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Sparsity-promoting optimal control of systems with symmetries, consensus and synchronization networks. (English) Zbl 1370.93035

Summary: Optimal control problems in systems with symmetries and consensus/synchronization networks are characterized by structural constraints that arise either from the underlying group structure or the lack of absolute measurements for part of the state vector. Our objective is to design controller structures and resulting control strategies that utilize limited information exchange between subsystems in large-scale networks. To obtain controllers with low communication requirements, we seek solutions to regularized versions of the \(\mathcal{H}_2\) optimal control problem. Non-smooth regularization terms are introduced to tradeoff network performance with sparsity of the feedback-gain matrix. In contrast to earlier results, our framework allows state-space representations that are used to quantify the system’s performance and sparsity of the static output-feedback controller to be expressed in different sets of coordinates. We show how alternating direction method of multipliers can be leveraged to exploit the underlying structure and compute sparsity-promoting controllers. In particular, for spatially-invariant systems, the computational complexity of our algorithm scales linearly with the number of subsystems. We also identify a class of optimal control problems that can be cast as semidefinite programs and provide an example to illustrate our developments.

MSC:

93A14 Decentralized systems
93A15 Large-scale systems
90C22 Semidefinite programming
49N90 Applications of optimal control and differential games
Full Text: DOI

References:

[1] Bamieh, B.; Paganini, F.; Dahleh, M. A., Distributed control of spatially invariant systems, IEEE Trans. Automat. Control, 47, 7, 1091-1107 (2002) · Zbl 1364.93363
[2] Rotkowitz, M.; Lall, S., A characterization of convex problems in decentralized control, IEEE Trans. Automat. Control, 51, 2, 274-286 (2006) · Zbl 1366.93017
[3] Fardad, M.; Lin, F.; Jovanović, M. R., Sparsity-promoting optimal control for a class of distributed systems, (Proceedings of the 2011 American Control Conference (2011)), 2050-2055
[4] Lin, F.; Fardad, M.; Jovanović, M. R., Sparse feedback synthesis via the alternating direction method of multipliers, (Proceedings of the 2012 American Control Conference (2012)), 4765-4770
[5] Lin, F.; Fardad, M.; Jovanović, M. R., Design of optimal sparse feedback gains via the alternating direction method of multipliers, IEEE Trans. Automat. Control, 58, 9, 2426-2431 (2013) · Zbl 1369.93215
[6] Matni, N.; Chandrasekaran, V., Regularization for design, IEEE Trans. Automat. Control, 61, 12, 3991-4006 (2016) · Zbl 1359.93148
[7] Mesbahi, M.; Egerstedt, M., Graph Theoretic Methods in Multiagent Networks (2010), Princeton University Press · Zbl 1203.93001
[8] Xiao, L.; Boyd, S.; Kim, S.-J., Distributed average consensus with least-mean-square deviation, J. Parallel Distrib. Comput., 67, 1, 33-46 (2007) · Zbl 1109.68019
[9] Ghosh, A.; Boyd, S.; Saberi, A., Minimizing effective resistance of a graph, SIAM Rev., 50, 1, 37-66 (2008) · Zbl 1134.05057
[10] Ren, W., Synchronization of coupled harmonic oscillators with local interaction, Automatica, 44, 3195-3200 (2008) · Zbl 1153.93421
[11] Zelazo, D.; Mesbahi, M., Edge agreement: Graph-theoretic performance bounds and passivity analysis, IEEE Trans. Automat. Control, 56, 3, 544-555 (2011) · Zbl 1368.93162
[12] Bamieh, B.; Jovanović, M. R.; Mitra, P.; Patterson, S., Coherence in large-scale networks: dimension dependent limitations of local feedback, IEEE Trans. Automat. Control, 57, 9, 2235-2249 (2012) · Zbl 1369.93685
[13] Dörfler, F.; Bullo, F., Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators, SIAM J. Control Optim., 50, 1616-1642 (2012) · Zbl 1264.34105
[14] Mauroy, A.; Sacré, P.; Sepulchre, R. J., Kick synchronization versus diffusive synchronization, (Proceedings of the 51st IEEE Conference on Decision and Control (2012)), 7171-7183
[15] Arcak, M., Synchronization and pattern formation in diffusively coupled systems, (Proceedings of the 51st IEEE Conference on Decision and Control (2012)), 7184-7192
[16] Zelazo, D.; Schuler, S.; Allgöwer, F., Performance and design of cycles in consensus networks, Systems Control Lett., 62, 1, 85-96 (2013) · Zbl 1257.93010
[17] Fardad, M.; Lin, F.; Jovanović, M. R., Design of optimal sparse interconnection graphs for synchronization of oscillator networks, IEEE Trans. Automat. Control, 59, 9, 2457-2462 (2014) · Zbl 1360.93245
[18] Wu, X.; Jovanović, M. R., Sparsity-promoting optimal control of consensus and synchronization networks, (Proceedings of the 2014 American Control Conference (2014)), 2948-2953
[19] Dörfler, F.; Jovanović, M. R.; Chertkov, M.; Bullo, F., Sparse and optimal wide-area damping control in power networks, (Proceedings of the 2013 American Control Conference (2013)), 4295-4300
[20] Dörfler, F.; Jovanović, M. R.; Chertkov, M.; Bullo, F., Sparsity-promoting optimal wide-area control of power networks, IEEE Trans. Power Syst., 29, 5, 2281-2291 (2014)
[21] Wu, X.; Dörfler, F.; Jovanović, M. R., Input-output analysis and decentralized optimal control of inter-area oscillations in power systems, IEEE Trans. Power Syst., 31, 3, 2434-2444 (2016)
[22] 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., 3, 1, 1-124 (2011)
[23] Kundur, P., Power System Stability and Control (1994), McGraw-Hill
[24] Zoltowski, D. M.; Dhingra, N. K.; Lin, F.; Jovanović, M. R., Sparsity-promoting optimal control of spatially-invariant systems, (Proceedings of the 2014 American Control Conference (2014)), 1261-1266
[25] Candès, E. J.; Wakin, M. B.; Boyd, S. P., Enhancing sparsity by reweighted \(\ell_1\) minimization, J. Fourier Anal. Appl, 14, 877-905 (2008) · Zbl 1176.94014
[26] Polyak, B.; Khlebnikov, M.; Shcherbakov, P., An LMI approach to structured sparse feedback design in linear control systems, (Proceedings of the 2013 European Control Conference (2013)), 833-838
[27] Dhingra, N. K.; Jovanović, M. R.; Luo, Z. Q., An ADMM algorithm for optimal sensor and actuator selection, (Proceedings of the 53rd IEEE Conference on Decision and Control (2014)), 4039-4044
[28] Dhingra, N. K.; Jovanović, M. R., Convex synthesis of symmetric modifications to linear systems, (Proceedings of the 2015 American Control Conference (2015)), 3583-3588
[29] Dhingra, N. K.; Colombino, M.; Jovanović, M. R., On the convexity of a class of structured optimal control problems for positive systems, (Proceedings of the 2016 European Control Conference (2016)), 825-830
[30] Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 1, 183-202 (2009) · Zbl 1175.94009
[31] Barzilai, J.; Borwein, J. M., Two-point step size gradient methods, IMA J. Numer. Anal., 8, 1, 141-148 (1988) · Zbl 0638.65055
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