**4. H**∞ **non-fragile control**

8 Advances in Discrete Time Systems

+*ε*−1*QSTHcHT*

  �

*ε* > 0 such that

 

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

−ΨΛ−1*BT*Γ−1(*A* + *ε*−1*HcH<sup>T</sup>*

By Schur complement formula, we obtain

*<sup>c</sup> SQ<sup>T</sup>* = *QS<sup>T</sup> A* + *ATSQT* − *ETPE* + (*A* + *ε*−1*HcH<sup>T</sup>*

+*εGTG* − [*QS<sup>T</sup>* + (*A* + *ε*−1*HcH<sup>T</sup>*

−ΨΛ−1*BT*[*SQ<sup>T</sup>* + Γ−1(*A* + *ε*−1*HcH<sup>T</sup>*

= *QS<sup>T</sup> A* + *ATSQT* − *ETPE* + *εGTG* + *ε*−1*QSTHcHT*

*QST*(*A* − *B*Λ−1Ψ*T*)+(*A* − *B*Λ−1Ψ*T*)*TSQT* −*ETPE* + *εGTG* + *ε*ΨΛ−1Λ−1Ψ*<sup>T</sup>*

*QST*(*A* + *BK*)+(*A* + *BK*)*TSQT* − *ETPE* + *εG<sup>T</sup>*

*H<sup>T</sup>*

<sup>−</sup>1*QSTHcHT*

Similarly, we can prove the following theorem by using Theorem 3.6.

*if there exist matrix P and scalar ε* > 0 *such that (10), (12), and*

It follows from Schur complement formula that

*QS<sup>T</sup> A* + *ATSQT* − *ETPE* + *ε*

By Lemma 3.4, we have

(14).

*in (14).*

*H<sup>T</sup>*

= *QS<sup>T</sup> A* + *ATSQT* − *ETPE* + (*A* + *ε*−1*HcH<sup>T</sup>*

+*εGTG* − *QSTB*Λ−1Ψ*<sup>T</sup>* − (*A* + *ε*−1*HcH<sup>T</sup>*

*QST*(*A* − *B*Λ−1Ψ*T*)+(*A* − *B*Λ−1Ψ*T*)*TSQT* − *ETPE* + *εG<sup>T</sup>*

*<sup>c</sup> SQT*)*T*Γ−1(*<sup>A</sup>* − *<sup>B</sup>*Λ−1Ψ*<sup>T</sup>* + *<sup>ε</sup>*−1*HcH<sup>T</sup>*

�

*P*(*A* + *BK*) −*P PHc*

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

*KT*Ψ*<sup>T</sup>* + Ψ*K* + *KT*Λ*K* ≥ −ΨΛ−1Ψ*T*. Therefore, we obtain the conditions (12) and (13), and the feedback gain *K* is calculated as in

**Theorem 3.8.** *There exists a controller (4) that makes the descriptor system (1) robustly admissible*

*where* Γ, Λ *are given in Theorem 3.7. In this case, the feedback gain in the controller (4) is given as*

−*ETPE* + *εGTG* + *AT*Γ−1*A* − *AT*Γ−1*B*Λ−1*BT*Γ−1*A* < 0 (15)

*P*(*A* − *B*Λ−1Ψ*T*) −*P PHc*

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

Hence, by Theorem 3.5 we can show that the closed-loop system (6) is robustly admissible. (Necessity) Assume there exists a feedback control of the form (4) which makes the descriptor system (1) robustly admissible. Then, it follows from Theorem 3.5 that there exists a scalar

*<sup>c</sup> SQT*)*T*Γ−1]*B*Λ−1Ψ*<sup>T</sup>*

*<sup>c</sup> SQT*)*T*Γ−1(*<sup>A</sup>* + *<sup>ε</sup>*−1*HcH<sup>T</sup>*

*<sup>c</sup> SQT*)*T*Γ−1(*<sup>A</sup>* + *<sup>ε</sup>*−1*HcH<sup>T</sup>*

*<sup>c</sup> SQT*)] + ΨΛ−1Ψ*<sup>T</sup>* + *<sup>ε</sup>*−1*QSTHcHT*

*<sup>c</sup> SQT*) + ΨΛ−1*BT*Γ−1*B*Λ−1Γ*<sup>T</sup>* + *<sup>ε</sup>*ΨΛ−1Λ−1Ψ*<sup>T</sup>*

*<sup>c</sup> SQT*)*T*Γ−1*B*Λ−1Ψ*<sup>T</sup>* − ΨΛ−1*BTSQT*

*<sup>c</sup> Gc* + *<sup>ε</sup>*−1*QSTHcHT*

*<sup>c</sup> SQT*)

*<sup>c</sup> SQT*)

*<sup>c</sup> SQT*)

*<sup>c</sup> SQ<sup>T</sup>* + <sup>Θ</sup>*T*Γ−1<sup>Θ</sup> − ΨΛ−1Ψ*<sup>T</sup>* < 0.

*<sup>c</sup> P* −*εI*

*<sup>c</sup> P* −*εI*

(*A* − *B*Λ−1Ψ*T*)*TP QSTHc*

*<sup>c</sup> Gc* (*A* + *BK*)*TP QSTHc*

*<sup>c</sup> SQ<sup>T</sup>* <sup>+</sup> <sup>Θ</sup>*T*Γ−1<sup>Θ</sup> <sup>+</sup> *<sup>K</sup>T*Ψ*<sup>T</sup>* <sup>+</sup> <sup>Ψ</sup>*<sup>K</sup>* <sup>+</sup> *<sup>K</sup>T*Λ*<sup>K</sup>* <sup>&</sup>lt; 0.

*<sup>c</sup> SQ<sup>T</sup>*

 < 0.

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

*<sup>c</sup> SQ<sup>T</sup>*

In this section, we consider the robust admissibility with H∞ disturbance attenuation. H∞ disturbance attenuation problem plays an important role in control systems. Section 4.1 discusses the analysis of H∞ disturbance attenuation, and Section 4.2 gives design methods of the H∞ non-fragile controllers.

#### **4.1. Robust H**∞ **disturbance attenuation for uncertain systems**

First, we consider the robust H∞ disturbance attenuation *γ* for the following uncertain descriptor system

$$\begin{array}{l} \text{Ex}(k+1) = (A + \Delta A)\text{x}(k) + B\_1\text{w}(k) + B\_2\text{u}(k),\\ \text{z}(k) = \text{Cx}(k) + \text{Du}(k) \end{array} \tag{16}$$

where *w*(*k*) ∈ ℜ*m*<sup>1</sup> is the disturbances and *z*(*k*) ∈ ℜ*<sup>p</sup>* is the controlled output. *A*, *B*1, *B*2, *C* and *D* are system matrices with appropriate dimensions. Uncertain matrix ∆*A* is assumed to be of the form (3).

The controller is assumed to be of the form (4). Applying the controller (4) to the system (16), we have the closed-loop system

$$\begin{array}{l} \text{Ex}(k+1) = (A + B\_2K + H\_\mathcal{E}F\_\mathcal{c}(k)\mathbf{G}\_\mathcal{c})\mathbf{x}(k) + B\_1w(k),\\ \mathbf{z}(k) = (\mathbf{C} + DK + \alpha D\Phi(k)\mathbf{K})\mathbf{x}(k) \end{array} \tag{17}$$

where

$$H\_{\mathfrak{c}} = [H \mid \mathfrak{a}B\_2], \; F\_{\mathfrak{c}}(k) = \text{diag}[F(k) \; \Phi(k)], \; G\_{\mathfrak{c}} = \begin{bmatrix} G \\ K \end{bmatrix}.$$

Define the cost function

$$J = \sum\_{k=0}^{\infty} (z^T(k)z(k) - \gamma^2 w^T(k)w(k)). \tag{18}$$

The problem is to find a controller (4) which makes the system (16) with *w*(*k*) = 0 robustly admissible, and makes it satisfy *J* < 0 in (18). If there exists such a controller, it is said to be an H∞ non-fragile controller and the closed-loop system is said to be robustly admissible with H∞ disturbance attenuation *γ*.

The following is a well-known result for the admissibility with H∞ disturbance attenuation *γ* of linear descriptor systems.

