# American Institute of Mathematical Sciences

September  2019, 9(3): 283-295. doi: 10.3934/naco.2019019

## Optimal sparse output feedback for networked systems with parametric uncertainties

 1 School of Electrical Engineering and Telecommunications, University of New South Wales, NSW 2052 Australia 2 Faculty of Engineering and Information Technology, University of Technology, Sydney (UTS), PO Box 123, Broadway, NSW 2007, Australia

* Corresponding author: steven.su@uts.edu.au

Received  May 2018 Revised  April 2019 Published  May 2019

Fund Project: An earlier version of the paper appears in the proceedings of American Control Conference 2018

This paper investigates the design of block row/column-sparse distributed static output ${H}_2$ feedback control for interconnected systems with polytopic uncertainties. The proposed approach is applicable to the networked systems with publisher/subscriber communication topology. We added two additional terms into the optimisation index function to penalise the number of publishers and subscribers. To optimally select a subset of available publishers and/or subscribers in the network, we introduced both an explicit scheme and an iterative process to handle this problem. We demonstrated the effectiveness by using a numerical example. The example showed that the simultaneous identification of favourable networks topologies and design of controller strategy can be achieved by using the proposed method.

Citation: Ahmadreza Argha, Steven W. Su, Lin Ye, Branko G. Celler. Optimal sparse output feedback for networked systems with parametric uncertainties. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 283-295. doi: 10.3934/naco.2019019
##### References:
 [1] R. Arastoo, Y. GhaedSharaf, M. V. Kothare and N. Motee, Optimal state feedback controllers with strict row sparsity constraints, in American Control Conference (ACC), IEEE, (2016), 1948–1953.Google Scholar [2] A. Argha, S. W. Su, A. Savkin and B. Celler, A framework for optimal actuator/sensor selection in a control system, International Journal of Control, 92 (2019), 242-260. doi: 10.1080/00207179.2017.1350755. Google Scholar [3] A. Argha, S. W. Su and A. Savkin, Optimal actuator/sensor selection through dynamic output feedback, in Decision and Control (CDC), IEEE, (2016), 3624–3629.Google Scholar [4] A. Argha, L. Li and S. W. Su, Design of $H_2$ ($H_{\infty}$)-based optimal structured and sparse static output feedback gains, Journal of the Franklin Institute, 354 (2017), 4156-4178. doi: 10.1016/j.jfranklin.2017.03.011. Google Scholar [5] A. Argha, S. W. Su, A. Savkin and B. G. Celler, Mixed $H_2/H_{\infty}$-based actuator selection for uncertain polytopic systems with regional pole placement, International Journal of Control, 91 (2018), 320-336. doi: 10.1080/00207179.2017.1279753. Google Scholar [6] A. Argha, L. Li and S. W. Su, $H_2$-based optimal sparse sliding mode control for networked control systems, International Journal of Robust and Nonlinear Control, 28 (2018), 16-30. doi: 10.1002/rnc.3852. Google Scholar [7] E. J. Candes, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. Google Scholar [8] M. Fardad, F. Lin and M. R. Jovanovic, Sparsity-promoting optimal control for a class of distributed systems, in Proc. the American Control Conference, San Francisco, CA, USA, (2011), 2050–2055.Google Scholar [9] M. Fardad and M. R. Jovanovic, On the design of optimal structured and sparse feedback gains via sequential convex programming, in American Control Conference (ACC), IEEE, (2014), 2426–2431.Google Scholar [10] F. Lin, M. Fardad and M. Jovanovic, Augmented lagrangian approach to design of structured optimal state feedback gains, IEEE Trans. Autom. Control, 56 (2011), 2923-2929. doi: 10.1109/TAC.2011.2160022. Google Scholar [11] G. Pipeleers, B. Demeulenaere, J. Swevers and L. Vandenberghe, Extended LMI characterizations for stability and performance of linear systems, Systems & Control Letters, 58 (2009), 510-518. doi: 10.1016/j.sysconle.2009.03.001. Google Scholar [12] M. Razeghi-Jahromi and A. Seyedi, Stabilization of networked control systems with sparse observer-controller networks, IEEE Transactions on Automatic Control, 60 (2015), 1686-1691. doi: 10.1109/TAC.2014.2360310. Google Scholar [13] J. Rubió-Massegú, J. M. Rossell, H. R. Karimi and F. Palacios-Quiñonero, Static output-feedback control under information structure constraints, Automatica, 49 (2013), 313-316. doi: 10.1016/j.automatica.2012.10.012. Google Scholar [14] D. D. Šiljak and A. Zečević, Control of large-scale systems: Beyond decentralized feedback, Annual Reviews in Control, 29 (2005), 169-179. Google Scholar [15] D. Siljak, Decentralized Control of Complex Systems, Dover Publications, 2012. Google Scholar [16] M. Staroswiecki and A. M. Amani, Fault-tolerant control of distributed systems by information pattern reconfiguration, International Journal of Adaptive Control and Signal Processing, 2014. doi: 10.1002/acs.2497. Google Scholar [17] S. Schuler, U. Münz and F. Allgöwer, Decentralized state feedback control for interconnected systems with application to power systems, Journal of Process Control, 24 (2014), 379-388. Google Scholar [18] M. Van De Wal and B. De Jager, A review of methods for input/output selection, Automatica, 37 (2001), 487-510. doi: 10.1016/S0005-1098(00)00181-3. Google Scholar [19] X. Wang and M. Lemmon, Event-triggering in distributed networked control systems, IEEE Transactions on Automatic Control, 56 (2011), 586-601. doi: 10.1109/TAC.2010.2057951. Google Scholar [20] A. Zecevic and D. Siljak, Global low-rank enhancement of decentralized control for large-scale systems, IEEE Transactions on Automatic Control, 50 (2005), 740-744. doi: 10.1109/TAC.2005.847054. Google Scholar [21] D. M. Zoltowski, N. Dhingra, F. Lin and M. R. Jovanovic, Sparsity-promoting optimal control of spatially-invariant systems, in American Control Conference (ACC) IEEE, (2014), 1255–1260.Google Scholar

