Static Output-feedback Controller Synthesis for Positive Systems under ℓ ∞ Performance

This paper investigates the l∞-induced output-feedback controller synthesis problem for discrete-time positive systems. For positive linear systems, a necessary and sufficient condition for stability and l∞-induced performance is first put forward using a group of linear inequalities. Then, on the basis of these inequalities, the conditions on which static output-feedback controllers exist are set up, and to solve these conditions, a method of iterative convex optimization is provided. Moreover, a complete solution to the synthesis problem of the controller is proposed based on linear programming for positive systems with single output or input. Finally, two examples are presented to illustrate and verify the effective and applicable methods derived in this paper.

[1]  Hamid Reza Karimi,et al.  Stability and l1-gain analysis for positive 2D T-S fuzzy state-delayed systems in the second FM model , 2014, Neurocomputing.

[2]  James Lam,et al.  Internal positivity preserved model reduction , 2010, Int. J. Control.

[3]  Sanjit A. Seshia,et al.  Symbolic control design for monotone systems with directed specifications , 2017, Autom..

[4]  Wassim M. Haddad,et al.  Stability theory for nonnegative and compartmental dynamical systems with time delay , 2004, Proceedings of the 2004 American Control Conference.

[5]  Lou Caccetta,et al.  A positive linear discrete-time model of capacity planning and its controllability properties , 2004, Math. Comput. Model..

[6]  James Lam,et al.  Positive filtering for continuous-time positive systems under L1 performance , 2014, Int. J. Control.

[7]  W. Haddad,et al.  Nonnegative and Compartmental Dynamical Systems , 2010 .

[8]  Dimitri Peaucelle,et al.  Optimal L1-controller synthesis for positive systems and its robustness properties , 2012, 2012 American Control Conference (ACC).

[9]  Yonggui Kao,et al.  Stochastic Stability, ℒ1-gain and Control Synthesis for Positive Semi-Markov Jump Systems , 2018, International Journal of Control, Automation and Systems.

[10]  Hao Xing,et al.  Input-output Finite-time Control of Uncertain Positive Impulsive Switched Systems with Time-varying and Distributed Delays , 2018 .

[11]  Xingwen Liu,et al.  Stability Criterion of 2‐D Positive Systems with Unbounded Delays Described by Roesser Model , 2015 .

[12]  M. E. Valcher.  Controllability and reachability criteria for discrete time positive systems , 1996 .

[13]  Xiaoming Chen,et al.  A novel approach to L1-induced controller synthesis for positive systems with interval uncertainties , 2017, J. Frankl. Inst..

[14]  Shengyuan Xu,et al.  Control for stability and positivity: equivalent conditions and computation , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[15]  J. Kurek.  Stability of positive 2-D system described by the Roesser model , 2002 .

[16]  Jun Shen,et al.  Input-output gain analysis for linear systems on cones , 2017, Autom..

[17]  Min Meng,et al.  Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization , 2013, J. Frankl. Inst..

[18]  P. Varaiya,et al.  Analysis of fixed-time control☆ , 2014, 1408.4229.

[19]  Dimitri Peaucelle,et al.  LMI approach to linear positive system analysis and synthesis , 2014, Syst. Control. Lett..

[20]  F. Tadeo,et al.  Controller Synthesis for Positive Linear Systems With Bounded Controls , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[21]  Shen Yin,et al.  Improved results on stability of continuous-time switched positive linear systems , 2014, Autom..

[22]  Fei Hao,et al.  Asynchronous decentralised event-triggered control of multi-agent systems , 2014, Int. J. Control.

[23]  Luca Benvenuti,et al.  A tutorial on the positive realization problem , 2004, IEEE Transactions on Automatic Control.

[24]  Corentin Briat,et al.  Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1‐gain and L∞‐gain characterization , 2012, ArXiv.

[25]  James Lam,et al.  ℓ1ℓ1-induced Norm and Controller Synthesis of Positive Systems , 2013, Autom..

[26]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[27]  Yonggui Kao,et al.  Positive observer design for positive Markovian jump systems with mode-dependent time-varying delays and incomplete transition rates , 2017 .

[28]  Chunyu Yang,et al.  Positive observer design for discrete-time positive system with missing data in output , 2015, The 27th Chinese Control and Decision Conference (2015 CCDC).

[29]  Dong Wang,et al.  Fault detection of switched positive systems based on lossy networks , 2014, Proceedings of the 33rd Chinese Control Conference.

[30]  Calin Belta,et al.  Formal Synthesis of Control Strategies for Positive Monotone Systems , 2017, IEEE Transactions on Automatic Control.

[31]  Ahmed El Hajjaji,et al.  Stabilization of LPV positive systems , 2014, 53rd IEEE Conference on Decision and Control.

[32]  Wenhai Qi,et al.  Positive L1-gain filter design for positive continuous-time Markovian jump systems with partly known transition rates , 2016 .

[33]  T. Raghavan,et al.  Nonnegative Matrices and Applications , 1997 .

[34]  Dimitri Peaucelle,et al.  Dominant pole of positive systems with time-delays , 2014, 2014 European Control Conference (ECC).

[35]  James Lam,et al.  Controller synthesis for positive Takagi-Sugeno fuzzy systems under ℓ1 performance , 2017, Int. J. Syst. Sci..

[36]  H. Caswell.  Matrix population models : construction, analysis, and interpretation , 2001 .

[37]  Maria Pia Fanti,et al.  Controllability of multi-input positive discrete-time systems , 1990 .

[38]  Long Wang,et al.  Containment control of heterogeneous multi-agent systems , 2014, Int. J. Control.

[39]  Xiaoming Chen,et al.  ℓ1-induced state-bounding observer design for positive Takagi-Sugeno fuzzy systems , 2017, Neurocomputing.

[40]  Xudong Zhao,et al.  Discussions on observer design of nonlinear positive systems via T-S fuzzy modeling , 2015, Neurocomputing.