Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1-gain and L∞-gain characterization

International Journal of Robust and Nonlinear Control

Volumn 23, Issue 17, 2013, Pages 1932-1954

  Briat, Corentin ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

integral linear constraints; positive linear systems; relaxation; robust control; robust linear programming; robustness

Indexed keywords

LINEAR PROGRAMMING; LINEAR SYSTEMS; LYAPUNOV FUNCTIONS; ROBUST CONTROL; ROBUSTNESS (CONTROL SYSTEMS); SYSTEM STABILITY; UNCERTAINTY ANALYSIS;

INTEGRAL LINEAR CONSTRAINTS; PERFORMANCE ANALYSIS; PERFORMANCE OPTIMIZATIONS; POSITIVE LINEAR SYSTEMS; RELAXATION; ROBUST LINEAR PROGRAMMING; ROBUST STABILITY ANALYSIS; ROBUST STABILIZATION;

STABILIZATION;

EID: 84886588324     PISSN: 10498923     EISSN: 10991239     Source Type: Journal    
DOI: 10.1002/rnc.2859     Document Type: Article

References (52)

1. Haddad WM, Chellaboina V, Hui Q,. Nonnegative and Compartmental Dynamical Systems, Vol. 41. Princeton University Press: William Street, Princeton, New Jersey, USA, 2010.
2. Murray JD,. Mathematical Biology Part I. An Introduction, (3rd edn). Springer-Verlag: Berlin Heidelberg, 2002.
3. Shorten R, Wirth F, Leith D,. A positive systems model of TCP-like congestion control: asymptotic results. IEEE Transactions on Networking 2006; 14 (3): 616-629.
4. Farina L, Rinaldi S,. Positive Linear Systems: Theory and Applications. John Wiley & Sons, 2000.
5. Leenheer PD, Aeyels D,. Stabilization of positive linear systems. Systems & Control Letters 2001; 44: 259-271.
6. Haddad WM, Chellaboina V,. Stability and dissipativity theory for nonnegative dynamic systems: a unified analysis framework for biological and physiological systems. Nonlinear Analysis: Real World Applications 2005; 6: 35-65.
7. Ait Rami M, Tadeo F,. Controller synthesis for positive linear systems with bounded controls. IEEE Transactions on Circuits and Systems-II. Express Briefs 2007; 54 (2): 151-155.
8. Bru R, Romero-Vivõ S,. Positive systems-Proceedings of the 3rd Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2009). Springer-Verlag: Berlin Heidelberg, 2009.
9. Shorten R, Mason O, King C,. An alternative proof of the Barker, Berman, Plemmons result on diagonal stability and extensions. Linear Algebra and Its Applications 2009; 430: 34-40.
10. Tanaka T, Langbort C,. KYP Lemma for internally positive systems and a tractable class of distributed h-infinity control problems. In Americal Control Conference, Baltimore, Maryland, USA, 2010; 6238-6243.
11. Tanaka T, Langbort C,. Symmetric formulation of the Kalman-Yakubovich- Popov Lemma and exact losslessness condition. In 50th IEEE Conference on Decision and Control, Orlando, FLorida, USA, 2011; 5645-5652.
12. 1 gain analysis of linear positive systems and its applications. In 50th Conference on Decision and Control, Orlando, Florida, USA, 2011; 4029-4034.
13. Rantzer A,. Distributed control of positive systems. In 50th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2011; 6608-6611.
14. Scherer CW,. A full-block S -procedure with applications. In 36th IEEE Conference on Decision and Control, San Diego, California, 1997; 2602-2607.
15. Rantzer A, Megretski A,. System analysis via integral quadratic constraints. IEEE Transactions on Automatic Control 1997; 42 (6): 819-830.
16. Kalman R,. Lyapunov functions for the problem of Lur'e in automatic control. Proceedings of the National Academy of Sciences of the United States of America 1963; 49 (2): 201-205.
17. Rantzer A,. On the Kalman-Yakubovicth-Popov lemma. System & Control Letters 1996; 28 (1): 7-10.
18. Scherer CW, Weiland S,. Linear Matrix Inequalities in Control. 2005.
19. Safonov MG,. Stability, Robustness of Multivariable Feedback Systems. MIT Press: Cambridge, MA, USA, 2000.
20. Iwasaki T,. LPV system analysis with quadratic separator. In 37th IEEE Conference on Decision and Control, Tampa, Florida, USA, 1998; 3021-3026.
21. Peaucelle D, Arzelier D, Henrion D, Gouaisbaut F,. Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation. Automatica 2007; 43: 795-804.
22. Boyd S, El-Ghaoui L, Feron E, Balakrishnan V,. Linear Matrix Inequalities in Systems and Control Theory. PA, SIAM: Philadelphia, 1994.
23. Nesterov Y, Nemirovskii A,. Interior-Point Polynomial Algorithm in Convex Programming. PA, SIAM: Philadelphia, 1994.
24. Barker GP, Berman A, Plemmons RJ,. Positive diagonal solutions to the lyapunov equations. Linear and Multilinear Algebra 1978; 5 (3): 249-256.
