Solutions And Algorithms To Classes Of Mixed Lyapunov Equations Over Quaternion Field

With the development of science and technology, quaternion matrix is appliedwidely in the areas of the attitude control system of the spacecraft. Determined thestability and estimated the state of the system are the central technical procedure.However, the Lyapunov method is an important research toll to deal with the issues,which makes researchers'attention. For instance, mixed-typed Lyapunov equation cansimple the conclusions and enhance the system's identification, in this case, it playsan important role in stability analysis and optimal control of multi-variable processeswith time delays. so it is meaningful to discuss solution approaches of mixed Lyapunovequations.In the paper, the solutions of the unified algebraic Lyapunov equation based on deltaoperator and the mixed-typed Lyapunov equation over quaternion field are investigatedin di~*erent condition. the present paper is organized as follows:Firstly, using the structure-preserving property of real representation operationof quaternion matrix, the self-conjugate and positive-definite solutions of the unifiedalgebraic Lyapunov equation over quaternion field (AX + XA~* +θAXA~* =-B) arederived. Simultaneously, we construct iterative algorithms to find self-conjugate andpositive definite solutions of this matrix equation and analyze the convergence of thealgorithm. In order to increase convergent rate of iteration, we discuss how to choosethe sampling period and give the range.Secondly, we study the expression of general least square solution and least squareself-conjugate solution for(AX+XA~*+θAXA~* = -B,A∈SC_n(Q)), the correspondingsolutions with the minimum-norm are given contemporarily.Next, the inverse problem of unitary matrices are discussed by the discrete timealgebraic Lyapunov equation. The main idea: using F-norm is unitary invariance andHermitian matrices are unitary diagonalization, the necessary and su~*cient conditionsis derived, when solutions are existed; and a portion of least square solutions, whensolutions aren't existed.Finally, some necessary and su~*cient conditions for the existence of self-conjugateand positive solutions of the mixed-typed Lyapunov equation(AX+XA~*+BXB~* =-P) over quaternion field are derived in the consistent term. At the same time, we constructiterative algorithm to find self-conjugate and positive solutions of this matrix equation,then given the least square self-conjugate solutions.