© The Institution of Engineering and Technology
To characterise complete controllability of a time-varying linearised error system of a given non-linear system, this study focuses the attention on hyper-regularity of a polynomial matrix derived from the non-linear system and introduces the concept called controllable trajectory. It is shown that if a polynomial matrix derived from a given non-linear system is hyper-regular, then every linearised system along any controllable trajectory is completely controllable. Through a non-holonomic mobile robot example, it is demonstrated that if a polynomial matrix derived from a given non-linear system is hyper-regular, and if a reference trajectory is the state part of a periodic controllable trajectory, a trajectory tracking control is achieved by a two-degree-of-freedom controller design method.
References
-
-
1)
-
16. McCoy, N.H., Janusz, G.J.: ‘Introduction to modern algebra’ (Allyn and Bacon), 1987.
-
2)
-
17. Kotta, U., Tonso, M.: ‘Realization of discrete-time nonlinear input–output equations: polynomial approach’, Automatica, 2012, 48, (2), pp. 255–262 (doi: 10.1016/j.automatica.2011.07.010).
-
3)
-
2. Khalil, H.K.: ‘Nonlinear systems’ (Prentice-Hall, 2002, 3rd edn.).
-
4)
-
5. Fliess, M., Lévine, J., Martin, P., Rouchon, P.: ‘Flatness and defect of non-linear systems: introductory theory and examples’, Int. J. Control, 1995, 61, (6), pp. 1327–1361 (doi: 10.1080/00207179508921959).
-
5)
-
24. Zerz, E.: ‘An algebraic analysis approach to linear time-varying systems’, IMA J. Math. Control Inf., 2006, 23, (1), pp. 113–126 (doi: 10.1093/imamci/dni047).
-
6)
-
22. Rantzer, A.: ‘A dual to Lyapunov's stability theorem’, Syst. Control Lett., 2001, 42, (3), pp. 161–168 (doi: 10.1016/S0167-6911(00)00087-6).
-
7)
-
18. Sato, K.: ‘Algebraic controllability of nonlinear mechanical control systems’, SICE J. Control Meas. Syst. Integr., 2014, 7, (4), pp. 191–198 (doi: 10.9746/jcmsi.7.191).
-
8)
-
14. Cohn, P.: ‘Free ideal rings and localization in general rings’ (Cambridge University Press), 2006.
-
9)
-
8. Kalman, R.: ‘Contributions to the theory of optimal control’, Bol. Soc. Mat. Mex, 1960, 5, (1), pp. 102–119.
-
10)
-
9. Bittanti, S., Colaneri, P., Guardabassi, G.: ‘Analysis of the periodic Lyapunov and Riccati equations via canonical decomposition’, SIAM J. Control Optim., 1986, 24, (6), pp. 1138–1149 (doi: 10.1137/0324069).
-
11)
-
3. Bittanti, S., Colaneri, P., Guardabassi, G.: ‘Periodic solutions of periodic Riccati equations’, IEEE Trans. Autom. Control, 1984, 29, (7), pp. 665–667 (doi: 10.1109/TAC.1984.1103613).
-
12)
-
10. Gusev, S., Johansson, S., Kågström, B., et al: ‘A numerical evaluation of solvers for the periodic Riccati differential equation’, BIT Numer. Math., 2010, 50, (2), pp. 301–329 (doi: 10.1007/s10543-010-0257-5).
-
13)
-
6. Lévine, J.: ‘On necessary and sufficient conditions for differential flatness’, Appl. Algebra Eng. Commun. Comput., 2011, 22, (1), pp. 47–90 (doi: 10.1007/s00200-010-0137-x).
-
14)
-
23. Vaidya, U., Mehta, P.: ‘Lyapunov measure for almost everywhere stability’, IEEE Trans. Autom. Control, 2008, 53, (1), pp. 307–323 (doi: 10.1109/TAC.2007.914955).
-
15)
-
13. Polderman, J., Willems, J.: ‘Introduction to mathematical systems theory: a behavioral approach’ (Springer Verlag, 1998).
-
16)
-
21. Morin, P., Samson, C.: ‘Trajectory tracking for non-holonomic vehicles: overview and case study’, IEEE Proc. Fourth Int. Workshop on Robot Motion and Control, 2004. RoMoCo'04, pp. 139–153.
-
17)
-
5. Fliess, M., Lévine, J., Martin, P., et al: ‘A Lie–B"e;acklund approach to equivalence and flatness of nonlinear systems’, IEEE Trans. Autom. Control, 1999, 44, (5), pp. 922–937 (doi: 10.1109/9.763209).
-
18)
-
7. Anderson, B., Ilchmann, A., Wirth, F.R.: ‘Stabilizability of linear time-varying systems’, Syst. Control Lett., 2013, 62, (9), pp. 747–755 (doi: 10.1016/j.sysconle.2013.05.003).
-
19)
-
19. Stengel, R.F.: ‘Optimal control and estimation’ (Dover Publications), 1986.
-
20)
-
11. Hench, J., Kenney, C., Laub, A.: ‘Methods for the numerical integration of hamiltonian systems’, Circuits Syst. Signal Process., 1994, 13, (6), pp. 695–732 (doi: 10.1007/BF02523123).
-
21)
-
25. Sontag, E.: ‘Mathematical control theory: deterministic finite dimensional systems’, (Springer Verlag), 1998.
-
22)
-
12. Varga, A.: ‘On solving periodic Riccati equations’, Numer. Linear Algebra Appl., 2008, 15, (9), pp. 809–835 (doi: 10.1002/nla.604).
-
23)
-
20. Kanayama, Y., Kimura, Y., Miyazaki, F., et al: ‘A stable tracking control method for an autonomous mobile robot’, IEEE Int. Conf. on Robotics and Automation, pp. 384–389.
-
24)
-
15. Conte, G., Moog, C., Perdon, A.: ‘Algebraic methods for nonlinear control systems’, (Springer, 2007).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2015.0765
Related content
content/journals/10.1049/iet-cta.2015.0765
pub_keyword,iet_inspecKeyword,pub_concept
6
6