×

A randomised algorithm for computing static-output-feedbacks for large-scale systems. (English) Zbl 1485.93034

The article considers numerical algorithms for the computation of static-output-feedbacks for large-scale linear time-invariant control systems of the form \[\dot{x}(t)=Ax(t)+Bu(t), \ \ y(t)=Cx(t),\] where the number of inputs \(n_u\) and outputs \(n_y\) is small in comparison to the state space. In particular, it is assumed that \(n_u\cdot n_y \ll n_x^2\). The contribution of the manuscript is a modification of a method called “ray-shooting algorithm”. In more detail, the authors suggest the inclusion of random search directions within the algorithm. A considerable amount of numerical examples is used to demonstrate the applicability of the method. However, while the authors theoretically seem to aim at “large-scale” or “huge-scale” systems, the final examples do not go beyond medium-scale state space dimensions of  \(n_x=3600\) which can easily be handled by standard optimal control algorithms, e.g., LQG/LQR/\(\mathcal{H}_\infty\)-control.

MSC:

93A15 Large-scale systems
93B52 Feedback control
93D15 Stabilization of systems by feedback

Software:

HIFOO; PENBMI
Full Text: DOI

References:

[1] Apkarian, P.; Dao, M. N.; Noll, D., Parametric robust structured control design, IEEE Transactions on Automatic Control, 60, 7, 1857-1869 (2015) · Zbl 1360.93213 · doi:10.1109/TAC.2015.2396644
[2] Apkarian, P., Gahinet, P., & Buhr, C. (2014, July 24). Multi-model, multi-objective tuning of fixed-structure controllers.2014 European control conference (ECC), (pp. 856-861), Strasbourg, France, . IEEE.
[3] Apkarian, P.; Noll, D., Nonsmooth \(####\) synthesis, IEEE Transactions on Automatic Control, 51, 1, 71-86 (2006) · Zbl 1366.93148 · doi:10.1109/TAC.2005.860290
[4] Arzelier, D.; Deaconu, G.; Gumussoy, S.; Henrion, D. (2011)
[5] Arzelier, D., Gryazina, E. N., Peaucelle, D., & Polyak, B. T. (2010). Mixed LMI/randomized methods for static output feedback control. In Proceedings of the American Control Conference, (pp. 4683-4688). Baltimore, MD: IEEE Conference Publications.
[6] Berekmeri, M.; Serrano, D.; Bouchenak, S.; Marchand, N.; Robu, B., A control approach for performance of big data systems, IFAC Proceedings Volumes, 47, 3, 152-157 (2014) · doi:10.3182/20140824-6-ZA-1003.01319
[7] Blondel, V.; Tsitsiklis, J. N., NP-hardness of some linear control design problems, SIAM Journal on Control and Optimization, 35, 6, 2118-2127 (1997) · Zbl 0892.93050 · doi:10.1137/S0363012994272630
[8] Borges, R. A., Calliero, T. R., Oliveira, C. L. F., & Peres, P. L. D. (2011). Improved conditions for reduced-order \(####\) filter design as a static output feedback problem. In American Control Conference, , San-Francisco, CA, USA, , 2011.
[9] Fu, M., Pole placement via static output feedback is NP-hard, IEEE Transactions On Automatic Control, 49, 5, 855-857 (2004) · Zbl 1365.93196 · doi:10.1109/TAC.2004.828311
[10] Gumussoy, S., Henrion, D., Millstone, M., & Overton, M. L. (2009). Multiobjective Robust Control with HIFOO 2.0. In Proceedings of the IFAC symposium on robust control design, , Haifa, Israel, , 2009.
[11] Henrion, D., Loefberg, J., Kočvara, M., & Stingl, M. (2005). Solving polynomial static output feedback problems with PENBMI. In Proc. Joint IEEE conf. decision control and Europ. control conf., , Sevilla, Spain, , 2005.
[12] Horn, R. A.; Johnson, C. R., Matrix analysis (2012), New York, NY: Cambridge University Press
[13] Leibfritz, F. (2003a). \(####\) ib: Constrained Matrix-optimization Problem library – a collection of test examples for nonlinear semidefinite programs, control system design and related problems (Tech.-Report).
