Linear Positive Systems May Have a Reachable Subset from the Origin That is Either Polyhedral or Nonpolyhedral
Abstract
Positive systems with positive inputs and positive outputs are used in several branches of engineering, biochemistry, and economics. Both control theory and system theory require the concept of reachability of a time-invariant discrete-time linear positive system. The subset of the state set that is reachable from the origin is therefore of interest. The reachable subset is in general a cone in the positive vector space of the positive real numbers. It is established in this paper that the reachable subset can be either a polyhedral or a nonpolyhedral cone. For a single-input case, a characterization is provided of when the infinite-time and the finite-time reachable subsets are polyhedral. An example is provided for which the reachable subset is nonpolyhedral. Finally, for the case of polyhedral reachable subset(s), a method is provided to verify if a target set can be reached from the origin using positive inputs.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
I. Bárány and R. Karasev, Notes about the Carathéodory number, Discrete Comput. Geom., 48 (2012), pp. 783--792.
2.
Z. Bartosiewicz, Minimal polynomial realizations, Math. Control Signals Systems, 1 (1988), pp. 227--237.
3.
Z. Bartosiewicz, Linear positive control systems on time scales; controllability, Math. Control Signals Systems, 25 (2013), pp. 327--343, https://doi.org/10.1007/s00498-013-0106-6.
4.
L. Benvenuti and L. Farina, The geometry of the reachable set for linear discrete-time systems with positive controls, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 306--325.
5.
A. Berman, M. Neumann, and R. Stern, Nonnegative Matrices in Dynamic Systems, Wiley, New York, 1989.
6.
A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
7.
D. Bertsekas and I. Rhodes, Recursive state estimation for a set-membership description of uncertainty, IEEE Trans. Automat. Control, 16 (1971), pp. 117--128.
8.
R. Bru, S. Romero, and E. Sánchez, Canonical forms for positive discrete-time linear control systems, Linear Algebra Appl., 310 (2000), pp. 49--71, https://doi.org/10.1016/S0024-3795(00)00044-6.
9.
R. Brualdi and H. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991.
10.
L. Caccetta and V. Rumchev, A survey of reachability and controllability for positive linear systems, Ann. Oper. Res., 98 (2000), pp. 101--122.
11.
P. G. Coxson, L. C. Larson, and H. Schneider, Monomial patterns in the sequence $A^kb$, Linear Algebra Appl., 94 (1987), pp. 89--101.
12.
P. G. Coxson and H. Shapiro, Positive input reachability and controllability of positive systems, Linear Algebra Appl., 94 (1987), pp. 35--53, https://doi.org/10.1016/0024-3795(87)90076-0.
13.
R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear, Texts Appl. Math. 21, Springer, Berlin, 1995.
14.
C. Davis, Theory of positive dependence, Amer. J. Math., 76 (1954), pp. 733--746.
15.
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969.
16.
D. Donoho, High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension, Discrete Comput. Geom., 35 (2006), pp. 617--652.
17.
D. Donoho and J.Tanner, Counting the faces of randomly-projected hypercubes and orthants, with applications, Discrete Comput. Geom., 43 (2010), pp. 522--541.
18.
D. Donoho and J. Tanner, Sparse nonnegative solution of underdetermined linear equations by linear programming, Proc. Natl. Acad. Sci. USA, 102 (2005), pp. 9446--9451.
19.
M. Fanti, B. Maione, and B. Turchiano, Controllability of multi-input positive discrete-time systems, Internat. J. Control, 51 (1990), pp. 1295--1308, https://doi.org/10.1080/00207179008934134.
20.
L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Pure Appl. Math., Wiley, New York, 2000.
21.
D. Gale, Convex polyhedral cones and linear inequalities, in Activity Analysis of Production and Allocation, T. Koopmans, ed., Wiley, New York, 1951, pp. 287--297.
22.
K. Glover and F. Schweppe, Control of linear dynamic systems with set constrained disturbances, IEEE Trans. Automat. Control, 16 (1971), pp. 41--423.
23.
B. Grünbaum, Convex Polytopes, Pure Appl. Math. 16, Wiley, London, 1967.
24.
C. Guiver, D. Hodgson, and S. Townsley, Positive state controllability of positive linear systems, Systems Control Lett., 65 (2014), pp. 23--29.
25.
P.-O. Gutman and M. Cwikel, Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states, IEEE Trans. Automat. Control, 31 (1986), pp. 373--376.
26.
W. M. Haddad, V. Chellaboina, and Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, Princeton, NJ, 2010.
27.
H. M. Härdin and J. H. van Schuppen, Observers for linear positive systems, Linear Algebra Appl., 425 (2007), pp. 571--607.
28.
R. Horn and C. Johnson, Matrix Analysis, 2nd. ed., Cambridge University Press, Cambridge, 2013.
29.
A. Isidori, Nonlinear Control Systems, 3rd ed., Springer, Berlin, 1995.
30.
