New Upper Solution Bounds of the Continuous Algebraic Riccati Matrix Equation

Abstract

By utilizing some linear algebraic techniques, new upper bounds of the solution of the continuous algebraic Riccati equation (CARE) are derived. According to the present bounds, iterative procedures are also developed for obtaining more precise estimations. Comparing with existing results, the obtained bounds are less restrictive.

Two new upper bounds of the solution for the continuous algebraic Riccati equation and their application

“…This upper bound of eigenvalues is always solvable if there exists a unique, positive definite solution for CARE (1).…”

“…Here, we will discuss the existence condition of the upper bounds in this paper and [1][2][3][4][5][6][7]. The existence condition of the previous upper matrix bounds in [4][5][6] is that BB T must be nonsingular.…”

“…Although this assumption was improved in some literatures, it has some constraints. In [1,3,7], the upper bounds can be obtained with satisfying the condition (26). In [2], Q must be nonsingular.…”

“…We provide some examples to illustrate related problem. Example 1 in the following illustrates that the upper bounds in [1][2][3][4][5][6][7] are conservative. For any system, the upper bounds (3) and (18) are solvable.…”

“…where A ∈ R n×n is the state matrix, B ∈ R n×r is the input matrix, Q ∈ R n×n is a given positive (semi-) definite matrix, R ∈ R r×r is a given positive definite matrix, and V ∈ R n×n is the unique positive definite solution of CARE (1). The pair (A, B) and the pair (A, Q 1/2 ) are respectively assumed to be controllable and observable.…”