Comparing Performance of Vectorized High Order Expansions and SDRE Method for Vertical Landing Mission of Booster

Document Type : Dynamics, Vibrations, and Control

Authors

1 Ph.D. Student, Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Tehran, Iran

2 Corresponding author: Associate Professor, Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Tehran, Iran

3 Assistant Professor, Faculty of Electrical & Computer Engineering, Malek Ashtar University of Technology, Tehran, Iran

Abstract

In this research the main goal is to compare the performance of Vectorized High Order Expansions and SDRE method for vertical landing of booster. To this end, at first a comprehensive study of references related to both Vectorized High Order and SDRE method has been performed. Then, the Vectorized High Order Expansions method and the implementation in optimal control problems has been introduced. After that, the SDRE method has been reviewed briefly, and the landing problem has been solved using both methods. To evaluate the performance, a set of various simulations have been performed for both methods and with respect to different initial deviations. By means of simulation results, the performance of both method is studied with regard to landing point errors. To achieve this, statistical results of terminal state errors have been calculated for both method with respect to different initial deviation to evaluate and compare both method, quality-wise.

Highlights

[1] Jones H, editor The recent large reduction in space launch cost2018: 48th International Conference on Environmental Systems.##

[2] Banerjee A, Padhi R, editors. An optimal explicit guidance algorithm for terminal descent phase of lunar soft landing. AIAA Guidance, Navigation, and Control Conference; 2017.##

[3] Swaminathan S, Ghose D, editors. Real time powered descent guidance algorithm for mars pinpoint landing with inequality constraints. AIAA Scitech 2020 Forum; 2020.##

[4] Anglim KSG. Minimum-fuel optimal trajectory for reusable first-stage rocket landing using Particle Swarm Optimization: California State University, Long Beach; 2016.##

[5] Li Y, Chen W, Zhou H, Yang L. Conjugate gradient method with pseudospectral collocation scheme for optimal rocket landing guidance. Aerospace Science and Technology. 2020;104:105999.##

[6] Guibout VM, Scheeres DJ. Solving two-point boundary value problems using generating functions: Theory and Applications to optimal control and the study of Hamiltonian dynamical systems. arXiv preprint math/0310475. 2003.##

[7] Guibout VM, Scheeres DJ. Solving two-point boundary value problems using generating functions: Theory and applications to astrodynamics. Modern Astrodynamics. 2006:53-105.##

[8] Guibout VM, Scheeres DJ. Solving relative two-point boundary value problems: Spacecraft formulation flight transfers application. Journal of guidance, control, and dynamics. 2004;27(4):693-704.##

[9] Di Lizia P, Armellin R, Ercoli-Finzi A, Berz M. High-order robust guidance of interplanetary trajectories based on differential algebra. Journal of Aerospace Engineering, Sciences and Applications. 2008;1(1):43-57.##

[10] Di Lizia P, Armellin R, Bernelli-Zazzera F, Berz M. High order optimal control of space trajectories with uncertain boundary conditions. Acta Astronautica. 2014;93:217-29.##

[11] Di Lizia P, Armellin R, Morselli A, Bernelli-Zazzera F. High order optimal feedback control of space trajectories with bounded control. Acta Astronautica. 2014;94(1):383-94.##

[12] Wittig A, Armellin R. High order transfer maps for perturbed Keplerian motion. Celestial Mechanics and Dynamical Astronomy. 2015;122(4):333-58.##

[13] Vetrisano M, Vasile M. Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions. Advances in Space Research. 2017;60(1):38-56.##

[14] Sun Z-J, Di Lizia P, Bernelli-Zazzera F, Luo Y-Z, Lin K-P. High-order state transition polynomial with time expansion based on differential algebra. Acta Astronautica. 2019;163:45-55.##

[15] Morselli A, Armellin R, Di Lizia P, Zazzera FB. A high order method for orbital conjunctions analysis: sensitivity to initial uncertainties. Advances in Space Research. 2014;53(3):490-508.##

[16] Morselli A, Armellin R, Di Lizia P, Zazzera FB. A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation. Advances in Space Research. 2015;55(1):311-33.##

