© The Institution of Engineering and Technology
In this study, controllability of discrete-time bilinear systems is studied. Necessary conditions and sufficient conditions for the systems to be controllable are presented. In particular, the sufficient conditions improve some existing results and the necessary conditions are new and easy to verify. Examples and simulations are provided to demonstrate the results of the paper.
References
-
-
1)
-
7. Rink, R.E., Mohler, R.R.: ‘Completely controllable bilinear systems’, SIAM J. Control Optim., 1968, 6, (3), pp. 477–486 (doi: 10.1137/0306030).
-
2)
-
5. Kucˇera, J.: ‘Solution in large of the control problem: x′ = (A(1−u)+Bu)x′’, Czech. Math. J., 1966, 16, (91), pp. 600–623.
-
3)
-
3. Mohler, R.R., Kolodziej, W.J.: ‘An overview of bilinear system theory and applications’, IEEE Trans. Syst. Man Cybern., 1980, 10, (10), pp. 683–688 (doi: 10.1109/TSMC.1980.4308378).
-
4)
-
10. Tarn, T.J., Elliott, D.L., Goka, T.: ‘Controllability of discrete bilinear systems with bounded control’, IEEE Trans. Autom. Control, 1973, 18, (3), pp. 298–301 (doi: 10.1109/TAC.1973.1100317).
-
5)
-
11. Goka, T., Tarn, T.J., Zaborszky, J.: ‘On the controllability of a class of discrete bilinear systems’, Automatica, 1973, 9, (5), pp. 615–622 (doi: 10.1016/0005-1098(73)90046-0).
-
6)
-
13. Evans, M.E., Murthy, D.N.P.: ‘Controllability of a class of discrete time bilinear systems’, IEEE Trans. Autom. Control, 1977, 22, (1), pp. 78–83 (doi: 10.1109/TAC.1977.1101408).
-
7)
-
1. Mohler, R.R.: ‘Bilinear control processes’ (Academic, New York, 1973).
-
8)
-
15. Tie, L., Cai, K.-Y., Lin, Y.: ‘Study on the controllability of a class of discrete-time bilinear systems’, J. Control Theory Applic., 2010, 27, (5), pp. 648–652 ().
-
9)
-
19. Nwankwor, E., Nagar, A.K., Reid, D.C.: ‘Controllability and observability in non-linear flow reservoir model’, IET Control Theory Applic., 2012, 6, (10), pp. 1456–1467 (doi: 10.1049/iet-cta.2011.0494).
-
10)
-
9. Elliott, D.L.: ‘A consequence of controllability’, J. Differ. Equ., 1971, 10, (2), pp. 364–370 (doi: 10.1016/0022-0396(71)90059-3).
-
11)
-
22. Tie, L., Cai, K.-Y., Lin, Y.: ‘On uncontrollable discrete-time bilinear systems which are ‘nearly’ controllable’, IEEE Trans. Autom. Control, 2010, 55, (12), pp. 2853–2858 (doi: 10.1109/TAC.2010.2072551).
-
12)
-
6. Kucˇera, J.: ‘Solution in large of the control problem: x′ = (Au+Bv)x′’, Czech. Math. J., 1967, 17, (92), pp. 91–96.
-
13)
-
4. Elliott, D.L.: ‘Bilinear control systems: matrices in action’ (Springer, Dordrecht, 2009).
-
14)
-
20. Jakubczyk, B., Sontag, E.D.: ‘Controllability of nonlinear discrete-time systems: a Lie-algebraic approach’, SIAM J. Control Optim., 1990, 28, (1), pp. 1–33 (doi: 10.1137/0328001).
-
15)
-
24. Tie, L., Cai, K.-Y.: ‘On near-controllability of nonlinear control systems’. Proc. 30th Chinese Control Conf., Yantai, Shandong, 2011, pp. 131–136.
-
16)
-
12. Cheng, G.S.J., Tarn, T.J., Elliott, D.L.: ‘Controllability of bilinear systems’, in Ruberti, A., Mohler, R.R. (Eds): ‘Variable structure systems with application to economics and biology’ (Springer-Verlag, 1975) pp. 83–100.
-
17)
-
18. Dumitrescu, B., Sicleru, B.C., Stefan, R.: ‘Computing the controllability radius: a semi-definite programming approach’, IET Control Theory Applic., 2009, 3, (6), pp. 654–660 (doi: 10.1049/iet-cta.2008.0169).
-
18)
-
23. Tie, L., Cai, K.-Y.: ‘On near-controllability and stabilizability of a class of discrete-time bilinear systems’, Systems Control Lett., 2011, 60, (8), pp. 650–657 (doi: 10.1016/j.sysconle.2011.04.023).
-
19)
-
21. Sontag, E.D.: ‘Controllability is harder to decide than accessibility’, SIAM J. Control Optim., 1988, 26, (5), pp. 1106–1118 (doi: 10.1137/0326061).
-
20)
-
16. Tie, L., Cai, K.-Y., Lin, Y.: ‘On controllability of discrete-time bilinear systems’, J. Franklin Institute, 2011, 348, (5), pp. 933–940 (doi: 10.1016/j.jfranklin.2011.03.003).
-
21)
-
14. Elliott, D.L.: ‘A controllability counterexample’, IEEE Trans. Autom. Control, 2005, 50, (6), pp. 840–841 (doi: 10.1109/TAC.2005.849223).
