© The Institution of Engineering and Technology
An implicit iterative algorithm is established for solving a class of Lyapunov matrix equations appearing in the discretetime stochastic systems with multiplicative noise. This algorithm contains a tuning parameter which can be appropriately chosen such that it has better convergence performance. Some convergence conditions have been derived for the proposed iterative algorithm, and for a special case an approach is given to choose the optimal tuning parameter such that the algorithm has the best convergence performance. In addition, the properties of boundedness and monotonicity of the proposed algorithm are also investigated when the corresponding stochastic system is asymptotically meansquare stable.
References


1)

1. Gershon, E., Shaked, U., Yaesh, I.: ‘H∞ control and filtering of discretetime stochastic systems with multiplicative noise’, Bull. Inst. Politeh. ‘Gheorghe GheorghiuDej’ Bucur. Ser. Autom., 2001, 37, pp. 409–417.

2)

6. Feng, J.E., Lam, J., Xu, S., et al: ‘Optimal stabilizing controllers for linear discretetime stochastic systems’, Optim. Control Appl. Methods, 2008, 29, pp. 243–253.

3)

12. Ding, F., Chen, T.: ‘Gradient based iterative algorithms for solving a class of matrix equations’,IEEE Trans. Autom. Control, 2005, 50, (8), pp. 1216–1221.

4)

10. Li, Z.Y., Wang, Y., Zhou, B., et al: ‘Detectability and observability of discretetime stochastic systems and their application’, Automatica, 2009, 45, (5), pp. 1340–1346.

5)

8. Zhang, W., Chen, B.S.: ‘On stabilizability and exact observability of stochastic systems with their applications’, Automatica, 2004, 40, pp. 87–94.

6)

4. Huang, L., Hjalmarsson, H., Koeppl, H.: ‘Almost sure stability and stabilization of discretetime stochastic systems’, Syst. Control Lett., 2015, 82, pp. 26–32.

7)

5. Kubrusly, C.S., Costa, O.L.V.: ‘Mean square stability conditions for discrete stochastic bilinear systems’, IEEE Trans. Autom. Control, 1985, AC30, (11), pp. 1082–1087.

8)

7. Rami, M.A., Zhou, X.Y., Xu, S.: ‘Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls’,IEEE Trans. Autom. Control, 2000, 45, (6), pp. 1131–1143.

9)

13. Ding, F., Liu, P.X., Ding, J.: ‘Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle’, Appl. Math. Comput., 2008, 197, pp. 41–50.

10)

9. Zhang, W., Zhang, H., Chen, B.S.: ‘Generalized Lyapunov equation approach to statedependent stochastic stabilization/detectability criterion’, IEEE Trans. Autom. Control, 2008, 53, (7), pp. 1630–1642.

11)

2. Xu, S., Lam, J., Chen, T.: ‘Robust H∞ control for uncertain discrete stochastic timedelay systems’, Syst. Control Lett., 2004, 51, pp. 203–215.

12)

11. Deng, F., Feng, Z., Liu, Y.: ‘Solution to matrix equation ATP+PA+∑i=1mFiTPFi=−Q (in Chinese)’, Control Theory Appl., 1996, 13, (2), pp. 163–168.

13)

3. Sheng, L., Zhang, W., Gao, M.: ‘Mixed H2/H∞ control of timevarying stochastic discretetime systems under uniform detectability’, IET Control Theory Appl., 2014, 8, (17), pp. 1866–1874.
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