A. Isidori, Nonlinear Control Systems II, Comm. Control Engrg. Ser., Springer, London, 1999.
31.
J. Jacquez, Compartmental Analysis in Biology and Medicine, 2nd ed., The University of Michigan Press, Ann Arbor, MI, 1985.
32.
J.M. van den Hof, Positive linear observers for linear compartmental systems, SIAM J. Control Optim., 36 (1998), pp. 590--608.
33.
J.S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16 (1968), pp. 1208--1222.
34.
T. Kaczorek, Positive $1$D and $2$D systems, Comm. Control Engrg. Ser., Springer, Berlin, 2002.
35.
T. Kaczorek, Some recent developments in positive and compartmental systems, in Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments II, R. S. Romaniuk, ed., 5484, Proc. SPIE, Bellingham, WA, 2004, pp. 277--786, https://doi.org/10.1117/12.568841.
36.
R. Kalman, A new approach to linear filtering and prediction problems, J. Basic Eng., 82 (1960), pp. 35--45.
37.
R. Kalman, Mathematical description of linear dynamical systems, J. Soc. Ind. Appl. Math. Ser. A Control, 1 (1963), pp. 152--192.
38.
R. Kalman, P. Falb, and M. Arbib, Topics in Mathematical Systems Theory, McGraw-Hill, New York, 1969.
39.
S. Keerthi and E. Gilbert, Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints, IEEE Trans. Automat. Control, 32 (1987), pp. 432--435, https://doi.org/10.1109/TAC.1987.1104625.
40.
M. Khajehnejad, A. Dimakis, W. Xu, and B. Hassibi, Sparse recovery of nonnegative signals with minimal expansion, IEEE Trans. Signal Process., 59 (2011), pp. 196--208.
41.
E. Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967.
42.
W. Leontief, Input-Output Economics, 2nd. ed., Oxford University Press, Oxford, 1986.
43.
D. Luenberger, Introduction to Dynamic Systems - Theory, Models and Applications, Wiley, New York, 1979.
44.
H. Minc, Nonnegative Matrices, Wiley, New York, 1988.
45.
J. Němcová and J. van Schuppen, Realization theory for rational systems: Minimal rational realizations, Acta Appl. Math., 110 (2010), pp. 605--626, https://doi.org/10.1007/s10440-009-9464-y.
46.
M. Roitman and Z. Rubinstein, On linear recursions with nonnegative coefficients, Linear Algebra Appl., 167 (1992), pp. 151--155.
47.
R. Shorten, F. Wirth, and D. Leith, A positive systems model of TCP-like congestion control: Asymptotic results, IEEE/ACM Trans. Networking, 14 (2006), pp. 616--629, http://doi.org/10.1109/TNET.2006.876178.
48.
E. Sontag, Polynomial Response Maps, Lect. Notes Control Inf. Sci. 13, Springer, Berlin, 1979.
49.
E. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed., Text Appl. Math. 6, Springer, New York, 1998.
50.
M. Valcher, Controllability and reachability criteria for discrete time positive systems, Internat. J. Control, 65 (1996), pp. 511--536.
51.
M. Wang, W. Xu, and A. Tang, A unique “nonnegative” solution to an underdetermined system: From vectors to matrices, IEEE Trans. Signal Process., 59 (2011), pp. 1007--1016.
52.
W. Wonham, Linear Multivariable Control: A Geometric Approach, Springer, Berlin, 1979.
53.
Y. Zeinaly, B. de Schutter, and H. Hellendoorn, An integrated model predictive scheme for baggage-handling systems: Routing, line balancing, and empty-cart management, IEEE Trans. Control Systems Tech., 23 (2015), pp. 1536--1545, https://doi.org/10.1109/TCST.2014.2363135.
54.
G. Ziegler, Lectures on Polytopes, Grad. Texts in Math. 152, Springer, Berlin, 1995.
Information & Authors
Information
Published In
SIAM Journal on Matrix Analysis and Applications
Pages: 279 - 307
DOI: 10.1137/19M1268161
ISSN (online): 1095-7162
Copyright
© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 13 June 2019
Accepted: 28 October 2019
Published online: 3 March 2020
Keywords
MSC codes
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Regularized Identification with Internal Positivity Side-InformationSIAM Journal on Control and Optimization, Vol. 63, No. 1 | 6 January 2025
- A Review of Semi-Physical Simulation Techniques for Aircraft Ring Control Systems2024 International Conference on Electrical Drives, Power Electronics & Engineering (EDPEE) | 27 Feb 2024
- Stochastic SystemsControl and System Theory of Discrete-Time Stochastic Systems | 3 August 2021
- Inverse Filtering for Hidden Markov Models With Applications to Counter-Adversarial Autonomous SystemsIEEE Transactions on Signal Processing, Vol. 68 | 1 Jan 2020
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].