© The Institution of Engineering and Technology
This study investigates the tracking-control problems of the Lorenz, Chen and Lu chaotic systems. Note that the input–output linearisation method cannot solve these tracking-control problems because of the existence of singularities, at which such chaotic systems fail to have a well-defined relative degree. By combining Zhang dynamics and gradient dynamics, an effective controller-design method, termed Zhang-gradient (ZG) method, is proposed for tracking control of the three chaotic systems. This ZG method, with singularities conquered, is capable of solving the tracking-control problems of the chaotic systems. Both theoretical analyses and simulative verifications substantiate that the tracking controllers based on the ZG method can achieve satisfactory tracking accuracy and successfully conquer singularities encountered during the tracking-control process.
References
-
-
1)
-
8. Vanecek, A., Celikovsky, S.: ‘Control systems: from linear analysis to synthesis of chaos’ (Prentice-Hall, London, 1996).
-
2)
-
7. Radwan, A.G., Soliman, A.M., EL-Sedeek, A.-L.: ‘MOS realization of the modified Lorenz chaotic system’, Chaos Solitons Fractals, 2004, 21, pp. 553–561 (doi: 10.1016/S0960-0779(03)00077-8).
-
3)
-
12. Chen, A., Lu, J., Lu, J., Yu, S.: ‘Generating hyperchaotic Lu attractor via state feedback control’, Physica A, 2006, 364, (1), pp. 103–110 (doi: 10.1016/j.physa.2005.09.039).
-
4)
-
J. Hauser ,
S. Sastry ,
P. Kokotovic
.
Nonlinear control via approximate input–output linearization: the ball and beam example.
IEEE Trans. Autom. Control
,
3 ,
392 -
398
-
5)
-
2. Lin, D., Zhang, F., Liu, J.: ‘Symbolic dynamics-based error analysis on chaos synchronization via noisy channels’, Phys. Rev. E, 2014, 90, (1), pp. 012908 (doi: 10.1103/PhysRevE.90.012908).
-
6)
-
36. Zhang, Y., Yi, C.: ‘Zhang neural networks and neural-dynamic method’ (Nova Science Publishers, New York, 2011).
-
7)
-
19. Li, Y., Tang, W., Chen, G.: ‘Generating hyperchaos via state feedback control’, Int. J. Bifurcation Chaos, 2005, 15, (10), pp. 3367–3375 (doi: 10.1142/S0218127405013988).
-
8)
-
20. Li, Z., Chen, G., Halang, W.A.: ‘Homoclinic and heteroclinic orbits in a modified Lorenz system’, Inf. Sci., 2004, 165, (3–4), pp. 235–245 (doi: 10.1016/j.ins.2003.06.005).
-
9)
-
P. Li ,
J. Cao
.
Stabilisation and synchronisation of chaotic systems via hybrid control.
IET Control Theory Appl.
,
3 ,
795 -
801
-
10)
-
33. Zhang, Y., Ma, W., Cai, B.: ‘From Zhang neural network to Newton iteration for matrix inversion’, IEEE Trans. Circuits Syst. I, Regul. Pap., 2009, 56, (7), pp. 1405–1415 (doi: 10.1109/TCSI.2008.2007065).
-
11)
-
Y. Zhang ,
K. Chen ,
H.-Z. Tan
.
Performance analysis of gradient neural network exploited for online time-varying matrix inversion.
IEEE Trans. Autom. Control
,
8 ,
1940 -
1945
-
12)
-
24. Lin, D., Liu, H., Song, H., Zhang, F.: ‘Fuzzy neural control of uncertain chaotic systems with backlash nonlinearity’, Int. J. Mach. Learn. Cybern., 2014, 5, (5), pp. 721–728 (doi: 10.1007/s13042-013-0218-9).
-
13)
-
11. Zhu, F., Xu, J., Chen, M.: ‘The combination of high-gain sliding mode observers used as receivers in secure communication’, IEEE Trans. Circuits Syst. I, Regul. Pap., 2012, 59, (11), pp. 2702–2712 (doi: 10.1109/TCSI.2012.2190570).
-
14)
-
H.-N. Wu ,
M.-Z. Bai
.
Active fault-tolerant fuzzy control design of nonlinear model tracking with application to chaotic systems.
IET Control Theory Appl.
,
6 ,
642 -
653
-
15)
-
14. Iu, H.H.C., Yu, D.S., Fitch, A.L., Sreeram, V., Chen, H.: ‘Controlling chaos in a memristor based circuit using a Twin-T notch filter’, IEEE Trans. Circuits Syst. I, Reg. Pap., 2011, 58, (6), pp. 1337–1344 (doi: 10.1109/TCSI.2010.2097771).
-
16)
-
25. Lin, D., Wang, X., Yao, Y.: ‘Fuzzy neural adaptive tracking control of unknown chaotic systems with input saturation’, Nonlinear Dyn., 2012, 67, pp. 2889–2897 (doi: 10.1007/s11071-011-0196-y).
-
17)
-
26. Isidori, A.: ‘Nonlinear control systems: an introduction’ (Springer-Verlag, New York, 1989).
-
18)
-
34. Xiao, L., Zhang, Y.: ‘Solving time-varying nonlinear inequalities using continuous and discrete-time Zhang dynamics’, Int. J. Comput. Math., 2013, 90, (5), pp. 1114–1127 (doi: 10.1080/00207160.2012.750305).
-
19)
-
18. Apostolou, N., King, R.E.: ‘Design of globally stable controllers for a class of chaotic systems’, Int. J. Syst. Sci., 2002, 33, (5), pp. 379–386 (doi: 10.1080/00207720210123733).
