Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design
Abstract
:1. Introduction
Notation
2. Reduced-Order Controller Design Problems
2.1. Reduced-Order Stabilizing Controllers
2.2. Reduced-Order Controllers
3. Algorithms
3.1. Proposed Algorithm
3.2. Projection onto the Set of Rank-Constrained Structured Matrices
3.3. Projection onto the Set Described by LMIs
Algorithm 1 Algorithm to solve Problem 3 |
Require:
Initial guess Ensure: for do ▹ Projection onto for do end for ▹ Projection onto subject to , , end for |
4. Numerical Examples
4.1. Stabilizing Static Controllers
- 1
- Nonsmooth synthesis [36] with hinfstruct funciton in MATLAB (NS)
- 2
- Cone complementarity linearization algorithm [12] (CCL)
- 3
- Nuclear norm minimization [15] (NNM)
- 4
- Alternating projection with approximate projection onto [13] (GS96)
- 5
- Alternating projection with the proposed precise projection in Section 3 (Proposed)
4.2. Stabilizing Low-Order Controllers
4.3. Static Controllers
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Lemma 1
Appendix B. ADMM Algorithm
References
- Iwasaki, T.; Skelton, R.E. Parametrization of all stabilizing controllers via quadratic Lyapunov functions. J. Optim. Theory Appl. 1995, 85, 291–307. [Google Scholar] [CrossRef]
- Scherer, C.; Gahinet, P.; Chilali, M. Multiobjective output-feedback control via LMI optimization. IEEE Trans. Autom. Control 1997, 42, 896–911. [Google Scholar] [CrossRef]
- Sturm, J.F. Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 1999, 11, 625–653. [Google Scholar] [CrossRef]
- Toh, K.C.; Todd, M.J.; Tütüncü, R.H. SDPT3—A Matlab software package for semidefinite programming, Version 1.3. Optim. Methods Softw. 1999, 11, 545–581. [Google Scholar] [CrossRef]
- Available online: https://www.mosek.com/ (accessed on 20 July 2022).
- Lofberg, J. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the 2004 IEEE International Conference on Robotics and Automation, New Orleans, LA, USA, 26 April–1 May 2004; pp. 284–289. [Google Scholar]
- Grant, M.; Boyd, S. Graph implementations for nonsmooth convex programs. In Recent Advances in Learning and Control; Blondel, V., Boyd, S., Kimura, H., Eds.; Lecture Notes in Control and Information Sciences; Splinger: Berlin/Heidelberg, Germany, 2008; Volume 371, pp. 95–110. [Google Scholar]
- Diamond, S.; Boyd, S. CVXPY: A Python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 2016, 17, 1–5. [Google Scholar]
- Toker, O.; Özbay, H. On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback. In Proceedings of the 1995 American Control Conference (ACC’95), Seattle, WA, USA, 21–23 June 1995; Volume 4, pp. 2525–2526. [Google Scholar]
- Ghaoui, L.E.; Gahinet, P. Rank Minimization under LMI constraints: A Framework for Output Feedback Problems. In Proceedings of the European Control Conference 1993, Groningen, The Netherlands, 28 June–1 July 1993; Volume 3. [Google Scholar]
- Iwasaki, T.; Skelton, R.E. The XY-centring algorithm for the dual LMI problem: A new approach to fixed-order control design. Int. J. Control 1995, 62, 1257–1272. [Google Scholar] [CrossRef]
- El Ghaoui, L.; Oustry, F.; AitRami, M. A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans. Autom. Control 1997, 42, 1171–1176. [Google Scholar] [CrossRef]
- Grigoriadis, K.M.; Skelton, R.E. Low-order control design for LMI problems using alternating projection methods. Automatica 1996, 32, 1117–1125. [Google Scholar] [CrossRef]
- Yang, K.; Orsi, R.; Moore, J.B. A projective algorithm for static output feedback stabilization. In Proceedings of the 2nd IFAC Symposium on System, Structure and Control, Oaxaca, Mexico, 8–10 December 2004; pp. 263–268. [Google Scholar]
- Fazel, M.; Hindi, H.; Boyd, S. Rank minimization and applications in system theory. In Proceedings of the 2004 American Control Conference, Boston, MA, USA, 30 June–2 July 2004; Volume 4, pp. 3273–3278. [Google Scholar]
- Amirifar, R.; Sadati, N. Low-order H∞ controller design for an active suspension system via LMIs. IEEE Trans. Ind. Electron. 2006, 53. [Google Scholar] [CrossRef]
- Recht, B.; Xu, W.; Hassibi, B. Necessary and sufficient conditions for success of the nuclear norm heuristic for rank minimization. In Proceedings of the 2008 47th IEEE Conference on Decision and Control, Cancun, Mexico, 9–11 December 2008; pp. 3065–3070. [Google Scholar]
- Recht, B.; Fazel, M.; Parrilo, P.A. Guaranteed minimum-rank solutions of linear matrix equations via nulcear norm minimization. SIAM Rev. 2010, 52, 451–501. [Google Scholar] [CrossRef]
- Doelman, R.; Verhaegen, M. Sequential convex relaxation for robust static output feedback structured control. IFAC-PapersOnLine 2017, 50, 15518–15523. [Google Scholar] [CrossRef]
- Vidyasagar, M. An Introduction to Compressed Sensing; SIAM: Philadelphia, PA, USA, 2019. [Google Scholar]
- Mi, J.X.; Zhang, Y.N.; Lai, Z.; Li, W.; Zhou, L.; Zhong, F. Principal Component Analysis based on Nuclear norm Minimization. Neural Netw. 2019, 118, 1–16. [Google Scholar] [CrossRef]
- Liu, S.; Hu, Q.; Li, P.; Zhao, J.; Liu, M.; Zhu, Z. Speckle Suppression Based on Weighted Nuclear Norm Minimization and Grey Theory. IEEE Trans. Geosci. Remote Sens. 2019, 57, 2700–2708. [Google Scholar] [CrossRef]
- Tian, R.; Sun, G.; Liu, X.; Zheng, B. Sobel Edge Detection Based on Weighted Nuclear Norm Minimization Image Denoising. Electronics 2021, 10, 655. [Google Scholar] [CrossRef]
- Gunes, B.; van Wingerden, J.W.; Verhaegen, M. Tensor Nuclear Norm LPV Subspace Identification. IEEE Trans. Autom. Control 2018, 63, 3897–3903. [Google Scholar] [CrossRef]
- Varanasi, S.K.; Jampana, P. Nuclear norm subspace identification of continuous time state–space models with missing outputs. Control Eng. Pract. 2020, 95, 104239. [Google Scholar] [CrossRef]
- Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 2011, 3, 1–122. [Google Scholar] [CrossRef]
- Leibfritz, F. COMPLeib: Constraint Matrix Optimization Problem Library; Tech. Report; University of Trier: Trier, Germany, 2005. [Google Scholar]
- Available online: https://github.com/nagahara-masaaki/MR/ (accessed on 20 July 2022).
- Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B.A. State-space solutions to standard H2 and H∞ control problems. IEEE Trans. Autom. Control 1989, 34, 831–847. [Google Scholar] [CrossRef]
- Iwasaki, T. LMI and Control; Shokodo: Tokyo, Japan, 1997. [Google Scholar]
- Escalante, R.; Raydan, M. Alternating Projection Methods; SIAM: Philadelphia, PA, USA, 2011. [Google Scholar]
- Boyle, J.P.; Dykstra, R.L. A Method for Finding Projections onto the Intersection of Convex Sets in Hilbert Spaces. In Advances in Order Restricted Statistical Inference, Lecture Notes in Statistics; Dykstra, R., Robertson, T., Wright, F.T., Eds.; Springer: New York, NY, USA, 1986; Volume 37. [Google Scholar]
- Bai, J.; Li, J.; Dai, P. Novel alternating update method for low rank approximation of structured matrices. Appl. Numer. Math. 2017, 121, 223–233. [Google Scholar] [CrossRef]
- Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
- Nagahara, M.; Ogura, M.; Yamamoto, Y. Iterative Greedy LMI for Sparse Control. IEEE Contr. Syst. Lett. 2022, 6, 986–991. [Google Scholar] [CrossRef]
- Apkarian, P.; Noll, D. Nonsmooth H∞ Synthesis. IEEE Trans. Autom. Control 2006, 51, 71–86. [Google Scholar] [CrossRef] [Green Version]
- Fares, B.; Apkarian, P.; Noll, D. An augmented Lagrangian method for a lass of LMI-constrained problems in robust control theory. Int. J. Control 2001, 74, 348–360. [Google Scholar] [CrossRef]
- Miller, L.F.; Cohran, R.G.; Howze, J.W. Output feedback stabilization of a spectral radius functional. Int. J. Control 1978, 27, 455–462. [Google Scholar] [CrossRef]
- Rosenthal, J.; Wang, X.A. Output feedback pole placement with dynamic compensators. IEEE Trans. Autom. Control 1996, 41, 830–843. [Google Scholar] [CrossRef]
- Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control; John Wiley & Sons: Chichester, UK; New York, NY, USA; Brisbane, Australia; Toronto, ON, Canada; Singapore, 1996. [Google Scholar]
NS [36] | CCL [12] | NNM [15] | GS96 [13] | Proposed | |
---|---|---|---|---|---|
# success | 75 | 61 | 53 | 56 | 59 |
CPU time [s] | 0.365 | 56.5 | 4.58 | 663 | 666 |
NS [36] | CCL [12] | NNM [15] | GS96 [13] | Proposed | |
---|---|---|---|---|---|
HE6 | unstable | stable | unstable | stable | stable |
HE7 | stable | stable | unstable | stable | stable |
REA3 | unstable | stable | unstable | unstable | stable |
DIS1 | stable | stable | unstable | unstable | stable |
PAS | unstable | stable | stable | stable | stable |
TF1 | unstable | unstable | unstable | unstable | stable |
TF2 | unstable | stable | stable | stable | stable |
NN1 | stable | unstable | unstable | unstable | stable |
NN11 | stable | stable | unstable | stable | stable |
NN12 | unstable | unstable | unstable | stable | stable |
FS | unstable | stable | stable | stable | stable |
NS [36] | CCL [12] | NNM [15] | GS96 [13] | Proposed | |
---|---|---|---|---|---|
TF1 () | unstable | stable | stable | stable | stable |
CPU time [s] | 0.016910 | 0.214947 | 0.177307 | 0.194252 | 1.836873 |
TF1 () | unstable | stable | stable | stable | stable |
CPU time [s] | 0.016012 | 0.263448 | 0.205307 | 0.206729 | 4.250059 |
Model | AC4 | NN1 | NN12 | HE6 |
---|---|---|---|---|
1.000 | 74.72 | 28.09 | 520.0 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Nagahara, M.; Iwai, Y.; Sebe, N. Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design. Algorithms 2022, 15, 322. https://doi.org/10.3390/a15090322
Nagahara M, Iwai Y, Sebe N. Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design. Algorithms. 2022; 15(9):322. https://doi.org/10.3390/a15090322
Chicago/Turabian StyleNagahara, Masaaki, Yu Iwai, and Noboru Sebe. 2022. "Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design" Algorithms 15, no. 9: 322. https://doi.org/10.3390/a15090322