The Value of Communication in Synthesizing Controllers given an Information Structure

We consider computation of optimal distributed controllers with constraints on states and inputs. The problem of distributed control arises in several large-scale systems, such as transportation and power grid systems. Well-established results highlight the non-convexity of the corresponding optimization problem for many cases of interest. In particular, convex computation of structured controllers is not possible for any system whose dynamics propagate according to a strongly connected topology. Such limitation is due to the fact that given enough time, the decisions of each controller spread to all system states and thus affect the future decisions of other controllers. Based on this insight, we propose designing communication links between controllers to propagate relevant output information quickly enough for convexity. We show that a limited number of communication links between controllers can be used to restore convexity for strongly connected systems. The resulting distributed control policy is optimal and satisfies state and input constraints robustly. Additionally, we propose novel graph theoretic interpretations of the result.

[1]  Anders Rantzer,et al.  On Feasibility, Stability and Performance in Distributed Model Predictive Control , 2013, IEEE Transactions on Automatic Control.

[2]  H. Witsenhausen.  A Counterexample in Stochastic Optimum Control , 1968 .

[3]  Maryam Kamgarpour,et al.  Robust control of constrained systems given an information structure , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[4]  M. Rotkowitz.  Tractable problems in optimal decentralized control , 2005 .

[5]  Y. Ho,et al.  Team decision theory and information structures in optimal control problems--Part II , 1972 .

[6]  Javad Lavaei,et al.  Convex Relaxation for Optimal Distributed Control Problems , 2014, IEEE Transactions on Automatic Control.

[7]  Laurent Lessard,et al.  Optimal decentralized state-feedback control with sparsity and delays , 2013, Autom..

[8]  Yohji Kobayashi.  Decentralized robust optimal control of large flexible space structures by local proper controllers using displacement output , 2015 .

[9]  Nikolai Matni,et al.  Localized LQR optimal control , 2014, 53rd IEEE Conference on Decision and Control.

[10]  Nikolai Matni,et al.  System Level Parameterizations, constraints and synthesis , 2017, 2017 American Control Conference (ACC).

[11]  P. Voulgaris.  A convex characterization of classes of problems in control with specific interaction and communication structures , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[12]  Calin Belta,et al.  Distributed Robust Set-Invariance for Interconnected Linear Systems , 2017, 2018 Annual American Control Conference (ACC).

[13]  John Doyle,et al.  On the structure of state-feedback LQG controllers for distributed systems with communication delays , 2011, IEEE Conference on Decision and Control and European Control Conference.

[14]  John Doyle,et al.  Dynamic programming solutions for decentralized state-feedback LQG problems with communication delays , 2012, 2012 American Control Conference (ACC).

[15]  Ashutosh Nayyar,et al.  Structural results for partially nested LQG systems over graphs , 2015, 2015 American Control Conference (ACC).

[16]  James D. Turner,et al.  Optimal distributed control of a flexible spacecraft during a large-angle maneuver , 1984 .

[17]  Pablo A. Parrilo,et al.  ℋ2-optimal decentralized control over posets: A state space solution for state-feedback , 2010, 49th IEEE Conference on Decision and Control (CDC).

[18]  S. Lall,et al.  Quadratic invariance is necessary and sufficient for convexity , 2011, Proceedings of the 2011 American Control Conference.

[19]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[20]  J. Tsitsiklis,et al.  NP-hardness of some linear control design problems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[21]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[22]  Nuno C. Martins,et al.  Information structures in optimal decentralized control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[23]  R. S. Ryan,et al.  Dynamics and control of large space structures , 1984 .

[24]  Michael Rotkowitz.  Parametrization of all stabilizing controllers subject to any structural constraint , 2010, 49th IEEE Conference on Decision and Control (CDC).

[25]  J. Speyer,et al.  Centralized and decentralized solutions of the linear-exponential-Gaussian problem , 1994, IEEE Trans. Autom. Control..

[26]  Eric C. Kerrigan,et al.  Optimization over state feedback policies for robust control with constraints , 2006, Autom..

[27]  Nikolai Matni,et al.  Separable and Localized System-Level Synthesis for Large-Scale Systems , 2017, IEEE Transactions on Automatic Control.

[28]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[29]  Eilyan Bitar,et al.  Performance bounds for robust decentralized control , 2016, 2016 American Control Conference (ACC).

[30]  Sanjay Lall,et al.  A graph-theoretic approach to distributed control over networks , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[31]  John C. Doyle,et al.  The H2 Control Problem for Quadratically Invariant Systems with Delays , 2013 .

[32]  M. Rotkowitz,et al.  Q-Parametrization and an SDP for H∞-optimal Decentralized Control , 2013 .

[33]  Manfred Morari,et al.  Distributed synthesis and stability of cooperative distributed model predictive control for linear systems , 2016, Autom..

[34]  Randy Cogill,et al.  Convexity of optimal control over networks with delays and arbitrary topology , 2010 .

[35]  Andrew G. Lamperski,et al.  Optimal state-feedback control under sparsity and delay constraints , 2012 .

[36]  Eilyan Bitar,et al.  A Convex Information Relaxation for Constrained Decentralized Control Design Problems , 2017, IEEE Transactions on Automatic Control.

[37]  Sanjay Lall,et al.  A Characterization of Convex Problems in Decentralized Control$^ast$ , 2005, IEEE Transactions on Automatic Control.