[17] Moghadasian M, Roshanian J. Optimal Landing of Unmanned Aerial Vehicle Using Vectorised High Order Expansions Method. Modares Mechanical Engineering. 2019;19(11):2761-9.##

[18] Moghadasian M, Roshanian J. Continuous maneuver of unmanned aerial vehicle using High Order Expansions method for optimal control problem. Modares Mechanical Engineering. 2018;17(12):382-90.##

[19] Moghadasian M, Roshanian J. Approximately optimal manoeuvre strategy for aero-assisted space mission. Advances in Space Research. 2019;64(2):436-50.##

[20] Çimen T. State-dependent Riccati equation (SDRE) control: a survey. IFAC Proceedings Volumes. 2008;41(2):3761-75.##

[21] Pearson J. Approximation methods in optimal control I. Sub-optimal control. International Journal of Electronics. 1962;13(5):453-69.##

[22] Cloutier JR, D’Souza CN, Mracek CP, editors. Nonlinear regulation and nonlinear H∞ control via the state-dependent Riccati equation technique: Part 1, theory. Proceedings of the international conference on nonlinear problems in aviation and aerospace; 1996: Embry Riddle University.##

[23] Cloutier J, D'Souza C, Mracek C, editors. Nonlinear regulation and nonlinear $ H_ {infty} $ control via the state-dependent Riccati equation technique: Part2 Examples. Proceedings of the International Conference on Nonlinear Problems in Aviation and Aerospace; 1996.##

[24] Cloutier JR, editor State-dependent Riccati equation techniques: an overview. Proceedings of the 1997 American control conference (Cat No 97CH36041); 1997: IEEE.##

[25] Mracek CP, Cloutier JR. Control designs for the nonlinear benchmark problem via the state‐dependent Riccati equation method. International Journal of robust and nonlinear control. 1998;8(4‐5):401-33.##

[26] Cloutier JR, Stansbery DT, editors. The capabilities and art of state-dependent Riccati equation-based design. Proceedings of the 2002 American Control Conference (IEEE Cat No CH37301); 2002: IEEE.##

[27] Wernli A, Cook G. Suboptimal control for the nonlinear quadratic regulator problem. Automatica. 1975;11(1):75-84.##

[28] VADALI S, editor Examination of the optimal nonlinear regulator problem. Guidance, Navigation and Control Conference; 1988.##

[29] Banks S, Mhana K. Optimal control and stabilization for nonlinear systems. IMA journal of mathematical control and information. 1992;9(2):179-96.##

[30] Gong C, Thompson S. A comment on ‘Stabilization and optimal control for nonlinear systems’. IMA Journal of Mathematical Control and Information. 1995;12(4):395-8.##

[31] Tsiotras P, Corless M, Rotea M. Counterexample to a recent result on the stability of nonlinear systems. IMA Journal of Mathematical Control and Information. 1996;13(2):129-30.##

[32] Mracek CP, Cloutier JR, editors. Missile longitudinal autopilot design using the state-dependent Riccati equation method. Proceedings of the International Conference on Nonlinear Problems in Aviation and Aerospace; 1996.##

[33] Mracek C, Cloutier J, Cloutier J, Mracek C, editors. Full envelope missile longitudinal autopilot design using the state-dependent Riccati equation method. Guidance, Navigation, and Control Conference; 1997.##

[34] Wise KA, Sedwick JL, editors. Nonlinear control of agile missiles using state dependent Riccati equations. Proceedings of the 1997 American Control Conference (Cat No 97CH36041); 1997: IEEE.##

[35] Stansbery D, Cloutier J, editors. Nonlinear, hybrid bank-to-turn/skid-to-turn missile autopilot design. AIAA guidance, navigation, and control conference and exhibit; 2001.##

[36] Mracek CP. SDRE autopilot for dual controlled missiles. IFAC Proceedings Volumes. 2007;40(7):750-5.##

[37] ÇLimen T. A generic approach to missile autopilot design using state-dependent nonlinear control. IFAC Proceedings Volumes. 2011;44(1):9587-600.##

