American Institute of Mathematical Sciences

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July  2015, 35(7): 2997-3013. doi: 10.3934/dcds.2015.35.2997

Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron

Zoltán Horváth 1, , Yunfei Song 2, and  Tamás Terlaky 2,

1. 

Department of Mathematics and Computational Sciences, Széchenyi István University, 9026 Győr, Egyetem tér 1, Hungary
2. 

Department of Industrial and Systems Engineering, Lehigh University, 200 West Packer Avenue, Bethlehem, PA, 18015-1582, United States, United States

Received  June 2014 Revised  October 2014 Published  January 2015

Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the steplength threshold. For rational function type discretization methods we derive a valid steplength threshold for invariance preserving, which can be computed by using an analogous algorithm as in the first case. The relationship between the previous two types of discretization methods and the forward Euler method is studied. Finally, we show that, for the forward Euler method, the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems.
Keywords: invariance preserving., Dynamical system, discretization method, invariant set, polyhedron.
Mathematics Subject Classification: Primary: 34A30, 37M25, 65K0.
Citation: Zoltán Horváth, Yunfei Song, Tamás Terlaky. Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2997-3013. doi: 10.3934/dcds.2015.35.2997
References:
[1]

D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, Nashua, 1998. Google Scholar

[2]

G. Bitsoris, On the positive invariance of polyhedral sets for discrete-time systems, System and Control Letters, 11 (1998), 243-248. doi: 10.1016/0167-6911(88)90065-5.  Google Scholar

[3]

F. Blanchini, Set invariance in control, Automatica, 35 (1999), 1747-1767. doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar

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F. Blanchini and S. Miani, Constrained stabilization of continuous-time linear systems, Systems and Control Letters, 28 (1996), 95-102. doi: 10.1016/0167-6911(96)00013-8.  Google Scholar

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F. Blanchini, S. Miani, C. E. T. Dórea and J. C. Hennet, Discussion on: '(A, B)- invariance conditions of polyhedral domains for continuous-time systems by C. E. T. Dórea and J.-C. Hennet', European Journal of Control, 5 (1999), 82-86. doi: 10.1016/S0947-3580(99)70142-1.  Google Scholar

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S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

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E. B. Castelan and J. C. Hennet, On invariant polyhedra of continuous-time linear systems, IEEE Transactions on Automatic Control, 38 (1993), 1680-1685. doi: 10.1109/9.262058.  Google Scholar

[8]

C. E. T. Dórea and J. C. Hennet, (A, B)-invariance conditions of polyhedral domains for continuous-time systems, European Journal of Control, 5 (1999), 70-81. Google Scholar

[9]

C. E. T. Dórea and J. C. Hennet, (A,B)-invariant polyhedral sets of linear discrete time systems, Journal of Optimization Theory and Applications, 103 (1999), 521-542. doi: 10.1023/A:1021727806358.  Google Scholar

[10]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, New York, 1993.  Google Scholar

[11]

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, 2008. doi: 10.1137/1.9780898717778.  Google Scholar

[12]

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.  Google Scholar

[13]

Z. Horváth, Invariant cones and polyhedra for dynamical systems, in Proceeding of the International Conference in Memoriam Gyula Farkas, Cluj Univ. Press, Cluj-Napoca, 2006, 65-74.  Google Scholar

[14]

Z. Horváth, On the positivity step size threshold of Runge-Kutta methods, Applied Numerical Mathematics, 53 (2005), 341-356. doi: 10.1016/j.apnum.2004.08.026.  Google Scholar

[15]

Z. Horváth, Y. Song and T. Terlaky, Invariance Preserving Discretization Methods of Dynamical Systems, Lehigh University, Department of Industrial and Systems Engineering, Technical Report 14T-009, 2014. Google Scholar

[16]

Z. Horváth, Y. Song and T. Terlaky, A Novel Unified Approach to Invariance in Control, Lehigh University, Department of Industrial and Systems Engineering, Technical Report 14T-003, 2014. Google Scholar

[17]

J. F. B. M. Kraaijevanger, Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems, Numerische Mathematik, 48 (1986), 303-322. doi: 10.1007/BF01389477.  Google Scholar

[18]

R. Loewy and H. Schneider, Positive operators on the $n$-dimensional ice cream cone, Journal of Mathematical Analysis and Applications, 49 (1975), 375-392. doi: 10.1016/0022-247X(75)90186-9.  Google Scholar

[19]

C. Roos, T. Terlaky and J.-Ph. Vial, Interior Point Methods for Linear Optimization, Springer Science, Heidelberg, 2006.  Google Scholar

[20]

