An approach to nonlinear optimisation via set stabilisation of dynamical systems with an application to synchronous machines

References

• Aghili, F. (2018). Optimal feedback linearization control of interior PM synchronous motors subject to time-varying operation conditions minimizing power loss. IEEE Transactions on Industrial Electronics (1982), 65(7), 5414–5421. https://doi.org/10.1109/TIE.2017.2784348
   Web of Science ®Google Scholar
• Antonello, R., Carraro, M., & Zigliotto, M. (2014). Maximum-torque-per-ampere operation of anisotropic synchronous permanent-magnet motors based on extremum seeking control. IEEE Transactions on Industrial Electronics (1982), 61(9), 5086–5093. https://doi.org/10.1109/TIE.2013.2278518
   Web of Science ®Google Scholar
• Arrow, K. J., Hurwicz, L., & Uzawa, H. (1958). Studies in linear and non-linear programming. Stanford University Press.
   Google Scholar
• Bertsekas, D. P. (2016). Nonlinear programming (3rd ed.). Athena Scientific.
   Google Scholar
• Böck, M., & Kugi, A. (2014). Manifold stabilization and path-following control for flat systems with application to a laboratory tower crane. In 53rd IEEE Conf. on Decision and Control (CDC) (pp. 4529–4535). IEEE.
   Google Scholar
• Böck, M., & Kugi, A. (2016). Constrained model predictive manifold stabilization based on transverse normal forms. Automatica, 74), 315–326. https://doi.org/10.1016/j.automatica.2016.07.046
   Web of Science ®Google Scholar
• Bolognani, S., Calligaro, S., & Petrella, R. (2014). Adaptive flux-weakening controller for interior permanent magnet synchronous motor drives. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2(2), 236–248. https://doi.org/10.1109/JESTPE.2014.2299153
   Web of Science ®Google Scholar
• Bonifacio, J., & Kennel, R. M. (2018). On considering saturation and cross-coupling effects for copper loss minimization on highly anisotropic synchronous machines. IEEE Transactions on Industry Applications, 54(5), 4177–4185. https://doi.org/10.1109/TIA.2018.2836326
   Web of Science ®Google Scholar
• Braun, T., Bonifacio, J., & Lüttgens, A. (2023). A feedback systems approach to copper-loss minimization of anisotropic synchronous machines. IEEE Transactions on Control Systems Technology, 31(1), 410–417. https://doi.org/10.1109/TCST.2022.3167613
   Web of Science ®Google Scholar
• Brown, G. W., & Von Neumann, J. (1951). Solutions of games by differential equations. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games (am-24) (Vol. I, pp. 73–80). Princeton University Press.
   Google Scholar
• Brunner, F. D., Dürr, H. B., & Ebenbauer, C. (2012). Feedback design for multi-agent systems: A saddle point approach. In 51st IEEE Conf. on Decision and Control (CDC) (pp. 3783–3789). IEEE.
   Google Scholar
• Cavallaro, C., Tommaso, A. O. D., Miceli, R., Raciti, A., Galluzzo, G. R., & Trapanese, M. (2005). Efficiency enhancement of permanent-magnet synchronous motor drives by online loss minimization approaches. IEEE Transactions on Industrial Electronics (1982), 52(4), 1153–1160. https://doi.org/10.1109/TIE.2005.851595
   Web of Science ®Google Scholar
• Cherukuri, A., Gharesifard, B., & Cortés, J. (2017). Saddle-point dynamics: Conditions for asymptotic stability of saddle points. SIAM Journal on Control and Optimization, 55(1), 486–511. https://doi.org/10.1137/15M1026924
   Web of Science ®Google Scholar
• Choi, K., Kim, Y., Kim, K. S., & Kim, S. K. (2021). Real-time optimal torque control of interior permanent magnet synchronous motors based on a numerical optimization technique. IEEE Transactions on Control Systems Technology, 29(4), 1815–1822. https://doi.org/10.1109/TCST.2020.3006900
   Web of Science ®Google Scholar
• Consoli, A., Scarcella, G., Scelba, G., & Testa, A. (2010). Steady-state and transient operation of IPMSMs under maximum-torque-per-ampere control. IEEE Transactions on Industry Applications, 46(1), 121–129. doi:10.1109/TIA.2009.2036665
   Web of Science ®Google Scholar
• Dürr, H. B., & Ebenbauer, C. (2011). A smooth vector field for saddle point problems. In 50th IEEE Conf. on Decision and Control (CDC) and European Control Conf. (ECC) (pp. 4654–4660). IEEE.
