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.

In many areas of optimal control, filter design, and stability analysis, the continuous algebraic Riccati equation plays an important role (see [

Besides, from [

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 [

In this paper, we propose new upper and lower matrix bounds for the solution of the continuous algebraic Riccati equation. And, using the upper and lower matrix bounds we obtain the trace, the eigenvalue, and the determinant bounds. In certain cases, these lower bounds improve and extend the previous results. Finally, we give a numerical example to illustrate the effectiveness of our results.

In the following, let

The following lemmas are used to prove the main results.

The symmetric positive semidefinite solution

For any symmetric matrix

Let

Let

Let

Let

From Lemma

The following matrix inequality:

Choi and Kuc in [

In this section, we will give new lower matrix bounds for the solution of the continuous algebraic Riccati equation which improve (

Assume that

By adding and subtracting

By using Theorem

Assume that

For CARE (

From Remark

It is known to us that in most of the previous results much attention had been payed to derive the bounds for the maximum eigenvalue; the minimum eigenvalue; the trace; the determinant for the exact solution of the CARE, while there have been little work focusing on the matrix solution bounds. However, matrix bounds are the most general and desirable as they can infer all other types of bounds mentioned above. The matrix bounds yields less conservative results than eigenvalue bounds in the practical applications of the solution bounds (Mori and Derese 1984 [

From Section

Mori and Derese 1984 [

Viewing the literatures, we know that lower matrix bounds for the solution of CARE (

In this section, we will give new upper matrix bounds for the solution of the continuous algebraic Riccati equation.

Assume that

Applying Lemmas

By using Theorem

Assume that

As Chen and Lee 2009 [

Consider the following example.

Let

Choose

Using Theorem

Using Theorem

Using Theorem

By using the method of Kwon and Pearson 1977 [

By computation, it is obvious that

In Table

Numerical results of the lower eigenvalue bounds.

Method | Eigenvalue bounds |
---|---|

Ours ( | |

Choi and Kuc 2002 [ | |

Chen and Lee 2009 [ | |

Lee 1997 [ | |

Kwon and Pearson 1977 [ | |

Yasuda and Hirai 1979 [ | |

Karanam 1983 [ | |

Patel and Toda 1978 [ | |

Kwon et al. 1985 [ | |

Wang et al. 1986 [ |

Using (

In this paper, we have proposed new lower and upper bounds for the solution of the continuous algebraic Riccati equation (CARE). The numerical example has illustrated that in certain cases our lower bounds are tighter than the previous results.

The authors would like to thank Professor John Burns and the referees for the very helpful comments and suggestions to improve the contents and presentation of this paper. The work was supported in part by the National Natural Science Foundation of China (10971176), young scientist project of National Natural Science Foundation of China (11001233) and the key project of Hunan Provincial Natural Science Foundation of China (10JJ2002).