Optimal Nonlinear Feedback Control Design Using a Waypoint Method

Abstract

This paper discusses an innovative idea of blending the notion of a waypoint scheme with a series solution method developed by the authors for solving the Hamilton-Jacobi-Bellman equation in the context of designing optimal feedback control laws for nonlinear dynamic systems subject to terminal constraints. The overall time interval of the given problem is partitioned into smaller segments, and the series solution method is applied within each segment using stored gains that are computed from one segment only. The methodology is applied to highly nonlinear systems including a minimum orbit transfer problem. Several examples are demonstrated, and the results are compared with the corresponding open-loop solutions to demonstrate the efficacy of the proposed method.

New near-optimal feedback guidance algorithms for space missions

“…The zero-effort-miss (ZEM) distance and zero-effort-velocity (ZEV) error denote, respectively, the differences between the desired final position and velocity and the projected final position and velocity if no additional control is commanded after the current time. For the assumed gravitational acceleration g (t), the ZEM and ZEV have the following expressions [22,28]:…”

“…For these highly nonlinear cases, a general way to improve the performance of the ZEM/ZEV feedback algorithm is to divide the total flight time into one or more segments and somehow determine optimal or near-optimal waypoints to connect the different segments. Such a waypoint concept was considered by Sharma et al, [28] and the computational method was provided to solve nonlinear optimal control problems with terminal constraints.…”

“…Reference [21] showed that in a uniform gravitational field, the ZEM/ZEV algorithm is equivalent to well-known classical optimal feedback controllers for problems such as intercept or rendezvous, [18] terminal guidance, [3] and planetary landing. [28,40] All of the optimal feedback guidance algorithms discussed above are concerned only with the specified boundary conditions. For the Mars landing problem, this is not sufficient.…”

Closed-form solution to finite-horizon suboptimal control of nonlinear systems

“…x; t f / D S (5) and the control is then calculated in a feedback form as [17] investigates this issue and provides some remedies for factorization of non-polynomial terms in the dynamics, for example, sin.x/. Considering the degree of freedom in factorization of f .x/, matrix A.x/ has to be selected such that it satisfies the conditions given in Assumption 1.…”

“…The drawback of this 2688 A. HEYDARI AND S. N. BALAKRISHNAN method is the limited domain of convergence of the series, such as other series-based solutions to the optimal control problems. Authors of [5] defined some waypoints to break the main problem into several simpler problems to avoid divergence when using series-based method for solving the problem. Finite-horizon optimal control using generating functions [6] also can be classified as series-based solutions.…”