**5. Time-delay systems**

This section investigates the robust admissibility of uncertain descriptor delay systems and provides a non-fragile control design method for such systems. Section 5.1 gives a robust admissibility condition for uncertain descriptor delay systems, and Section 5.2 proposes non-fragile control design methods.

#### **5.1. Robust admissibility for uncertain systems**

Consider the following descriptor system with time-delay and uncertainties:

$$\operatorname{Ex}(k+1) = (A + \Delta A)\mathbf{x}(k) + (A\_d + \Delta A\_d)\mathbf{x}(k-d) + Bu(k) \tag{34}$$

where *x*(*k*) ∈ ℜ*<sup>n</sup>* is the state and *u*(*k*) ∈ ℜ*<sup>m</sup>* is the control. *A*, *Ad* and *B* are system matrices with appropriate dimensions. *d* is a constant delay and may be unknown. Uncertain matrices are given by

$$\begin{bmatrix} \Delta A \ \Delta A\_d \end{bmatrix} = HF(k) \begin{bmatrix} G \ G\_d \end{bmatrix}$$

where *F*(*k*) ∈ ℜ*l*×*<sup>j</sup>* is an unknown time-varying matrix satisfying *FT*(*k*)*F*(*k*) ≤ *I* and *H*, *G* and *Gd* are constant matrices of appropriate dimensions. Unforced nominal descriptor system (34) with *<sup>u</sup>*(*k*) = 0 and <sup>∆</sup>*<sup>A</sup>* = <sup>∆</sup>*Ad* = 0 is denoted by the triplet (*E*, *<sup>A</sup>*, *Ad*).

**Definition 5.1.** *(i) The triplet* (*E*, *A*, *Ad*) *is said to be regular if det*(*zd*+1*E* − *zdA* − *Ad*) *is not identically zero.*

*(ii) The triplet* (*E*, *A*, *Ad*) *is said to be causal if it is regular and deg*(*znddet*(*zE* − *A* − *zdAd*)) = *nd* + *rank*(*E*)*.*

*(iii) Define the generalized spectral radius as <sup>ρ</sup>*(*E*, *<sup>A</sup>*, *Ad*) = max *<sup>z</sup>*|*λ*∈{*det*(*zd*<sup>+</sup>1*E*−*zdA*−*Ad* )=0} |*λ*|*. The*

*triplet* (*E*, *A*, *Ad*) *is said to be stable if ρ*(*E*, *A*, *Ad*) < 1*. (iv) The triplet* (*E*, *A*, *Ad*) *is said to be admissible if it is regular, causal and stable.* Applying the controller (4) to the system (34), we have the closed-loop system

$$\operatorname{Ext}(k+1) = \left(A + BK + H\_{\mathcal{C}} F\_{\mathcal{C}}(k) \mathcal{G}\_{\mathcal{C}}\right) \mathbf{x}(k) + \left(A\_d + HF(k) \mathcal{G}\_d\right) \mathbf{x}(k-d) \tag{35}$$

where

$$H\_{\mathfrak{c}} = [H \ \mathfrak{a}B]\_{\prime} \ \ F\_{\mathfrak{c}}(k) = \text{diag}[F(k) \ \Phi(k)]\_{\prime} \ \ G\_{\mathfrak{c}} = \begin{bmatrix} G \\ K \end{bmatrix}.$$

First, we consider the robust admissibility of the closed-loop system (35). In order to show a robust admissibility condition, we need the following theorem.

**Theorem 5.2.** *(Xu & Lam [32]) (i) The descriptor delay system (34) with u*(*k*) = 0 *and* ∆*A* = <sup>∆</sup>*Ad* = <sup>0</sup> *is admissible if and only if there exist matrices P* > 0, *<sup>Q</sup>* > 0, *X such that*

$$
\begin{bmatrix}
\boldsymbol{A}^T \boldsymbol{P} \boldsymbol{A} - \boldsymbol{E}^T \boldsymbol{P} \boldsymbol{E} + \boldsymbol{Q} & \boldsymbol{A}^T \boldsymbol{P} \boldsymbol{A}\_d \\
\boldsymbol{A}\_d^T \boldsymbol{P} \boldsymbol{A} & \boldsymbol{A}\_d^T \boldsymbol{P} \boldsymbol{A}\_d - \boldsymbol{Q}
\end{bmatrix} + \begin{bmatrix}
\boldsymbol{A}^T \\
\boldsymbol{A}\_d^T
\end{bmatrix} \boldsymbol{S} \boldsymbol{X}^T + \boldsymbol{X} \boldsymbol{S}^T \begin{bmatrix}
\boldsymbol{A}^T \\
\boldsymbol{A}\_d^T
\end{bmatrix}^T < 0\tag{36}
$$

*where S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*. (ii) The descriptor delay system (34) with u*(*k*) = <sup>0</sup> *and* <sup>∆</sup>*<sup>A</sup>* = <sup>∆</sup>*Ad* = <sup>0</sup> *is admissible if and only if there exist matrices P*, *Q* > 0 *such that*

$$E^T P E \geq 0,\tag{37}$$

**Proof:** Suppose that there exist matrices *P* > 0, *Q* > 0, *X* = [*X<sup>T</sup>*

*HTSXT*

*<sup>K</sup>SX<sup>T</sup>*

<sup>1</sup> *<sup>H</sup>TSXT*

*P*(*AK* + *HcFc*(*k*)*Gc*) *P*(*Ad* + *HF*(*k*)*Gd*) −*P*

+ 

*<sup>K</sup>SX<sup>T</sup>*

*<sup>K</sup>SX<sup>T</sup>*

<sup>∆</sup><sup>11</sup> <sup>=</sup> *<sup>X</sup>*1*ST*(*<sup>A</sup>* <sup>+</sup> *BK* <sup>+</sup> *HcFc*(*k*)*Gc*)+(*<sup>A</sup>* <sup>+</sup> *BK* <sup>+</sup> *HcFc*(*k*)*Gc*)*TSXT*

