© The Institution of Engineering and Technology
This study addresses the control synthesis of positive switched systems in both continuous and discretetime contexts. First, a novel design approach to the statefeedback controller of continuoustime positive switched systems without time delay is presented by virtue of multiple linear copositive Lyapunov functions associated with linear programming. Then, the design approach is extended to discretetime positive switched systems. It is shown that the presented approach is less conservative than existing ones through comparisons. Furthermore, the approach is developed to the stabilisation of positive switched systems with time delay. Some discussions on the presented approach are provided to show potential applications in corresponding control issues of positive switched systems. Finally, two examples are given to illustrate the theoretical findings.
References


1)

24. Briat, C.: ‘Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1gain and L∞gain characterization’, Int. J. Robust Nonlinear Control, 2013, 23, (17), pp. 1932–1954.

2)

21. Ait Rami, M.: ‘Solvability of static outputfeedback stabilization for LTI positive systems’, Syst. Control Lett., 2011, 60, pp. 704–708.

3)

29. Liu, J., Lian, J., Zhuang, Y.: ‘Output feedback L1 finitetime control of switched positive delayed systems with MDADT’, Nonlinear Anal. Hybrid Syst., 2015, 15, pp. 11–22.

4)

5. Jadbabaie, A., Lin, J., Morse, A.S.: ‘Coordination of groups of mobile autonomous agents using nearest neighbor rules’, IEEE Trans. Autom. Control, 2003, 48, (6), pp. 988–1001.

5)

18. Lian, J., Liu, J.: ‘New results on stability of switched positive systems: an average dwelltime approach’, IET Control Theory Appl., 2013, 7, (12), pp. 1651–1658.

6)

7. Zorzan, I.: ‘An introduction to positive switched systems and their application to HIV treatment modeling’ (Universitá degli Studi di Padova, 2014).

7)

27. Xiang, M., Xiang, Z.: ‘Stability, L1gain and control synthesis for positive switched systems with timevarying delay’, Nonlinear Anal. Hybrid Syst., 2013, 9, pp. 9–17.

8)

26. Yu, Z., Zhang, Z., Cheng, S., et al: ‘Approach to stabilisation of continuoustime switched positive systems’, IET Control Theory Appl., 2014, 8, (13), pp. 1207–1214.

9)

19. Ait Rami, M., Tadeo, F.: ‘Controller synthesis for positive linear systems with bounded controls’, IEEE Trans. Circuits Syst. II Expr. Briefs, 2007, 54, (2), pp. 151–155.

10)

33. Hespanha, J.P.: ‘Stability of switched systems with average dwelltime’. Proc. 38th IEEE Conf. Decision Control, 1999, vol. 3, pp. 2655–2660.

11)

20. Ait Rami, M., Tadeo, F., Benzaouia, A.: ‘Control of constrained positive discrete systems’. Proc. 2007 American Control Conf., Marriott Marquis, New York, USA, 2007, pp. 5851–5856.

12)

28. Qi, W., Gao, X.: ‘State feedback controller design for singular positive Markovian jump systems with partly known transition rates’, Appl. Math. Lett., 2015, 46, pp. 111–116.

13)

14. Zhao, X., Zhang, L., Shi, P.: ‘Stability of a class of switched positive linear timedelay systems’, Int. J. Robust Nonlinear Control, 2013, 23, (5), pp. 578–589.

14)

2. Farina, L., Rinaldi, S.: ‘Positive linear systems: theory and applications’ (Wiley, New York, 2000).

15)

30. Liu, T., Wu, B., Liu, L., et al: ‘Asynchronously finitetime control of discrete impulsive switched positive timedelay systems’, J. Franklin Inst., 2015, 352, (10), pp. 4503–4514.

16)

31. Zhang, J., Han, Z., Zhu, F., et al: ‘Stability and stabilization of positive switched systems with modedependent average dwell time’, Nonlinear Anal. Hybrid Syst., 2013, 9, pp. 42–55.

17)

4. Shorten, R., Wirth, F., Leith, D.: ‘A positive systems model of TCPlike congestion control: asymptotic results’, IEEE/ACM Trans. Netw., 2006, 14, (3), pp. 616–629.

18)

11. Knorn, F., Mason, O., Shorten, R.: ‘On linear copositive Lyapunov functions for sets of linear positive systems’, Automatica, 2009, 45, (8), pp. 1943–1947.

19)

10. Mason, O., Shorten, R.: ‘On linear copositive Lyapunov functions and the stability of switched positive linear systems’, IEEE Trans. Autom. Control, 2007, 52, (7), pp. 1346–1349.

20)

12. Fornasini, E., Valcher, M.E.: ‘Linear copositive Lyapunov functions for continuoustime positive switched systems’, IEEE Trans. Autom. Control, 2010, 55, (8), pp. 1933–1937.

21)

8. Arneson, H.: ‘Control design techniques for constrained positive compartmental systems with applications to air traffic flow management’ (University of Illinois, UrbanaChampaign, 2012).

22)

15. Gurvits, L., Shorten, R., Mason, O.: ‘On the stability of switched positive linear systems’, IEEE Trans. Autom. Control, 2007, 52, (6), pp. 1099–1103.

23)

13. Zhao, X., Zhang, L., Shi, P., et al: ‘Stability of switched positive linear systems with average dwell time switching’, Automatica, 2012, 48, (6), pp. 1132–1137.

24)

9. Liu, X., Lam, J.: ‘Stability analysis of discretetime positive switched linear delay systems’. 2012 American Control Conf., Montreal, Canada, 27–29 June 2012, pp. 5444–5449.

25)

3. Kaczorek, T.: ‘Positive 1D and 2D systems’ (SpringerVerlag, London, 2002).

26)

32. Xiang, M., Xiang, Z., Karimi, H.R.: ‘Asynchronous L1 control of delayed switched positive systems with modedependent average dwell time’, Inf. Sci., 2014, 278, pp. 703–714.

27)

34. Zhao, X., Zhang, L., Shi, P., et al: ‘Stability and stabilization of switched linear systems with modedependent average dwell time’, IEEE Trans. Autom. Control, 2012, 57, (7), pp. 1809–1815.

28)

22. Zhang, J., Zhao, X., Zhang, R.: ‘An improved approach to controller design of positive systems using controller gain decomposition’, J. Franklin Inst., 2017, 354, pp. 1356–1373.

29)

6. HernandezVargas, E., Middleton, R., Colaneri, P., et al: ‘Discretetime control for switched positive systems with application to mitigating viral escape’, Int. J. Robust Nonlinear Control, 2011, 21, (10), pp. 1093–1111.

30)

25. Zhang, J., Han, Z., Zhu, F., et al: ‘Feedback control for switched positive linear systems’, IET Control Theory Appl., 2013, 7, (3), pp. 464–469.

31)

16. Liu, X., Dang, C.: ‘Stability analysis of positive switched linear systems with delays’, IEEE Trans. Autom. Control, 2011, 56, (7), pp. 1684–1690.

32)

17. Blanchini, F., Colaneri, P., Valcher, M.E.: ‘Copositive Lyapunov functions for the stabilization of positive switched systems’, IEEE Trans. Autom. Control, 2012, 57, (12), pp. 3038–3050.

33)

1. Luenberger, E.: ‘Introduction to dynamic systems: theory, models, and applications’ (Wiley, New York, 1979).

34)

23. Chen, X., Lam, J., Li, P., et al: ‘ℓ1induced norm and controller synthesis of positive systems’, Automatica, 2013, 49, (5), pp. 1377–1385.
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