[38] Cloutier J, Stansbery D, editors. All-aspect acceleration-limited homing guidance. Guidance, Navigation, and Control Conference and Exhibit; 1999.##

[39] Ratnoo A, Ghose D. State-dependent Riccati-equation-based guidance law for impact-angle-constrained trajectories. Journal of Guidance, Control, and Dynamics. 2009;32(1):320-6.##

[40] Vaddi S, Menon PK, Ohlmeyer EJ. Numerical state-dependent Riccati equation approach for missile integrated guidance control. Journal of guidance, control, and dynamics. 2009;32(2):699-703.

[41] Ewing CM. An analysis of a new nonlinear estimation technique: The state-dependent Ricatti equation method: University of Florida; 1999.##

[42] Parrish DK, Ridgely DB, editors. Attitude control of a satellite using the SDRE method. Proceedings of the 1997 American Control Conference (Cat No 97CH36041); 1997: IEEE.##

[43] Hammett KD, Hall CD, Ridgely DB. Controllability issues in nonlinear state-dependent Riccati equation control. Journal of guidance, control, and dynamics. 1998;21(5):767-73.##

[44] Stansbery DT, Cloutier JR, editors. Position and attitude control of a spacecraft using the state-dependent Riccati equation technique. Proceedings of the 2000 American Control Conference ACC (IEEE Cat No 00CH36334); 2000: IEEE.##

[45] Cloutier JR, Zipfel PH, editors. Hypersonic guidance via the state-dependent Riccati equation control method. Proceedings of the 1999 IEEE International Conference on Control Applications (Cat No 99CH36328); 1999: IEEE.##

[46] Harman RR, Bar-Itzhack IY. Pseudolinear and state-dependent Riccati equation filters for angular rate estimation. Journal of Guidance, Control, and Dynamics. 1999;22(5):723-5.##

[47] Chang I, Park S-Y, Choi K-H. Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique. International Journal of Non-Linear Mechanics. 2009;44(8):891-904.##

[48] Lin L-G, Xin M. Computational enhancement of the SDRE scheme: general theory and robotic control system. IEEE Transactions on Robotics. 2020;36(3):875-93.##

[49] Lin L-G. Computationally Improved State-Dependent Riccati Equation Scheme for Nonlinear Benchmark System. IEEE/ASME Transactions on Mechatronics. 2020;26(2):1064-75.##

