**Author details**

Ivan Ivanov *Sofia University "St.Kl.Ohridski", Bulgaria*

### **6. References**

22 Will-be-set-by-IN-TECH

*B*0(3)(6, 3) = 13.498; *B*0(3)(5, 3) = 1.285; *B*1(1)(2, 3) = 6.525; *B*1(1)(4, 3) = −5.2; *B*1(2)(2, 1) = −22.99; *B*1(2)(4, 1) = 3.25; *B*1(3)(6, 1) = 6.8466; *B*1(3)(6, 2) = 2.5 . *This choice of the matrices B*0(*i*), *B*1(*i*), *i* = 1, 2, 3 *guaranteed that the matrices R*(*i*, **X**) *are positive semidefinite, i.e. there are symmetric matrices* **X** *which belongs to dom P*†*. The remain coefficient*

We find the maximal solution to (3) - (4) for the constructed example with iterative method (12) and the LMI approach applied to optimization problems (16) and (18). The results are the following. Iteration (12) needs 15 iteration steps to achieve the maximal solution. The

Eig *W*(1)=(4.9558*e* − 5; 0.50058; 0.50004; 0.50002; 0.50001; 0.5),

Eig *W*(3)=(9.33*e* − 5; 0.05041; 0.05005; 0.050018; 0.050011; 0.050003).

The LMI approach for optimization problem (16) does not give a satisfactory result. After 32 iteration step the calculations stop with the computed maximal solution **V**. However, the norm of the difference between two solutions **W** and **V** is �*W*(1) − *V*(1)� = 1.0122*e* −

The LMI approach for optimization problem (18) needs 28 iteration steps to compute the maximal solution to (1). This solution **Z** has the same eigenvalues as in (19). The norm of the difference between two solutions **W** and **Z** is �*W*(1) − *Z*(1)� = 7.4105*e* − 12, �*W*(2) −

The results from this example show that the LMI approach applied to optimization problem (18) gives the more accurate results than the LMI method for (16). Moreover, the results obtained for problem (16) are not applicability. A researcher has to be careful when applied the LMI approach for solving a set of general discrete time equations in positive semidefinite

This chapter presents a survey on the methods for numerical computation the maximal solution of a wide class of coupled Riccati-type equations arising in the optimal control of discrete-time stochastic systems corrupted with state-dependent noise and with Markovian jumping. In addition, computational procedures to compute this solution for a set of discrete-time generalized Riccati equations (7)-(8) are derived. Moreover, the LMI solvers for this case are implemented and numerical simulations are executed. The results are compared

(19)

Eig *W*(2)=(0.00019562; 1.0007; 1.0001; 1; 1; 1),

9, �*W*(2) − *V*(2)� = 2.8657*e* − 6, �*W*(3) − *V*(3)� = 3.632*e* − 6.

*Z*(2)� = 2.8982*e* − 11, �*W*(3) − *Z*(3)� = 3.0796*e* − 11.

and the usefulness of the proposed solvers are commented.

*Sofia University "St.Kl.Ohridski", Bulgaria*

*matrices are already in place.*

case.

**5. Conclusion**

**Author details**

Ivan Ivanov

computed maximal solution **W** has the eigenvalues

	- [17] do Val, J.B.R, Basar, T. (1999). Receding horizon control of jump linear systems and a macroeconomic policy problem, *Journal of Economic Dynamics and Control*, Vol. 23, 1099-1131, ISSN: 0165-1889.
	- [18] Yao, D.D., Zhang, S. & Zhou, X. (2006). Tracking a financial benchmark using a few assets. *Operations Research*, Vol. 54, 232-246, ISSN: 0030-364X.

© 2012 Sokolov, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

 (1)

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Sokolov, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Stochastic Observation Optimization on** 

Sergey V. Sokolov

**1. Introduction** 

particularly as follows.

functions;

**2. Description of the task** 

differential equation in the symmetrized form

of the vector nonlinear observer of form:

http://dx.doi.org/10.5772/39266

Additional information is available at the end of the chapter

**the Basis of the Generalized Probabilistic Criteria** 

Till now the synthesis problem of the optimum control of the observation process has been considered and solved satisfactorily basically for the linear stochastic objects and observers by optimization of the *quadratic* criterion of quality expressed, as a rule, through the a posteriori dispersion matrix [1-4]. At the same time, the statement of the synthesis problem for the optimum observation control in a more general case assumes, first, a nonlinear character of the object and observer, and, second, the application of the non-quadratic criteria of quality,

In connection with the fact that the solution of the given problem in such a statement generalizing the existing approaches, represents the obvious interest, we formulate it more

Let the Markovian vector process *t*, described generally by the nonlinear stochastic

<sup>0</sup> 0 0 , , , , *t t*

where *f*, *f*0 are known *N* – dimensional vector and *N M* – dimensional matrix nonlinear

*nt* is the white Gaussian normalized *M* – dimensional vector - noise; be observed by means

 , , *<sup>t</sup> ZH t W* 

*f t f tn t*

 

which, basically, can provide the potentially large estimation accuracy[3-6].
