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
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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 |
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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