An implicit iterative algorithm is established for solving a class of Lyapunov matrix equations appearing in the discrete-time 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 mean-square stable.

References

1. 1)
  - 1. Gershon, E., Shaked, U., Yaesh, I.: ‘H∞ control and filtering of discrete-time stochastic systems with multiplicative noise’, Bull. Inst. Politeh. ‘Gheorghe Gheorghiu-Dej’ Bucur. Ser. Autom., 2001, 37, pp. 409–417.
2. 2)
  - 6. Feng, J.E., Lam, J., Xu, S., et al: ‘Optimal stabilizing controllers for linear discrete-time stochastic systems’, Optim. Control Appl. Methods, 2008, 29, pp. 243–253.
3. 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. 4)
  - 10. Li, Z.Y., Wang, Y., Zhou, B., et al: ‘Detectability and observability of discrete-time stochastic systems and their application’, Automatica, 2009, 45, (5), pp. 1340–1346.
5. 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. 6)
  - 4. Huang, L., Hjalmarsson, H., Koeppl, H.: ‘Almost sure stability and stabilization of discrete-time stochastic systems’, Syst. Control Lett., 2015, 82, pp. 26–32.
7. 7)
  - 5. Kubrusly, C.S., Costa, O.L.V.: ‘Mean square stability conditions for discrete stochastic bilinear systems’, IEEE Trans. Autom. Control, 1985, AC-30, (11), pp. 1082–1087.
8. 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. 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. 10)
  - 9. Zhang, W., Zhang, H., Chen, B.S.: ‘Generalized Lyapunov equation approach to state-dependent stochastic stabilization/detectability criterion’, IEEE Trans. Autom. Control, 2008, 53, (7), pp. 1630–1642.
11. 11)
  - 2. Xu, S., Lam, J., Chen, T.: ‘Robust H∞ control for uncertain discrete stochastic time-delay systems’, Syst. Control Lett., 2004, 51, pp. 203–215.
12. 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. 13)
  - 3. Sheng, L., Zhang, W., Gao, M.: ‘Mixed H2/H∞ control of time-varying stochastic discrete-time systems under uniform detectability’, IET Control Theory Appl., 2014, 8, (17), pp. 1866–1874.

Errata

An Erratum has been published for this content:

Errata: ‘Implicit iterative algorithms with a tuning parameter for discrete stochastic Lyapunov matrix equations’

Implicit iterative algorithms with a tuning parameter for discrete stochastic Lyapunov matrix equations

References

Related content