<sup>∆</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*ST*(*Ad* <sup>+</sup> *HF*(*k*)*Gd*)+(*Ad* <sup>+</sup> *HF*(*k*)*Gd*)*TSXT*

*<sup>X</sup>*1*S<sup>T</sup> AK* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc*

*<sup>X</sup>*1*STH <sup>X</sup>*2*STH PH*

∆*<sup>T</sup>*

*<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc*

*<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc*

> 

> 

<sup>∆</sup><sup>12</sup> = (*AK* <sup>+</sup> *HcFc*(*k*)*Gc*)*TSXT*

*<sup>X</sup>*1*S<sup>T</sup> AK* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

 *Fc*(*k*) � *Gc* 0 0�

 *<sup>F</sup>*(*k*) � 0 *Gd* 0 � + 0 *G<sup>T</sup> d* 0

*<sup>X</sup>*1*S<sup>T</sup> AK* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc* 0

*<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc* 0

*<sup>X</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

 � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *ε*<sup>1</sup> 

 � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *ε*<sup>2</sup> 0 *G<sup>T</sup> d* 0

*<sup>X</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*<sup>X</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

 � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *ε*<sup>1</sup> 

 �

 

+*ε* −1 1

+*ε* −1 2

Now, using Lemma 3.2, we have

 

+ 

+ 

+*ε* −1 1

+*ε* −1 2

where

≤ 

=   

 

0, *ε*<sup>2</sup> > 0 such that the condition (39) holds. Then, by Schur complement formula, we have

<sup>1</sup> <sup>+</sup> *Q A<sup>T</sup>*

<sup>2</sup> *<sup>H</sup>TP* � + *ε*<sup>2</sup> 0 *G<sup>T</sup> d* 0

<sup>∆</sup><sup>11</sup> <sup>∆</sup><sup>12</sup> (*AK* <sup>+</sup> *HcFc*(*k*)*Gc*)*TP*

<sup>12</sup> <sup>∆</sup><sup>22</sup> (*Ad* <sup>+</sup> *HF*(*k*)*Gd*)*TP*

*<sup>d</sup> SX<sup>T</sup>* <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*S<sup>T</sup> Ad* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*PAK PAd* −*P*

*<sup>K</sup>SX<sup>T</sup>*

*<sup>K</sup>SX<sup>T</sup>*

<sup>1</sup> <sup>+</sup> *Q A<sup>T</sup>*

*G<sup>T</sup> c* 0 0

 *<sup>F</sup><sup>T</sup> <sup>c</sup>* (*k*) � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* �

 *<sup>F</sup>T*(*k*) � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* �

<sup>1</sup> <sup>+</sup> *Q A<sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*ST*(*Ad* <sup>+</sup> *HF*(*k*)*Gd*),

*<sup>d</sup> SX<sup>T</sup>* <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*S<sup>T</sup> Ad* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*PAK PAd* −*P*

*<sup>d</sup> SX<sup>T</sup>* <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*S<sup>T</sup> Ad* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*PAK PAd* −*P*

*<sup>K</sup>SX<sup>T</sup>*

<sup>1</sup> *<sup>X</sup><sup>T</sup>* 2 ]

*<sup>d</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>A</sup><sup>T</sup> d P*

*Gc* 0 0�

*KP*

 

*KP*

*KP*

<sup>1</sup> <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>Q</sup>*,

 

 

 

http://dx.doi.org/10.5772/ 51538

<sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup>*

*G<sup>T</sup> c* 0 0

Robust Control Design of Uncertain Discrete-Time Descriptor Systems with Delays

 �

 � 0 *Gd* 0 � < 0.

<sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup>*

*G<sup>T</sup> c* 0 0

 �

 � 0 *Gd* 0 �

2 .

*<sup>d</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>A</sup><sup>T</sup> d P*

*Gc* 0 0�

*<sup>d</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>A</sup><sup>T</sup> d P*

*<sup>T</sup>* and scalars *<sup>ε</sup>*<sup>1</sup> <sup>&</sup>gt;

(40)

45

$$
\begin{bmatrix}
A^T P A - E^T P E + Q & A^T P A\_d \\
A\_d^T P A & A\_d^T P A\_d - Q
\end{bmatrix} < 0. \tag{38}
$$

Robust admissibility conditions for uncertain descriptor delay system (35) are given in the following theorems.

**Theorem 5.3.** *Given γ and K, the descriptor system (35) is robustly admissible if there exist matrices P* > 0, *Q* > 0, *X* = [*X<sup>T</sup>* <sup>1</sup> *<sup>X</sup><sup>T</sup>* 2 ] *<sup>T</sup> and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that*

$$
\begin{bmatrix}
\begin{pmatrix}
X\_{1}S^{T}A\_{K} - E^{T}PE + Q \\
+ A\_{K}^{T}SX\_{1}^{T} + \varepsilon\_{1}G\_{c}^{T}G\_{c}
\end{pmatrix} & A\_{K}^{T}SX\_{2}^{T} + X\_{1}S^{T}A\_{d} & A\_{K}^{T}P \ X\_{1}S^{T}H\_{c} \ X\_{1}S^{T}H \\
X\_{2}S^{T}A\_{K} + A\_{d}^{T}SX^{T} & \begin{pmatrix}
+ A\_{d}^{T}SX\_{2}^{T} + \varepsilon\_{2}G\_{d}^{T}G\_{d}
\end{pmatrix} & A\_{d}^{T}P \ X\_{2}S^{T}H\_{c} \ X\_{2}S^{T}H \\
PA\_{K} & PA\_{d} & -P & PH\_{\mathcal{C}} & PH \\
H\_{\mathcal{C}}^{T}SX\_{1}^{T} & H\_{\mathcal{C}}^{T}SX\_{2}^{T} & H\_{\mathcal{C}}^{T}P & -\varepsilon\_{1}I & 0 \\
H^{T}SX\_{1}^{T} & H^{T}SX\_{2}^{T} & H^{T}P & 0 & -\varepsilon\_{2}I
\end{pmatrix} < 0
$$

*where AK* = *A* + *BK and S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*.*

