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Robust gain-scheduled autopilot design with anti-windup compensation for a guided projectile

CEAS Aeronautical Journal volume 14, pages 765–786 (2023)Cite this article

Abstract

This article deals with the control design of a dual-spin projectile concept, characterized by highly nonlinear parameter-dependent and coupled dynamics, and subject to uncertainties and actuator saturations. An open-loop nonlinear model stemming from flight mechanics is first developed. It is subsequently linearized and decomposed into a linear parameter-varying system for the roll channel, and a quasi-linear parameter-varying system for the pitch/yaw channels. The obtained models are then used to design gain-scheduled \(\mathcal {H}_\infty\) baseline autopilots, which do not take the saturations into account. As a major contribution of this paper, the saturation nonlinearities are addressed in a second step through anti-windup augmentation. Three anti-windup schemes are proposed, which are evaluated and compared through time-domain simulations and integral quadratic constraints analysis. Finally, complete guided flight scenarios involving a wind disturbance, perturbed launch conditions, or aerodynamic uncertainties, are analyzed by means of nonlinear Monte Carlo simulations to evaluate the improvements brought by the proposed anti-windup compensators.

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Notes

  1. Using a more standard \(\mathcal {H}_\infty\) soft goal of the form \(\Vert {W_M(s) \mathcal T_{\varvec{n}_{zy,g} \rightarrow \varvec{e}_{zy,ref}}(s)}\Vert_\infty\), as with the roll channel, led to less robust margins, as well as less smooth gain surfaces when performing the synthesis over the flight envelope.

  2. Thus, the \(\mathcal {H}_2\) norm of G(s)/s corresponds to the energy of the step response of G(s).

References

  1. Jitpraphai, T., Costello, M.: Dispersion reduction of a direct fire rocket using lateral pulse jets. J. Spacecr. Rockets 38(6), 929–936 (2001). https://doi.org/10.2514/2.3765

    Article  Google Scholar 

  2. Burchett, B., Peterson, A., Costello, M.: Prediction of swerving motion of a dual-spin projectile with lateral pulse jets in atmospheric flight. Math. Comput. Model. 35(7–8), 821–834 (2002). https://doi.org/10.1016/S0895-7177(02)00053-5

    Article  MathSciNet  MATH  Google Scholar 

  3. Hahn, P.V., Frederick, R.A., Slegers, N.: Predictive guidance of a projectile for hit-to-kill interception. IEEE Trans. Control Syst. Technol. 17(4), 745–755 (2009). https://doi.org/10.1109/tcst.2008.2004440

    Article  Google Scholar 

  4. Frost, G., Costello, M.: Control authority of a projectile equipped with an internal unbalanced part. J. Dyn. Syst. Meas. Contr. 128(4), 1005 (2006). https://doi.org/10.1115/1.2363205

    Article  Google Scholar 

  5. Rogers, J., Costello, M.: A variable stability projectile using an internal moving mass. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 223(7), 927–938 (2009). https://doi.org/10.1243/09544100jaero509

    Article  Google Scholar 

  6. Cooper, G.R., Costello, M.: Trajectory prediction of spin-stabilized projectiles with a liquid payload. J. Spacecr. Rockets 48(4), 664–670 (2011). https://doi.org/10.2514/1.52564

    Article  Google Scholar 

  7. Theodoulis, S., Wernert, P.: Flight dynamics & control for smart munition: the ISL contribution. IFAC-PapersOnLine 50(1), 15512–15517 (2017). https://doi.org/10.1016/j.ifacol.2017.08.2127

    Article  Google Scholar 

  8. Fresconi, F.: Guidance and control of a projectile with reduced sensor and actuator requirements. J. Guid. Control. Dyn. 34(6), 1757–1766 (2011). https://doi.org/10.2514/1.53584

    Article  Google Scholar 

  9. Costello, M., Peterson, A.: Linear theory of a dual-spin projectile in atmospheric flight. J. Guid. Control. Dyn. 23(5), 789–797 (2000). https://doi.org/10.2514/2.4639

    Article  Google Scholar 

  10. Gagnon, E., Lauzon, M.: Maneuverability analysis of the conventional 155 mm gunnery projectile. In: AIAA Guidance, Navigation and Control Conference and Exhibit, American Institute of Aeronautics and Astronautics, 2007. https://doi.org/10.2514/6.2007-6784