Keywords

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References

[1] Jones H, editor The recent large reduction in space launch cost2018: 48th International Conference on Environmental Systems.##
[2] Banerjee A, Padhi R, editors. An optimal explicit guidance algorithm for terminal descent phase of lunar soft landing. AIAA Guidance, Navigation, and Control Conference; 2017.##
[3] Swaminathan S, Ghose D, editors. Real time powered descent guidance algorithm for mars pinpoint landing with inequality constraints. AIAA Scitech 2020 Forum; 2020.##
[4] Anglim KSG. Minimum-fuel optimal trajectory for reusable first-stage rocket landing using Particle Swarm Optimization: California State University, Long Beach; 2016.##
[5] Li Y, Chen W, Zhou H, Yang L. Conjugate gradient method with pseudospectral collocation scheme for optimal rocket landing guidance. Aerospace Science and Technology. 2020;104:105999.##
[6] Guibout VM, Scheeres DJ. Solving two-point boundary value problems using generating functions: Theory and Applications to optimal control and the study of Hamiltonian dynamical systems. arXiv preprint math/0310475. 2003.##
[7] Guibout VM, Scheeres DJ. Solving two-point boundary value problems using generating functions: Theory and applications to astrodynamics. Modern Astrodynamics. 2006:53-105.##
[8] Guibout VM, Scheeres DJ. Solving relative two-point boundary value problems: Spacecraft formulation flight transfers application. Journal of guidance, control, and dynamics. 2004;27(4):693-704.##
[9] Di Lizia P, Armellin R, Ercoli-Finzi A, Berz M. High-order robust guidance of interplanetary trajectories based on differential algebra. Journal of Aerospace Engineering, Sciences and Applications. 2008;1(1):43-57.##
[10] Di Lizia P, Armellin R, Bernelli-Zazzera F, Berz M. High order optimal control of space trajectories with uncertain boundary conditions. Acta Astronautica. 2014;93:217-29.##
[11] Di Lizia P, Armellin R, Morselli A, Bernelli-Zazzera F. High order optimal feedback control of space trajectories with bounded control. Acta Astronautica. 2014;94(1):383-94.##
[12] Wittig A, Armellin R. High order transfer maps for perturbed Keplerian motion. Celestial Mechanics and Dynamical Astronomy. 2015;122(4):333-58.##
[13] Vetrisano M, Vasile M. Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions. Advances in Space Research. 2017;60(1):38-56.##
[14] Sun Z-J, Di Lizia P, Bernelli-Zazzera F, Luo Y-Z, Lin K-P. High-order state transition polynomial with time expansion based on differential algebra. Acta Astronautica. 2019;163:45-55.##
[15] Morselli A, Armellin R, Di Lizia P, Zazzera FB. A high order method for orbital conjunctions analysis: sensitivity to initial uncertainties. Advances in Space Research. 2014;53(3):490-508.##
[16] Morselli A, Armellin R, Di Lizia P, Zazzera FB. A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation. Advances in Space Research. 2015;55(1):311-33.##
[17] Moghadasian M, Roshanian J. Optimal Landing of Unmanned Aerial Vehicle Using Vectorised High Order Expansions Method. Modares Mechanical Engineering. 2019;19(11):2761-9.##
[18] Moghadasian M, Roshanian J. Continuous maneuver of unmanned aerial vehicle using High Order Expansions method for optimal control problem. Modares Mechanical Engineering. 2018;17(12):382-90.##
[19] Moghadasian M, Roshanian J. Approximately optimal manoeuvre strategy for aero-assisted space mission. Advances in Space Research. 2019;64(2):436-50.##
[20] Çimen T. State-dependent Riccati equation (SDRE) control: a survey. IFAC Proceedings Volumes. 2008;41(2):3761-75.##
[21] Pearson J. Approximation methods in optimal control I. Sub-optimal control. International Journal of Electronics. 1962;13(5):453-69.##
[22] Cloutier JR, D’Souza CN, Mracek CP, editors. Nonlinear regulation and nonlinear H∞ control via the state-dependent Riccati equation technique: Part 1, theory. Proceedings of the international conference on nonlinear problems in aviation and aerospace; 1996: Embry Riddle University.##
[23] Cloutier J, D'Souza C, Mracek C, editors. Nonlinear regulation and nonlinear $ H_ {infty} $ control via the state-dependent Riccati equation technique: Part2 Examples. Proceedings of the International Conference on Nonlinear Problems in Aviation and Aerospace; 1996.##
[24] Cloutier JR, editor State-dependent Riccati equation techniques: an overview. Proceedings of the 1997 American control conference (Cat No 97CH36041); 1997: IEEE.##
[25] Mracek CP, Cloutier JR. Control designs for the nonlinear benchmark problem via the state‐dependent Riccati equation method. International Journal of robust and nonlinear control. 1998;8(4‐5):401-33.##
