Block-wise discretization accounting for structural constraints

This paper presents an approximate discretization method, named Mixed Euler-ZOH (mE-ZOH), which can be applied to any continuous-time linear system. This method has been explicitly developed to preserve the system sparsity, a property that is particularly important when dealing with the analysis and design of distributed controllers for large-scale systems. In terms of stability preservation as a function of the sampling interval, we show that mE-ZOH outperforms the classical forward Euler (fE) approach, which is the only known discretization method guaranteeing the preservation of sparsity for all possible sampling times. It is then shown that this new discretization method is capable of preserving stability for all sampling times for a wide classes of dynamical systems, including the important class of positive systems. Besides stability, also positivity of the resulting discrete-time system is preserved, contrarily to what happens for the fE approach. A couple of examples are reported to illustrate the main theoretical results of the paper.

[1]  Riccardo Scattolini,et al.  Architectures for distributed and hierarchical Model Predictive Control - A review , 2009 .

[2]  Stefano Riverso,et al.  Tube-based distributed control of linear constrained systems , 2012, Autom..

[3]  Anders Rantzer,et al.  Distributed control of positive systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[4]  Francis J. Doyle,et al.  Structural analysis of large-scale systems for distributed state estimation and control applications , 2003 .

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

[6]  Stephen J. Wright,et al.  Cooperative distributed model predictive control , 2010, Syst. Control. Lett..

[7]  Marcello Farina,et al.  A Note on Discretization of Sparce Linear Systems , 2012, ROCOND.

[8]  Marcello Farina,et al.  Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems , 2012, Autom..

[9]  Eduardo Camponogara,et al.  Distributed model predictive control , 2002 .

[10]  Xingwen Liu,et al.  Constrained control of delayed discrete-time positive systems , 2008, 2008 International Conference on Communications, Circuits and Systems.

[11]  Cédric Langbort,et al.  A linear programming approach to routing control in networks of constrained linear positive systems , 2012, Autom..

[12]  H.M. Arneson,et al.  Distributed sliding mode control design for a class of positive compartmental systems , 2008, 2008 American Control Conference.

[13]  Patrizio Colaneri,et al.  Essentially negative news about positive systems , 2012 .

[14]  R. Scattolini.  Self-tuning control of systems with infrequent and delayed output sampling , 1988 .

[15]  Franco Blanchini,et al.  Rational 𝓛1 suboptimal compensators for continuous-time systems , 1994, IEEE Trans. Autom. Control..

[16]  Y. Ohta,et al.  Overlapping block diagonal dominance and existence of Liapunov functions , 1985 .

[17]  Cédric Langbort,et al.  Distributed control design for systems interconnected over an arbitrary graph , 2004, IEEE Transactions on Automatic Control.

[18]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.