**Proof:** Suppose that there exist matrices *P* > 0, *Q* > 0, *X* = [*X<sup>T</sup>* <sup>1</sup> *<sup>X</sup><sup>T</sup>* 2 ] *<sup>T</sup>* and scalars *<sup>ε</sup>*<sup>1</sup> <sup>&</sup>gt; 0, *ε*<sup>2</sup> > 0 such that the condition (39) holds. Then, by Schur complement formula, we have

 *<sup>X</sup>*1*S<sup>T</sup> AK* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>K</sup>SX<sup>T</sup>* <sup>1</sup> <sup>+</sup> *Q A<sup>T</sup> <sup>K</sup>SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup> KP <sup>X</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>d</sup> SX<sup>T</sup>* <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*S<sup>T</sup> Ad* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>d</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>A</sup><sup>T</sup> d P PAK PAd* −*P* +*ε* −1 1 *<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc* � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *ε*<sup>1</sup> *G<sup>T</sup> c* 0 0 � *Gc* 0 0� +*ε* −1 2 *<sup>X</sup>*1*STH <sup>X</sup>*2*STH PH* � *HTSXT* <sup>1</sup> *<sup>H</sup>TSXT* <sup>2</sup> *<sup>H</sup>TP* � + *ε*<sup>2</sup> 0 *G<sup>T</sup> d* 0 � 0 *Gd* 0 � < 0. (40)

Now, using Lemma 3.2, we have

16 Advances in Discrete Time Systems

�

following theorems.

  �

*P* > 0, *Q* > 0, *X* = [*X<sup>T</sup>*

+*A<sup>T</sup> KSX<sup>T</sup>*

where

Applying the controller (4) to the system (34), we have the closed-loop system

<sup>∆</sup>*Ad* = <sup>0</sup> *is admissible if and only if there exist matrices P* > 0, *<sup>Q</sup>* > 0, *X such that*

*<sup>d</sup> PAd* − *Q*

*ATPA* − *ETPE* + *Q ATPAd*

*<sup>d</sup> PA A<sup>T</sup>*

*where S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*.*

*A<sup>T</sup>*

robust admissibility condition, we need the following theorem.

*ATPA* − *ETPE* + *Q ATPAd*

�

<sup>1</sup> *<sup>X</sup><sup>T</sup>* 2 ]

*<sup>c</sup> Gc*

*<sup>d</sup> SX<sup>T</sup>*

�

<sup>1</sup> *<sup>H</sup><sup>T</sup>*

<sup>1</sup> *<sup>H</sup>TSXT*

*A<sup>T</sup> KSX<sup>T</sup>*

+*A<sup>T</sup> <sup>d</sup> SX<sup>T</sup>*

� −*Q* + *X*2*S<sup>T</sup> Ad*

*X*1*S<sup>T</sup> AK* − *ETPE* + *Q*

*X*2*S<sup>T</sup> AK* + *A<sup>T</sup>*

*H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>*

*HTSXT*

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup>*

*<sup>d</sup> PA A<sup>T</sup>*

*A<sup>T</sup>*

*there exist matrices P*, *Q* > 0 *such that*

*Hc* = [*H αB*], *Fc*(*k*) = diag[*F*(*k*) Φ(*k*)], *Gc* =

First, we consider the robust admissibility of the closed-loop system (35). In order to show a

**Theorem 5.2.** *(Xu & Lam [32]) (i) The descriptor delay system (34) with u*(*k*) = 0 *and* ∆*A* =

� + � *A<sup>T</sup> A<sup>T</sup> d* �

*(ii) The descriptor delay system (34) with u*(*k*) = <sup>0</sup> *and* <sup>∆</sup>*<sup>A</sup>* = <sup>∆</sup>*Ad* = <sup>0</sup> *is admissible if and only if*

Robust admissibility conditions for uncertain descriptor delay system (35) are given in the

**Theorem 5.3.** *Given γ and K, the descriptor system (35) is robustly admissible if there exist matrices*

<sup>2</sup> <sup>+</sup> *<sup>ε</sup>*2*G<sup>T</sup>*

*PAK PAd* −*P PHc PH*

*where AK* = *A* + *BK and S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*.*

*<sup>c</sup> SX<sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup>*

*<sup>d</sup> Gd* � *A<sup>T</sup>*

<sup>2</sup> *<sup>H</sup><sup>T</sup>*

*<sup>T</sup> and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that*

*<sup>d</sup> PAd* − *Q*

�

*Ex*(*k* + 1)=(*A* + *BK* + *HcFc*(*k*)*Gc*)*x*(*k*)+(*Ad* + *HF*(*k*)*Gd*)*x*(*k* − *d*) (35)

� *G K* � .

� *A<sup>T</sup> A<sup>T</sup> d*

*<sup>K</sup>P X*1*STHc <sup>X</sup>*1*STH*

*<sup>d</sup> P X*2*STHc <sup>X</sup>*2*STH*

*<sup>c</sup> P* −*ε*<sup>1</sup> *I* 0

<sup>2</sup> *<sup>H</sup>TP* <sup>0</sup> <sup>−</sup>*ε*<sup>2</sup> *<sup>I</sup>*

�*T*

*ETPE* ≥ 0, (37)

< 0. (38)

 

< 0 (39)

< 0 (36)