**Lemma 4.1.** *(Xu & Lam [32]) Consider the system*

$$\begin{array}{l} \text{Ex}(k+1) = A\text{x}(k) + B\text{w}(k),\\ z(k) = \text{Cx}(k) + Dw(k) \end{array} \tag{19}$$

*where A*, *B*, *C and D are matrices of appropriate dimensions.*

*(i) Given a scalar γ* > 0*. The descriptor system (19) is robustly admissible with H*<sup>∞</sup> *disturbance attenuation γ if and only if there exist matrices P* > 0, *Q such that*

$$
\begin{bmatrix}
\boldsymbol{A}^T \boldsymbol{P} \mathbf{A} - \boldsymbol{E}^T \boldsymbol{P} \mathbf{E} + \mathbf{C}^T \mathbf{C} & \boldsymbol{A}^T \boldsymbol{P} \mathbf{B} + \mathbf{C}^T \boldsymbol{D} \\
\boldsymbol{B}^T \boldsymbol{P} \mathbf{A} + \boldsymbol{D}^T \mathbf{C} & \boldsymbol{B}^T \boldsymbol{P} \mathbf{B} + \boldsymbol{D}^T \boldsymbol{D} - \gamma^2 \boldsymbol{I}
\end{bmatrix} + \begin{bmatrix}
\boldsymbol{A}^T \\
\boldsymbol{B}^T
\end{bmatrix} \boldsymbol{S} \mathbf{Q}^T + \boldsymbol{Q} \mathbf{S}^T \begin{bmatrix} \boldsymbol{A} \ \boldsymbol{B} \end{bmatrix} < \boldsymbol{0} \tag{20}
$$

*where S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*. (ii) Given a scalar γ* > 0*. The descriptor system (19) is robustly admissible with H*<sup>∞</sup> *disturbance attenuation γ if and only if there exist matrix P such that*

$$E^{T}PE \geq 0,\tag{21}$$

*Q*¯ =

*H*¯ <sup>1</sup> = �

*H*¯ <sup>2</sup> = �

*G*¯ <sup>1</sup> = �

*G*¯ <sup>2</sup> = �

where

holds:

It follows from (24) that

 

 

 

> *H<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>2</sup> *<sup>H</sup><sup>T</sup> <sup>c</sup> P* 0 �*T* ,

<sup>000</sup> *<sup>α</sup>DT*�*<sup>T</sup>* ,

 

≤ *Q*¯ + *ε*

,

∆*<sup>T</sup>*

∆*<sup>T</sup>*

= *Q*¯ + *H*¯ <sup>1</sup>*Fc*(*k*)*G*¯

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

<sup>∆</sup><sup>12</sup> = (*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>+</sup> *HcFc*(*k*)*Gc*)*TSQT*

∆*<sup>T</sup>*

∆*<sup>T</sup>*

is robustly admissible with H∞ disturbance attenuation *γ*.

∆*<sup>T</sup>*

∆*<sup>T</sup>*

<sup>∆</sup><sup>13</sup> = (*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>+</sup> *HcFc*(*k*)*Gc*)*TP*, <sup>∆</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>+</sup> *<sup>Q</sup>*2*STB*<sup>1</sup> <sup>+</sup> *<sup>B</sup><sup>T</sup>*

.

*Gc* 000�

*K* 000�

Now, using Lemma 3.2, we have

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

*<sup>Q</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>B</sup><sup>T</sup>*

*<sup>K</sup>SQ<sup>T</sup>*

<sup>12</sup> <sup>∆</sup><sup>22</sup> *<sup>B</sup><sup>T</sup>*

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

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

<sup>1</sup> *SQ<sup>T</sup>* 2 .

<sup>12</sup> <sup>∆</sup><sup>22</sup> *<sup>B</sup><sup>T</sup>*

<sup>12</sup> <sup>∆</sup><sup>22</sup> *<sup>B</sup><sup>T</sup>*

*C* + *DK* + *D*Φ(*k*)*K* 0 0 −*I*

*C* + *DK* + *D*Φ(*k*)*K* 0 0 −*I*

*C* + *DK* + *D*Φ(*k*)*K* 0 0 −*I*

1 *G*¯ <sup>1</sup> + *ε* −1 <sup>2</sup> *<sup>H</sup>*¯ <sup>2</sup>*H<sup>T</sup>*

<sup>∆</sup><sup>11</sup> <sup>=</sup> *<sup>Q</sup>*1*ST*(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>+</sup> *HcFc*(*k*)*Gc*)+(*<sup>A</sup>* <sup>+</sup> *<sup>B</sup>*2*<sup>K</sup>* <sup>+</sup> *HcFc*(*k*)*Gc*)*TSQT*

<sup>1</sup> *<sup>F</sup>T*(*k*)*H*¯ *<sup>T</sup>*

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

<sup>1</sup> *SQ<sup>T</sup>* <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>+</sup> *<sup>Q</sup>*2*STB*<sup>1</sup> <sup>+</sup> *<sup>B</sup><sup>T</sup>*

*<sup>K</sup>SQ<sup>T</sup>*

<sup>∆</sup><sup>11</sup> <sup>∆</sup><sup>12</sup> <sup>∆</sup><sup>13</sup> (*<sup>C</sup>* <sup>+</sup> *DK* <sup>+</sup> *<sup>D</sup>*Φ(*k*)*K*)*<sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>Q</sup>*1*STB*1,

<sup>∆</sup><sup>11</sup> <sup>∆</sup><sup>12</sup> <sup>∆</sup><sup>13</sup> (*<sup>C</sup>* <sup>+</sup> *DK* <sup>+</sup> *<sup>D</sup>*Φ(*k*)*K*)*<sup>T</sup>*

This implies by Schur complement formula and Lemma 4.1(i) that the descriptor system (17)

(Necessity) Assume that the descriptor system (17) is robustly admissible with H∞ disturbance attenuation *γ*. Then, it follows from Lemma 4.1 that there exist matrices *P* > 0, *Q* and scalars *ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 such that (20) with *A* replaced by *A* + *BK* + *HcFc*(*k*)*Gc* and *C* by *C* + *DK* + *α*Φ(*k*)*K* holds. Thus, for all admissible *F*(*k*) and Φ(*k*), the following inequality

<sup>∆</sup><sup>11</sup> <sup>∆</sup><sup>12</sup> <sup>∆</sup><sup>13</sup> (*<sup>C</sup>* <sup>+</sup> *DK* <sup>+</sup> *<sup>D</sup>*Φ(*k*)*K*)*<sup>T</sup>*

<sup>13</sup> *PB*<sup>1</sup> <sup>−</sup>*<sup>P</sup>* <sup>0</sup>

<sup>1</sup> *<sup>P</sup>* <sup>0</sup>

<sup>13</sup> *PB*<sup>1</sup> <sup>−</sup>*<sup>P</sup>* <sup>0</sup>

<sup>1</sup> *<sup>P</sup>* <sup>0</sup>

<sup>13</sup> *PB*<sup>1</sup> <sup>−</sup>*<sup>P</sup>* <sup>0</sup>

<sup>1</sup> *<sup>P</sup>* <sup>0</sup>

<sup>1</sup> <sup>+</sup> *<sup>H</sup>*¯ <sup>2</sup>Φ(*k*)*G*¯

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

*PAK PB*<sup>1</sup> −*P* 0 *C* + *DK* 0 0 −*I*

<sup>2</sup> <sup>+</sup> *<sup>Q</sup>*1*STB*<sup>1</sup> *<sup>A</sup><sup>T</sup>*

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

<sup>1</sup> *SQ<sup>T</sup>* <sup>2</sup> *<sup>B</sup><sup>T</sup>*

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

2 *G*¯ 2 *<sup>K</sup><sup>P</sup>* (*<sup>C</sup>* <sup>+</sup> *DK*)*<sup>T</sup>*

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

 , 39

<sup>1</sup> *<sup>P</sup>* <sup>0</sup>

 

<sup>2</sup> <sup>Φ</sup>*T*(*k*)*H*¯ *<sup>T</sup>* 2

> < 0.

> < 0.

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

$$
\begin{bmatrix}
\begin{bmatrix}
A^T P A - E^T P E + \mathbb{C}^T \mathbb{C} & A^T P B + \mathbb{C}^T D \\
B^T P A + D^T \mathbb{C} & B^T P B + D^T D - \gamma^2 I
\end{bmatrix} < 0. \tag{22}
$$

The following theorem provides a necessary and sufficient condition for the robust admissibility with H∞ disturbance attenuation of (17).