   Google Scholar
• Dürr, H. B., & Ebenbauer, C. (2012). On a class of smooth optimization algorithms with applications in control. IFAC Proceedings, 45(17), 291–298.
   Google Scholar
• Eldeeb, H., Hackl, C. M., Horlbeck, L., & Kullick, J. (2018). A unified theory for optimal feedforward torque control of anisotropic synchronous machines. International Journal of Control, 91(10), 2273–2302. https://doi.org/10.1080/00207179.2017.1338359
   Web of Science ®Google Scholar
• El-Hawwary, M. I., & Maggiore, M. (2008). Global path following for the unicycle and other results. In 2008 American Control Conf. (pp. 3500–3505). IEEE.
   Google Scholar
• El-Hawwary, M. I., & Maggiore, M. (2011). Case studies on passivity-based stabilisation of closed sets. International Journal of Control, 84(2), 336–350. https://doi.org/10.1080/00207179.2010.551303
   Web of Science ®Google Scholar
• Gallegos-López, G., Gunawan, F. S., & Walters, J. E. (2005). Optimum torque control of permanent-magnet AC machines in the field-weakened region. IEEE Transactions on Industry Applications, 41(4), 1020–1028. https://doi.org/10.1109/TIA.2005.851588
   Web of Science ®Google Scholar
• Gramlich, D., Ebenbauer, C., & Scherer, C. W. (2022). Synthesis of accelerated gradient algorithms for optimization and saddle point problems using Lyapunov functions and LMIs. Systems & Control Letters, 165, 105271. https://doi.org/10.1016/j.sysconle.2022.105271
   Web of Science ®Google Scholar
• Grune, R. (2013). Verlustoptimaler Betrieb einer elektrisch erregten Synchronmaschine für den Einsatz in Elektrofahrzeugen [Unpublished doctoral dissertation]. TU Berlin.
   Google Scholar
• Haala, O., Wagner, B., Hofmann, M., & März, M. (2013). Optimal current control of externally excited synchronous machines in automotive traction drive applications. International Journal of Electrical and Computer Engineering, 7(9), 1133–1139.
   Google Scholar
• Hackl, C. M., Kullick, J., & Monzen, N. (2021). Generic loss minimization for nonlinear synchronous machines by analytical computation of optimal reference currents considering copper and iron losses. In 22nd IEEE Int. Conf. on Industrial Technology (ICIT) (Vol. 1, pp. 1348–1355). IEEE.
   Google Scholar
• Haddad, W. M., & Chellaboina, V. (2008). Nonlinear dynamical systems and control – Lyapunov-based approach. Princeton University Press.
   Google Scholar
• Hairer, E., & Wanner, G. (1996). Solving ordinary differential equations II, stiff and differential-algebraic problems (2nd ed.). Springer.
   Google Scholar
• Horn, R. A., & Johnson, C. R. (1990). Matrix analysis. Cambridge University Press.
   Google Scholar
• Iacchetti, M. F., Perini, R., Carmeli, M. S., Castelli-Dezza, F., & Bressan, N. (2012). Numerical integration of ODEs in real-time systems like state observers: Stability aspects. IEEE Transactions on Industry Applications, 48(1), 132–141. https://doi.org/10.1109/TIA.2011.2175471
   Web of Science ®Google Scholar
• Isidori, A. (1995). Nonlinear control systems. Springer.
   Google Scholar
• Jeong, Y., Sul, S., Hiti, S., & Rahman, K. (2006). Online minimum-copper-loss control of an interior permanent-magnet synchronous machine for automotive applications. IEEE Transactions on Industry Applications, 42(5), 1222–1229. https://doi.org/10.1109/TIA.2006.880910
   Web of Science ®Google Scholar
• Jokic, A., & Lazar, M. (2009). On constrained steady-state regulation: Dynamic KKT controllers. IEEE Transactions on Automatic Control, 54(9), 2250–2254. https://doi.org/10.1109/TAC.2009.2026856
   Web of Science ®Google Scholar
• Jung, S. Y., Hong, J., & Nam, K. (2013a). Copper loss minimizing torque control of IPMSM based on flux variables. In 2013 Int. Electric Machines & Drives Conf. (pp. 1174–1179). IEEE.
   Google Scholar
• Jung, S. Y., Hong, J., & Nam, K. (2013b). Current minimizing torque control of the IPMSM using ferrari's method. IEEE Transactions on Power Electronics, 28(12), 5603–5617. https://doi.org/10.1109/TPEL.2013.2245920
   Web of Science ®Google Scholar
• Khalil, H. K. (2002). Nonlinear systems (3rd ed.). Prentice-Hall.