*SX<sup>T</sup>* + *XS<sup>T</sup>*

 <sup>∆</sup><sup>11</sup> <sup>∆</sup><sup>12</sup> (*AK* <sup>+</sup> *HcFc*(*k*)*Gc*)*TP* ∆*<sup>T</sup>* <sup>12</sup> <sup>∆</sup><sup>22</sup> (*Ad* <sup>+</sup> *HF*(*k*)*Gd*)*TP P*(*AK* + *HcFc*(*k*)*Gc*) *P*(*Ad* + *HF*(*k*)*Gd*) −*P* = *<sup>X</sup>*1*S<sup>T</sup> AK* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>K</sup>SX<sup>T</sup>* <sup>1</sup> <sup>+</sup> *Q A<sup>T</sup> <sup>K</sup>SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup> KP <sup>X</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>d</sup> SX<sup>T</sup>* <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*S<sup>T</sup> Ad* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>d</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>A</sup><sup>T</sup> d P PAK PAd* −*P* + *<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc Fc*(*k*) � *Gc* 0 0� + *G<sup>T</sup> c* 0 0 *<sup>F</sup><sup>T</sup> <sup>c</sup>* (*k*) � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc <sup>F</sup>*(*k*) � 0 *Gd* 0 � + 0 *G<sup>T</sup> d* 0 *<sup>F</sup>T*(*k*) � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � ≤ *<sup>X</sup>*1*S<sup>T</sup> AK* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>K</sup>SX<sup>T</sup>* <sup>1</sup> <sup>+</sup> *Q A<sup>T</sup> <sup>K</sup>SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup> KP <sup>X</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>d</sup> SX<sup>T</sup>* <sup>−</sup>*<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*2*S<sup>T</sup> Ad* <sup>+</sup> *<sup>A</sup><sup>T</sup> <sup>d</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>A</sup><sup>T</sup> d P PAK PAd* −*P* +*ε* −1 1 *<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc* 0 � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *ε*<sup>1</sup> *G<sup>T</sup> c* 0 0 � *Gc* 0 0� +*ε* −1 2 *<sup>X</sup>*1*STHc <sup>X</sup>*2*STHc PHc* 0 � *H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* � + *ε*<sup>2</sup> 0 *G<sup>T</sup> d* 0 � 0 *Gd* 0 �

where

$$\begin{array}{l} \Delta\_{11} = X\_1 \mathbf{S}^T (A + BK + H\_\mathcal{C} \mathbf{F}\_\mathbf{c}(k) \mathbf{G}\_\mathbf{c}) + (A + BK + H\_\mathcal{C} \mathbf{F}\_\mathbf{c}(k) \mathbf{G}\_\mathbf{c})^T \mathbf{S} \mathbf{X}\_1^T - \mathbf{E}^T P \mathbf{E} + \mathbf{Q}\_\mathcal{C} \mathbf{A}\_1 \mathbf{A}\_2 = (A\_K + H\_\mathcal{C} \mathbf{F}\_\mathbf{c}(k) \mathbf{G}\_\mathbf{c})^T \mathbf{S} \mathbf{X}\_2^T + \mathbf{X}\_1 \mathbf{S}^T (A\_d + HF(k) \mathbf{G}\_d),\\ \Delta\_{12} = (A\_K + H\_\mathcal{C} \mathbf{F}\_\mathbf{c}(k) \mathbf{G}\_\mathbf{c})^T \mathbf{S} \mathbf{X}\_2^T + \mathbf{X}\_1 \mathbf{S}^T (A\_d + HF(k) \mathbf{G}\_d) + (A\_d + HF(k) \mathbf{G}\_d)^T \mathbf{S} \mathbf{X}\_2^T. \end{array}$$

It follows from (40) that

$$
\begin{bmatrix}
\Delta\_{11} & \Delta\_{12} & (A\_K + H\_c F\_c(k) G\_c)^T P \\
\Delta\_{12}^T & \Delta\_{22} & (A\_d + HF(k) G\_d)^T P \\
P(A\_K + H\_c F\_c(k) G\_c) & P(A\_d + HF(k) G\_d) & -P
\end{bmatrix} < 0.
$$

This implies by Schur complement formula and Theorem 5.2(i) that the descriptor system (35) is robustly admissible.

Similarly, we can prove the following theorem by using Theorem 5.2(ii).

**Theorem 5.4.** *Given γ and K, the descriptor system (35) is robustly admissible if there exist matrices P* > 0, *Q* > 0 *and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that (37), and*

$$
\begin{bmatrix}
0 & -Q + \varepsilon\_2 \mathbf{G}\_d^T \mathbf{G}\_d & A\_d^T P & 0 & 0 \\
P A\_K & P A\_d & -P & P H\_c & P H \\
0 & 0 & H\_c^T P - \varepsilon\_1 I & 0 \\
0 & 0 & 0 & H^T P & 0 & -\varepsilon\_2 I
\end{bmatrix} < 0 \tag{41}
$$

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

�*T*

−1 <sup>1</sup> *HcH<sup>T</sup>*

*<sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>−</sup> ΨΛ−1*BT*Γ−1Θ*<sup>A</sup>* <sup>−</sup> <sup>Θ</sup>*<sup>T</sup>*

*X*1*ST*(*A* − *B*Λ−1Ψ*T*)+(*A* − *B*Λ−1Ψ*T*)*TSXT*

*<sup>c</sup>* + *ε* −1

<sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*W*−1(*SX<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*1*ST*(*<sup>ε</sup>*

*A<sup>T</sup> KSX<sup>T</sup>*

+*A<sup>T</sup> <sup>d</sup> SX<sup>T</sup>*

*X*1*ST*(*A* − *B*Λ−1Ψ*T*)+(*A* − *B*Λ−1Ψ*T*)*TSXT*

*<sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>+</sup> *YW*−1*Y<sup>T</sup>* <sup>−</sup> ΨΛ−1Ψ*<sup>T</sup>* <sup>&</sup>lt; 0.

