**7. Conclusions**

20 Advances in Discrete Time Systems

(35) is robustly admissible.

*where* Γ *is given in (42), and*

**6. Numerical examples**

Consider an uncertain system:

non-fragile control gain *K* in (4):

*x*(*k* + 1) =

Next, we consider a uncertain time-delay system

*x*(*k* + 1) =

 1 0 0 0 

non-fragile control gain *K* in (4):

 1 0 0 0 

where *AK* = *A* − *B*Λ−1Ψ*T*. Hence, by Theorem 5.3 we can show that the closed-loop system

**Theorem 5.6.** *There exists a controller (4) that makes the descriptor system (34) robustly admissible*

*W* = *Q* − *ε*2*G<sup>T</sup>*

*<sup>d</sup>* <sup>Γ</sup>−1*B*,

*<sup>d</sup>* <sup>Γ</sup>−1*B*.

*x*(*k*) + 0.2 0.3 *u*(*k*).

*<sup>d</sup> Gd* <sup>−</sup> *<sup>A</sup><sup>T</sup>*

*<sup>d</sup>* <sup>Γ</sup>−<sup>1</sup>*Ad* <sup>&</sup>gt; 0, (47)

*<sup>d</sup>* <sup>Γ</sup>−1*<sup>A</sup>* <sup>−</sup> ΨΛ−1Ψ*<sup>T</sup>* <sup>&</sup>lt; 0 (48)

The following theorem can similarly be obtained from Theorem 5.4.

−*ETPE* + *<sup>Q</sup>* + *<sup>ε</sup>*1*GTG* + *<sup>A</sup>T*Γ−1*<sup>A</sup>* + *<sup>A</sup>T*Γ−<sup>1</sup>*AdW*<sup>−</sup>1*A<sup>T</sup>*

*In this case, a feedback gain in the controller (4) is given as in (45).*

0 1 1 0.2 + 0.06 0.08 *F*(*k*) 0.2 0 

*K* =

+

*K* =

 1 0.2 0.1 0.3

> 1 0.2 0.1 0.3

*if there exist matrices P* > 0, *Q* > 0 *and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that (37), (42),*

<sup>Ψ</sup> = *<sup>A</sup>T*Γ−1*<sup>B</sup>* + *<sup>A</sup>T*Γ−<sup>1</sup>*AdW*<sup>−</sup>1*A<sup>T</sup>*

<sup>Λ</sup> = *<sup>B</sup>T*Γ−1*<sup>B</sup>* + *<sup>ε</sup>*<sup>1</sup> *<sup>I</sup>* + *<sup>B</sup>T*Γ−<sup>1</sup>*AdW*<sup>−</sup>1*A<sup>T</sup>*

In order to illustrate our control design methods, we consider the following two examples. The first one shows a non-fragile controller design method for an uncertain system, and the second gives the same class of a controller design method for a time-delay counterpart.

Assuming the measure of non-fragility *α* = 0.3, we apply Theorem 3.7 or 3.8, which gives a

 + 0.06 0.08 *F*(*k*) 0.1 0 *x*(*k*)

Assuming the measure of non-fragility *α* = 0.35, we apply Theorem 5.5 or 5.6, which gives a

−0.6349 −1.9048

*x*(*k* − 3) +

−4.0650 −0.8131

.

 0.1 0.2 *u*(*k*).

.

In this chapter, we investigated non-fragile control system analysis and design for uncertain discrete-time descriptor systems when the controller has some uncertainty in gain matrix. The controller is assumed to have multiplicative uncertainty in gain matrix. First, the robust admissibility of uncertain descriptor systems was discussed and the non-fragile control design methods were proposed. Then, theory was developed to the robust admissibility with H∞ disturbance attenuation. Necessary and Sufficient conditions for the robust admissibility with H∞ disturbance attenuation were obtained. Based on such conditions, the H∞ non-fragile controller design methods were proposed. Next, uncertain descriptor systems with delay were considered. Based on system analysis of such systems, the non-fragile control design methods were proposed. Numerical examples were finally given to illustrate our controller design methods.