  11. Gagnon, E., Lauzon, M.: Course correction fuze concept analysis for in-service 155 mm spin-stabilized gunnery projectiles. In: AIAA Guidance, Navigation and Control Conference and Exhibit, American Institute of Aeronautics and Astronautics, 2008. https://doi.org/10.2514/6.2008-6997

  12. Theodoulis, S., Sève, F., Wernert, P.: Robust gain-scheduled autopilot design for spin-stabilized projectiles with a course-correction fuze. Aerosp. Sci. Technol. 42, 477–489 (2015). https://doi.org/10.1016/j.ast.2014.12.027

    Article  Google Scholar 

  13. Sève, F., Theodoulis, S., Wernert, P., Zasadzinski, M., Boutayeb, M.: Gain-scheduled \({H}_\infty\) loop-shaping autopilot design for spin-stabilized canard-guided projectiles. AerospaceLab J. (2017). https://doi.org/10.12762/2017.AL13-03. (ISSN: 2107-6596)

    Article  Google Scholar 

  14. Sève, F., Theodoulis, S., Wernert, P., Zasadzinski, M., Boutayeb, M.: Flight dynamics modeling of dual-spin guided projectiles. IEEE Trans. Aerosp. Electron. Syst. 53(4), 1625–1641 (2017). https://doi.org/10.1109/taes.2017.2667820

    Article  Google Scholar 

  15. Rugh, W.J., Shamma, J.S.: Research on gain scheduling. Automatica 36(10), 1401–1425 (2000). https://doi.org/10.1016/s0005-1098(00)00058-3

    Article  MathSciNet  MATH  Google Scholar 

  16. Leith, D.J., Leithead, W.E.: Survey of gain-scheduling analysis and design. Int. J. Control 73(11), 1001–1025 (2000). https://doi.org/10.1080/002071700411304

    Article  MathSciNet  MATH  Google Scholar 

  17. Skogestad, S., Postlethwaite, I.: Multivariable Feedback Control: Analysis and Design. Wiley-Interscience (2005)

    MATH  Google Scholar 

  18. Thai, S.: Advanced anti-windup flight control algorithms for fast time-varying aerospace systems. Ph.D. thesis, Toulouse, ISAE (2021)

  19. Galeani, S., Tarbouriech, S., Turner, M., Zaccarian, L.: A tutorial on modern anti-windup design. Eur. J. Control. 15(3–4), 418–440 (2009). https://doi.org/10.3166/ejc.15.418-440

    Article  MathSciNet  MATH  Google Scholar 

  20. Megretski, A., Rantzer, A.: System analysis via integral quadratic constraints. IEEE Trans. Autom. Control 42(6), 819–830 (1997). https://doi.org/10.1109/9.587335

    Article  MathSciNet  MATH  Google Scholar 

  21. Zipfel, P.H.: Modeling and Simulation of Aerospace Vehicle Dynamics, 2nd edn. American Institute of Aeronautics and Astronautics (2007). https://doi.org/10.2514/4.862182

    Book  MATH  Google Scholar 

  22. McCoy, R.L.: Modern Exterior Ballistics: The Launch and Flight Dynamics of Symmetric Projectiles. Schiffer Publishing (1999)

    Google Scholar 

  23. Siouris, G.M.: Missile Guidance and Control Systems. Springer (2004). https://doi.org/10.1007/b97614

    Book  Google Scholar 

  24. Theodoulis, S., Gassmann, V., Brunner, T., Wernert, P: Fixed structure robust control design for the 155mm canard-guided projectile roll-channel autopilot. In: Proceedings of the 21st Mediterranean Conference on Control and Automation, IEEE, 2013. https://doi.org/10.1109/med.2013.6608714

  25. Apkarian, P., Noll, D.: Nonsmooth \({H}_\infty\) Synthesis. IEEE Trans. Autom. Control 51(1), 71–86 (2006). https://doi.org/10.1109/tac.2005.860290

    Article  MathSciNet  MATH  Google Scholar 

  26. Doyle, J.: Analysis of feedback systems with structured uncertainties. IEE Proc. D Control Theory Appl. 129(6), 242 (1982). https://doi.org/10.1049/ip-d.1982.0053