*<sup>c</sup> Gc*

*<sup>d</sup> SX<sup>T</sup>*

�

<sup>1</sup> *<sup>H</sup><sup>T</sup>*

<sup>1</sup> *<sup>H</sup>TSXT*

*<sup>c</sup>* + *ε* −1

×Γ−1[*A* − *B*Λ−1Ψ*<sup>T</sup>* + (*ε*

×[(*A* − *B*Λ−1Ψ*T*)*TSXT*

By Schur complement formula, we obtain

*X*1*S<sup>T</sup> AK* − *ETPE* + *Q*

*X*2*S<sup>T</sup> AK* + *A<sup>T</sup>*

*H<sup>T</sup> <sup>c</sup> SX<sup>T</sup>*

*HTSXT*

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup>*

×[*A* − *B*Λ−1Ψ*<sup>T</sup>* + (*ε*

where *Gc* = �

= Θ*<sup>T</sup>*

*G<sup>T</sup>* −ΨΛ−<sup>1</sup>

[*A* − *B*Λ−1Ψ*<sup>T</sup>* + (*ε*

[(*A* − *B*Λ−1Ψ*T*)*TSXT*

= *YW*−1*Y<sup>T</sup>* − ΨΛ−1*BT*(*SX<sup>T</sup>*

Thus, it can be verified with (44) that

−1 <sup>1</sup> *HcH<sup>T</sup>*

+[(*A* − *B*Λ−1Ψ*T*)*TSXT*

+ΨΛ−1*BT*(*SX<sup>T</sup>*

+*X*1*ST*(*ε*

+Θ*<sup>T</sup>*

  �

= *X*1*S<sup>T</sup> A* + *ATSXT*

+*A<sup>T</sup> KSX<sup>T</sup>*

= *X*1*S<sup>T</sup> A* − *X*1*STB*Λ−1Ψ*<sup>T</sup>* + *ATSXT*

×[(*A* − *B*Λ−1Ψ*T*)*TSXT*

**Proof:** The closed-loop system (35) with the feedback gain (45) is given by

*Ex*(*<sup>k</sup>* + <sup>1</sup>)=(*<sup>A</sup>* − *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* + *HcFc*(*k*)*Gc*)*x*(*k*)+(*Ad* + *HF*(*k*)*Gd*)*x*(*<sup>k</sup>* − *<sup>d</sup>*) (46)

. Then, by some mathematical manipulation, we have

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup>*

<sup>1</sup> <sup>−</sup> ΨΛ−1*BTSXT*

<sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>*

<sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*TW*<sup>−</sup>1*Y<sup>T</sup>* <sup>−</sup> *YW*−1(*SX<sup>T</sup>*

<sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*TB*Λ−1Ψ*T*.

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>X</sup>*1*S<sup>T</sup> Ad <sup>A</sup><sup>T</sup>*

*<sup>d</sup> Gd* � *A<sup>T</sup>*

<sup>2</sup> *<sup>H</sup><sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>ε</sup>*2*G<sup>T</sup>*

*PAK PAd* −*P PHc PH*

*<sup>c</sup> SX<sup>T</sup>*

<sup>1</sup> + [*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* + (*<sup>ε</sup>*

*<sup>c</sup>* + *ε* −1

<sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>*

� −*Q* + *X*2*S<sup>T</sup> Ad*

<sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>*

1 ] *T*Γ−1

−1 <sup>1</sup> *HcH<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

<sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>*

*<sup>c</sup> Gc*

*<sup>c</sup>* + *ε* −1

*<sup>A</sup>*Γ−1*B*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> ΨΛ−1*BT*Γ−1*B*Λ−1Ψ*T*,

−1 <sup>1</sup> *HcH<sup>T</sup>*

*K* = −Λ−1Ψ*T*. (45)

Robust Control Design of Uncertain Discrete-Time Descriptor Systems with Delays

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>+</sup> *<sup>ε</sup>*1ΨΛ−1Λ−1Ψ*T*,

1 ]

*<sup>c</sup> SX<sup>T</sup>*

*<sup>c</sup>* + *ε* −1

*<sup>c</sup> SX<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

*<sup>c</sup> SX<sup>T</sup>*

*<sup>K</sup>P X*1*STHc <sup>X</sup>*1*STH*

*<sup>d</sup> P X*2*STHc <sup>X</sup>*2*STH*

*<sup>c</sup> P* −*ε*<sup>1</sup> *I* 0

<sup>2</sup> *<sup>H</sup>TP* <sup>0</sup> <sup>−</sup>*ε*<sup>2</sup> *<sup>I</sup>*

<sup>1</sup> )*T*Γ−1)Θ*D*]*W*−<sup>1</sup>

http://dx.doi.org/10.5772/ 51538

47

<sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*TB*Λ−1Ψ*<sup>T</sup>*

<sup>1</sup> )*T*Γ−1)Θ*D*)]*<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

<sup>1</sup> )*T*Γ−1)Θ*D*]*W*−<sup>1</sup>

1

 

< 0

<sup>1</sup> )*T*Γ−1)Θ*D*)]*<sup>T</sup>*

1 ] *T*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

*<sup>c</sup> SX<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

*<sup>c</sup> Gc* − *ETPE* + *Q*

−1 <sup>1</sup> *HcH<sup>T</sup>*

> −1 <sup>1</sup> *HcH<sup>T</sup>*

*<sup>c</sup>* + *ε* −1

1 ]

−1 <sup>1</sup> *HcH<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

*where AK* = *A* + *BK.*

#### **5.2. Control design for time-delay systems**

Now, we are ready to propose control design methods for uncertain descriptor delay systems. The following theorems propose design methods of a non-fragile controller that makes the system (34) robustly admissible.

**Theorem 5.5.** *There exists a controller (4) that makes the descriptor system (34) robustly admissible if there exist matrices P* > 0, *Q* > 0, *X* = [*X<sup>T</sup>* <sup>1</sup> *<sup>X</sup><sup>T</sup>* 2 ] *<sup>T</sup> and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that*

$$\Gamma = P^{-1} - \varepsilon\_1^{-1} H\_c H\_c^T - \varepsilon\_2^{-1} H H^T > 0,\quad \text{(42)}$$

$$\begin{split} W = Q - X\_2 S^T A\_d - A\_d^T S X\_2^T - \varepsilon\_2 G\_d^T G\_d - \Theta\_D^T \Gamma^{-1} \Theta\_D \\ - X\_2 S^T (\varepsilon\_1^{-1} H\_c H\_c^T + \varepsilon\_2^{-1} H H^T) S X\_2^T > 0, \quad \text{(43)} \end{split}$$

$$\begin{aligned} X\_1 \mathbf{S}^T \mathbf{A} + A^T \mathbf{S} \mathbf{X}\_1^T - E^T \mathbf{P} \mathbf{E} + \mathbf{Q} + \varepsilon\_1 \mathbf{G}^T \mathbf{G} + \mathbf{X}\_1 \mathbf{S}^T (\varepsilon\_1^{-1} H\_c H\_c^T + \varepsilon\_2^{-1} H \mathbf{H}^T) \mathbf{S} \mathbf{X}\_1^T \\ + \Theta\_A^T \Gamma^{-1} \Theta\_A + \mathbf{Y} \mathbf{W}^{-1} \mathbf{Y}^T - \mathbf{Y} \Lambda^{-1} \mathbf{F}^T < 0 \end{aligned} \tag{44}$$

