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In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature.

It is known that the Lyapunov equation is widely used in various control systems. Furthermore, solution bounds of the above equation can also treat many control problems. For example, robust stability analysis for time-delay systems, robust root clustering for linear systems, determination of the size of the estimation error for multiplicative systems, and others can be solved by the mentioned solution bounds. Gajic and Qureshi [

Consider the discrete algebraic Lyapunov Equation (DALE) which are represented by

where

Before developing the main results, we review the following useful result.

Lemma 1 [

where

Then, by utilizing lemma 1 and some linear algebraic techniques, new lower matrix bounds of the solution of the DALE (1) are derived as follows.

Theorem 1. The solution P of the DALE (1) has the following bounds.

and

where the positive semi-definite matrices

and

Proof. Let a positive semi-definite matrix R is defined as

Then we have

which infers

It is seen that by using the positive semi-definite matrix R, the DALE (1) can be transformed into a continuous-type Lyapunov Equation (9). Then, by (9), we rewrite the DALE (1) as

Since A is stable, we have

and

Then, according to Lemma 1, it is seen that the right-hand side of Equation (10) is negative semi-definite. Therefore, Equation (10) is a continuous Lyapunov equation and its solution is positive semi-definite. That is,

Define

The DALE (1) now can be rewritten as

Then, we have

Due to the facts that

and

the right-hand side of (16) then is negative semi-definite. Therefore, the solution of the Lyapunov Equation (16) is positive semi-definite. We have

Substituting

and

Thus, the proof is completed.

Remark 1. According to the proof of Theorem 1, it is seen that if

Corollary 1. The solution P of the DALE (1) satisfies

and

where matrices

Remark 2.It is found that if

Algorithm 1.

Step 1. Set

Step 2. Compute

Then, comparing to

Proof. Let

Now, we assume

Then the definition of

By the inductive method, one can conclude that

Algorithm 2.

Step 1. Set

Step 2. Compute

Then, solution bounds

Proof. Let

Now, we assume

Then the definition of

By the inductive method, one can conclude that

Remark 3. Surveying the literature, existing lower matrix bounds of the solution of the DALE (1) are summarized as follows.

where

From the above conditions, it is seen that most of them contain points of weakness. The matrix Q in [

Example 1. Consider the DALE (1). Matrices A and Q are chosen as

where matrix A is diagonalizable and normal and Q is positive definite. In this case, we choose

Then, from the obtained results and (30)-(35), solution bounds of the DALE (1) for this case are shown below.

For this case, it is seen that

Obviously our result

In this paper, the lower matrix bounds of the solution for the DALE have been discussed. By transform the DALE into a continuous-type Lyapunov equation, we have established several concise lower solution bounds of the DALE. All proposed bounds are new and less restrictive than the majority of those appeared in the literature. According to some of these results, iterative algorithms have also been developed for obtaining sharper lower matrix bounds. Finally, we give a numerical example to demonstrate the applicability of the presented schemes.

The author would like to thank the National Science Council for financial support of this research under the grant MOST 104-2221-E-230-009.

Chien-Hua Lee, (2016) On the Measurement of Lower Solution Bounds of the Discrete Algebraic Lyapunov Equation. Journal of Applied Mathematics and Physics,04,655-661. doi: 10.4236/jamp.2016.44075