25. Blondel V, Tsitsiklis J,. NP-hardness of some linear control problems. SIAM Journal on Control and Optimization 1997; 35 (6): 2118-2127.
26. Mason O, Shorten RN,. On linear copositive lyapunov functions and the stability of switched positive linear systems. IEEE Transactions on Automatic Control 2007; 52 (7): 1346-1349.
27. Ait Rami M,. Stability analysis and synthesis for linear positive systems with time-varying delays. In Positive systems-Proceedings of the 3rd Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2009). Springer-Verlag: Berlin Heidelberg, 2009; 205-216.
28. Knorn F, Mason O, Shorten RN,. On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica 2009; 45 (8): 1943-1947.
29. Fornasini E, Valcher M,. Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Transactions on Automatic Control 2010; 55 (8): 1933-1937.
30. Willems J,. Dissipative dynamical systems i & ii. Rational Mechanics and Analysis 1972; 45 (5): 321-393.
31. Zhou K, Doyle JC, Glover K,. Robust and Optimal Control. Prentice Hall: Upper Saddle River, New Jersey, USA, 1996.
32. ∞-gains characterizations. In 50th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2011; 3122-3129.
33. Handelman D,. Representing polynomials by positive linear functions on compact convex polyhedra. Pacific Journal of Mathematics 1988; 132 (1): 35-62.
34. Desoer CA, Vidyasagar M,. Feedback Systems: Input-Output Properties. Academic Press: New York, 1975.
35. Zappavigna A, Colaneri P, Geromel JC, Middleton RH,. Stabilization of continuous-time switched linear positive systems. In American Control Conference, Baltimore, Maryland, USA, 2010; 3275-3280.
36. Yakubovich V,. S-procedure in nonlinear control theory (english translation). Vestnik Leningrad University 1977; 4: 73-93.
37. Jönsson UT,. A lecture on the S-procedure Division of Optimization and Systems Theory. Technical Report, Royal Institute of Technology, Stockholm, Sweden. [Online] Available: http://www.math.kth.se/ulfj/5B5746/Lecture.ps.
38. Peaucelle D, Fradkov A,. Robust adaptive L 2 -gain control of polytopic MIMO LTI systems-LMI results. Systems & Control Letters 2008; 57: 881-887.
39. Briat C, Martinez JJ,. Design of H ∞ bounded non-fragile controllers for discrete-time systems. In 48th IEEE Conference on Decision and Control, Shanghai, China, 2009; 2192-2197.
40. Ait Rami M, Tadeo F,. Positive observation problem for linear discrete positive systems. In 45th IEEE Conference on Decision and Control, San Diego, California, USA, 2006; 4729-4733.
41. Iwasaki T, Hara S,. Well-posedness of feedback systems: insight into exact robustness analysis and approximate computations. IEEE Transactions on Automatic Control 1998; 43: 619-630.
42. ∞ norm computations of linear continuous-time periodic systems via the skew analysis of frequency response operators. Automatica 2002; 38: 1381-1387.
43. Scherer CW, Köse IE,. On robust controller synthesis with dynamic D-scalings. In 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, 2007; 6316-6321.
44. Powers V, Reznick B,. A new bound for Põlya Theorem with applications to polynomials positive on polyhedra. Journal of Pure and Applied Algebra 2001; 164: 221-229.
45. Scheiderer C,. Positivity and sums of squares: A guide to recent results. In Emerging Applications of Algebraic Geometry, Vol. 149, Putinar M, Sullivant S, (eds), ser. IMA Volumes Math. Appl.: Springer Science, New York, 2009; 271-324.
46. Hardy G, Littlewood J, Pölya G,. Inequalities. Cambridge University Press, 1978.
47. Parrilo P,. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. dissertation, California Institute of Technology, Pasadena, California, 2000.
48. Haddad WM, Chellaboina V,. Stability theory for nonnegative and compartmental dynamical systems with time delay. Systems & Control Letters 2004; 51 (5): 355-361.
49. Gu K, Kharitonov V, Chen J,. Stability of Time-Delay Systems. Birkhäuser: Boston, 2003.
50. Kaczorek T,. Stability of positive continuous-time linear systems with delays. Bulletin of the Polish Academy of Sciences-Technical sciences 2009; 57 (4): 395-398.
51. Briat C,. Robust stability analysis in the * -norm and Lyapunov-Razumikhin functions for the stability analysis and control of time-delay systems. In 50th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2011; 6319-6324.
52. Khammash M,. Modeling and analysis of stochastic biochemical networks. In Control Theory and Systems Biology, Iglesias PA, Ingalls BP, (eds). MIT Press: Cambridge, 2010; 29-44.