*where S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*, and*

$$\begin{array}{l} \Psi = (\boldsymbol{X}\_{1}\boldsymbol{S}^{T} + \boldsymbol{\Theta}\_{A}^{T}\boldsymbol{\Gamma}^{-1})\boldsymbol{B} + \boldsymbol{Y}\boldsymbol{W}^{-1}(\boldsymbol{S}\boldsymbol{X}\_{2}^{T} + \boldsymbol{\Theta}\_{D}^{T}\boldsymbol{\Gamma}^{-1})\boldsymbol{B}, \\ \boldsymbol{\Lambda} = \boldsymbol{B}^{T}\boldsymbol{\Gamma}^{-1}\boldsymbol{B} + \boldsymbol{\epsilon}\_{1}\boldsymbol{I} + \boldsymbol{B}^{T}(\boldsymbol{S}\boldsymbol{X}\_{2}^{T} + \boldsymbol{\Gamma}^{-1}\boldsymbol{\Theta}\_{D})\boldsymbol{W}^{-1}(\boldsymbol{X}\_{2}\boldsymbol{S}^{T} + \boldsymbol{\Theta}\_{D}^{T}\boldsymbol{\Gamma}^{-1})\boldsymbol{B}, \\ \boldsymbol{Y} = \boldsymbol{A}^{T}\boldsymbol{S}\boldsymbol{X}\_{2}^{T} + (\boldsymbol{X}\_{1}\boldsymbol{S}^{T} + \boldsymbol{\Theta}\_{A}^{T}\boldsymbol{\Gamma}^{-1})\boldsymbol{\Theta}\_{D}, \\ \boldsymbol{\Theta}\_{A} = \boldsymbol{A} + (\boldsymbol{\varepsilon}\_{1}^{-1}\boldsymbol{H}\_{c}\boldsymbol{H}\_{c}^{T} + \boldsymbol{\varepsilon}\_{2}^{-1}\boldsymbol{H}\boldsymbol{H}^{T})\boldsymbol{S}\boldsymbol{X}\_{1}^{T}, \\ \boldsymbol{\Theta}\_{D} = \boldsymbol{A}\_{d} + (\boldsymbol{\varepsilon}\_{1}^{-1}\boldsymbol{H}\_{c}\boldsymbol{H}\_{c}^{T} + \boldsymbol{\varepsilon}\_{2}^{-1}\boldsymbol{H}\boldsymbol{H}^{T})\boldsymbol{S}\boldsymbol{X}\_{2}^{T}. \end{array}$$

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

18 Advances in Discrete Time Systems

It follows from (40) that

 

(35) is robustly admissible.

 

system (34) robustly admissible.

*X*1*S<sup>T</sup> A* + *ATSXT*

*where AK* = *A* + *BK.*

∆*T*

∆<sup>11</sup> ∆<sup>12</sup> (*AK* + *HcFc*(*k*)*Gc*)*TP*

 <sup>&</sup>lt; 0.

<sup>12</sup> <sup>∆</sup><sup>22</sup> (*Ad* <sup>+</sup> *HF*(*k*)*Gd*)*TP*

This implies by Schur complement formula and Theorem 5.2(i) that the descriptor system

**Theorem 5.4.** *Given γ and K, the descriptor system (35) is robustly admissible if there exist matrices*

*<sup>c</sup> Gc* 0 *A<sup>T</sup>*

Now, we are ready to propose control design methods for uncertain descriptor delay systems. The following theorems propose design methods of a non-fragile controller that makes the

**Theorem 5.5.** *There exists a controller (4) that makes the descriptor system (34) robustly admissible*

<sup>1</sup> *<sup>X</sup><sup>T</sup>* 2 ]

*W* = *Q* − *X*2*S<sup>T</sup> Ad* − *A<sup>T</sup>*

<sup>1</sup> <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>+</sup> *<sup>X</sup>*1*ST*(*<sup>ε</sup>*

*where S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*, and*

*<sup>A</sup>*Γ−1)*<sup>B</sup>* <sup>+</sup> *YW*−1(*SX<sup>T</sup>*

*<sup>A</sup>*Γ−1)Θ*D*,

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

*PAK PAd* −*P PHc PH*

0 0 *HTP* 0 −*ε*<sup>2</sup> *I*

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

Γ = *P*−<sup>1</sup> − *ε*

−1 <sup>1</sup> *HcH<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

*<sup>d</sup> SX<sup>T</sup>*

<sup>2</sup> <sup>+</sup> <sup>Θ</sup>*<sup>T</sup>*

1 ,

2 .

<sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*W*−1(*X*2*S<sup>T</sup>* <sup>+</sup> <sup>Θ</sup>*<sup>T</sup>*

−*X*2*ST*(*ε*

+Θ*<sup>T</sup>*

*<sup>K</sup>P* 0 0

 

< 0 (41)

<sup>2</sup> *HH<sup>T</sup>* <sup>&</sup>gt; 0, (42)

<sup>2</sup> > 0, (43)

*<sup>d</sup> P* 0 0

*<sup>c</sup> P* −*ε*<sup>1</sup> *I* 0

*<sup>T</sup> and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that*

*<sup>c</sup>* + *ε* −1

*<sup>c</sup>* + *ε* −1

*<sup>c</sup>* − *ε* −1

*<sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>+</sup> *YW*−1*Y<sup>T</sup>* <sup>−</sup> ΨΛ−1Ψ*<sup>T</sup>* <sup>&</sup>lt; 0 (44)

*<sup>D</sup>*Γ−1Θ*<sup>D</sup>*

1

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

<sup>2</sup> *HHT*)*SX<sup>T</sup>*

*<sup>D</sup>*Γ−1)*B*,

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

−1 <sup>1</sup> *HcH<sup>T</sup>*

<sup>2</sup> <sup>−</sup> *<sup>ε</sup>*2*G<sup>T</sup>*

*<sup>D</sup>*Γ−1)*B*,

*P*(*AK* + *HcFc*(*k*)*Gc*) *P*(*Ad* + *HF*(*k*)*Gd*) −*P*

0 −*Q* + *ε*2*G<sup>T</sup>*

0 0 *H<sup>T</sup>*

Similarly, we can prove the following theorem by using Theorem 5.2(ii).

