On the upper and lower bounds of the solution for the continuous Riccati matrix equation

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

“…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.…”

“…So all upper bounds cannot work for this case in [1][2][3][4][5][6][7]. However, according to Lemma 2 and condition (4), we let η = 12, α = 1.5.…”

“…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.…”

“…The existence condition of the previous upper matrix bounds in [4][5][6] is that BB T must be nonsingular. However, BB T is singular in most control problems, this assumption is very conservative.…”

New lower solution bounds for the continuous algebraic Riccati equation

“…It is difficult to compare the sharpness of them, and we will present two examples in Section 3 to illustrate this. On the other hand, from the literature, we know that lower matrix bounds for the solution of the CARE (1.2) have been presented in Kwon and Pearson (1977) [16], Lee (1997) [17], Choi and Kuc (2002) [4], Davies, Shi, and Wiltshire (2007) [6], and Chen and Lee (2009) [3]. As Chen and Lee (2009) [3] pointed out, to give a general comparison between any parallel lower bounds for the same measure is hard.…”