-
20)
-
Y. Zhang ,
J. Wang
.
Global exponential stability of recurrent neural networks for synthesizing linear feedback control systems via pole assignment.
IEEE Trans. Neural Netw.
,
3 ,
633 -
644
-
21)
-
35. Zhang, Y., Yi, C., Guo, D., Zheng, J.: ‘Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation’, Neural Comput. Appl., 2011, 20, (1), pp. 1–7 (doi: 10.1007/s00521-010-0452-y).
-
22)
-
32. Hirschorn, R.M.: ‘Output tracking through singularities’, SIAM J. Control Optim., 2002, 40, (4), pp. 993–1010 (doi: 10.1137/S0363012999354879).
-
23)
-
J. Lü ,
G. Chen
.
A new chaotic attractor coined.
Int. J. Bifurcation Chaos
,
3 ,
659 -
661
-
24)
-
30. Tomlin, C.J., Sastry, S.S.: ‘Switching through singularities’, Syst. Control Lett., 1998, 35, (3), pp. 145–154 (doi: 10.1016/S0167-6911(98)00046-2).
-
25)
-
W. Chen ,
D.J. Balance
.
On a switching control scheme for nonlinear systems with ill-defined relative degree.
Syst. Control Lett.
,
159 -
166
-
26)
-
7. Yu, Y., Zhang, S.: ‘Controlling uncertain Lu system using backstepping design’, Chaos Solitons Fractals, 2003, 15, (5), pp. 897–902 (doi: 10.1016/S0960-0779(02)00205-9).
-
27)
-
E. Ott ,
C. Greboggi ,
J.A. Yorke
.
Controlling chaos.
Phys. Rev. Lett.
,
1196 -
1199
-
28)
-
10. Addabbo, T., Fort, A., Kocarev, L., Rocchi, S., Vignoli, V.: ‘Pseudo-chaotic lossy compressors for true random number generation’, IEEE Trans. Circuits Syst. I, Regul. Pap., 2011, 58, (8), pp. 1897–1909 (doi: 10.1109/TCSI.2011.2108050).
-
29)
-
27. Slotine, J.E., Li, W.: ‘Applied nonlinear control’ (Prentice-Hall, New Jersey, 1991).
-
30)
-
29. Kulkarni, A., Purwar, S.: ‘Wavelet based adaptive backstepping controller for a class of nonregular systems with input constraints’, Expert Syst. Appl., 2009, 36, (3), pp. 6686–6696 (doi: 10.1016/j.eswa.2008.08.070).
-
31)
-
17. Li, W.: ‘Tracking control of chaotic coronary artery system’, Int. J. Syst. Sci., 2012, 43, (1), pp. 21–30 (doi: 10.1080/00207721003764125).
-
32)
-
13. Wang, F., Liu, C.: ‘Hyperchaos evolved from the Liu chaotic system’, Chin. Phys., 2006, 15, (5), pp. 963–968 (doi: 10.1088/1009-1963/15/5/016).
-
33)
-
29. Gan, Q.: ‘Synchronisation of chaotic neural networks with unknown parameters and random time-varying delays based on adaptive sampled-data control and parameter identification’, IET Control Theory Appl., 2012, 6, pp. 1508–1515 (doi: 10.1049/iet-cta.2011.0426).
-
34)
-
22. Xu, Y., Zhou, W., Deng, L., Lu, H.: ‘Modified projective synchronization among three modified chen chaotic systems with unicoupled response system’. Proc. Int. Conf. Young Computer Scientists, Hunan, China, November 2008, pp. 2903–2907.
-
35)
-
41. Abramowitz, M., Stegun, I.A.: ‘Handbook of mathematical function with formulas, graphs, and mathematical tables’ (Dover Publications, New York, 1972).
-
36)
-
S. Čelikovský ,
G. Chen
.
On a generalized Lorenz canonical form of chaotic systems.
Int. J. Bifurcation Chaos
,
8 ,
1789 -
1812
-
37)
-
5. Chen, G., Ueta, T.: ‘Yet another chaotic attractor’, Int. J. Bifurcation Chaos, 1999, 9, (7), pp. 1465–1466 (doi: 10.1142/S0218127499001024).
-
38)
-
42. Chu, S., Metcalf, F.: ‘On Gronwall's inequality’, Proc. Am. Math. Soc., 1967, 18, (3), pp. 439–440.
-
39)
-
38. Zhang, Y., Yang, Y., Ruan, G.: ‘Performance analysis of gradient neural network exploited for online time-varying quadratic minimization and equality-constrained quadratic programming’, Neurocomputing, 2011, 74, (10), pp. 1710–1719 (doi: 10.1016/j.neucom.2011.02.007).
-
40)
-
1. Wu, Z., Fu, X.: ‘Structure identification of uncertain dynamical networks coupled with complex-variable chaotic systems’, IET Control Theory Appl., 2013, 7, (9), pp. 2285–2292 (doi: 10.1049/iet-cta.2013.0201).
-
41)
-
40. Zhang, Z., Zhang, Y.: ‘Design and experimentation of acceleration-level drift-free scheme aided by two recurrent neural networks’, IET Control Theory Appl., 2013, 7, (1), pp. 25–42 (doi: 10.1049/iet-cta.2011.0573).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2014.0931
Related content
content/journals/10.1049/iet-cta.2014.0931
pub_keyword,iet_inspecKeyword,pub_concept
6
6