A model predictive control framework for constrained uncertain positive systems

ABSTRACT In this paper, a new model predictive control framework is proposed for positive systems subject to input/state constraints and interval/polytopic uncertainty. Instead of traditional quadratic performance index, simple linear performance index, linear Lyapunov function, cone invariant set with linear form and linear computation tool are first adopted. Then, a control law that can handle the constraints and robustly stabilise the systems is proposed. The advantages of the new framework lie in the following facts: (1) an equivalent linear problem is formulated that can be easily solved than other problems including the quadratic ones, (2) simple linear index and linear tool can be used based on the essential property of positive systems to achieve the desired control performance and (3) a general model predictive control law without sign restriction is designed. Finally, an attempt of application on mitigating viral escape is provided to verify the effectiveness of the proposed approach.

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

[2]  Guochen Pang,et al.  Robust stabilisation for constrained discrete-time switched positive linear systems with uncertainties , 2015 .

[3]  Richard D. Braatz,et al.  Switched model predictive control of switched linear systems: Feasibility, stability and robustness , 2016, Autom..

[4]  Manfred Morari,et al.  Robust Model Predictive Control , 1987, 1987 American Control Conference.

[5]  Jun Huang,et al.  Robust Stabilization of Discrete-Time Positive Switched Systems with Uncertainties and Average Dwell Time Switching , 2014, Circuits Syst. Signal Process..

[6]  David Clarke,et al.  Application of generalised predictive control to the paper machine benchmark , 1995 .

[7]  Basil Kouvaritakis,et al.  Efficient robust predictive control , 2000, IEEE Trans. Autom. Control..

[8]  Rainer Manuel Schaich,et al.  Robust model predictive control , 2017 .

[9]  Xudong Zhao,et al.  Linear programming-based robust model predictive control for positive systems , 2016 .

[10]  J. Rawlings,et al.  The stability of constrained receding horizon control , 1993, IEEE Trans. Autom. Control..

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

[12]  J. Geromel,et al.  A new discrete-time robust stability condition , 1999 .

[13]  Anders Rantzer,et al.  Scalable control of positive systems , 2012, Eur. J. Control.

[14]  Mayuresh V. Kothare,et al.  Efficient robust constrained model predictive control with a time varying terminal constraint set , 2003, Syst. Control. Lett..

[15]  Yu-Geng Xi,et al.  Design of Robust Model Predictive Control Based on Multi-step Control Set , 2009 .

[16]  S. Joe Qin,et al.  An Overview of Nonlinear Model Predictive Control Applications , 2000 .

[17]  David Q. Mayne,et al.  Model predictive control: Recent developments and future promise , 2014, Autom..

[18]  Wei Zhang,et al.  Robust model predictive control with ℓ1-gain performance for positive systems , 2015, J. Frankl. Inst..

[19]  Zhengrong Xiang,et al.  Exponential L1 output tracking control for positive switched linear systems with time-varying delays , 2014 .

[20]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .

[21]  Henk B. Verbruggen,et al.  Optimizing the end-point state-weighting matrix in model-based predictive control , 2002, Autom..

[22]  S. Joe Qin,et al.  A survey of industrial model predictive control technology , 2003 .

[23]  Jon Rigelsford,et al.  Positive 1D and 2D Systems , 2002 .

[24]  ZhangLixian,et al.  Switched model predictive control of switched linear systems , 2016 .

[25]  Xiaohua Xia,et al.  Switched Model Predictive Control for Energy Dispatching of a Photovoltaic-Diesel-Battery Hybrid Power System , 2015, IEEE Transactions on Control Systems Technology.

[26]  D. G. Lewis,et al.  Application of predictive control techniques to a distillation column , 1991 .

[27]  Wei-Jie Mao,et al.  Robust stabilization of uncertain time-varying discrete systems and comments on "an improved approach for constrained robust model predictive control" , 2003, Autom..

[28]  Manfred Morari,et al.  An improved approach for constrained robust model predictive control , 2002, Autom..

[29]  M Morari,et al.  Energy efficient building climate control using Stochastic Model Predictive Control and weather predictions , 2010, Proceedings of the 2010 American Control Conference.

[30]  Long Wang,et al.  Constrained Control of Positive Discrete-Time Systems With Delays , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[32]  Franco Blanchini,et al.  Discrete‐time control for switched positive systems with application to mitigating viral escape , 2011 .

[33]  Xiaoming Chen,et al.  Dynamic output-feedback control for continuous-time interval positive systems under L1 performance , 2016, Appl. Math. Comput..

[34]  James Lam,et al.  L∞-gain analysis for positive systems with distributed delays , 2014, Autom..

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

[36]  Manfred Morari,et al.  Model predictive control: Theory and practice - A survey , 1989, Autom..

[37]  F. Tadeo,et al.  Control of constrained positive discrete systems , 2007, 2007 American Control Conference.

[38]  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.

[39]  James Lam,et al.  Positive Observers and Dynamic Output-Feedback Controllers for Interval Positive Linear Systems , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.

[40]  Marko Bacic,et al.  Model predictive control , 2003 .

[41]  Patrizio Colaneri,et al.  Switching Strategies to Mitigate HIV Mutation , 2014, IEEE Transactions on Control Systems Technology.

[42]  Xudong Zhao,et al.  Stability of a class of switched positive linear time‐delay systems , 2013 .

[43]  Fabian R. Wirth,et al.  A positive systems model of TCP-like congestion control: asymptotic results , 2006, IEEE/ACM Transactions on Networking.

[44]  M. Kothare,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[45]  Bertrand Cottenceau,et al.  Model predictive control for discrete event systems with partial synchronization , 2016, Autom..

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

[47]  Leonidas G. Bleris,et al.  A System-on-a-Chip Implementation for Embedded Real-Time Model Predictive Control , 2009, IEEE Transactions on Control Systems Technology.