An L 1 adaptive closed-loop guidance law for an orbital injection problem

Abstract

The current paper presents the determination of a closed-loop guidance law for an orbital injection problem using two different approaches and, considering the existing time-optimal open-loop trajectory as the nominal solution, compares the advantages of the two proposed strategies. In the first method, named neighbouring optimal control (NOC), the perturbation feedback method is utilized to determine the closed-loop trajectory in an analytical form for the non-linear system. This law, which produces feedback gains, is in general a function of small perturbations appearing in the states and constraints separately. The second method uses an L₁ adaptive strategy in determination of the non-linear closed-loop guidance law. The main advantages of this method include characteristics such as improvement of asymptotic tracking, guaranteed time-delay margin, and smooth control input. The accuracy of the two methods is compared by introducing a high-frequency sinusoidal noise. The simulation results indicate that the L₁ adaptive strategy has a better performance than the NOC method to track the nominal trajectory when the noise amplitude is increased. On the other hand, the main advantage of the NOC method is its ability to solve a non-linear, two-point, boundary-value problem in the minimum time.

Keywords

optimal control optimal guidance

Get full access to this article

View all access options for this article.

References

Bryson

A. E.

Y. C.

Applied optimal control: Optimization, estimation, and control, 1975, pp. 177–203 (Hemisphere, New York).

Kirk

D. E.

Optimal control theory: An introduction, 1970, pp. 202–209 (Prentice-Hall, Englewood Cliffs, New Jersey).

Lee

A. Y.

Smyth

‘Synthesis of minimum-time feedback laws for dynamics systems using neural networks’

J. Guid. Contr. Dyn., 1993, 17(4) 868–870.

Jardin

M. R.

Bryson

A. E.

‘Neighboring optimal aircraft guidance in winds’

J. Guid. Contr. Dyn. 24 (4) (2001): 710–715

Lee

A. Y.

Bryson

A. E.

‘Neighboring extremals of dynamic optimization problems with parameter variations’

Optim. Contr. Appl. Meth. 10 (1) (1989): 39–52

Pourtakdoust

S. H.

Rahbar

Novinzadeh

A. B.

‘Nonlinear feedback optimal control law for minimum-time injection problem using fuzzy system’

Aircraft Engng Aerospace Technol. 77 (5) (2005): 376–383

Chun-Liang

Yu-Ping

Tian-Lin

‘A fuzzy guidance law for vertical launch interceptors’

Control Engng Practice, 2009 17(8), 914–923

Palma

Magni

‘On optimality of nonlinear model predictive control’

Syst. Contr. Lett. 5 (6) (2005): 58–61

Afshari

H. H.

Roshanian

Novinzadeh

A. B.

‘A perturbation approach in determination of a closed-loop optimal-fuzzy control policy for planetary landing mission’

Proc. IMechE, Part G: J. Aerospace Engineering, 2009, 233-243 DOI: 10.1243/09544100JAERO429.

10.

Afshari

H. H.

Zareh

Rezaei

Roshanian

Novinzadeh

A. B.

‘A new adaptive approach in determination of closed-loop guidance law for injection problem.’ In Proceedings of the

2009 IEEE Aerospace conference, Big Sky, Montana, 7-14 March 2009, p. 1705 (International Astronautical Federation, Paris).

11.

Cao

Hovakimyan

‘Design and analysis of a novel L₁ adaptive control architecture, Part 1: Control signal and asymptotic stability.’ In Proceedings of the 2006 American control conference, Minneapolis, Minnesota, 14-16 June 2006, pp. 3397–3402 (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey).

12.

Cao

Hovakimyan

‘Design and analysis of a novel L₁ adaptive controller, Part 2: Guaranteed transient performance.’ In Proceedings of the 2006 American control conference, Minneapolis, Minnesota, 14-16 June 2006, pp. 3403–3408 (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey).

13.

Cao

Hovakimyan

‘Guaranteed transient performance with L₁ adaptive controller for systems with unknown time-varying parameters and bounded disturbance: Part 2.’ In Proceedings of the 2007 American control conference, New York, New York, 9-13 July 2007, pp. 3925–3930 (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey).

14.

Beard

W. R.

Knoebel

Cao

Hovakimyan

Matthews

‘An L₁ adaptive pitch controller for miniature air vehicles.’ AIAA Guidance, Navigation, and Control Conference, AIAA paper 2006–6777

15.

Slotine

J. J.

Applied nonlinear control, (Prentice Hall, Englewood Cliffs, New Jersey 1991)

16.

Åström

Wittenmark

Adaptive control, edition 2, (Addison Wesley, Reading, Massachusetts). 1995)

17.

Cao

Hovakimyan

‘Novel L₁ neural network adaptive control architecture with guaranteed transient performance’

IEEE Trans. Neural Network. 18 (4) (2007): 1160–1171

18.

Wang

Cao

Hovakimyan

Lavretsky

‘Novel L₁ adaptive control approach to autonomous aerial refueling with guaranteed transient performance.’ In Proceedings of the 2006 American control conference, Minneapolis, Minnesota, 14-16 June 2006, pp. 3569–3574 (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey).

19.

Cao

Hovakimyan

‘Stability margins of L₁ adaptive controller.’ In Proceedings of the 2007 American control conference, New York, New York, 9-13 July 2007, pp. 3931–3936 (Institute of Electrical and Electronics Engineers, Piscataway, New Jersey).

20.

Cao

Patel

Reddy

C. K.

Hovakimyan

Lavretsky

‘Are the phase and time-delay margins always adversely affected by high-gain?’ AIAA Guidance, Navigation, and Control Conference, AIAA paper 2006–6347

21.

Wang

Patel

V. V.

Cao

Hovakimyan

Lavretsky

‘L₁ adaptive neural network controller for autonomous aerial refueling with guaranteed transient performance.’ AIAA Guidance, Navigation, and Control Conference, AIAA paper 2006–6206

22.

Pomet

Praly

‘Adaptive nonlinear regulation: Estimation from the Lyapunov equation’

IEEE Trans. Automat. Contr. 37 (6) (1992): 729–740