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Automated Nonlinear Control Structure Design by Domain of Attraction Maximization with Eigenvalue and Frequency Domain Specifications

  • Elias ReichensdörferEmail author
  • Dirk Odenthal
  • Dirk Wollherr
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 613)

Abstract

This work proposes a new method for nonlinear control structure design for nonlinear dynamical systems by grammatical evolution. The optimization is based on a novel fitness function which implements three different design objectives. First, the closed loop eigenvalues of the linearized system dynamics are restricted to be located in a predefined region of the complex plane. Second, design specifications on the frequency magnitude of the sensitivity transfer functions of the linearized system dynamics are imposed. Third, the estimated domain of attraction of the nonlinear closed loop system dynamics is maximized by evaluating a quadratic Lyapunov function, obtained by the solution of the Lyapunov equation, with a Monte-Carlo sampling algorithm. Additionally, a general formula for the centroid of a design region for pole placement is derived analytically and embedded in the optimization framework. The generated controllers are evaluated on a common, nonlinear benchmark system. It is shown that the proposed method can efficiently generate control laws which meet the imposed design specifications on the closed loop system while maximizing the domain of attraction.

Keywords

Nonlinear control structure design Lyapunov equation Grammatical evolution Robust control Domain of attraction maximization Frequency domain specifications Stability centroid 

