Design of controllers for hybrid linear systems with impulsive inputs and periodic jumps

In this study, the problem of designing a controller for a hybrid system with impulsive input and periodic jumps is addressed. In particular, it is shown that any hybrid system with impulsive inputs and periodic jumps can be recast into a discrete-time, linear, time-invariant system, which, in turn, can be used to design a controller by using classical methods. Furthermore, it is shown that, once such a controller has been designed, it can be readily used to control the hybrid system by mean of an interfacing system that is based just on the continuous-time dynamics of the plant to be controlled. Several examples, spanning from aerospace to biomedical applications, are reported in order to corroborate the theoretical results.

[1]  Bruce A. Francis,et al.  The internal model principle of control theory , 1976, Autom..

[2]  John E. Prussing,et al.  Optimal four-impulse fixed-time rendezvous in the vicinity of a circular orbit. , 1969 .

[3]  Laura Menini,et al.  Robust Hybrid Output Regulation for Linear Systems With Periodic Jumps: Semiclassical Internal Model Design , 2017, IEEE Transactions on Automatic Control.

[4]  Corrado Possieri,et al.  LQ optimal control for a class of hybrid systems , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[5]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[6]  O. M. Grasselli,et al.  A canonical decomposition of linear periodic discrete-time systems , 1984 .

[7]  Corrado Possieri,et al.  Zeros and poles of transfer functions for linear hybrid systems with periodic jumps , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[8]  G. Stein,et al.  Multivariable feedback design: Concepts for a classical/modern synthesis , 1981 .

[9]  Laura Menini,et al.  A case study for hybrid regulation: Output tracking for a spinning and bouncing disk , 2013, 21st Mediterranean Conference on Control and Automation.

[10]  Corrado Possieri,et al.  Structural Properties of a Class of Linear Hybrid Systems and Output Feedback Stabilization , 2017, IEEE Transactions on Automatic Control.

[11]  Laura Menini,et al.  Output regulation for a class of linear hybrid systems. Part 2: stabilization , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[12]  Xinyu Song,et al.  Modeling Impulsive Injections of Insulin: Towards Artificial Pancreas , 2012, SIAM J. Appl. Math..

[13]  Paolo Bolzern,et al.  Stabilizability and detectability of linear periodic systems , 1985 .

[14]  Tao Yang,et al.  In: Impulsive control theory , 2001 .

[15]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[16]  Enmin Feng,et al.  Bilevel parameters identification for the multi-stage nonlinear impulsive system in microorganisms fed-batch cultures , 2008 .

[17]  Jonas Gloeckner,et al.  Impulsive Differential Equations , 2016 .

[18]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[19]  A. Poularikas.  The transforms and applications handbook , 2000 .

[20]  Corrado Possieri,et al.  On the notion of transfer function for linear hybrid systems with periodic jumps , 2017, 2017 25th Mediterranean Conference on Control and Automation (MED).

[21]  Alberto Bemporad,et al.  Model Predictive Control for Linear Impulsive Systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[22]  Patrizio Colaneri,et al.  Stabilization of multirate sampled-data linear systems , 1990, Autom..

[23]  Thomas Carter,et al.  Optimal impulsive space trajectories based on linear equations , 1991 .

[24]  Ju H. Park,et al.  Quantized Static Output Feedback Control For Discrete-Time Systems , 2018, IEEE Transactions on Industrial Informatics.

[25]  Manuela Herman,et al.  Aided Navigation Gps With High Rate Sensors , 2016 .

[26]  Enrique A. Medina,et al.  Reachability and observability of linear impulsive systems , 2008, Autom..

[27]  Claude H. Moog,et al.  Impulsive control of single-input nonlinear systems with application to HIV dynamics , 2012, Appl. Math. Comput..

[28]  Laura Menini,et al.  A comparison between classical robust stability conditions , 2004 .

[29]  Laura Menini,et al.  Output regulation for a class of linear hybrid systems. Part 1: trajectory generation , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[30]  Pantelis Sopasakis,et al.  MPC for Sampled-Data Linear Systems: Guaranteeing Constraint Satisfaction in Continuous-Time , 2014, IEEE Transactions on Automatic Control.

[31]  Guang-Hong Yang,et al.  New Results on Output Feedback $H_{\infty} $ Control for Linear Discrete-Time Systems , 2014, IEEE Transactions on Automatic Control.

[32]  Elena Zattoni,et al.  Disturbance Decoupling in Hybrid Linear Systems With State Jumps , 2017, IEEE Transactions on Automatic Control.

[33]  Alan J. Laub,et al.  Control System Toolbox User''s Guide , 1990 .

[34]  Ilya V. Kolmanovsky,et al.  Model Predictive Control for Spacecraft Rendezvous and Docking: Strategies for Handling Constraints and Case Studies , 2015, IEEE Transactions on Control Systems Technology.

[35]  M. Araki,et al.  Multivariable multirate sampled-data systems: State-space description, transfer characteristics, and Nyquist criterion , 1986 .

[36]  E Comets,et al.  An in vivo Pharmacokinetic/Pharmacodynamic Model for Antiretroviral Combination , 2003, HIV clinical trials.

[37]  Katsuhiko Ogata,et al.  Discrete-time control systems , 1987 .

[38]  Hyungbo Shim,et al.  A control theoretic approach to malaria immunotherapy with state jumps , 2011, Autom..

[39]  Antonio Ferramosca,et al.  Control Strategies for Nonzero Set-Point Regulation of Linear Impulsive Systems , 2018, IEEE Transactions on Automatic Control.

[40]  Laura Menini,et al.  Hybrid Output Regulation for Linear Systems With Periodic Jumps: Solvability Conditions, Structural Implications and Semi-Classical Solutions , 2016, IEEE Transactions on Automatic Control.

[41]  Richard Bellman,et al.  Topics in pharmacokinetics, III: Repeated dosage and impulse control , 1971 .

[42]  Chris Clausen,et al.  Lithium pharmacokinetics: Single-dose experiments and analysis using a physiological model , 1980, Journal of Pharmacokinetics and Biopharmaceutics.

[43]  Charles A. Desoer,et al.  Necessary and sufficient condition for robust stability of linear distributed feedback systems , 1982 .

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

[45]  Tomomichi Hagiwara,et al.  Design of a stable state feedback controller based on the multirate sampling of the plant output , 1988 .

[46]  Hai Lin,et al.  Switched Linear Systems: Control and Design , 2006, IEEE Transactions on Automatic Control.

[47]  W. Wonham,et al.  Synthesis of multivariable regulators: The internal model principle , 1974 .

[48]  Michael J. Piovoso,et al.  Optimal Antiviral Switching to Minimize Resistance Risk in HIV Therapy , 2011, PloS one.

[49]  D. Jezewski,et al.  An analytic approach to optimal rendezvous using Clohessy-Wiltshire equations , 1979 .

[50]  Guangdeng Zong,et al.  Improved stability criteria for switched positive linear systems with average dwell time switching , 2017, J. Frankl. Inst..

[51]  Hongmin Li,et al.  Fuzzy-Approximation-Based Adaptive Output-Feedback Control for Uncertain Nonsmooth Nonlinear Systems , 2018, IEEE Transactions on Fuzzy Systems.

[52]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[53]  Alessandro Astolfi,et al.  A control systems analysis of HIV prevention model using impulsive input , 2014, Biomed. Signal Process. Control..

[54]  Rafal Goebel,et al.  Solutions to hybrid inclusions via set and graphical convergence with stability theory applications , 2006, Autom..