Abstract
In process control it is essential that disturbances and parameter uncertainties do not affect the process in a negative way. Simultaneously optimizing an objective function for different scenarios can be solved in theory by evaluating candidate solutions on all scenarios. This is not feasible in real-world applications, where the scenario space often forms a continuum. A traditional approach is to approximate this evaluation using Monte Carlo sampling. To overcome the difficulty of choosing an appropriate sampling count and to reduce evaluations of low-quality solutions, a novel approach using Wilson scoring and criticality ranking within a grammatical evolution framework is presented. A nonlinear spring mass system is considered as benchmark example from robust control. The method is tested against Monte Carlo sampling and the results are compared to a backstepping controller. It is shown that the method is capable of outperforming state of the art methods.
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See http://www.boost.org.
References
Ackermann, J., Blue, P., Bünte, T., Güvenc, L., Kaesbauer, D., Kordt, M., Muhler, M., Odenthal, D.: Robust Control: The Parameter Space Approach. Springer Science+Business Media, Heidelberg (2002)
Ahnert, K., Mulansky, M.: Odeint-solving ordinary differential equations in C++ (2011). arXiv preprint arXiv:1110.3397
Chen, P., Lu, Y.Z.: Automatic design of robust optimal controller for interval plants using genetic programming and Kharitonov theorem. Int. J. Comput. Intell. Syst. 4(5), 826–836 (2011)
Cupertino, F., Naso, D., Salvatore, L., Turchiano, B.: Design of cascaded controllers for DC drives using evolutionary algorithms. In: Proceedings of the 2002 Congress on Evolutionary Computation, 2002, CEC 2002, vol. 2, pp. 1255–1260. IEEE (2002)
Dasgupta, D., Michalewicz, Z.: Evolutionary Algorithms in Engineering Applications. Springer Science+Business Media, Heidelberg (2013)
Dempsey, I., O’Neill, M., Brabazon, A.: Foundations in Grammatical Evolution for Dynamic Environments, vol. 194. Springer, Heidelberg (2009)
Doyle, J.C., Francis, B.A., Tannenbaum, A.R.: Feedback Control Theory. Courier Corporation, New York (2013)
Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications. CRC Press, Boca Raton (1998)
Francis, B.A.: A Course in \(\cal{H}_{\infty }\) Control Theory. Springer, New York (1987)
Gholaminezhad, I., Jamali, A., Assimi, H.: 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), pp. 041–046. IEEE (2014)
Holland, J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. University of Michigan Press, Ann Arbor (1975)
Hornby, G.S., Globus, A., Linden, D.S., Lohn, J.D.: Automated antenna design with evolutionary algorithms. In: AIAA Space, pp. 19–21 (2006)
Kharitonov, V.: Asympotic stability of an equilibrium position of a family of systems of linear differntial equations. Differntia Uravn. 14(11), 1483–1485 (1978)
Knuth, D.E.: The Art of Computer Programming: Sorting and Searching, vol. 3. Pearson Education, Upper Saddle River (1998)
Koza, J.R.: Genetic Programming: on The Programming of Computers by Means of Natural Selection, vol. 1. MIT Press, Cambridge (1992)
Koza, J.R., Keane, M.A., Yu, J., Bennett, F.H., Mydlowec, W., Stiffelman, O.: Automatic synthesis of both the topology and parameters for a robust controller for a non-minimal phase plant and a three-lag plant by means of genetic programming. In: Proceedings of the 38th IEEE Conference on Decision and Control, 1999, vol. 5, pp. 5292–5300. IEEE (1999)
Koza, J.R., Keane, M.A., Yu, J., Mydlowec, W., Bennett, F.H.: Automatic synthesis of both the topology and parameters for a controller for a three-lag plant with a five-second delay using genetic programming. In: Cagnoni, S. (ed.) EvoWorkshops 2000. LNCS, vol. 1803, pp. 168–177. Springer, Heidelberg (2000). doi:10.1007/3-540-45561-2_17
Mitchell, M.: An Introduction to Genetic Algorithms. MIT Press, Cambridge (1998)
Ryan, C., Collins, J.J., Neill, M.O.: Grammatical evolution: evolving programs for an arbitrary language. In: Banzhaf, W., Poli, R., Schoenauer, M., Fogarty, T.C. (eds.) EuroGP 1998. LNCS, vol. 1391, pp. 83–96. Springer, Heidelberg (1998). doi:10.1007/BFb0055930
Shimooka, H., Fujimoto, Y.: 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, pp. 491–495. Morgan Kaufmann Publishers Inc. (2000)
Soltoggio, A.: A comparison of genetic programming and genetic algorithms in the design of a robust, saturated control system. In: Deb, K. (ed.) GECCO 2004. LNCS, vol. 3103, pp. 174–185. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24855-2_16
Wang, Q., Stengel, R.F.: Robust control of nonlinear systems with parametric uncertainty. Automatica 38(9), 1591–1599 (2002)
Wang, Q., Stengel, R.F.: Robust nonlinear flight control of a high-performance aircraft. IEEE Trans. Control Syst. Technol. 13(1), 15–26 (2005)
Wie, B., Bernstein, D.S.: Benchmark problems for robust control design. J. Guid. Control Dyn. 15(5), 1057–1059 (1992)
Wills, A.G., Bates, D., Fleming, A.J., Ninness, B., Moheimani, S.R.: Model predictive control applied to constraint handling in active noise and vibration control. IEEE Trans. Control Syst. Technol. 16(1), 3–12 (2008)
Wilson, E.B.: Probable inference, the law of succession, and statistical inference. J. Am. Statist. Assoc. 22(158), 209–212 (1927)
Zames, G.: Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans. Autom. Control 26(2), 301–320 (1981)
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Appendix: Stability Analysis
Appendix: Stability Analysis
We calculate the Jacobian of the system from Eq. (9), substituting u with the GE controller from Eq. (14). The Jacobian for this system is given by \(\varvec{A}(\varvec{x}) = \begin{bmatrix} \frac{\partial \varvec{f}}{\partial \varvec{x}} \end{bmatrix}\). The characteristic polynomial can be calculated by
Looking at the intervals from Eq. (10) it is trivial to see that all coefficients of this polynomial are strictly positive. We can thus continue the analysis by looking at the Hurwitz matrix of the polynomial:
From the 4 principal minors of \(\varvec{H}\) we get the Hurwitz conditions for stability
Again since these conditions only depend on \(k_1, m_1\) and \(m_2\) which are strictly positive, the stability conditions hold for any parameter combination \(\varvec{q} \in Q\). The controller \(u_{\text {GE}}(\varvec{x})\) is thus a (local) robust stabilizer of the system.
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Reichensdörfer, E., Odenthal, D., Wollherr, D. (2017). Grammatical Evolution of Robust Controller Structures Using Wilson Scoring and Criticality Ranking. In: McDermott, J., Castelli, M., Sekanina, L., Haasdijk, E., GarcÃa-Sánchez, P. (eds) Genetic Programming. EuroGP 2017. Lecture Notes in Computer Science(), vol 10196. Springer, Cham. https://doi.org/10.1007/978-3-319-55696-3_13
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