*P* > 0, *Q* > 0 *and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that (37), and*

−*ETPE* + *Q* + *ε*1*G<sup>T</sup>*

**5.2. Control design for time-delay systems**

*if there exist matrices P* > 0, *Q* > 0, *X* = [*X<sup>T</sup>*

Ψ = (*X*1*S<sup>T</sup>* + Θ*<sup>T</sup>*

*Y* = *ATSXT*

Θ*<sup>A</sup>* = *A* + (*ε*

<sup>Θ</sup>*<sup>D</sup>* = *Ad* + (*<sup>ε</sup>*

Λ = *BT*Γ−1*B* + *ε*<sup>1</sup> *I* + *BT*(*SX<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

<sup>2</sup> + (*X*1*S<sup>T</sup>* <sup>+</sup> <sup>Θ</sup>*<sup>T</sup>*

*<sup>c</sup>* + *ε* −1

*<sup>c</sup>* + *ε* −1

$$K = -\Lambda^{-1} \Psi^T.\tag{45}$$

**Proof:** The closed-loop system (35) with the feedback gain (45) is given by

$$\operatorname{Ext}(k+1) = (A - B\Lambda^{-1}\mathbf{\bar{y}}^T + H\_\mathsf{C}\mathbf{F}\_\mathsf{C}(k)\mathbf{G}\_\mathsf{C})\mathbf{x}(k) + (A\_d + HF(k)\mathbf{G}\_d)\mathbf{x}(k-d) \tag{46}$$

where *Gc* = � *G<sup>T</sup>* −ΨΛ−<sup>1</sup> �*T* . Then, by some mathematical manipulation, we have

*X*1*ST*(*A* − *B*Λ−1Ψ*T*)+(*A* − *B*Λ−1Ψ*T*)*TSXT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup> <sup>c</sup> Gc* = *X*1*S<sup>T</sup> A* − *X*1*STB*Λ−1Ψ*<sup>T</sup>* + *ATSXT* <sup>1</sup> <sup>−</sup> ΨΛ−1*BTSXT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>+</sup> *<sup>ε</sup>*1ΨΛ−1Λ−1Ψ*T*, [*A* − *B*Λ−1Ψ*<sup>T</sup>* + (*ε* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup>* + *ε* −1 <sup>2</sup> *HHT*)*SX<sup>T</sup>* 1 ] *T*Γ−1 ×[*A* − *B*Λ−1Ψ*<sup>T</sup>* + (*ε* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup>* + *ε* −1 <sup>2</sup> *HHT*)*SX<sup>T</sup>* 1 ] = Θ*<sup>T</sup> <sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>−</sup> ΨΛ−1*BT*Γ−1Θ*<sup>A</sup>* <sup>−</sup> <sup>Θ</sup>*<sup>T</sup> <sup>A</sup>*Γ−1*B*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> ΨΛ−1*BT*Γ−1*B*Λ−1Ψ*T*, [(*A* − *B*Λ−1Ψ*T*)*TSXT* <sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*D*]*W*−<sup>1</sup> ×[(*A* − *B*Λ−1Ψ*T*)*TSXT* <sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*D*)]*<sup>T</sup>* = *YW*−1*Y<sup>T</sup>* − ΨΛ−1*BT*(*SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*TW*<sup>−</sup>1*Y<sup>T</sup>* <sup>−</sup> *YW*−1(*SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*TB*Λ−1Ψ*<sup>T</sup>* +ΨΛ−1*BT*(*SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*W*−1(*SX<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*D*)*TB*Λ−1Ψ*T*.

Thus, it can be verified with (44) that

*X*1*ST*(*A* − *B*Λ−1Ψ*T*)+(*A* − *B*Λ−1Ψ*T*)*TSXT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup> <sup>c</sup> Gc* − *ETPE* + *Q* +*X*1*ST*(*ε* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup>* + *ε* −1 <sup>2</sup> *HHT*)*SX<sup>T</sup>* <sup>1</sup> + [*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* + (*<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup>* + *ε* −1 <sup>2</sup> *HHT*)*SX<sup>T</sup>* 1 ] *T* ×Γ−1[*A* − *B*Λ−1Ψ*<sup>T</sup>* + (*ε* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup>* + *ε* −1 <sup>2</sup> *HHT*)*SX<sup>T</sup>* 1 ] +[(*A* − *B*Λ−1Ψ*T*)*TSXT* <sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*D*]*W*−<sup>1</sup> ×[(*A* − *B*Λ−1Ψ*T*)*TSXT* <sup>2</sup> + (*X*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SX<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*D*)]*<sup>T</sup>* = *X*1*S<sup>T</sup> A* + *ATSXT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>Q</sup>* <sup>+</sup> *<sup>X</sup>*1*ST*(*<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup>* + *ε* −1 <sup>2</sup> *HHT*)*SX<sup>T</sup>* 1 +Θ*<sup>T</sup> <sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>+</sup> *YW*−1*Y<sup>T</sup>* <sup>−</sup> ΨΛ−1Ψ*<sup>T</sup>* <sup>&</sup>lt; 0.