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##### References:
 [1] R. Arastoo, Y. GhaedSharaf, M. V. Kothare and N. Motee, Optimal state feedback controllers with strict row sparsity constraints, in American Control Conference (ACC), IEEE, (2016), 1948–1953.Google Scholar [2] A. Argha, S. W. Su, A. Savkin and B. Celler, A framework for optimal actuator/sensor selection in a control system, International Journal of Control, 92 (2019), 242-260. doi: 10.1080/00207179.2017.1350755. Google Scholar [3] A. Argha, S. W. Su and A. Savkin, Optimal actuator/sensor selection through dynamic output feedback, in Decision and Control (CDC), IEEE, (2016), 3624–3629.Google Scholar [4] A. Argha, L. Li and S. W. Su, Design of $H_2$ ($H_{\infty}$)-based optimal structured and sparse static output feedback gains, Journal of the Franklin Institute, 354 (2017), 4156-4178. doi: 10.1016/j.jfranklin.2017.03.011. Google Scholar [5] A. Argha, S. W. Su, A. Savkin and B. G. Celler, Mixed $H_2/H_{\infty}$-based actuator selection for uncertain polytopic systems with regional pole placement, International Journal of Control, 91 (2018), 320-336. doi: 10.1080/00207179.2017.1279753. Google Scholar [6] A. Argha, L. Li and S. W. Su, $H_2$-based optimal sparse sliding mode control for networked control systems, International Journal of Robust and Nonlinear Control, 28 (2018), 16-30. doi: 10.1002/rnc.3852. Google Scholar [7] E. J. Candes, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_1$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. Google Scholar [8] M. Fardad, F. Lin and M. R. Jovanovic, Sparsity-promoting optimal control for a class of distributed systems, in Proc. the American Control Conference, San Francisco, CA, USA, (2011), 2050–2055.Google Scholar [9] M. Fardad and M. R. Jovanovic, On the design of optimal structured and sparse feedback gains via sequential convex programming, in American Control Conference (ACC), IEEE, (2014), 2426–2431.Google Scholar [10] F. Lin, M. Fardad and M. Jovanovic, Augmented lagrangian approach to design of structured optimal state feedback gains, IEEE Trans. Autom. Control, 56 (2011), 2923-2929. doi: 10.1109/TAC.2011.2160022. Google Scholar [11] G. Pipeleers, B. Demeulenaere, J. Swevers and L. Vandenberghe, Extended LMI characterizations for stability and performance of linear systems, Systems & Control Letters, 58 (2009), 510-518. doi: 10.1016/j.sysconle.2009.03.001. Google Scholar [12] M. Razeghi-Jahromi and A. Seyedi, Stabilization of networked control systems with sparse observer-controller networks, IEEE Transactions on Automatic Control, 60 (2015), 1686-1691. doi: 10.1109/TAC.2014.2360310. Google Scholar [13] J. Rubió-Massegú, J. M. Rossell, H. R. Karimi and F. Palacios-Quiñonero, Static output-feedback control under information structure constraints, Automatica, 49 (2013), 313-316. doi: 10.1016/j.automatica.2012.10.012. Google Scholar [14] D. D. Šiljak and A. Zečević, Control of large-scale systems: Beyond decentralized feedback, Annual Reviews in Control, 29 (2005), 169-179. Google Scholar [15] D. Siljak, Decentralized Control of Complex Systems, Dover Publications, 2012. Google Scholar [16] M. Staroswiecki and A. M. Amani, Fault-tolerant control of distributed systems by information pattern reconfiguration, International Journal of Adaptive Control and Signal Processing, 2014. doi: 10.1002/acs.2497. Google Scholar [17] S. Schuler, U. Münz and F. Allgöwer, Decentralized state feedback control for interconnected systems with application to power systems, Journal of Process Control, 24 (2014), 379-388. Google Scholar [18] M. Van De Wal and B. De Jager, A review of methods for input/output selection, Automatica, 37 (2001), 487-510. doi: 10.1016/S0005-1098(00)00181-3. Google Scholar [19] X. Wang and M. Lemmon, Event-triggering in distributed networked control systems, IEEE Transactions on Automatic Control, 56 (2011), 586-601. doi: 10.1109/TAC.2010.2057951. Google Scholar [20] A. Zecevic and D. Siljak, Global low-rank enhancement of decentralized control for large-scale systems, IEEE Transactions on Automatic Control, 50 (2005), 740-744. doi: 10.1109/TAC.2005.847054. Google Scholar [21] D. M. Zoltowski, N. Dhingra, F. Lin and M. R. Jovanovic, Sparsity-promoting optimal control of spatially-invariant systems, in American Control Conference (ACC) IEEE, (2014), 1255–1260.Google Scholar
Networked System Architecture
Structure of block row/column-sparse SOF gains for different values of $\Psi_s$ and $\Psi_p$
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