**Theorem 4.2.** *Given γ and K, the descriptor system (17) is robustly admissible with H*<sup>∞</sup> *disturbance attenuation γ if and only if there exist matrices P* > 0, *Q* = [*Q<sup>T</sup>* <sup>1</sup> *<sup>Q</sup><sup>T</sup>* 2 ] *<sup>T</sup> and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that*

$$
\begin{bmatrix}
\Pi\_{11} & A\_{1}^{T}S\mathbf{Q}\_{2}^{T} + Q\_{1}S^{T}B\_{1} & A\_{K}^{T}P \ (\mathcal{C} + DK)^{T}B\_{1}S^{T}H\_{c} & 0 \\
Q\_{2}S^{T}A\_{K} + B\_{1}^{T}S\mathbf{Q}^{T} & \Pi\_{22} & B\_{1}^{T}P & 0 & Q\_{2}S^{T}H\_{c} & 0 \\
P A\_{K} & P B\_{1} & -P & 0 & P H\_{c} & 0 \\
\mathcal{C} + DK & 0 & 0 & -I & 0 & aD \\
H\_{t}^{T}S Q\_{1}^{T} & H\_{t}^{T}S Q\_{2}^{T} & H\_{t}^{T}P & 0 & -\varepsilon\_{1}I & 0 \\
0 & 0 & 0 & aD^{T} & 0 & -\varepsilon\_{2}I
\end{bmatrix} < 0 \tag{23}
$$

*where AK* = *A* + *B*2*K*, *S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*, and*

$$\begin{array}{l} \Pi\_{11} = Q\_1 \mathbf{S}^T A\_K - \mathbf{E}^T P \mathbf{E} + A\_K^T \mathbf{S} \mathbf{Q}\_1^T + \varepsilon\_1 \mathbf{G}\_\mathbf{c}^T \mathbf{G}\_\mathbf{c} + \varepsilon\_2 \mathbf{K}^T \mathbf{K}\_\mathbf{c},\\ \Pi\_{22} = -\gamma^2 I + Q\_2 \mathbf{S}^T B\_1 + \mathbf{B}\_1^T \mathbf{S} \mathbf{Q}\_2^T. \end{array}$$

**Proof:** (Sufficiency) Suppose that there exist matrices *P* > 0, *Q* = [*Q<sup>T</sup>* <sup>1</sup> *<sup>Q</sup><sup>T</sup>* 2 ] *<sup>T</sup>* and scalars *ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 such that the condition (23) holds. Then, by Schur complement formula, we have

$$
\ddot{Q} + \varepsilon\_1^{-1} \boldsymbol{H}\_1 \boldsymbol{H}\_1^T + \varepsilon\_1 \boldsymbol{G}\_2^T \boldsymbol{G}\_1 + \varepsilon\_2^{-1} \boldsymbol{H}\_2 \boldsymbol{H}\_2^T + \varepsilon\_2 \boldsymbol{G}\_2^T \boldsymbol{G}\_2 < 0. \tag{24}
$$

where

$$
\begin{split}
\bar{Q} &= \begin{bmatrix}
\boldsymbol{Q}\_{1}\boldsymbol{S}^{T}\boldsymbol{A}\_{K} - \boldsymbol{E}^{T}\boldsymbol{P}\boldsymbol{E} + \boldsymbol{A}\_{K}^{T}\boldsymbol{S}\boldsymbol{Q}\_{1}^{T} & \boldsymbol{A}\_{K}^{T}\boldsymbol{S}\boldsymbol{Q}\_{2}^{T} + \boldsymbol{Q}\_{1}\boldsymbol{S}^{T}\boldsymbol{B}\_{1} & \boldsymbol{A}\_{K}^{T}\boldsymbol{P}\left(\mathbb{C} + \boldsymbol{D}\boldsymbol{K}\right)^{T} \\
\boldsymbol{Q}\_{2}\boldsymbol{S}^{T}\boldsymbol{A}\_{K} + \boldsymbol{B}\_{1}^{T}\boldsymbol{S}\boldsymbol{Q}^{T} & -\gamma^{2}\boldsymbol{I} + \boldsymbol{Q}\_{2}\boldsymbol{S}^{T}\boldsymbol{B}\_{1} + \boldsymbol{B}\_{1}^{T}\boldsymbol{S}\boldsymbol{Q}\_{2}^{T} & \boldsymbol{B}\_{1}^{T}\boldsymbol{P} & \boldsymbol{0} \\
\boldsymbol{P}\boldsymbol{A}\_{K} & \boldsymbol{P}\boldsymbol{B}\_{1} & -\boldsymbol{P} & \boldsymbol{0} \\
\boldsymbol{C} + \boldsymbol{D}\boldsymbol{K} & \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{-I}
\end{bmatrix}, \\
\bar{H}\_{1} &= \begin{bmatrix}
\boldsymbol{H}\_{c}^{T}\boldsymbol{S}\boldsymbol{Q}\_{1}^{T}\boldsymbol{H}\_{c}^{T}\boldsymbol{S}\boldsymbol{Q}\_{2}^{T}\boldsymbol{H}\_{c}^{T}\boldsymbol{P}\boldsymbol{0} \end{bmatrix}^{T}, \\
\bar{H}\_{2} &= \begin{bmatrix}
\boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0} \boldsymbol{D} \boldsymbol{T}^{T} \\
\boldsymbol{G}\_{1} = \begin{bmatrix}
\boldsymbol{G}\_{c} \boldsymbol{0} \ \boldsymbol{0} \ \boldsymbol{0} \end{bmatrix}^{T}, \\
\bar{G}\_{2} & \mathbb{K} \$$

Now, using Lemma 3.2, we have

$$
\begin{bmatrix}
\Delta\_{11} & \Delta\_{12} & \Delta\_{13} & (\mathbb{C} + DK + D\Phi(k)K)^{T} \\
\Delta\_{12}^{T} & \Delta\_{22} & B\_{1}^{T}P & 0 \\
\Delta\_{13}^{T} & PB\_{1} - P & 0 \\
\mathbb{C} + DK + D\Phi(k)K & 0 & -I \\
= \bar{Q} + H\_{1}F\_{c}(k)\bar{\mathcal{G}}\_{1} + \bar{G}\_{1}^{T}F^{T}(k)\bar{H}\_{1}^{T} + \bar{H}\_{2}\Phi(k)\bar{\mathcal{G}}\_{2} + \bar{G}\_{2}^{T}\Phi^{T}(k)\bar{H}\_{2}^{T} \\
\leq \bar{Q} + \varepsilon\_{1}^{-1}H\_{1}H\_{1}^{T} + \varepsilon\_{1}\bar{G}\_{1}^{T}\bar{G}\_{1} + \varepsilon\_{2}^{-1}H\_{2}H\_{2}^{T} + \varepsilon\_{2}\bar{G}\_{2}^{T}\bar{G}\_{2}
\end{bmatrix}
$$

where

10 Advances in Discrete Time Systems

�

*such that*

*and*

have

where

 

*Q*2*S<sup>T</sup> AK* + *B<sup>T</sup>*

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

*where A*, *B*, *C and D are matrices of appropriate dimensions.*

*ATPA* − *ETPE* + *CTC ATPB* + *CTD BTPA* + *DTC BTPB* + *DTD* − *γ*<sup>2</sup> *I*

*attenuation γ if and only if there exist matrix P such that*

admissibility with H∞ disturbance attenuation of (17).

Π<sup>11</sup> *A<sup>T</sup>*

*Q*¯ + *ε* −1 <sup>1</sup> *<sup>H</sup>*¯ <sup>1</sup>*H*¯ *<sup>T</sup>*

*attenuation γ if and only if there exist matrices P* > 0, *Q* = [*Q<sup>T</sup>*

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

*KSQ<sup>T</sup>*

Π<sup>11</sup> = *Q*1*S<sup>T</sup> AK* − *ETPE* + *A<sup>T</sup>*

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

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

2 *G*¯ <sup>1</sup> + *ε* −1 <sup>2</sup> *<sup>H</sup>*¯ <sup>2</sup>*H*¯ *<sup>T</sup>*

Π<sup>22</sup> = −*γ*<sup>2</sup> *I* + *Q*2*STB*<sup>1</sup> + *B<sup>T</sup>*

<sup>1</sup> *SQ<sup>T</sup>* <sup>Π</sup><sup>22</sup> *<sup>B</sup><sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>*

�

*attenuation γ if and only if there exist matrices P* > 0, *Q such that*

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

*(i) Given a scalar γ* > 0*. The descriptor system (19) is robustly admissible with H*<sup>∞</sup> *disturbance*

*(ii) Given a scalar γ* > 0*. The descriptor system (19) is robustly admissible with H*<sup>∞</sup> *disturbance*

The following theorem provides a necessary and sufficient condition for the robust

**Theorem 4.2.** *Given γ and K, the descriptor system (17) is robustly admissible with H*<sup>∞</sup> *disturbance*

*PAK PB*<sup>1</sup> −*P* 0 *PHc* 0 *C* + *DK* 0 0 −*I* 0 *αD*

0 00 *αD<sup>T</sup>* 0 −*ε*<sup>2</sup> *I*

*KSQ<sup>T</sup>*

<sup>1</sup> *SQ<sup>T</sup>* 2 .

*ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 such that the condition (23) holds. Then, by Schur complement formula, we

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

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

2 *G*¯

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

*where AK* = *A* + *B*2*K*, *S* ∈ ℜ*n*×(*n*−*r*) *is any matrix with full column rank and satisfies ETS* = 0*,*

<sup>2</sup> <sup>+</sup> *<sup>Q</sup>*1*STB*<sup>1</sup> *<sup>A</sup><sup>T</sup>*

*ATPA* − *ETPE* + *CTC ATPB* + *CTD BTPA* + *DTC BTPB* + *DTD* − *γ*<sup>2</sup> *I*

� + � *A<sup>T</sup> BT* �

*SQ<sup>T</sup>* + *QS<sup>T</sup>* �

�

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

*<sup>K</sup><sup>P</sup>* (*<sup>C</sup>* <sup>+</sup> *DK*)*<sup>T</sup> <sup>Q</sup>*1*STHc* <sup>0</sup>

<sup>1</sup> *<sup>P</sup>* <sup>0</sup> *<sup>Q</sup>*2*STHc* <sup>0</sup>

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

*<sup>c</sup> Gc* + *ε*2*KTK*,

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

<sup>2</sup> < 0. (24)

*A B*�

*ETPE* ≥ 0, (21)

< 0. (22)

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

 

< 0 (23)

*<sup>T</sup>* and scalars

< 0 (20)

$$\begin{split} \Delta\_{11} &= \mathcal{Q}\_1 \mathcal{S}^T (A + \mathcal{B}\_2 K + H\_c F\_c(k) \mathcal{G}\_c) + (A + \mathcal{B}\_2 K + H\_c F\_c(k) \mathcal{G}\_c)^T \mathcal{S} \mathcal{Q}\_1^T - E^T \mathcal{P} E\_1 \\ \Delta\_{12} &= (A + \mathcal{B}\_2 K + H\_c F\_c(k) \mathcal{G}\_c)^T \mathcal{S} \mathcal{Q}\_2^T + \mathcal{Q}\_1 \mathcal{S}^T \mathcal{B}\_1 \\ \Delta\_{13} &= (A + \mathcal{B}\_2 K + H\_c F\_c(k) \mathcal{G}\_c)^T \mathcal{P}\_1 \\ \Delta\_{22} &= -\gamma^2 I + \mathcal{Q}\_2 \mathcal{S}^T \mathcal{B}\_1 + \mathcal{B}\_1^T \mathcal{S} \mathcal{Q}\_2^T. \end{split}$$

It follows from (24) that

$$
\begin{bmatrix}
\Delta\_{11} & \Delta\_{12} & \Delta\_{13} & (\mathbb{C} + DK + D\Phi(k)K)^T \\
\Delta\_{12}^T & \Delta\_{22} & B\_1^T P & 0 \\
\Delta\_{13}^T & PB\_1 & -P & 0 \\
\mathbb{C} + DK + D\Phi(k)K & 0 & 0 & -I
\end{bmatrix} < 0.
$$

This implies by Schur complement formula and Lemma 4.1(i) that the descriptor system (17) is robustly admissible with H∞ disturbance attenuation *γ*.

(Necessity) Assume that the descriptor system (17) is robustly admissible with H∞ disturbance attenuation *γ*. Then, it follows from Lemma 4.1 that there exist matrices *P* > 0, *Q* and scalars *ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 such that (20) with *A* replaced by *A* + *BK* + *HcFc*(*k*)*Gc* and *C* by *C* + *DK* + *α*Φ(*k*)*K* holds. Thus, for all admissible *F*(*k*) and Φ(*k*), the following inequality holds:

$$
\begin{bmatrix}
\Delta\_{11} & \Delta\_{12} & \Delta\_{13} & (\mathbb{C} + DK + D\Phi(k)K)^T \\
\Delta\_{12}^T & \Delta\_{22} & B\_1^T P & 0 \\
\Delta\_{13}^T & PB\_1 & -P & 0 \\
\mathbb{C} + DK + D\Phi(k)K & 0 & 0 & -I
\end{bmatrix} < 0.
$$

That is,

$$\dot{Q} + \mathcal{H}\_1 \mathcal{F}\_\mathcal{c}(k) \bar{\mathcal{G}}\_1 + \bar{\mathcal{G}}\_1^T \mathcal{F}\_\mathcal{c}^T(k) \mathcal{H}\_1^T + \mathcal{H}\_2 \Phi(k) \bar{\mathcal{G}}\_2 + \bar{\mathcal{G}}\_2^T \Phi^T(k) \mathcal{H}\_2^T < 0$$

is satisfied for all admissible *F*(*k*) and Φ(*k*). It follows from Lemma 3.2 that

$$
\ddot{Q} + \varepsilon\_1^{-1} \ddot{H}\_1 \ddot{H}\_1^T + \varepsilon\_1 \ddot{G}\_1^T \ddot{G}\_1 + \varepsilon\_2^{-1} \ddot{H}\_2 \ddot{H}\_2^T + \varepsilon\_2 \ddot{G}\_2^T \ddot{G}\_2 < 0,
$$

which leads to the condition (23). This completes the proof.

Based on Lemma 4.1(ii), we have the following theorem. The proof is similar to that of Theorem 4.2, and is thus omitted.

**Theorem 4.3.** *Given γ and K, the descriptor system (17) is robustly admissible with H*∞ *disturbance attenuation γ if and only if there exist matrix P and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that (21), and*

$$
\begin{bmatrix}
\mathbf{0} & -\gamma^2 I & \mathbf{B}\_1^T \mathbf{P} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
P(\mathbf{A} + \mathcal{B}\_2 \mathbf{K}) & P \mathcal{B}\_1 & -P & \mathbf{0} & P \mathcal{H}\_\mathcal{E} & \mathbf{0} \\
\mathbf{C} + \mathcal{D} \mathbf{K} & \mathbf{0} & \mathbf{0} & -I & \mathbf{0} & a \mathcal{D} \\
\mathbf{0} & \mathbf{0} & H\_\mathcal{C}^T \mathbf{P} & \mathbf{0} & -\varepsilon\_1 I & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & a \mathcal{D}^T & \mathbf{0} & -\varepsilon\_2 I
\end{bmatrix} < \mathbf{0}.\tag{25}
$$

#### **4.2. H**∞ **non-fragile control design for uncertain systems**

Now, we are at the position to propose design methods of H∞ non-fragile controller for uncertain descriptor systems.

**Theorem 4.4.** *There exists a controller (4) that makes the descriptor system (16) robustly admissible with H*<sup>∞</sup> *disturbance attenuation γ if there exist matrices P* > 0, *Q* = [*Q<sup>T</sup>* <sup>1</sup> *<sup>Q</sup><sup>T</sup>* 2 ] *<sup>T</sup> and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that*

$$
\Gamma = P^{-1} - \varepsilon\_1^{-1} H\_\mathfrak{c} H\_\mathfrak{c}^T > 0,\tag{26}
$$

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

where *Gc* =

= Θ*<sup>T</sup>*

*G<sup>T</sup>* −ΨΛ−<sup>1</sup>

= *Q*1*S<sup>T</sup> A* − *Q*1*STB*2Λ−1Ψ*<sup>T</sup>* + *ATSQT*

(*C* − *D*Λ−1Ψ*T*)*TZ*<sup>−</sup>1(*C* − *D*Λ−1Ψ*T*)

−1 <sup>1</sup> *HcH<sup>T</sup>*

<sup>2</sup> (*SQ<sup>T</sup>*

*Q*1*ST*(*A* − *B*2Λ−1Ψ*T*)+(*A* − *B*2Λ−1Ψ*T*)*TSQT*

−1 <sup>1</sup> *HcH<sup>T</sup>*

+(*C* − *D*Λ−1Ψ*T*)*TZ*<sup>−</sup>1(*C* − *D*Λ−1Ψ*T*) − *ETPE* + *ε*

*<sup>c</sup> SQ<sup>T</sup>*

<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>ε</sup>*

<sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*B*)*W*−1(*SQ<sup>T</sup>*

+(*ε*<sup>1</sup> + *ε*2)ΨΛ−1Λ−1Ψ*T*,

(*A* − *B*2Λ−1Ψ*<sup>T</sup>* + *ε*

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

[(*A* − *B*2Λ−1Ψ*T*)*TSQT*

<sup>2</sup> (*SQ<sup>T</sup>*

Thus, it can be verified with (29) that

+(*A* − *B*2Λ−1Ψ*<sup>T</sup>* + *ε*

= *Q*1*S<sup>T</sup> A* + *ATSQT*

+[(*A* − *B*2Λ−1Ψ*T*)*TSQT*

×[(*A* − *B*2Λ−1Ψ*T*)*TSQT*

+*YW*−1*Y<sup>T</sup>* − ΨΛ−1Ψ*<sup>T</sup>* < 0.

