Optimal trajectory, guidance, and conjugate points

Lee, Imsong

doi:10.1016/s0019-9958(65)90417-1

Cited by 22 publications

(18 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where zðtÞ is the Gaussian white noise with the correlation matrix zðt 1 Þzðt 2 Þ h i¼ H À1 dðt 1 À t 2 Þ: Derivation of Equation (14) for the covariance matrix of the linear Gaussian Markovian process (16) can be found, for example, in Reference [10]. The density of the probability distribution for the random process xðtÞ is determined by…”

Section: Reduction Of Dimensionality In Choosing Optimal Control Formentioning

confidence: 99%

“…Introduce the state subspace R contr ðt; D 1 ð0ÞÞ that is the set of current points xðtÞ of trajectories of system (16) provided that zðsÞ (for 04s4t) is a limited piecewise continuous arbitrary control input and that initial values xð0Þ satisfy the inequality xð0Þ T D þ 1 ð0Þxð0Þ > 0: Recall that DðT À tÞ ¼ D 1 ðtÞ; 04t4T : Introduce also the state subspace R x ðt; D 1 ð0ÞÞ in which the probability density for the random Gaussian vector xðtÞ is localized, provided that xðtÞ satisfies Equation (16) with zðsÞ represented by the Gaussian white noise with the correlation matrix zðs 1 Þzðs 2 Þ h i¼ H À1 dðs 1 À s 2 Þ: It is also assumed that the initial vector xð0Þ is the random Gaussian vector with the covariance matrix D 1 ð0Þ and that the random process zðsÞ (for 04s4t) and the random vector xð0Þ are independent.…”

Section: Proof Of Statement 1 Of Theoremmentioning

confidence: 99%

“…A solution of the linear system of differential equation (16) can be written in the well-known form (see e.g. References [1,11]):…”

Section: Proof Of Statement 2 Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Reduction of dimensionality in choosing optimal control

Khazen¹

2003

Adaptive Control & Signal

View full text Add to dashboard Cite

Given that xðtÞ ¼ ðx 1 ðtÞ; . . . ; x n ðtÞÞ represents a trajectory of the considered dynamical system, we demonstrate that reduction of dimensionality of xðtÞ by using its principal components in choosing the optimal control uðtÞ is necessary to overcome the computational difficulties that might arise in obtaining a solution of the optimal control problem. We obtain explicit solutions for linear time-varying systems with quadratic cost functional and show how to overcome difficulties appearing in non-linear matrix differential equations of Riccati type. A gradient method of searching for the close-to-optimal trajectory and control in non-linear optimal control problems is also considered.

show abstract

Section: Reduction Of Dimensionality In Choosing Optimal Control Formentioning

confidence: 99%

Section: Proof Of Statement 1 Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Reduction of dimensionality in choosing optimal control

Khazen¹

2003

Adaptive Control & Signal

View full text Add to dashboard Cite

show abstract

“…Moreover, it is possible to derive a continuous linear feedback law [24] similar to those neighboring optimum feedback laws which are obtained using the backward sweep method to solve the linear boundary value problem (see e.g. [3], [21], [22], [23], Dyer, McReynolds [8], and Wood [31]). Because of the numerical instability, the use of such feedback laws is not recommended.…”

Section: Si~- C ~ Si B Cbmentioning

confidence: 99%

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

Pesch¹

1979

Appl Math Optim

View full text Add to dashboard Cite

Abstract.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.

show abstract

“…As it is known the boundedness of the solution of the matrix Riccati equation is equivalent to the "no-conjugate point to the final time" [1, 7,10].…”

Section: Introductionmentioning

confidence: 99%

A computational solution for a Matrix Riccati differential equation

Razzaghi

1979

Numer. Math.

View full text Add to dashboard Cite

Summary. This paper is concerned with the solution of the finite time Riccati equation. The solution to the Riccati equation is given in terms of the partition of the transition matrix. Matrix differential equations for the partition of the transition matrix are derived and are solved using computational methods. Examples illustrating the method are presented and the computational algorithms are given.

show abstract

Optimal trajectory, guidance, and conjugate points

Cited by 22 publications

References 5 publications

Reduction of dimensionality in choosing optimal control

Reduction of dimensionality in choosing optimal control

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

A computational solution for a Matrix Riccati differential equation

Contact Info

Product

Resources

About