References

  1. 1.
    Ackermann J (1980) Parameter space design of robust control systems. IEEE Trans Autom Control 25(6):1058–1072zbMATHCrossRefGoogle Scholar
  2. 2.
    Ackermann J, Blue P, Bünte T, Güvenc L, Kaesbauer D, Kordt M, Muhler M, Odenthal D (2002) Robust control: the parameter space approach. Springer Science & Business Media, HeidelbergCrossRefGoogle Scholar
  3. 3.
    Artstein Z (1983) Stabilization with relaxed controls. Nonlinear Anal TMA 7(11):1163–1173MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Backus JW (1959) The syntax and semantics of the proposed international algebraic language of the Zurich ACM-GAMM conference. Proceedings of the international conference on information processing (1959)Google Scholar
  5. 5.
    Banks C (2002) Searching for Lyapunov functions using genetic programming. Virginia Polytech Institute and State University, Blacksburg, Virginia. http://www.aerojockey.com/files/lyapunovgp.pdf. Accessed 24 May 2018
  6. 6.
    Bartels RH, Stewart G (1972) Solution of the matrix equation \(AX+ XB= C\): algorithm 432. Commun ACM 15(9):820–826zbMATHCrossRefGoogle Scholar
  7. 7.
    Blackwell C (1984) On obtaining the coefficients of the output transfer function from a state-space model and an output model of a linear constant coefficient system. IEEE Trans Autom Control 29(12):1122–1125MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bobiti R, Lazar M (2018) Automated sampling-based stability verification and DOA estimation for nonlinear systems. IEEE Trans Autom Control 63(11):3659–3674MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brayton R, Tong C (1979) Stability of dynamical systems: a constructive approach. IEEE Trans Circuits Syst 26(4):224–234MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bünte T (2000) Mapping of Nyquist/Popov theta-stability margins into parameter space. IFAC Proc Vol 33(14):519–524CrossRefGoogle Scholar
  11. 11.
    Castro LN, De Castro LN, Timmis J (2002) Artificial immune systems: a new computational intelligence approach. Springer Science & Business Media, LondonzbMATHGoogle Scholar
  12. 12.
    Chen P, Lu YZ (2011) Automatic design of robust optimal controller for interval plants using genetic programming and Kharitonov theorem. Int J Comput Intell Syst 4(5):826–836CrossRefGoogle Scholar
  13. 13.
    Chen Z, Yuan X, Ji B, Wang P, Tian H (2014) Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Convers Manag 84:390–404CrossRefGoogle Scholar
  14. 14.
    Chesi G (2005) Domain of attraction: estimates for non-polynomial systems via LMIs. In: Proceedings of 16th IFAC world congress, Prague, Czech RepublicGoogle Scholar
  15. 15.
    Chilali M, Gahinet P (1996) \(H_{\infty }\) design with pole placement constraints: an LMI approach. IEEE Trans Autom Control 41(3):358–367MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B (Cybernetics) 26(1):29–41CrossRefGoogle Scholar
  17. 17.
    Doyle J, Stein G (1981) Multivariable feedback design: concepts for a classical/modern synthesis. IEEE Trans Autom Control 26(1):4–16zbMATHCrossRefGoogle Scholar
  18. 18.
    Doyle JC, Francis BA, Tannenbaum AR (2013) Feedback control theory. Courier CorporationGoogle Scholar
  19. 19.
    Duriez T, Brunton SL, Noack BR (2017) Machine learning control-taming nonlinear dynamics and turbulence. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  20. 20.
    Freeman R, Kokotovic PV (2008) Robust nonlinear control design: state-space and Lyapunov techniques. Springer Science & Business Media, BaselzbMATHGoogle Scholar
  21. 21.
    Gholaminezhad I, Jamali A, Assimi H (2014) Automated synthesis of optimal controller using multi-objective genetic programming for two-mass-spring system. In: 2014 second RSI/ISM international conference on robotics and mechatronics (ICRoM). IEEE, pp 041–046Google Scholar
  22. 22.
    Giesl P (2007) Construction of global Lyapunov functions using radial basisfunctions, vol 1904. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  23. 23.
    Giesl P, Hafstein S (2015) Review on computational methods for Lyapunov functions. Discrete Continuous Dyn Syst Ser B 20(8):2291–2337MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Glassman E, Desbiens AL, Tobenkin M, Cutkosky M, Tedrake R (2012) Region of attraction estimation for a perching aircraft: a Lyapunov method exploiting barrier certificates. In: 2012 IEEE international conference on robotics and automation (ICRA). IEEE, pp 2235–2242Google Scholar
  25. 25.
    Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 13(5):533–549MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Glover FW, Kochenberger GA (2006) Handbook of metaheuristics, vol 57. Springer Science & Business Media, HeidelbergzbMATHGoogle Scholar
  27. 27.
    Graichen K, Kugi A, Petit N, Chaplais F (2010) Handling constraints in optimal control with saturation functions and system extension. Syst Control Lett 59(11):671–679MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Grosman B, Lewin DR (2005) Automatic generation of Lyapunov functions using genetic programming. IFAC Proc Vol 38(1):75–80CrossRefGoogle Scholar
  29. 29.
    Hafstein SF (2007) An algorithm for constructing Lyapunov functions. Electron J Differ Eq 2007:101zbMATHGoogle Scholar
  30. 30.
    Haupt RL, Haupt SE (1998) Practical genetic algorithms, vol 2. Wiley, New YorkzbMATHGoogle Scholar
  31. 31.
    Holland JH (1975) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, OxfordzbMATHGoogle Scholar
  32. 32.
    Hosen MA, Khosravi A, Nahavandi S, Creighton D (2015) Improving the quality of prediction intervals through optimal aggregation. IEEE Trans Industr Electron 62(7):4420–4429CrossRefGoogle Scholar
  33. 33.
    Isidori A (1989) Nonlinear control systems, 2nd edn. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  34. 34.
    Johansson M, Rantzer A (1997) Computation of piecewise quadratic Lyapunov functions for hybrid systems. In: 1997 european control conference (ECC). IEEE, pp 2005–2010Google Scholar
  35. 35.
    Jones M, Mohammadi H, Peet MM (2017) Estimating the region of attraction using polynomial optimization: a converse Lyapunov result. In: 2017 IEEE 56th annual conference on decision and control (CDC). IEEE, pp 1796–1802Google Scholar
  36. 36.
    Kapinski J, Deshmukh JV, Sankaranarayanan S, Arechiga N (2014) Simulation-guided Lyapunov analysis for hybrid dynamical systems. In: Proceedings of the 17th international conference on Hybrid systems: computation and control. ACM, pp 133–142Google Scholar
  37. 37.
    Khalil HK (1996) Nonlinear systems, 3rd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  38. 38.
    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Koza JR (1992) Genetic programming: on the programming of computers by means of natural selection, vol 1. MIT Press, CambridgezbMATHGoogle Scholar
  40. 40.
    Koza JR, Keane MA, Yu J, Bennett FH, Mydlowec W, Stiffelman O (1999) Automatic synthesis of both the topology and parameters for a robust controller for a nonminimal phase plant and a three-lag plant by means of genetic programming. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol 5, IEEE, pp 5292–5300Google Scholar
  41. 41.
    Li C, Ryoo CS, Li N, Cao L (2009) Estimating the domain of attraction via moment matrices. Bull Korean Math Soc 46(6):1237–1248MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Lyapunov AM (1992) The general problem of the stability of motion. Taylor & Francis, London. English translation by A.T. Fuller, original work published in Russian in 1892Google Scholar
  43. 43.
    Maciejowski JM (1989) Multivariable feedback design. Electronic systems engineering series, vol 6. Addison-Wesley, Wokingham, pp 85–90Google Scholar
  44. 44.
    Materassi D, Salapaka MV (2009) Attraction domain estimates combining Lyapunov functions. In: American control conference, ACC 2009. IEEE, pp 4007–4012Google Scholar
  45. 45.
    Matthews ML, Williams C (2012) Region of attraction estimation of biological continuous boolean models. In: 2012 IEEE International Conference on Systems, Man, and Cybernetics (SMC). IEEE, pp 1700–1705Google Scholar
  46. 46.
    McGough JS, Christianson AW, Hoover RC (2010) Symbolic computation of Lyapunov functions using evolutionary algorithms. In: Proceedings of the 12th IASTED international conference, vol 15, pp 508–515Google Scholar
  47. 47.
    Mitchell M (1998) An introduction to genetic algorithms. MIT Press, CambridgezbMATHGoogle Scholar
  48. 48.
    Najafi E, Babuška R, Lopes GA (2016) A fast sampling method for estimating the domain of attraction. Nonlinear Dyn 86(2):823–834MathSciNetCrossRefGoogle Scholar
  49. 49.
    Naur P, Backus JW, Bauer FL, Green J, Katz, C., McCarthy J, Perlis AJ (1969) Revised report on the algorithmic language. In: ALGOL 60. SpringerGoogle Scholar
  50. 50.
    Neath MJ, Swain AK, Madawala UK, Thrimawithana DJ (2014) An optimal PID controller for a bidirectional inductive power transfer system using multiobjective genetic algorithm. IEEE Trans Power Electron 29(3):1523–1531CrossRefGoogle Scholar
  51. 51.
    Odenthal D, Blue P (2000) Mapping of frequency response performance specifications into parameter space. IFAC Proc Vol 33(14):531–536CrossRefGoogle Scholar