= *YW*−1*Y<sup>T</sup>* − ΨΛ−1*B<sup>T</sup>*

+ΨΛ−1*B<sup>T</sup>*

×[(*A* − *B*2Λ−1Ψ*T*)*TSQT*

*T*

*Q*1*ST*(*A* − *B*2Λ−1Ψ*T*)+(*A* − *B*2Λ−1Ψ*T*)*TSQT*

**Proof:** (Sufficiency) The closed-loop system (17) with the feedback gain (30) is given by

*Ex*(*k* + 1)=(*A* − *B*2Λ−1Ψ*<sup>T</sup>* + *HcFc*(*k*)*Gc*)*x*(*k*) + *B*1*w*(*k*),

<sup>1</sup> <sup>−</sup> ΨΛ−1*B<sup>T</sup>*

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

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

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

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

*<sup>A</sup>*Γ−1*B*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> ΨΛ−1*B<sup>T</sup>*

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

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

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

<sup>1</sup> *<sup>Q</sup>*1*STHcHT*

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

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

−1

−1

<sup>1</sup> *<sup>Q</sup>*1*STHcHT*

−1 <sup>1</sup> *HcH<sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

= *CTZ*<sup>−</sup>1*C* − ΨΛ−1*DZ*−1*C* − *CTZ*<sup>−</sup>1*D*Λ−1Ψ*<sup>T</sup>* + ΨΛ−1*DZ*−1*D*Λ−1Ψ*T*,

*<sup>c</sup> SQ<sup>T</sup>*

<sup>2</sup> <sup>Γ</sup>−1Θ*<sup>A</sup>* <sup>−</sup> <sup>Θ</sup>*<sup>T</sup>*

*K* = −Λ−1Ψ*T*. (30)

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

41

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

*<sup>c</sup> Gc* + *<sup>ε</sup>*2ΨΛ−1Λ−1Ψ*<sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>* 1 )

<sup>2</sup> <sup>Γ</sup>−1*B*2Λ−1Ψ*T*,

*<sup>c</sup> SQ<sup>T</sup>*

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

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

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

*<sup>c</sup> SQ<sup>T</sup>*

*<sup>c</sup> Gc* + *<sup>ε</sup>*2ΨΛ−1Λ−1Ψ*<sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>* 1

*<sup>c</sup> SQ<sup>T</sup>*

<sup>1</sup> <sup>+</sup> *<sup>C</sup>TZ*<sup>−</sup>1*<sup>C</sup>* <sup>+</sup> <sup>Θ</sup>*<sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>* 1 )

*<sup>c</sup> SQ<sup>T</sup>*

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

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

*<sup>A</sup>*Γ−1Θ*<sup>A</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

*<sup>z</sup>*(*k*)=(*<sup>C</sup>* <sup>−</sup> *<sup>D</sup>*Λ−1Ψ*<sup>T</sup>* <sup>−</sup> *<sup>α</sup>D*Φ(*k*)Λ−1Ψ*T*)*x*(*k*) (31)

. 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> *<sup>ε</sup>*1*GTG*

−1 <sup>1</sup> *HcH<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

<sup>2</sup> *SQ<sup>T</sup>*

$$Z = I - \varepsilon\_2^{-1} \mathfrak{a}^2 D D^T > 0,\tag{27}$$

$$\mathcal{W} = \gamma^2 I - Q\_2 \mathcal{S}^T B\_1 - B\_1^T \mathcal{S} \mathcal{Q}\_2^T - \varepsilon\_1^{-1} Q\_2 \mathcal{S}^T H\_c H\_c^T \mathcal{S} \mathcal{Q}\_2^T - \Theta\_B^T \Gamma^{-1} \Theta\_B > 0,\tag{28}$$

$$\mathcal{O}\_1 \mathcal{S}^T A + A^T \mathcal{S} \mathcal{O}\_1^T - F^T P F + \varepsilon\_1 \mathcal{G}^T G + \varepsilon\_1^{-1} \mathcal{O}\_1 \mathcal{S}^T H\_c H^T \mathcal{S} \mathcal{O}\_1^T$$

$$\begin{aligned} Q\_1 \mathbf{S}^T A + A^T \mathbf{S} \mathbf{Q}\_1^T - E^T P \mathbf{E} + \varepsilon\_1 \mathbf{G}^T \mathbf{G} + \varepsilon\_1^{-1} \mathbf{Q}\_1 \mathbf{S}^T H\_c H\_c^T \mathbf{S} \mathbf{Q}\_1^T \\ + \mathbf{C}^T \mathbf{Z}^{-1} \mathbf{C} + Y \mathbf{W}^{-1} \mathbf{Y}^T + \Theta\_A^T \boldsymbol{\Gamma}^{-1} \boldsymbol{\Theta}\_A - \mathbf{Y} \boldsymbol{\Lambda}^{-1} \mathbf{Y}^T < 0 \end{aligned} \tag{29}$$

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

$$\begin{array}{l} \Psi = (\mathbf{Q}\_1 \mathbf{S}^T + \mathbf{G}\_A^T \Gamma^{-1}) \mathbf{B}\_2 + \mathbf{C}^T \mathbf{Z}^{-1} \mathbf{D} + \mathbf{Y} \mathbf{W}^{-1} (\mathbf{Q}\_2 \mathbf{S}^T + \mathbf{G}\_B^T \Gamma^{-1}) \mathbf{B}\_{2'} \\ \Lambda = \mathbf{B}\_2^T \Gamma^{-1} \mathbf{B}\_2 + \mathbf{D}^T \mathbf{Z}^{-1} \mathbf{D} + (\varepsilon\_1 + \varepsilon\_2) \mathbf{I} + \mathbf{B}\_2^T (\mathbf{S} \mathbf{Q}\_2^T + \Gamma^{-1} \mathbf{G}\_B^T) \mathbf{W}^{-1} (\mathbf{Q}\_2 \mathbf{S}^T + \mathbf{G}\_B^T \Gamma^{-1}) \mathbf{B}\_{2'} \\ \Theta\_A = A + \varepsilon\_1^{-1} H\_\mathcal{E} H\_\mathcal{E}^T \mathbf{S} \mathbf{Q}\_1^T, \\ \Theta\_B = B\_1 + \varepsilon\_1^{-1} H\_\mathcal{E} H\_\mathcal{E}^T \mathbf{S} \mathbf{Q}\_2^T \\ \mathbf{Y} = A^T \mathbf{S} \mathbf{Q}\_2^T + (\mathbf{Q}\_1 \mathbf{S}^T + \mathbf{G}\_A^T \Gamma^{-1}) \Theta\_B. \end{array}$$

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

12 Advances in Discrete Time Systems

*Q*¯ + *H*¯ <sup>1</sup>*Fc*(*k*)*G*¯

*Q*¯ + *ε* −1 <sup>1</sup> *<sup>H</sup>*¯ <sup>1</sup>*H*¯ *<sup>T</sup>*

Theorem 4.2, and is thus omitted.

−*ETPE* + *ε*1*G<sup>T</sup>*

uncertain descriptor systems.