By Schur complement formula, we obtain

$$
\begin{bmatrix}
\begin{bmatrix}
X\_{1}S^{T}A\_{K}-E^{T}PE+Q\\+A\_{K}^{T}SX\_{1}^{T}+\varepsilon\_{1}G\_{c}^{T}G\_{c}
\end{bmatrix} & A\_{K}^{T}SX\_{2}^{T}+X\_{1}S^{T}A\_{d} & A\_{K}^{T}P \ X\_{1}S^{T}H\_{c} \ X\_{1}S^{T}H\\ 
& X\_{2}S^{T}A\_{K}+A\_{d}^{T}SX^{T} & \begin{pmatrix}
\end{pmatrix} & A\_{d}^{T}P \ X\_{2}S^{T}H\_{c} \ X\_{2}S^{T}H\\ 
& P A\_{K} & -P & P H\_{c} & P H\\ 
& H\_{c}^{T}SX\_{1}^{T} & H\_{c}^{T}P & -\varepsilon\_{1}I & 0\\ 
& H^{T}SX\_{1}^{T} & H^{T}SX\_{2}^{T} & H^{T}P & 0 & -\varepsilon\_{2}I
\end{bmatrix} & < 0
$$

where *AK* = *A* − *B*Λ−1Ψ*T*. Hence, by Theorem 5.3 we can show that the closed-loop system (35) is robustly admissible.

The following theorem can similarly be obtained from Theorem 5.4.

**Theorem 5.6.** *There exists a controller (4) that makes the descriptor system (34) robustly admissible if there exist matrices P* > 0, *Q* > 0 *and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that (37), (42),*

$$\mathcal{W} = \mathcal{Q} - \varepsilon\_2 \mathbf{G}\_d^T \mathbf{G}\_d - A\_d^T \Gamma^{-1} A\_d > 0,\qquad(47)$$

**7. Conclusions**

our controller design methods.

Aoyama Gakuin University, Japan

2003;150 325-330.

2003;16 667-680.

Jun Yoneyama, Yuzu Uchida and Ryutaro Takada

and Applications, 2003;150 295-302.

**Author details**

**References**

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

Robust Control Design of Uncertain Discrete-Time Descriptor Systems with Delays

http://dx.doi.org/10.5772/ 51538

49

Department of Electronics and Electrical Engineering, College of Science and Engineering,

[1] Boukas, E.K. & Liu, Z.K. Delay-dependent stability analysis of singular linear continuous-time system, IEE Proceedings on Control Theory and Applications,

[2] Chen, S.-H. & Chou, J.-H. Stability robustness of linear discrete singular time-delay systems with structured parameter uncertainties, IEE Proceedings on Control Theory

[3] Chen, S.-J. & Lin, J.-L. Robust stability of discrete time-delay uncertain singular systems, IEE Proceedings on Control Theory and Applications, 2004;151 45-52.

[5] Du, H.: Lam, J. and Sze, K. Y. Non-fragile output feedback H∞ vehicle suspension control using genetic algorithm, Engineering Applications of Artificial Intelligence,

[6] Fridman, E. Stability of linear descriptor system with delay: a Lyapunov-based approach, Journal of Mathematical Analysis and Applications, 2002;273 24-44.

[7] Fridman, E. & Shaked, U. H∞-control of linear state-delay descriptor systems: an LMI

[8] Fridman, E. & Shaked, U. Stability and guaranteed cost control of uncertain discrete

[4] Dai, L. Singular Control Systems, Springer, Berlin, Germany; 1981.

approach, Linear Algebra and its Applications, 2002;351 271-302.

delay systems, International Journal of Control, 2005;78 235-246.

$$-E^T P E + Q + \varepsilon\_1 G^T G + A^T \Gamma^{-1} A + A^T \Gamma^{-1} A\_d W^{-1} A\_d^T \Gamma^{-1} A - \Psi \Lambda^{-1} \Psi^T < 0 \tag{48}$$

*where* Γ *is given in (42), and*

$$\begin{cases} \mathbb{Y} = A^T \Gamma^{-1} B + A^T \Gamma^{-1} A\_d W^{-1} A\_d^T \Gamma^{-1} B\_{\prime} \\ \Lambda = B^T \Gamma^{-1} B + \varepsilon\_1 I + B^T \Gamma^{-1} A\_d W^{-1} A\_d^T \Gamma^{-1} B. \end{cases}$$

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

#### **6. Numerical examples**

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.

Consider an uncertain system:

$$
\begin{bmatrix} 1 \ 0 \\ 0 \ 0 \end{bmatrix} \mathbf{x}(k+1) = \left( \begin{bmatrix} 0 & 1 \\ 1 & 0.2 \end{bmatrix} + \begin{bmatrix} 0.06 \\ 0.08 \end{bmatrix} F(k) \begin{bmatrix} 0.2 \ 0 \end{bmatrix} \right) \mathbf{x}(k) + \begin{bmatrix} 0.2 \\ 0.3 \end{bmatrix} u(k).
$$

Assuming the measure of non-fragility *α* = 0.3, we apply Theorem 3.7 or 3.8, which gives a non-fragile control gain *K* in (4):

$$K = \begin{bmatrix} -4.0650 \ -0.8131 \end{bmatrix}.$$

Next, we consider a uncertain time-delay system

$$
\begin{array}{c}
\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \mathbf{x}(k+1) = \left( \begin{bmatrix} 1 & 0.2 \\ 0.1 & 0.3 \end{bmatrix} + \begin{bmatrix} 0.06 \\ 0.08 \end{bmatrix} F(k) \begin{bmatrix} 0.1 & 0 \end{bmatrix} \right) \mathbf{x}(k) \\\ + \begin{bmatrix} 1 & 0.2 \\ 0.1 & 0.3 \end{bmatrix} \mathbf{x}(k-3) + \begin{bmatrix} 0.1 \\ 0.2 \end{bmatrix} u(k).
\end{array}
$$

Assuming the measure of non-fragility *α* = 0.35, we apply Theorem 5.5 or 5.6, which gives a non-fragile control gain *K* in (4):

$$K = \begin{bmatrix} -0.6349 \ -1.9048 \end{bmatrix}.$$