    Article  MathSciNet  Google Scholar 

  27. Biannic, J.M., Roos, C.: Generalized State Space: a new Matlab class to model uncertain and nonlinear systems as Linear Fractional Representations (2012-2021). http://w3.onera.fr/smac/gss. Accessed 25 Nov 2021

  28. Roos, C.: Systems modeling, analysis and control (SMAC) toolbox: an insight into the robustness analysis library. In: Proceedings of the 2013 Conference on Computer Aided Control System Design (CACSD), IEEE, 2013. https://doi.org/10.1109/cacsd.2013.6663479

  29. Thai, S., Roos, C., Biannic, J.M.: Probabilistic μ-analysis for stability and H∞ performance verification. In: Proceedings of the 2019 American Control Conference (ACC), IEEE, 2019. https://doi.org/10.23919/acc.2019.8814315

  30. Gomes da Silva, J.M., Tarbouriech, S.: Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Trans. Autom. Control 50(1), 106–111 (2005). https://doi.org/10.1109/tac.2004.841128

    Article  MathSciNet  MATH  Google Scholar 

  31. Biannic, J.M., Tarbouriech, S.: Optimization and implementation of dynamic anti-windup compensators with multiple saturations in flight control systems. Control. Eng. Pract. 17(6), 703–713 (2009). https://doi.org/10.1016/j.conengprac.2008.11.002

    Article  Google Scholar 

  32. Thai, S., Theodoulis, S., Roos, C., Biannic, J.M., Proff, M.: Gain-scheduled autopilot design with anti-windup compensator for a dual-spin canard-guided projectile. In: Proceedings of the 2020 IEEE Conference on Control Technology and Applications (CCTA), IEEE, 2020. https://doi.org/10.1109/ccta41146.2020.9206311

  33. Kelly, J., Evers, J.: An interpolation strategy for scheduling dynamic compensators. Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics (1997). https://doi.org/10.2514/6.1997-3764

    Book  Google Scholar 

  34. Thai, S., Theodoulis, S., Roos, C., Biannic, J.M.: An interpolated model recovery anti-windup for a canard-guided projectile subject to uncertainties. In: Proceedings of the 2021 European Control Conference (ECC), IEEE, 2021. https://doi.org/10.23919/ECC54610.2021.9655059

  35. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. MOSEK ApS, The MOSEK optimization toolbox for MATLAB manual. Version 9.0 (2019). http://docs.mosek.com/9.0/toolbox/index.html. Accessed 25 Nov 2021

  37. Veenman, J., Scherer, C.W., Köroğlu, H.: Robust stability and performance analysis based on integral quadratic constraints. Eur. J. Control. 31, 1–32 (2016). https://doi.org/10.1016/j.ejcon.2016.04.004

    Article  MathSciNet  MATH  Google Scholar 

  38. Fetzer, M., Scherer, C.W.: Full-block multipliers for repeated, slope-restricted scalar nonlinearities. Int. J. Robust Nonlinear Control 27(17), 3376–3411 (2017). https://doi.org/10.1002/rnc.3751

    Article  MathSciNet  MATH  Google Scholar 

  39. Proff, M., Theodoulis, S.: Study of impact point prediction methods for zero-effort-miss guidance: application to a 155mm spin-stabilized guided projectile. In: Proceedings of the 5th CEAS Conference on Guidance, Navigation and Control (2019)

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Acknowledgements

This research is supported by the French-German Research Institute of Saint-Louis, and ONERA, the French Aerospace Lab.

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Author notes

  1. Sovanna Thai

    Present address: Airbus Defence and Space, 31 Rue des Cosmonautes, 31400, Toulouse, France

  2. Spilios Theodoulis

    Present address: Delft University of Technology, Kluyverweg 1, 2629 HS, Delft, The Netherlands

Authors and Affiliations

  1. French-German Research Institute of Saint-Louis, 5 Rue du Général Cassagnou, 68300, Saint-Louis, France

    Sovanna Thai & Spilios Theodoulis

  2. ONERA The French Aerospace Lab, 2 Avenue Edouard Belin, 31400, Toulouse, France

    Clément Roos & Jean-Marc Biannic

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  1. Sovanna Thai
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Correspondence to Sovanna Thai.

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Thai, S., Theodoulis, S., Roos, C. et al. Robust gain-scheduled autopilot design with anti-windup compensation for a guided projectile. CEAS Aeronaut J 14, 765–786 (2023). https://doi.org/10.1007/s13272-023-00668-9

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