Variable–Time–Domain Neighboring Optimal Guidance, Part 1: Algorithm Structure

Abstract

This paper presents a general purpose neighboring optimal guidance algorithm that is capable of driving a dynamical system along a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of a closed-loop guidance scheme. Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Original analytical developments, based on optimal control theory, in conjunction with the use of a normalized time scale, constitute the theoretical foundation for three relevant features: (i) a new, efficient law for the real-time update of the time of flight (the so called time-to-go), (ii) a new termination criterion, and (iii) a new analytical formulation of the sweep method. This new guidance, termed variable–time–domain neighboring optimal guidance, is rather general, avoids the usual numerical difficulties related to the occurrence of singularities for the gain matrices, and is exempt from the main disadvantages of similar algorithms proposed in the past. For these reasons, the variable–time–domain neighboring optimal guidance has all the ingredients for being successfully applied to problems of practical interest.

Appendix: Notation and Matrices with the Related Dimensions

This appendix is focused on clarifying some aspects of matrix and vector notation, and collects most of the matrices used to implement VTD-NOG, with the respective dimensions.

First of all, if a matrix $\varvec{\Gamma }_{\mathbf {bp}}$ is defined as the result of the successive derivation of a scalar variable $\Gamma$ with respect to two column vectors b and p (with sizes $n_{b} \times 1$ and $n_{p} \times 1)$, then its dimension is $n_{b} \times n_{p}$. It is apparent that $\varvec{\Gamma }_{\mathbf {bp}} =\Gamma _{\mathbf {pb}}^T$. Moreover, the symbol ${\mathbf {I}}_{l\times l}$ denotes the identity matrix of order $l$, whereas ${\mathbf {0}}_{l\times r}$ represents a ($l\times r)$-matrix composed of zeros. Lastly, the symbol 0 is employed several times throughout the paper, and denotes a column vector of zeros, with dimension appropriate to the context.

If the guidance problem involves a state vector with $n$ components, a control vector with $m$ components, $p$ unknown parameters and $q$ boundary conditions, the matrices extensively used in VTD-NOG have the following dimensions:

$$\begin{aligned} \begin{array}{l} {\mathbf {f}}_{\mathbf {x}} \quad ( {1\times n}) \quad {\mathbf {f}}_{\mathbf {u}} \quad ( {1\times m}) \quad {\mathbf {f}}_{\mathbf {a}} \quad ( {1\times p}) \\ H_{\mathbf {u}} \quad ( {1\times m}) \quad H_{{\mathbf {xx}}} \quad ( {n\times n}) \quad H_{\mathbf {xu}} \quad ( {n\times m}) \quad H_{\mathbf {xa}} \quad ( {n\times p}) \quad H_{{\mathbf {uu}}} \quad ( {m\times m}) \quad \\ \qquad H_{\mathbf {ua}} \quad ( {m\times p})\\ {\varvec{\psi }}_{{\mathbf {x}}_0 } \quad ( {q\times n}) \quad {\varvec{\psi }}_{{\mathbf {x}}_\mathrm{{f}} } \quad ( {q\times n}) \quad {\varvec{\psi }}_{\mathbf {a}} \quad ( {q\times p}) \\ \Phi _{{\mathbf {x}}_0 {\mathbf {x}}_0 } \quad ( {n\times n}) \quad \Phi _{{\mathbf {x}}_\mathrm{{f}} {\mathbf {x}}_\mathrm{{f}} } \quad ( {n\times n}) \quad \Phi _{{\mathbf {x}}_0 {\mathbf {a}}} \quad ( {n\times p}) \quad \Phi _{{\mathbf {x}}_\mathrm{{f}} {\mathbf {a}}} \quad ( {n\times p}) \quad \Phi _{\mathbf {aa}} \quad ( {p\times p}) \\ {\mathbf {A}} \quad ( {n\times n}) \quad {\mathbf {B}} \quad ( {n\times n}) \quad {\mathbf {C}} \quad ( {n\times n}) \quad {\mathbf {D}} \quad ( {n\times p}) \quad {\mathbf {E}} \quad ( {n\times p}) \quad {\mathbf {F}} \quad ( {p\times p}) \\ {\mathbf {U}} \quad ( {n\times ( {q+p})}) \quad {\mathbf {V}} \quad ( {( {q+p})\times ( {q+p})}) \quad {\mathbf {W}} \quad ( {n\times p}) \quad \Theta \quad ( {( {q+p})\times p}) \\ {\mathbf {S}} \quad ( {n\times n}) \quad \hat{\mathrm{S}} \quad ( {n\times n}) \quad {\mathbf {R}} \quad ( {n\times q}) \quad {\mathbf {Q}} \quad ( {q\times q}) \quad {\mathbf {m}} \quad ( {n\times p}) \quad {\mathbf {n}} \quad ( {q\times p}) \quad \\ \qquad \varvec{\alpha } \quad ( {p\times p})\\ \end{array} \end{aligned}$$

Finally, it is worth remarking that, due to definition (7),

$$\begin{aligned} H_{\varvec{\lambda } \mathbf {g}} ={\mathbf {f}}_{\mathbf {g}} \quad ( {\mathbf {g}={\mathbf {x}} \text{ or } {\mathbf {u}} \text{ or } {\mathbf {a}}}) \end{aligned}$$

(74)