Near Optimal LQR Performance in the Decentralized Setting

In this paper, we consider the use of a linear periodic controller (LPC) for the control of linear time-invariant (LTI) plants in the decentralized setting. If a plant has an unstable decentralized fixed mode (DFM), it is well known that no decentralized LTI controller can stabilize it, let alone provide good performance. Here we show that, if the plant is centrally controllable and observable and the graph associated with the plant is strongly connected, then even if the plant has an unstable DFM, we can still design a decentralized LPC to provide LQR performance as close to the centralized optimal performance as desired; the proposed controller in each channel consists of a sampler, a zero-order-hold, and a discrete-time linear periodic compensator, which makes it easy to implement.

[1]  E. Davison,et al.  Decentralized stabilization and pole assignment for general proper systems , 1990 .

[2]  E. Davison,et al.  Sampling and Decentralized Fixed Modes , 1985, 1985 American Control Conference.

[3]  Petros G. Voulgaris.  Control under structural constraints: An input-output approach , 1998, Robustness in Identification and Control.

[4]  Daniel E. Miller,et al.  Model Reference Adaptive Control Using Simultaneous Probing, Estimation, and Control , 2010, IEEE Transactions on Automatic Control.

[5]  Daniel E. Miller,et al.  Pole placement adaptive control with persistent jumps in the plant parameters , 2010, CDC.

[6]  Bruce A. Francis,et al.  Optimal Sampled-Data Control Systems , 1996, Communications and Control Engineering Series.

[7]  Edward J. Davison,et al.  An algebraic characterization of quotient decentralized fixed modes , 2012, Autom..

[8]  Sanjay Lall,et al.  Optimal synthesis and explicit state-space solution for a decentralized two-player linear-quadratic regulator , 2010, 49th IEEE Conference on Decision and Control (CDC).

[9]  Mohammad Aldeen,et al.  Stabilization of decentralized control systems , 1997 .

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

[11]  Pablo A. Parrilo,et al.  $ {\cal H}_{2}$-Optimal Decentralized Control Over Posets: A State-Space Solution for State-Feedback , 2010, IEEE Transactions on Automatic Control.

[12]  Murti V. Salapaka,et al.  Structured optimal and robust control with multiple criteria: a convex solution , 2004, IEEE Transactions on Automatic Control.

[13]  Pramod P. Khargonekar,et al.  Decentralized control and periodic feedback , 1994, IEEE Trans. Autom. Control..

[14]  Daniel E. Miller,et al.  Near optimal LQR performance for uncertain first order systems , 2004 .

[15]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[16]  Design of Low Order Compensators for Large Scale Decentralized Systems , 1995 .

[17]  T. Yoshikawa,et al.  Graph-theoretic approach to controllability and localizability of decentralized control systems , 1982 .

[18]  S. Lall,et al.  An explicit state-space solution for a decentralized two-player optimal linear-quadratic regulator , 2010, Proceedings of the 2010 American Control Conference.

[19]  Edward J. Davison,et al.  Structural modification of systems using discretization and generalized sampled-data hold functions , 2006, Autom..

[20]  Edward J. Davison,et al.  Discrete-time control of continuous systems with approximate decentralized fixed modes , 2008, Autom..

[21]  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).

[22]  A. Stephen Morse,et al.  Decentralized control of linear multivariable systems , 1976, Autom..

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

[24]  Daniel E. Miller.  Near Optimal LQR Performance for a Compact Set of Plants , 2006, IEEE Transactions on Automatic Control.

[25]  B. O. Anderson,et al.  Time-varying feedback laws for decentralized control , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[26]  E. Davison,et al.  On the stabilization of decentralized control systems , 1973 .

[27]  Edward J. Davison,et al.  Near optimal LQR control in the decentralized setting , 2011, IEEE Conference on Decision and Control and European Control Conference.

[28]  Jacques L. Willems.  Time-varying feedback for the stabilization of fixed modes in decentralized control systems , 1989, Autom..

[29]  Sanjay Lall,et al.  A state-space solution to the two-player decentralized optimal control problem , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[30]  Shih-Ho Wang,et al.  Stabilization of decentralized control systems via time-varying controllers , 1982 .

[31]  Daniel E. Miller,et al.  Simultaneous stabilization with near optimal LQR performance , 2001, IEEE Trans. Autom. Control..