[26] Cloutier JR, Stansbery DT, editors. The capabilities and art of state-dependent Riccati equation-based design. Proceedings of the 2002 American Control Conference (IEEE Cat No CH37301); 2002: IEEE.##
[27] Wernli A, Cook G. Suboptimal control for the nonlinear quadratic regulator problem. Automatica. 1975;11(1):75-84.##
[28] VADALI S, editor Examination of the optimal nonlinear regulator problem. Guidance, Navigation and Control Conference; 1988.##
[29] Banks S, Mhana K. Optimal control and stabilization for nonlinear systems. IMA journal of mathematical control and information. 1992;9(2):179-96.##
[30] Gong C, Thompson S. A comment on ‘Stabilization and optimal control for nonlinear systems’. IMA Journal of Mathematical Control and Information. 1995;12(4):395-8.##
[31] Tsiotras P, Corless M, Rotea M. Counterexample to a recent result on the stability of nonlinear systems. IMA Journal of Mathematical Control and Information. 1996;13(2):129-30.##
[32] Mracek CP, Cloutier JR, editors. Missile longitudinal autopilot design using the state-dependent Riccati equation method. Proceedings of the International Conference on Nonlinear Problems in Aviation and Aerospace; 1996.##
[33] Mracek C, Cloutier J, Cloutier J, Mracek C, editors. Full envelope missile longitudinal autopilot design using the state-dependent Riccati equation method. Guidance, Navigation, and Control Conference; 1997.##
[34] Wise KA, Sedwick JL, editors. Nonlinear control of agile missiles using state dependent Riccati equations. Proceedings of the 1997 American Control Conference (Cat No 97CH36041); 1997: IEEE.##
[35] Stansbery D, Cloutier J, editors. Nonlinear, hybrid bank-to-turn/skid-to-turn missile autopilot design. AIAA guidance, navigation, and control conference and exhibit; 2001.##
[36] Mracek CP. SDRE autopilot for dual controlled missiles. IFAC Proceedings Volumes. 2007;40(7):750-5.##
[37] ÇLimen T. A generic approach to missile autopilot design using state-dependent nonlinear control. IFAC Proceedings Volumes. 2011;44(1):9587-600.##
[38] Cloutier J, Stansbery D, editors. All-aspect acceleration-limited homing guidance. Guidance, Navigation, and Control Conference and Exhibit; 1999.##
[39] Ratnoo A, Ghose D. State-dependent Riccati-equation-based guidance law for impact-angle-constrained trajectories. Journal of Guidance, Control, and Dynamics. 2009;32(1):320-6.##
[40] Vaddi S, Menon PK, Ohlmeyer EJ. Numerical state-dependent Riccati equation approach for missile integrated guidance control. Journal of guidance, control, and dynamics. 2009;32(2):699-703.
[41] Ewing CM. An analysis of a new nonlinear estimation technique: The state-dependent Ricatti equation method: University of Florida; 1999.##
[42] Parrish DK, Ridgely DB, editors. Attitude control of a satellite using the SDRE method. Proceedings of the 1997 American Control Conference (Cat No 97CH36041); 1997: IEEE.##
[43] Hammett KD, Hall CD, Ridgely DB. Controllability issues in nonlinear state-dependent Riccati equation control. Journal of guidance, control, and dynamics. 1998;21(5):767-73.##
[44] Stansbery DT, Cloutier JR, editors. Position and attitude control of a spacecraft using the state-dependent Riccati equation technique. Proceedings of the 2000 American Control Conference ACC (IEEE Cat No 00CH36334); 2000: IEEE.##
[45] Cloutier JR, Zipfel PH, editors. Hypersonic guidance via the state-dependent Riccati equation control method. Proceedings of the 1999 IEEE International Conference on Control Applications (Cat No 99CH36328); 1999: IEEE.##
[46] Harman RR, Bar-Itzhack IY. Pseudolinear and state-dependent Riccati equation filters for angular rate estimation. Journal of Guidance, Control, and Dynamics. 1999;22(5):723-5.##
[47] Chang I, Park S-Y, Choi K-H. Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique. International Journal of Non-Linear Mechanics. 2009;44(8):891-904.##
[48] Lin L-G, Xin M. Computational enhancement of the SDRE scheme: general theory and robotic control system. IEEE Transactions on Robotics. 2020;36(3):875-93.##
[49] Lin L-G. Computationally Improved State-Dependent Riccati Equation Scheme for Nonlinear Benchmark System. IEEE/ASME Transactions on Mechatronics. 2020;26(2):1064-75.##
Aerospace Mechanics
Volume 18, Issue 3 - Serial Number 69
Serial No. 69, Autumn Quarterly
August 2022
Pages 69-85

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History

  • Receive Date: 29 May 2022
  • Revise Date: 23 June 2022
  • Accept Date: 15 August 2022
  • Publish Date: 23 September 2022

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