .

[21]  H. J. Pesch,et al.  Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting , 1976 .

[22]  R. England,et al.  Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations , 1969, Comput. J..

[23]  W. Schmitendorf,et al.  On the applicability of the sweep method to optimal control problems , 1969 .

[24]  Sanjoy K. Mitter,et al.  Successive approximation methods for the solution of optimal control problems , 1966, Autom..

[25]  Hans Josef Pesch,et al.  Numerische Berechnung optimaler Flugbahnkorrekturen in Echtzeitrechnung , 1978 .