*W* = *γ*<sup>2</sup> *I* − *Q*2*STB*<sup>1</sup> − *B<sup>T</sup>*

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

<sup>2</sup> <sup>Γ</sup>−1*B*<sup>2</sup> <sup>+</sup> *<sup>D</sup>TZ*<sup>−</sup>1*<sup>D</sup>* + (*ε*<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*2)*<sup>I</sup>* <sup>+</sup> *<sup>B</sup><sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>* 1 ,

*<sup>c</sup> SQ<sup>T</sup>* 2 ,

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

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

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

−1 <sup>1</sup> *HcH<sup>T</sup>*

−1 <sup>1</sup> *HcH<sup>T</sup>*

Λ = *B<sup>T</sup>*

Θ*<sup>A</sup>* = *A* + *ε*

Θ*<sup>B</sup>* = *B*<sup>1</sup> + *ε*

*Y* = *ATSQT*

<sup>1</sup> + *G*¯ *<sup>T</sup>* <sup>1</sup> *<sup>F</sup><sup>T</sup> <sup>c</sup>* (*k*)*H*¯ *<sup>T</sup>*

which leads to the condition (23). This completes the proof.

0 −*γ*<sup>2</sup> *I B<sup>T</sup>*

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

**4.2. H**∞ **non-fragile control design for uncertain systems**

*with H*<sup>∞</sup> *disturbance attenuation γ if there exist matrices P* > 0, *Q* = [*Q<sup>T</sup>*

<sup>1</sup> *SQ<sup>T</sup>* <sup>2</sup> − *ε* −1

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

*<sup>A</sup>*Γ−1)Θ*B*.

*<sup>A</sup>*Γ−1)*B*<sup>2</sup> <sup>+</sup> *<sup>C</sup>TZ*<sup>−</sup>1*<sup>D</sup>* <sup>+</sup> *YW*−1(*Q*2*S<sup>T</sup>* <sup>+</sup> <sup>Θ</sup>*<sup>T</sup>*

is satisfied for all admissible *F*(*k*) and Φ(*k*). It follows from Lemma 3.2 that

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

1 *G*¯ <sup>1</sup> + *ε* −1 <sup>2</sup> *<sup>H</sup>*¯ <sup>2</sup>*H*¯ *<sup>T</sup>*

Based on Lemma 4.1(ii), we have the following theorem. The proof is similar to that of

**Theorem 4.3.** *Given γ and K, the descriptor system (17) is robustly admissible with H*∞ *disturbance attenuation γ if and only if there exist matrix P and scalars ε*<sup>1</sup> > 0,*ε*<sup>2</sup> > 0 *such that (21), and*

> *P*(*A* + *B*2*K*) *PB*<sup>1</sup> −*P* 0 *PHc* 0 *C* + *DK* 0 0 −*I* 0 *αD*

*<sup>c</sup> Gc* + *ε*2*KTK* 0 (*A* + *B*2*K*)*TP* (*C* + *DK*)*<sup>T</sup>* 0 0

0 00 *αD<sup>T</sup>* 0 −*ε*<sup>2</sup> *I*

Now, we are at the position to propose design methods of H∞ non-fragile controller for

**Theorem 4.4.** *There exists a controller (4) that makes the descriptor system (16) robustly admissible*

<sup>1</sup> <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>+</sup> *<sup>ε</sup>*

+*CTZ*<sup>−</sup>1*C* + *YW*−1*Y<sup>T</sup>* + Θ*<sup>T</sup>*

<sup>1</sup> *<sup>Q</sup>*2*STHcHT*

<sup>2</sup> (*SQ<sup>T</sup>*

<sup>1</sup> <sup>+</sup> *<sup>H</sup>*¯ <sup>2</sup>Φ(*k*)*G*¯

<sup>2</sup> + *G*¯ *<sup>T</sup>*

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

<sup>1</sup> *P* 0 00

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

Γ = *P*−<sup>1</sup> − *ε*

*Z* = *I* − *ε*

*<sup>c</sup> SQ<sup>T</sup>*

−1

−1 <sup>1</sup> *HcH<sup>T</sup>*

> *<sup>c</sup> SQ<sup>T</sup>* 1

*<sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>−</sup> ΨΛ−1Ψ*<sup>T</sup>* <sup>&</sup>lt; 0 (29)

−1

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

<sup>1</sup> *<sup>Q</sup>*1*STHcHT*

*<sup>B</sup>*Γ−1)*B*2,

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

2 *G*¯ <sup>2</sup> < 0,

<sup>2</sup> <sup>Φ</sup>*T*(*k*)*H*¯ *<sup>T</sup>*

<sup>2</sup> <sup>&</sup>lt; <sup>0</sup>

 

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

< 0. (25)

*<sup>T</sup> and scalars*

*<sup>c</sup>* > 0, (26)

*<sup>B</sup>*Γ−1)*B*2,

<sup>2</sup> *<sup>α</sup>*2*DD<sup>T</sup>* <sup>&</sup>gt; 0, (27)

*<sup>B</sup>*Γ−1Θ*<sup>B</sup>* <sup>&</sup>gt; 0, (28)

That is,

 

$$K = -\Lambda^{-1} \Psi^T.\tag{30}$$

**Proof:** (Sufficiency) The closed-loop system (17) with the feedback gain (30) is given by

$$\begin{array}{l} \text{Ex}(k+1) = (A - B\_2 \Lambda^{-1} \mathbf{\varPi}^T + H\_\mathbf{c} \mathbf{F}\_\mathbf{c}(k) \mathbf{G}\_\mathbf{c}) \mathbf{x}(k) + B\_1 \mathbf{w}(k), \\ \mathbf{z}(k) = (\mathbf{C} - D \Lambda^{-1} \mathbf{\varPi}^T - \mathbf{a} D \boldsymbol{\varPhi}(k) \Lambda^{-1} \mathbf{\varPi}^T) \mathbf{x}(k) \end{array} \tag{31}$$

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

*Q*1*ST*(*A* − *B*2Λ−1Ψ*T*)+(*A* − *B*2Λ−1Ψ*T*)*TSQT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup> <sup>c</sup> Gc* + *<sup>ε</sup>*2ΨΛ−1Λ−1Ψ*<sup>T</sup>* = *Q*1*S<sup>T</sup> A* − *Q*1*STB*2Λ−1Ψ*<sup>T</sup>* + *ATSQT* <sup>1</sup> <sup>−</sup> ΨΛ−1*B<sup>T</sup>* <sup>2</sup> *SQ<sup>T</sup>* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* +(*ε*<sup>1</sup> + *ε*2)ΨΛ−1Λ−1Ψ*T*, (*C* − *D*Λ−1Ψ*T*)*TZ*<sup>−</sup>1(*C* − *D*Λ−1Ψ*T*) = *CTZ*<sup>−</sup>1*C* − ΨΛ−1*DZ*−1*C* − *CTZ*<sup>−</sup>1*D*Λ−1Ψ*<sup>T</sup>* + ΨΛ−1*DZ*−1*D*Λ−1Ψ*T*, (*A* − *B*2Λ−1Ψ*<sup>T</sup>* + *ε* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> )*T*Γ−1(*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* 1 ) = Θ*<sup>T</sup> <sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>−</sup> ΨΛ−1*B<sup>T</sup>* <sup>2</sup> <sup>Γ</sup>−1Θ*<sup>A</sup>* <sup>−</sup> <sup>Θ</sup>*<sup>T</sup> <sup>A</sup>*Γ−1*B*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> ΨΛ−1*B<sup>T</sup>* <sup>2</sup> <sup>Γ</sup>−1*B*2Λ−1Ψ*T*, [(*A* − *B*2Λ−1Ψ*T*)*TSQT* <sup>2</sup> + (*Q*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*B*]*W*−<sup>1</sup> ×[(*A* − *B*2Λ−1Ψ*T*)*TSQT* <sup>2</sup> + (*Q*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*B*)]*<sup>T</sup>* = *YW*−1*Y<sup>T</sup>* − ΨΛ−1*B<sup>T</sup>* <sup>2</sup> (*SQ<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*B*)*TW*<sup>−</sup>1*Y<sup>T</sup>* <sup>−</sup> *YW*−1(*SQ<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*B*)*TB*2Λ−1Ψ*<sup>T</sup>* +ΨΛ−1*B<sup>T</sup>* <sup>2</sup> (*SQ<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*B*)*W*−1(*SQ<sup>T</sup>* <sup>2</sup> <sup>+</sup> <sup>Γ</sup>−1Θ*B*)*TB*2Λ−1Ψ*T*.

