Matrix Bounds for the Solution of the Continuous Algebraic Riccati Equation

We propose new upper and lower matrix bounds for the solution of the continuous algebraic Riccati equation (CARE). In certain cases, these lower bounds improve and extend the previous results. Finally, we give a corresponding numerical example to illustrate the effectiveness of our results. 1. Introduction In many areas of optimal control, filter design, and stability analysis, the continuous algebraic Riccati equation plays an important role (see [1–5]). For example, consider the following linear system (see [5]): where , , is the initial state. The state feedback control and the performance index of the system (1.1), respectively, are where is the symmetric positive semidefinite solution of the continuous algebraic Riccati equation (CARE) with and are symmetric positive semidefinite matrices. Assume that the pair is stabilizable. Then the above CARE has a unique symmetric positive semidefinite stabilizing solution if the pair is observable (detectable). Besides, from [1, 6], we know that in the optimal regulator problem, the optimal cost can be written as where is the initial state of the system (1.1) and is the symmetric positive semidefinite solution of CARE (1.3). An interpretation of is that is the average value of the cost as varies over the surface of a unit sphere. Considering these applications, deriving the solution of the CARE has become a heated topic in the recent years. However, as we all know, for one thing, the analytical solution of this equation is often computational difficult and time-consuming as the dimensions of the system matrices increase, and we can only solve some special Riccati matrix equations and design corresponding algorithms (see [7, 8]). For another, in practice, the solution bounds can also be used as approximations of the exact solution or initial guesses in the numerical algorithms for the exact solution (Barnett and Storey 1970 [9]; Patel and Toda 1984 [10]; Mori and Derese 1984 [11]; Kwon et al. 1996 [12]). Therefore, during the past two and three decades, many scholars payed attention to estimate the bounds for the solution of the continuous algebraic Riccati equation (Kwon and Pearson 1977 [13]; Patel and Toda 1978 [14]; Yasuda and Hirai 1979 [15]; Karanam 1983 [16]; Kwon et al. 1985 [17]; Wang et al. 1986 [6]; Saniuk and Rhodes 1987 [18]; Kwon et al. 1996 [12]; Lee, 1997 [19]; Choi and Kuc, 2002 [20]; Chen and Lee, 2009 [21]). The previous results during 1974-1994 have been summarized in Kwon et al. 1996 [12]. In this paper, we propose new upper and lower matrix bounds for the solution of the continuous