In this section, we consider the augmented system (7)-(8). We give the design method based on LMI to compute the controller gain and the observer gain simultaneously.
Theorem 1 Given scalars , . For any delay satisfying and , system (7)-(8) is asymptotically stable and satisfies for any nonzero under the zero initial condition if there exist matrices , , , , and such that the following inequalities hold:
Proof The first step is to analyze the asymptotic stability of system (7). Consider system (7) in the absence of , that is,
Choose the following Lyapunov-Krasovskii functional:
Then along the solution of the system in (7), the time derivative of is given by
It is noted that
Letting and , from Lemma 1, one obtains
According to (12), (13), (14), it is clear that
where and , , , , are defined in (9a)-(9b). Applying the Schur complement to (9a)-(9b) gives
which implies . Hence, system (7) is asymptotically stable. Next, we shall establish the performance of the time-delay system (7)-(8) under the zero initial condition. Let
It can be shown that for any nonzero and ,
It is noted that
where and . According to (21) and (22), follows from (9a)-(9b), which implies that holds for any nonzero . □
In Theorem 1, a new Lyapunov-Krasovskii functional is constructed by employing slack variables and . It is noted that and are useless for reducing the conservatism of stability conditions in . However, they can provide a more relaxed design of the controller later on since need only be an invertible matrix rather than a positive definite matrix.
On the basis of Theorem 1, we will present a design method of the disturbance observer based controller in the following.
Theorem 2 Given scalars , , and . For any delay satisfying , system (7) is asymptotically stable and satisfies for any nonzero under the zero initial condition if there exist matrices , , , , , , , and invertible matrix and any matrices M, N, such that the following inequalities hold:
Moreover, the controller gain matrix K and observer gain matrix L are given by
Proof Suppose the inequality (9a)-(9b) holds and let
Substitute (25) into (9a)-(9b) and let
Thus, we have the following inequalities hold:
where , , and are defined in (23a)-(23b). Then pre-multiply (26a)-(26b) by and post-multiply by and define some matrices as follows:
From (9a)-(9b), it is clear that (23a)-(23b) holds. As a result, the closed-loop system (7)-(8) is asymptotically stable and satisfies . The proof is thus completed. □
Compared with the design method in , the matrices and are invertible matrices instead of positive definite matrices, which makes the design more flexible. Moreover, the augmented Lyapunov functional method also can be extended to the systems without time delay.
Using Theorem 2, a feasible design algorithm can be summarized as follows:
For given , , , and , solve LMI (23a)-(23b) with , , , , , , , ;
Compute K and L through , ;
Construct the controller and observer as (4) and (6).