M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numerische Mathematik, 42 (1983), 271-290. doi: 10.1007/BF01389573.  Google Scholar

[21]

R. Stern and H. Wolkowicz, Exponential nonnegativity on the ice cream cone, SIAM Journal on Matrix Analysis and Applications, 12 (1991), 160-165. doi: 10.1137/0612012.  Google Scholar

[22]

B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Lectures Series, American Mathematical Society, 2002.  Google Scholar

[23]

J. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM Journal on Applied Mathematics, 16 (1968), 1208-1222. doi: 10.1137/0116101.  Google Scholar

show all references

References:
[1]

D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, Nashua, 1998. Google Scholar

[2]

G. Bitsoris, On the positive invariance of polyhedral sets for discrete-time systems, System and Control Letters, 11 (1998), 243-248. doi: 10.1016/0167-6911(88)90065-5.  Google Scholar

[3]

F. Blanchini, Set invariance in control, Automatica, 35 (1999), 1747-1767. doi: 10.1016/S0005-1098(99)00113-2.  Google Scholar

[4]

F. Blanchini and S. Miani, Constrained stabilization of continuous-time linear systems, Systems and Control Letters, 28 (1996), 95-102. doi: 10.1016/0167-6911(96)00013-8.  Google Scholar

[5]

F. Blanchini, S. Miani, C. E. T. Dórea and J. C. Hennet, Discussion on: '(A, B)- invariance conditions of polyhedral domains for continuous-time systems by C. E. T. Dórea and J.-C. Hennet', European Journal of Control, 5 (1999), 82-86. doi: 10.1016/S0947-3580(99)70142-1.  Google Scholar

[6]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[7]

E. B. Castelan and J. C. Hennet, On invariant polyhedra of continuous-time linear systems, IEEE Transactions on Automatic Control, 38 (1993), 1680-1685. doi: 10.1109/9.262058.  Google Scholar

[8]

C. E. T. Dórea and J. C. Hennet, (A, B)-invariance conditions of polyhedral domains for continuous-time systems, European Journal of Control, 5 (1999), 70-81. Google Scholar

[9]

C. E. T. Dórea and J. C. Hennet, (A,B)-invariant polyhedral sets of linear discrete time systems, Journal of Optimization Theory and Applications, 103 (1999), 521-542. doi: 10.1023/A:1021727806358.  Google Scholar

[10]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, New York, 1993.  Google Scholar

[11]

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, 2008. doi: 10.1137/1.9780898717778.  Google Scholar

[12]

R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.  Google Scholar

[13]

Z. Horváth, Invariant cones and polyhedra for dynamical systems, in Proceeding of the International Conference in Memoriam Gyula Farkas, Cluj Univ. Press, Cluj-Napoca, 2006, 65-74.  Google Scholar

[14]

Z. Horváth, On the positivity step size threshold of Runge-Kutta methods, Applied Numerical Mathematics, 53 (2005), 341-356. doi: 10.1016/j.apnum.2004.08.026.  Google Scholar

[15]

Z. Horváth, Y. Song and T. Terlaky, Invariance Preserving Discretization Methods of Dynamical Systems, Lehigh University, Department of Industrial and Systems Engineering, Technical Report 14T-009, 2014. Google Scholar

[16]

Z. Horváth, Y. Song and T. Terlaky, A Novel Unified Approach to Invariance in Control, Lehigh University, Department of Industrial and Systems Engineering, Technical Report 14T-003, 2014. Google Scholar

[17]

J. F. B. M. Kraaijevanger, Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems, Numerische Mathematik, 48 (1986), 303-322. doi: 10.1007/BF01389477.  Google Scholar

[18]

R. Loewy and H. Schneider, Positive operators on the $n$-dimensional ice cream cone, Journal of Mathematical Analysis and Applications, 49 (1975), 375-392. doi: 10.1016/0022-247X(75)90186-9.  Google Scholar

[19]

C. Roos, T. Terlaky and J.-Ph. Vial, Interior Point Methods for Linear Optimization, Springer Science, Heidelberg, 2006.  Google Scholar

[20]

M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numerische Mathematik, 42 (1983), 271-290. doi: 10.1007/BF01389573.  Google Scholar

[21]

R. Stern and H. Wolkowicz, Exponential nonnegativity on the ice cream cone, SIAM Journal on Matrix Analysis and Applications, 12 (1991), 160-165. doi: 10.1137/0612012.  Google Scholar

[22]

B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Lectures Series, American Mathematical Society, 2002.  Google Scholar

[23]

J. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM Journal on Applied Mathematics, 16 (1968), 1208-1222. doi: 10.1137/0116101.  Google Scholar

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