Max–plus matrix method and cycle time assignability and feedback stabilizability for min–max–plus systems Authors Yuegang Tao College of Mathematics and Information Science Hebei Normal University Key Lab of Computational Mathematics and Applications Guo-Ping Liu Faculty of Advanced Technology University of Glamorgan CTGT Center Harbin Institute of Technology Xiaowu Mu Department of Mathematics Zhengzhou University Original Article

First Online: 26 October 2012 Received: 06 April 2009 Accepted: 09 October 2012 DOI :
10.1007/s00498-012-0098-7

Cite this article as: Tao, Y., Liu, G. & Mu, X. Math. Control Signals Syst. (2013) 25: 197. doi:10.1007/s00498-012-0098-7
Abstract A variety of problems arising in nonlinear systems with timing constraints such as manufacturing plants, digital circuits, scheduling managements, etc., can be modeled as min–max–plus systems described by the expressions in which the operations minimum, maximum and addition appear. This paper applies the max–plus matrix method to analyze the cycle time assignability and feedback stabilizability of min–max–plus systems with min–max–plus inputs and max–plus outputs, which are nonlinear extensions of the systems studied in recent years. The max–plus projection matrix representation of closed-loop systems is introduced to establish some structural and quantitative relationships between reachability, observability, cycle time assignability and feedback stabilizability. The necessary and sufficient conditions for the cycle time assignability with respect to a state feedback and an output feedback, respectively, and the sufficient condition for the feedback stabilizability with respect to an output feedback are derived. Furthermore, one output feedback stabilization policy is designed so that the closed-loop systems take the maximal Lyapunov exponent as an eigenvalue. The max–plus matrix method based on max–plus algebra and directed graph is constructive and intuitive, and several numerical examples are given to illustrate this method.

Keywords Cycle time assignability Feedback stabilizability Max–plus matrix method Min–max–plus system

References 1.

Akian M, Gaubert S, Lakhoua A (2008) The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis. SIAM J Control Optim 47:817–848

MathSciNet MATH CrossRef 2.

Astolfi A, Colaneri P (2005) Hankel/Toeplitz matrices and the static output feedback stabilization problem. Math Control Signals Syst 17:231–268

MathSciNet MATH CrossRef 3.

Baccelli F, Cohen G, Olsder GJ, Quadrat J-P (1992) Synchronization and linearity, Wiley, New York.

4.

Bertoluzza S, Falletta S, Manzini G (2008) Efficient design of residual-based stabilization techniques for the three fields domain decomposition method. Math Models Methods Appl Sci 18:973–999

MathSciNet MATH CrossRef 5.

Chakraborty S, Yun KY, Dill DL (1999) Timing analysis of asynchronous systems using time separation of events. IEEE Trans Comput Aided Des Integr Circuits Syst 18:1061–1076

CrossRef 6.

Chen W, Tao Y (2000) Observability and reachability for nonlinear discrete event dynamic systems and colored graphs. Chin Sci Bull 45:2457–2461

MathSciNet 7.

Cochet-Terrasson J, Gaubert S, Gunawardena J (1999) A constructive fixed point theorem for min-max functions. Dyn Stab Syst 14:407–433

MathSciNet CrossRef 8.

Cohen G, Moller P, Quadrat J-P, Viot M (1992) Linear system theory for discrete event systems. In: Ho Y C (ed) Discrete event dynamic systems: analyzing complexity and performance in the modern world. A selected Reprint Volume, IEEE Control Systems Society

9.

Cohen G, Moller P, Quadrat J-P, Viot M (1985) A linear system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans Autom Control 30:210–220

MATH CrossRef 10.

Cohen G, Moller P, Quadrat J-P, Viot M (1992) Algebraic tools for the performance evaluation of discrete event systems. In: Ho YC (ed) Discrete event dynamic systems: analyzing complexity and performance in the modern world. A selected reprint volume, IEEE Control Systems Society

11.

Commault C (1998) Feedback stabilization of some event graph models. IEEE Trans Autom Control 43:1419–1423

MathSciNet MATH CrossRef 12.

Cottenceau B, Hardouin L, Boimond JL, Ferrier JL (1999) Synthesis of greatest linear feedback for timed event graphs in dioid. IEEE Trans Autom Control 44:1258–1262

MathSciNet MATH CrossRef 13.

Cottenceau B, Lhommeau M, Hardouin L, Boimond J (2003) On timed event graph stabilization by output feedback in dioid. Kybernetika 39:165–176

MathSciNet MATH 14.

Cuninghame-Green RA (1979) Minimax algebra, Number 166 in Lecture Notes in Economics and Mathematical Systems, Springer, Berlin

15.

Cuninghame-Green RA (1991) Minimax algebra and applications. Fuzzy Sets Syst 41:251–267

MathSciNet MATH CrossRef 16.