  52. 52.
    O’Neill M, Ryan C (1999) Under the hood of grammatical evolution. In: Proceedings of the 1st annual conference on genetic and evolutionary computation, vol 2. Morgan Kaufmann Publishers Inc., pp 1143–1148Google Scholar
  53. 53.
    Poli R, Langdon WB, McPhee NF (2018) A field guide to genetic programming. Published via http://lulu.com, http://www.gp-field-guide.org.uk. (Accessed 24 May 2018), (With contributions by J. R. Koza)
  54. 54.
    Prajna S, Papachristodoulou A, Parrilo PA (2002) Introducing SOSTOOLS: a general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE conference on decision and control, vol 1. IEEE, pp 741–746Google Scholar
  55. 55.
    Precup RE, Sabau MC, Petriu EM (2015) Nature-inspired optimal tuning of input membership functions of Takagi-Sugeno-Kang fuzzy models for anti-lock braking systems. Appl Soft Comput 27:575–589CrossRefGoogle Scholar
  56. 56.
    Rall LB (1981) Automatic differentiation: techniques and applications. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  57. 57.
    Reichensdörfer E, Odenthal D, Wollherr D (2017) Grammatical evolution of robust controller structures using Wilson scoring and criticality ranking. In: European conference on genetic programming. Springer, pp 194–209Google Scholar
  58. 58.
    Reichensdörfer E, Odenthal D, Wollherr D (2018) Nonlinear control structure design using grammatical evolution and Lyapunov equation based optimization. In: Proceedings of the 15th international conference on informatics in control, automation and robotics - Volume 1: ICINCO. INSTICC, SciTePress, pp 55–65Google Scholar
  59. 59.
    Ryan C, Collins J, Neill MO (1988) Grammatical evolution: evolving programs for an arbitrary language. In: European conference on genetic programming. Springer, pp 83–96Google Scholar
  60. 60.
    Saadat J, Moallem P, Koofigar H (2017) Training echo state neural network using harmony search algorithm. Int J Artif Intell TM 15(1):163–179Google Scholar
  61. 61.
    Saleme A, Tibken B, Warthenpfuhl SA, Selbach C (2011) Estimation of the domain of attraction for non-polynomial systems: a novel method. IFAC Proc Vol 44(1):10976–10981CrossRefGoogle Scholar
  62. 62.
    Shimooka H, Fujimoto Y (2000) Generating robust control equations with genetic programming for control of a rolling inverted pendulum. In: Proceedings of the 2nd annual conference on genetic and evolutionary computation. Morgan Kaufmann Publishers Inc., pp 491–495Google Scholar
  63. 63.
    Singh R, Kuchhal P, Choudhury S, Gehlot A (2015) Implementation and evaluation of heating system using PID with genetic algorithm. Indian J Sci Technol 8(5):413CrossRefGoogle Scholar
  64. 64.
    Sontag ED (1983) A Lyapunov-like characterization of asymptotic controllability. SIAM J Control Optim 21(3):462–471MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Szczepanski R, Tarczewski T, Erwinski K, Grzesiak LM (2018) Comparison of constraint-handling techniques used in artificial bee colony algorithm for auto-tuning of state feedback speed controller for PMSM. In: Proceedings of the 15th international conference on informatics in control, automation and robotics - Volume 1: ICINCO. INSTICC, SciTePress, pp 269–276Google Scholar
  66. 66.
    Tibken B (2000) Estimation of the domain of attraction for polynomial systems via LMIs. In: Proceedings of the 39th IEEE conference on decision and control, vol 4. IEEE, pp 3860–3864Google Scholar
  67. 67.
    Verdier C, Mazo Jr. M (2017) Formal controller synthesis via genetic programming. IFAC-PapersOnLine 50(1):7205–7210Google Scholar
  68. 68.
    Vrkalovic S, Teban TA, Borlea ID (2017) Stable Takagi-Sugeno fuzzy control designed by optimization. Int J Artif Intell 15:17–29Google Scholar
  69. 69.
    Warthenpfuhl S, Tibken B, Mayer S (2010) An interval arithmetic approach for the estimation of the domain of attraction. In: 2010 IEEE international symposium on computer-aided control system design (CACSD). IEEE, pp 1999–2004Google Scholar
  70. 70.
    Zhou K, Doyle JC, Glover K et al (1996) Robust and optimal control, vol 40. Prentice Hall, New JerseyzbMATHGoogle Scholar
  71. 71.
    Zubov VI (1964) Methods of A.M. Lyapunov and their Application. P. NoordhoffGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Elias Reichensdörfer
    • 1
    • 3
    Email author
  • Dirk Odenthal
    • 2
  • Dirk Wollherr
    • 3
  1. 1.BMW GroupMunichGermany
  2. 2.BMW M GmbHGarching near MunichGermany
  3. 3.Chair of Automatic Control EngineeringTechnical University of MunichMunichGermany

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