-
22)
-
17. Tie, L., Cai, K.-Y.: ‘On controllability of a class of discrete-time homogeneous bilinear systems with solvable controls’, Int. J. Robust Nonlinear Control, 2012, 22, (6), pp. 591–603 (doi: 10.1002/rnc.1707).
-
23)
-
8. Elliott, D.L., Tarn, T.J.: ‘Controllability and observability for bilinear systems’ SIAM National Meeting (Seattle, Washington, 1971).
-
24)
-
2. Bruni, C., Pillo, G.D., Koch, G.: ‘Bilinear systems: an appealing class of ‘nearly linear’ systems in theory and applications’, IEEE Trans. Autom. Control, 1974, 19, (4), pp. 334–348 (doi: 10.1109/TAC.1974.1100617).
-
25)
-
R.R. Mohler ,
W.J. Kolodziej
.
An overview of bilinear system theory and applications.
IEEE Trans. Syst. Man Cybern.
,
10 ,
683 -
688
-
26)
-
M.E. Evans ,
D.N.P. Murthy
.
Controllability of a class of discrete time bilinear systems.
IEEE Trans. Autom. Control
,
1 ,
78 -
83
-
27)
-
Tie, L., Cai, K.-Y.: `On near-controllability of nonlinear control systems', Proc. 30th Chinese Control Conf., 2011, Yantai, Shandong, p. 131–136.
-
28)
-
L. Tie ,
K.-Y. Cai ,
Y. Lin
.
On uncontrollable discrete-time bilinear systems which are ‘nearly’ controllable.
IEEE Trans. Autom. Control
,
12 ,
2853 -
2858
-
29)
-
R.E. Rink ,
R.R. Mohler
.
Completely controllable bilinear systems.
SIAM J. Control Optim.
,
3 ,
477 -
486
-
30)
-
J. Kučera
.
Solution in large of the control problem: x= (A(1−u)+Bu)x′.
Czech. Math. J.
,
91 ,
600 -
623
-
31)
-
E.D. Sontag
.
Controllability is harder to decide than accessibility.
SIAM J. Control Optim.
,
5 ,
1106 -
1118
-
32)
-
C. Bruni ,
G.D. Pillo ,
G. Koch
.
Bilinear systems: an appealing class of ‘nearly linear’ systems in theory and applications.
IEEE Trans. Autom. Control
,
4 ,
334 -
348
-
33)
-
T. Goka ,
T.J. Tarn ,
J. Zaborszky
.
On the controllability of a class of discrete bilinear systems.
Automatica
,
5 ,
615 -
622
-
34)
-
J. Kučera
.
Solution in large of the control problem: x′= (Au+Bv)x′.
Czech. Math. J.
,
92 ,
91 -
96
-
35)
-
T.J. Tarn ,
D.L. Elliott ,
T. Goka
.
Controllability of discrete bilinear systems with bounded control.
IEEE Trans. Autom. Control
,
3 ,
298 -
301
-
36)
-
D.L. Elliott
.
A controllability counterexample.
IEEE Trans. Autom. Control
,
6 ,
840 -
841
-
37)
-
B. Jakubczyk ,
E.D. Sontag
.
Controllability of nonlinear discrete-time systems: a Lie-algebraic approach.
SIAM J. Control Optim.
,
1 ,
1 -
33
-
38)
-
G.S.J. Cheng ,
T.J. Tarn ,
D.L. Elliott ,
A. Ruberti ,
R.R. Mohler
.
(1975)
Controllability of bilinear systems.
-
39)
-
Elliott, D.L., Tarn, T.J.: `Controllability and observability for bilinear systems', SIAM National Meeting, 1971, Seattle, Washington.
-
40)
-
B. Dumitrescu ,
B.C. Sicleru ,
R. Stefan
.
Computing the controllability radius: a semi-definite programming approach.
IET Control Theory Applic.
,
6 ,
654 -
660
-
41)
-
L. Tie ,
K.-Y. Cai
.
On near-controllability and stabilizability of a class of discrete-time bilinear systems.
Systems Control Lett.
,
8 ,
650 -
657
-
42)
-
L. Tie ,
K.-Y. Cai ,
Y. Lin
.
On controllability of discrete-time bilinear systems.
J. Franklin Institute
,
5 ,
933 -
940
-
43)
-
L. Tie ,
K.-Y. Cai
.
On controllability of a class of discrete-time homogeneous bilinear systems with solvable controls.
Int. J. Robust Nonlinear Control
,
6 ,
591 -
603
-
44)
-
R.R. Mohler
.
(1973)
Bilinear control processes.
-
45)
-
D.L. Elliott
.
A consequence of controllability.
J. Differ. Equ.
,
2 ,
364 -
370
-
46)
-
E. Nwankwor ,
A.K. Nagar ,
D.C. Reid
.
Controllability and observability in non-linear flow reservoir model.
IET Control Theory Applic.
,
10 ,
1456 -
1467
-
47)
-
L. Tie ,
K.-Y. Cai ,
Y. Lin
.
Study on the controllability of a class of discrete-time bilinear systems.
J. Control Theory Applic.
,
5 ,
648 -
652
-
48)
-
D.L. Elliott
.
(2009)
Bilinear control systems: matrices in action.
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