[14] Leibfritz, F. (2003b). Description of the benchmark examples in \(####\) ib 1.0 (Tech.-Report).
[15] Leibfritz, F., & Lipinski, W. (2004). \(####\) ib 1.0 - User manual and quick reference (Tech.-Report).
[16] Mesbahi, M., A semi-definite programming solution of the least order dynamic output feedback synthesis problem., 2, 1851-1856 (1999)
[17] Mohammadpour, J.; Grigoriadis, K. M., Efficient modeling and control of large-scale systems (2010), Berlin: Springer Science & Business Media · Zbl 1196.93004
[18] Nemirovskii, A., Several NP-hard problems arising in robust stability analysis, Mathematics of Control, Signals, and Systems, 6, 99-105 (1993) · Zbl 0792.93100 · doi:10.1007/BF01211741
[19] Osiadacz, A. J., 8, 8 (1987)
[20] Palacios-Quiñonero, F.; Rubió-Massegú, J.; Rossell, J. M.; Karimi, H. R., Recent advances in static output- feedback controller design with applications to vibration control of large structures, Modeling, Identification and Control, 35, 3, 169-190 (2014) · Zbl 1293.93357 · doi:10.4173/mic.2014.3.4
[21] Peretz, Y., A randomized approximation algorithm for the minimal-norm static-output-feedback problem, Automatica, 63, 221-234 (2016) · Zbl 1329.93112 · doi:10.1016/j.automatica.2015.10.001
[22] Peretz, Y., On applications of the Ray-Shooting method for structured and structured-sparse static-output-feedbacks, International Journal of Systems Science, 48, 9, 1902-1913 (2017) · Zbl 1371.93223 · doi:10.1080/00207721.2017.1290300
[23] Peretz, Y., On application of the Ray-Shooting method for LQR via static-output-feedback, MDPI Algorithms Journal, 11, 1, 1-13 (2018) · Zbl 1461.49038
[24] Peretz, Y., A randomized algorithm for optimal PID controllers, MDPI Algorithms Journal, 11, 81, 1-15 (2018) · Zbl 1460.93036
[25] Peretz, Y., Moyal, S., & Merzbach, O. (2018). A randomized algorithm for robust stabilization via static-output-feedback. In IEEE international conference on the science of electrical engineering in Israel (ICSEE), , Eilat, Israel, , December 12-14, 2018.
[26] Sadabadi, M. S.; Peaucelle, D., From static output feedback to structured robust static output feedback: A survey, Annual Reviews in Control, 42, 11-26 (2016) · doi:10.1016/j.arcontrol.2016.09.014
[27] Syrmos, V. L.; Abdallah, C.; Dorato, P.; Grigoradis, K., Static output feedback: A survey, Automatica, 33, 125-137 (1997) · Zbl 0872.93036 · doi:10.1016/S0005-1098(96)00141-0
[28] Tempo, R.; Calafiore, G.; Dabbene, F., Randomized algorithms for analysis and control of uncertain systems (2005), London: Springer-Verlag, London · Zbl 1079.93002
[29] Tempo, R.; Ishii, H., Monte Carlo and Las Vegas randomized algorithms for systems and control, European Journal of Control, 13, 189-203 (2007) · Zbl 1293.93698 · doi:10.3166/ejc.13.189-203
[30] Vidyasagar, M.; Blondel, V. D., Probabilistic solutions to some NP-hard matrix problems, Automatica, 37, 1397-1405 (2001) · Zbl 1031.93165 · doi:10.1016/S0005-1098(01)00089-9
[31] Yang, K.; Orsi, R., Generalized pole placement via static output feedback: A methodology based on projections, Automatica, 42, 2143-2150 (2006) · Zbl 1104.93036 · doi:10.1016/j.automatica.2006.06.021
[32] Zečević, A. I.; Šiljak, D. D., Design of robust static output feedback for large-scale systems, IEEE Transactions On Automatic Control, 49, 11, 2040-2044 (2004) · Zbl 1365.93028 · doi:10.1109/TAC.2004.837542
[33] Zheng, F.; Wang, Q. G.; Lee, T. H., On the design of multivariable PID controllers via LMI approach, Automatica, 38, 517-526 (2002) · Zbl 1064.93017 · doi:10.1016/S0005-1098(01)00237-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.