Thus, it can be verified with (29) that

*Q*1*ST*(*A* − *B*2Λ−1Ψ*T*)+(*A* − *B*2Λ−1Ψ*T*)*TSQT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup> <sup>c</sup> Gc* + *<sup>ε</sup>*2ΨΛ−1Λ−1Ψ*<sup>T</sup>* +(*C* − *D*Λ−1Ψ*T*)*TZ*<sup>−</sup>1(*C* − *D*Λ−1Ψ*T*) − *ETPE* + *ε* −1 <sup>1</sup> *<sup>Q</sup>*1*STHcHT <sup>c</sup> SQ<sup>T</sup>* 1 +(*A* − *B*2Λ−1Ψ*<sup>T</sup>* + *ε* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> )*T*Γ−1(*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* 1 ) +[(*A* − *B*2Λ−1Ψ*T*)*TSQT* <sup>2</sup> + (*Q*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*B*]*W*−<sup>1</sup> ×[(*A* − *B*2Λ−1Ψ*T*)*TSQT* <sup>2</sup> + (*Q*1*S<sup>T</sup>* + (*<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *HcH<sup>T</sup> <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> )*T*Γ−1)Θ*B*)]*<sup>T</sup>* = *Q*1*S<sup>T</sup> A* + *ATSQT* <sup>1</sup> <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>ε</sup>* −1 <sup>1</sup> *<sup>Q</sup>*1*STHcHT <sup>c</sup> SQ<sup>T</sup>* <sup>1</sup> <sup>+</sup> *<sup>C</sup>TZ*<sup>−</sup>1*<sup>C</sup>* <sup>+</sup> <sup>Θ</sup>*<sup>T</sup> <sup>A</sup>*Γ−1Θ*<sup>A</sup>* +*YW*−1*Y<sup>T</sup>* − ΨΛ−1Ψ*<sup>T</sup>* < 0.

By Schur complement formula, we obtain

$$
\begin{bmatrix}
\begin{pmatrix}
Q\_{1}S^{T}A\_{K} + A\_{K}^{T}SQ\_{1}^{T} \\
+\epsilon\_{2}\Psi\Lambda^{-1}\Lambda^{-1}\Psi^{T}
\end{pmatrix} & A\_{K}^{T}SQ\_{2}^{T} + Q\_{1}S^{T}B\_{1} & A\_{K}^{T}P & C\_{K}^{T}Q\_{1}S^{T}H\_{c} & 0 \\
+Q\_{2}S^{T}A\_{K} + B\_{1}^{T}SQ\_{1}^{T} & \begin{pmatrix}
+B\_{1}^{T}SQ\_{2}^{T}
\end{pmatrix} & B\_{1}^{T}P & 0 & Q\_{2}S^{T}H\_{c} & 0 \\
P A\_{K} & P B\_{1} & -P & 0 & PH\_{c} & 0 \\
A\_{K} & 0 & 0 & -I & 0 & aD \\
H\_{c}^{T}SQ\_{1}^{T} & H\_{c}^{T}SQ\_{2}^{T} & H\_{c}^{T}P & 0 & -\varepsilon\_{1}I & 0 \\
0 & 0 & 0 & aD^{T} & 0 & -\varepsilon\_{2}I
\end{bmatrix} & <0.
$$

**Theorem 4.5.** *There exists a controller (4) that makes the descriptor system (16) robustly admissible with H*<sup>∞</sup> *disturbance attenuation γ if there exist matrix P and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that (21),*

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

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

= *HF*(*k*)

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

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

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

 *G Gd* 

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*).

*(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}

*(iv) The triplet* (*E*, *A*, *Ad*) *is said to be admissible if it is regular, causal and stable.*

*Ex*(*<sup>k</sup>* + <sup>1</sup>)=(*<sup>A</sup>* + <sup>∆</sup>*A*)*x*(*k*)+(*Ad* + <sup>∆</sup>*Ad*)*x*(*<sup>k</sup>* − *<sup>d</sup>*) + *Bu*(*k*) (34)

−*ETPE* + *<sup>ε</sup>*1*GTG* + *<sup>C</sup>TZ*<sup>−</sup>1*<sup>C</sup>* + *<sup>A</sup>T*Γ−1*B*1*W*−1*B<sup>T</sup>*

*In this case, a feedback gain in the controller (4) is given as in (30).*

**5.1. Robust admissibility for uncertain systems**

Λ = *B<sup>T</sup>*

non-fragile control design methods.

**5. Time-delay systems**

Ψ = *AT*Γ−1*B*<sup>2</sup> + *CTZ*<sup>−</sup>1*D* + *AT*Γ−1*B*1*W*−1*B<sup>T</sup>*

<sup>2</sup> <sup>Γ</sup>−1*B*<sup>2</sup> <sup>+</sup> *<sup>D</sup>TZ*<sup>−</sup>1*<sup>D</sup>* + (*ε*<sup>1</sup> <sup>+</sup> *<sup>ε</sup>*2)*<sup>I</sup>* <sup>+</sup> *<sup>B</sup><sup>T</sup>*

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

 <sup>∆</sup>*<sup>A</sup>* <sup>∆</sup>*Ad*

*triplet* (*E*, *A*, *Ad*) *is said to be stable if ρ*(*E*, *A*, *Ad*) < 1*.*

*W* = *γ*<sup>2</sup> *I* − *B<sup>T</sup>*

<sup>1</sup> <sup>Γ</sup>−1*B*2,

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

<sup>2</sup> <sup>Γ</sup>−1*B*1*W*−1*B<sup>T</sup>*

<sup>1</sup> <sup>Γ</sup>−1*<sup>A</sup>* <sup>+</sup> *<sup>A</sup>T*Γ−1*<sup>A</sup>* <sup>−</sup> ΨΛ−1Ψ*<sup>T</sup>* <sup>&</sup>lt; 0 (33)

<sup>1</sup> <sup>Γ</sup>−1*B*2.

<sup>1</sup> <sup>Γ</sup>−1*B*<sup>1</sup> <sup>&</sup>gt; 0, (32)

43

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


*(26), (27) and*

*where*

are given by

*identically zero.*

*nd* + *rank*(*E*)*.*

where *AK* <sup>=</sup> *<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* and *CK* <sup>=</sup> *<sup>C</sup>* <sup>−</sup> *<sup>D</sup>*Λ−1Ψ*T*. Hence, by Theorem 4.2 we can show that the closed-loop system (17) is robustly admissible with H∞ disturbance attenuation *γ*. (Necessity) Assume there exists a feedback control of the form (4) which makes the descriptor system (16) robustly admissible with H∞ disturbance attenuation *γ*. Then, it follows from Theorem 4.2 that there exist scalars *ε*<sup>1</sup> > 0 and *ε*<sup>2</sup> > 0 such that

$$
\begin{bmatrix}
\Pi\_{11} & A\_K^T S Q\_2^T + Q\_1 S^T B\_1 & A\_K^T P \ (\mathbb{C} + D\mathbb{K})^T & Q\_1 S^T H\_\mathbb{C} & 0 \\
Q\_2 S^T A\_K + B\_1^T S Q^T & \Pi\_{22} & B\_1^T P & 0 & Q\_2 S^T H\_\mathbb{C} & 0 \\
P A\_K & P B\_1 & -P & 0 & P H\_\mathbb{C} & 0 \\
\mathbb{C} + D\mathbb{K} & 0 & 0 & -I & 0 & aD \\
H\_\mathbb{C}^T S Q\_1^T & H\_\mathbb{C}^T S Q\_2^T & H\_\mathbb{C}^T P & 0 & -\varepsilon\_1 I & 0 \\
0 & 0 & 0 & aD^T & 0 & -\varepsilon\_2 I
\end{bmatrix} < 0
$$

where

$$\begin{array}{l} \Pi\_{11} = Q\_1 S^T A\_K + A\_K^T S Q\_1^T - E^T P E + \varepsilon\_1 \mathbf{G}\_c^T \mathbf{G}\_c + \varepsilon\_2 \mathbf{K}^T \mathbf{K}\_c \\\Pi\_{22} = -\gamma^2 I + Q\_2 \mathbf{S}^T B\_1 + \mathbf{B}\_1^T \mathbf{S} \mathbf{Q}\_2^T. \end{array}$$

It follows from Schur complement formula that we obtain the conditions (26), (27), (28), and

$$\begin{cases} Q\_1 \mathbf{S}^T \mathbf{A} + \mathbf{A}^T \mathbf{S} \mathbf{Q}\_1^T - \mathbf{E}^T \mathbf{P} \mathbf{E} + \varepsilon\_1 \mathbf{G}^T \mathbf{G} + \varepsilon\_1^{-1} \mathbf{Q} \mathbf{S}^T \mathbf{H}\_\varepsilon \mathbf{H}\_\varepsilon^T \mathbf{S} \mathbf{Q}^T + \mathbf{C}^T \mathbf{Z}^{-1} \mathbf{C} + \mathbf{Y} \mathbf{W}^{-1} \mathbf{Y}^T \\ + \mathbf{S}\_A^T \mathbf{I}^{-1} \boldsymbol{\Theta}\_A + \mathbf{K}^T \mathbf{Y}^T + \mathbf{Y} \mathbf{K} + \mathbf{K}^T \boldsymbol{\Lambda} \mathbf{K} < 0. \end{cases}$$

By Lemma 3.4, we have

$$K^T \Psi^T + \Psi K + K^T \Lambda K \ge -\Psi \Lambda^{-1} \Psi^T.$$

Therefore, we finally obtain the condition (29), and the feedback gain *K* as in (30).