   Google Scholar
• Kolavennu, S., Palanki, S., & Cockburn, J. (2001). Nonlinear control of nonsquare multivariable systems. Chemical Engineering Science, 56(6), 2103–2110. https://doi.org/10.1016/S0009-2509(00)00470-X
   Web of Science ®Google Scholar
• Kose, T. (1956). Solutions of saddle value problems by differential equations. Econometrica, 24(1), 59–70. https://doi.org/10.2307/1905259
   Web of Science ®Google Scholar
• Lee, J., Nam, K., Choi, S., & Kwon, S. (2009). Loss-minimizing control of PMSM with the use of polynomial approximations. IEEE Transactions on Power Electronics, 24(4), 1071–1082. https://doi.org/10.1109/TPEL.2008.2010518
   Web of Science ®Google Scholar
• Luenberger, D. (1975). A nonlinear economic control problem with a linear feedback solution. IEEE Transactions on Automatic Control, 20(2), 184–191. https://doi.org/10.1109/TAC.1975.1100916
   Google Scholar
• Luenberger, D. G., & Ye, Y. (2016). Linear and nonlinear programming. Springer.
   Google Scholar
• McLain, R., Kurtz, M., Henson, M., & Doyle, F. (1996). Habituating control for non-square nonlinear processes. In Proc. 1996 IEEE Int. Conf. on Control Applications (pp. 273–278). IEEE.
   Google Scholar
• Meyer, M., & Böcker, J. (2006). Optimum control for interior permanent magnet synchronous motors (IPMSM) in constant torque and flux weakening range. In 2006 12th Int. Power Electronics and Motion Control Conf. (pp. 282–286). IEEE.
   Google Scholar
• Monzen, N., & Hackl, C. M. (2023). Artificial neural network based optimal feedforward torque control of electrically excited synchronous machines. In 2023 IEEE 32nd Int. Symposium on Industrial Electronics (ISIE) (pp. 1–8). IEEE.
   Google Scholar
• Morimoto, S., Sanada, M., & Takeda, Y. (1994). Wide-speed operation of interior permanent magnet synchronous motors with high-performance current regulator. IEEE Transactions on Industry Applications, 30(4), 920–926. https://doi.org/10.1109/28.297908
   Web of Science ®Google Scholar
• Müller, S., Maier, D., & Parspour, N. (2023). Inductive electrically excited synchronous machine for electrical vehicles – design, optimization and measurement. Energies, 16(4), 1657. https://doi.org/10.3390/en16041657
   Web of Science ®Google Scholar
• Nielsen, C. (2009). Set stabilization using transverse feedback linearization [Unpublished doctoral dissertation]. University of Toronto, Canada.
   Google Scholar
• Nielsen, C., Fulford, C., & Maggiore, M. (2010). Path following using transverse feedback linearization: Application to a maglev positioning system. Automatica, 46(3), 585–590. https://doi.org/10.1016/j.automatica.2010.01.009
   Web of Science ®Google Scholar
• Nielsen, C., & Maggiore, M. (2008). On local transverse feedback linearization. SIAM Journal on Control and Optimization, 47(5), 2227–2250. https://doi.org/10.1137/070682125
   Web of Science ®Google Scholar
• Nocedal, J., & Wright, S. J. (2006). Numerical optimization. Springer.
   Google Scholar
• Preindl, M., & Bolognani, S. (2015). Optimal state reference computation with constrained MTPA criterion for PM motor drives. IEEE Transactions on Power Electronics, 30(8), 4524–4535. https://doi.org/10.1109/TPEL.2014.2354299
   Web of Science ®Google Scholar
• Shiriaev, A. (2000). The notion of V-detectability and stabilization of invariant sets of nonlinear systems. Systems & Control Letters, 39(5), 327–338. https://doi.org/10.1016/S0167-6911(99)00112-7
   Web of Science ®Google Scholar
• Shiriaev, A., & Fradkov, A. (1999). Stabilization of invariant sets of cascaded nonlinear systems. In Proc. 38th IEEE Conf. on Decision and Control (CDC) (Vol. 2, pp. 1296–1301). IEEE.
   Google Scholar
• Shiriaev, A. S., & Fradkov, A. L. (2001). Stabilization of invariant sets for nonlinear systems with applications to control of oscillations. International Journal of Robust and Nonlinear Control, 11(3), 215–240. https://doi.org/10.1002/rnc.v11:3
   Web of Science ®Google Scholar
• Slotine, J. J. E., & Li, W. (1991). Applied nonlinear control. Prentice-Hall.
   Google Scholar