Optimization Algorithm of Matrix Riccati Differential Equation and Application

To the matrix Riccati differential equation, based on dynamic design Variables Optimization Method, making unknown element of Riccati matrix as design variables, square sum of defined summation matrix element as objective function, a kind of new optimization Method about element of Riccati matrix orders is built. Universal program is formed. Practical examples are computed. Effectiveness is shown through result. The method is a new thinking for computing high order matrix Riccati Differential equation and obtaining control parameters.


Introduction
Matrix Riccati differential equation is an important equation acquiring feedback control parameter [1][2] . Many algorithm in both theoretical and applicable problem are studied [3][4][5] . Theoretically, only under the special condition, analytical solutions can be acquired [6][7] . Numerical solution of the elements in Riccati matrix differential equation could be obtained accurately [8][9][10] . Applying mathematics software and using numerical analysis theory, the matrix elements in Riccati differential equations could be computed by iterative method. Therefore, with calculating order and complication increase, calculating result becomes worse. A series of problems such as convergence and accuracy are existed and need be further studied.
Dynamic Design Variables Optimization Method [11] (DVOM) is put forward by Author. The method is successfully applied into computing unknown quantities of the restraint forces in any statics systems and the currents in any DC [11][12][13] . It is also used establish the solution to Matrix Riccati differential equation problems well [14] . In the following text, the precise program computation of matrix Riccati differential equation will be discussed.
Optimization principle of solving matrix Riccati differential equation Matrix Riccati differential equation in control problem. In general the dynamic equation of a linear system is Expressed as: According to maximum principle, through the optimal value analysis of state regulator's target function: , a state feedback control can be selected to make J access to a minimum, in which P is a symmetry Riccati matrix. P could be satisfied with matrix Riccati differential equation： In Eq.1 state matrix A , control matrix B , weighted matrix Q and R are known matrix. Elements of P matrix are unknown and changed with time. When time is given, Eq.1 is differential equations with ( 1) / 2 n n + . Dynamic design variables optimization can be used to compute solve Eq.1. [11] . The core of dynamic design variable optimization is converting complex solving problem of unknown design variables about analyzed problem into optimization problem of computation and analysis about dynamic design variables. Compared with static optimization problem, dynamic design variables can be sorted and framework objective function is dynamically formed according to different input conditions. Dynamic design variable optimization method can effectively used to solve the theoretical and engineering practical problems with a large number of unknown variables, and the variables are subjected to certain distribution rules. The unknown elements of Riccati matrix is satisfied with certain distribution rules, so that the basic principle of the dynamic design variables optimization could be used to complete the unknown elements of matrix Riccati differential equation rapidly.

Outline of dynamic design variables optimization
Dynamic design variables optimization about matrix Riccati differential equation. In matrix Riccati differential Eq.1, let , It can be converted into non-linear equations composed of 2 n equations: Difference principle can be used and , i j P can be expressed: Computing Eq. 2 is equivalent to searching for, , , , to satisfy: Fp is defined as a summing matrix. , i j p is independent element and expressed as: A total of independent element is: ( 1) / 2 N n n = + . Dynamic design variables optimization problem is established: Design variables are: , 1, 2, , Objective function is: Sorting relation between elements of Riccati matrix and design variables.

Advanced Engineering Forum Vols. 2-3 803
The example is the problem with single input and single output, and in which  Table 2, the time sequence of ( ) p t are shown in Figure  1. When, 10 T = ( ) 10 P T S = = , drawing the time sequence of ( ) P t in Figure 2. The average absolute error of the function ( ) p t is only 0.007 by the program computation, and the optimal control ( ) ( ) K t P t = 。 ,the time sequence of P(t), the program computed solution is blue , the analytical results below is pink： ( ) 0.5 1.5cth( 1.5 1.643841) Mechatronics and Information Technology

Conclusion
In this paper, a new method is established to solve matrix Riccati differential equations, and a general computing program is designed. By comparing the solutions of two examples with the exact solutions, the effectiveness and feasibility are inspected and verified. It will be used to obtain the ideal feedback control in complex systems.