De Schutter B, van den Boom T (2001) Model predictive control for max-plus linear discrete event systems. Automatica 37:1049–1056

MATH CrossRef 17.

De Schutter B, van den Boom T (2000) Model predictive control for max-min-plus systems. Boel R, Stremersch G (eds) DES: analysis and control, vol 569. Kluwer Int. Series in Engineering and Computer Science, KAP, Boston, pp 201–208

18.

Farahani SS, van den Boom T, De Schutter B (2011) Model predictive control for stochastic max-min-plus-scaling systems-An approximation approach. In: Proceedings of the 50th IEEE conference on decision and control and European control conference. Orlando, Florida, Dec, pp 391–396

19.

Fleming WH, McEneaney WM (2000) A max-plus-based algorithm for a Hamilton-Jacobi-Bellman equation of nonlinear filtering. SIAM J Control Optim 38:683–710

MathSciNet MATH CrossRef 20.

Gaubert S (1992) Resource optimization and (min, +) spectral theory. IEEE Trans Autom Control 40:1931–1934

MathSciNet CrossRef 21.

Gaubert S, Butkovic P, Cuninghame-Green R (1998) Minimal (max,+) realization of convex sequences. SIAM J Control Optim 36:137–147

MathSciNet MATH CrossRef 22.

Gaubert S, Gunawardena J (1998) The duality theorem for min–max functions. CR Acad Sci Ser I Math 326:43–48

MathSciNet MATH CrossRef 23.

Gunawardena J (1994) Min-max functions. Discrete Event Dyn Syst 4:377–406

MATH CrossRef 24.

Gunawardena J (1994) Cycle times and fixed points of min-max functions. Lecture Notes in Control and Information Sciences, Springer, Berlin, vol 199, pp 266–272

25.

Hardouin L, Maia CA, Cottenceau B, Lhommeau M (2010) Observer design for (max, plus) linear systems. IEEE Trans Autom Control 55–2:538–543

MathSciNet CrossRef 26.

Heidergott B, Olsder GJ, van der Woude J (2006) Max Plus at work: modeling and analysis of synchronized systems. Princeton University Press, New Jersey

MATH 27.

Jean-Marie A, Olsder GJ (1996) Analysis of stochastic min–max systems: results and cinjectures. Math Model 23:175–189

MathSciNet MATH 28.

Karafyllis I (2006) Stabilization by means of time-varying hybrid feedback. Math Control Signals Syst 18:236–259

MathSciNet MATH CrossRef 29.

Maia CA, Andrade CR, Hardouin L (2011) On the control of max-plus linear system subject to state restriction. Automatica 47:988–992

MathSciNet MATH CrossRef 30.

McCaffrey D (2008) Policy iteration and the max-plus finite element method. SIAM J Control Optim 47:2912–2929

MathSciNet MATH CrossRef 31.

McEneaney WM (2004) Max-plus eigenvector methods for nonlinear

\(H_\infty \) problems: error analysis. SIAM J Control Optim 43:379–412

MathSciNet MATH CrossRef 32.

Mikkola KM (2008) Weakly coprime factorization and state-feedback stabilization of discrete-time systems. Math Control Signals Syst 20:321–350

MathSciNet MATH CrossRef 33.

Mu S, Chu T, Wang L, Yu W (2004) Output feedback control of networked systems. Int J Autom Comput 1:26–34

CrossRef 34.

Necoara I, De Schutter B, van den Boom T, Hellendoorn H (2008) Model predictive control for uncertain max-min-plus-scaling systems. Int J Control 81:701–713

MATH CrossRef 35.

Olsder GJ (1991) Eigenvalues of dynamic max-min systems. Discrete Event Dyn Syst 1:177–207

MATH CrossRef 36.

Olsder GJ (1997) Timetables in the max-plus algebra. Euclides 72:158–163

MATH 37.

Tao Y, Chen W (2003) Cycle time assignment of min–max systems. Int J Control 76:1790–1799

MathSciNet MATH CrossRef 38.

Tao Y, Liu G-P (2005) State feedback stabilization and majorizing achievement of min–max–plus systems. IEEE Trans Autom Control 50:2027–2033

MathSciNet CrossRef 39.

Tao Y, Liu G-P, Chen W (2007) Globally optimal solutions of max–min systems. J Glob Optim 39:347–363

MathSciNet MATH CrossRef 40.

van der Woude J (2001) A characterization of the eigenvalue of a general (min, max, +)-system. Discrete Event Dyn Syst 3:203–210

CrossRef 41.

van der Woude J, Subiono (2003) Conditions for structural existence of an eigenvalue of bipartite (min, max, +)-system. Theor Comput Sci 293:13–24.

42.

Yang L, Guan X, Long C, Luo X (2008) Feedback stabilization over wireless network using adaptive coded modulation. Int J Autom Comput 5:381–388

CrossRef © Springer-Verlag London 2012