Based on Theorem 4.3, we can deduce the following theorem.

**Theorem 4.5.** *There exists a controller (4) that makes the descriptor system (16) robustly admissible with H*<sup>∞</sup> *disturbance attenuation γ if there exist matrix P and scalars ε*<sup>1</sup> > 0, *ε*<sup>2</sup> > 0 *such that (21), (26), (27) and*

$$\begin{aligned} W &= \gamma^2 I - B\_1^T \Gamma^{-1} B\_1 > 0, \quad \text{(32)}\\ -E^T P E + \varepsilon\_1 G^T G + \mathcal{C}^T Z^{-1} \mathcal{C} + A^T \Gamma^{-1} B\_1 W^{-1} B\_1^T \Gamma^{-1} A + A^T \Gamma^{-1} A - \Psi \Lambda^{-1} \Psi^T &< 0 \quad \text{(33)} \end{aligned}$$

*where*

14 Advances in Discrete Time Systems

 

 

where

*<sup>Q</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>B</sup><sup>T</sup>*

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

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

+Θ*<sup>T</sup>*

By Lemma 3.4, we have

 

By Schur complement formula, we obtain

*<sup>K</sup>SQ<sup>T</sup>* 1

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

�

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

Theorem 4.2 that there exist scalars *ε*<sup>1</sup> > 0 and *ε*<sup>2</sup> > 0 such that

*<sup>K</sup>SQ<sup>T</sup>*

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

<sup>Π</sup><sup>11</sup> <sup>=</sup> *<sup>Q</sup>*1*S<sup>T</sup> AK* <sup>+</sup> *<sup>A</sup><sup>T</sup>*

*<sup>A</sup>*Γ−1Θ*<sup>A</sup>* <sup>+</sup> *<sup>K</sup>T*Ψ*<sup>T</sup>* <sup>+</sup> <sup>Ψ</sup>*<sup>K</sup>* <sup>+</sup> *<sup>K</sup>T*Λ*<sup>K</sup>* <sup>&</sup>lt; 0.

Based on Theorem 4.3, we can deduce the following theorem.

<sup>Π</sup><sup>22</sup> <sup>=</sup> <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>+</sup> *<sup>Q</sup>*2*STB*<sup>1</sup> <sup>+</sup> *<sup>B</sup><sup>T</sup>*

<sup>1</sup> <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>ε</sup>*1*GTG* <sup>+</sup> *<sup>ε</sup>*

<sup>1</sup> *SQ<sup>T</sup>* <sup>Π</sup><sup>22</sup> *<sup>B</sup><sup>T</sup>*

*<sup>c</sup> SQ<sup>T</sup>*

<sup>Π</sup><sup>11</sup> *<sup>A</sup><sup>T</sup>*

*<sup>K</sup>SQ<sup>T</sup>*

<sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>* <sup>+</sup> *<sup>Q</sup>*2*STB*<sup>1</sup> +*B<sup>T</sup>* <sup>1</sup> *SQ<sup>T</sup>* 2

*<sup>c</sup> SQ<sup>T</sup>*

<sup>2</sup> <sup>+</sup> *<sup>Q</sup>*1*STB*<sup>1</sup> *<sup>A</sup><sup>T</sup>*

*<sup>K</sup>SQ<sup>T</sup>*

*PAK PB*<sup>1</sup> −*P* 0 *PHc* 0 *C* + *DK* 0 0 −*I* 0 *αD*

0 00 *<sup>α</sup>D<sup>T</sup>* <sup>0</sup> <sup>−</sup>*ε*<sup>2</sup> *<sup>I</sup>*

<sup>1</sup> <sup>−</sup> *<sup>E</sup>TPE* <sup>+</sup> *<sup>ε</sup>*1*G<sup>T</sup>*

<sup>1</sup> *QSTHcHT*

<sup>1</sup> *SQ<sup>T</sup>* 2 .

It follows from Schur complement formula that we obtain the conditions (26), (27), (28), and

−1

*<sup>K</sup>T*Ψ*<sup>T</sup>* + <sup>Ψ</sup>*<sup>K</sup>* + *<sup>K</sup>T*Λ*<sup>K</sup>* ≥ −ΨΛ−1Ψ*T*.

Therefore, we finally obtain the condition (29), and the feedback gain *K* as in (30).

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

where *AK* <sup>=</sup> *<sup>A</sup>* <sup>−</sup> *<sup>B</sup>*2Λ−1Ψ*<sup>T</sup>* and *CK* <sup>=</sup> *<sup>C</sup>* <sup>−</sup> *<sup>D</sup>*Λ−1Ψ*T*. Hence, by Theorem 4.2 we can show that the closed-loop system (17) is robustly admissible with H∞ disturbance attenuation *γ*. (Necessity) Assume there exists a feedback control of the form (4) which makes the descriptor system (16) robustly admissible with H∞ disturbance attenuation *γ*. Then, it follows from

<sup>2</sup> <sup>+</sup> *<sup>Q</sup>*1*STB*<sup>1</sup> *<sup>A</sup><sup>T</sup>*

*PAK PB*<sup>1</sup> −*P* 0 *PHc* 0 *AK* 0 0 −*I* 0 *αD*

� *BT*

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

0 00 *<sup>α</sup>D<sup>T</sup>* <sup>0</sup> <sup>−</sup>*ε*<sup>2</sup> *<sup>I</sup>*

*<sup>K</sup>P C<sup>T</sup>*

*<sup>K</sup> <sup>Q</sup>*1*STHc* <sup>0</sup>

 

< 0.

 

< 0

<sup>1</sup> *<sup>P</sup>* <sup>0</sup> *<sup>Q</sup>*2*STHc* <sup>0</sup>

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

*<sup>K</sup><sup>P</sup>* (*<sup>C</sup>* <sup>+</sup> *DK*)*<sup>T</sup> <sup>Q</sup>*1*STHc* <sup>0</sup>

<sup>1</sup> *<sup>P</sup>* <sup>0</sup> *<sup>Q</sup>*2*STHc* <sup>0</sup>

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

*<sup>c</sup> Gc* <sup>+</sup> *<sup>ε</sup>*2*KTK*,

*<sup>c</sup> SQ<sup>T</sup>* + *CTZ*−1*C* + *YW*−1*Y<sup>T</sup>*

*<sup>c</sup> Gc*

<sup>1</sup> *SQ<sup>T</sup>* 1

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

*<sup>Q</sup>*2*S<sup>T</sup> AK* <sup>+</sup> *<sup>B</sup><sup>T</sup>*

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

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

<sup>+</sup>*ε*2ΨΛ−1Λ−1Ψ*<sup>T</sup>*

$$\begin{cases} \boldsymbol{\Psi} = \boldsymbol{A}^T \boldsymbol{\Gamma}^{-1} \boldsymbol{B}\_2 + \boldsymbol{\mathcal{C}}^T \boldsymbol{Z}^{-1} \boldsymbol{D} + \boldsymbol{A}^T \boldsymbol{\Gamma}^{-1} \boldsymbol{B}\_1 \boldsymbol{W}^{-1} \boldsymbol{B}\_1^T \boldsymbol{\Gamma}^{-1} \boldsymbol{B}\_2, \\ \boldsymbol{\Lambda} = \boldsymbol{B}\_2^T \boldsymbol{\Gamma}^{-1} \boldsymbol{B}\_2 + \boldsymbol{D}^T \boldsymbol{Z}^{-1} \boldsymbol{D} + (\boldsymbol{\varepsilon}\_1 + \boldsymbol{\varepsilon}\_2) \boldsymbol{I} + \boldsymbol{B}\_2^T \boldsymbol{\Gamma}^{-1} \boldsymbol{B}\_1 \boldsymbol{W}^{-1} \boldsymbol{B}\_1^T \boldsymbol{\Gamma}^{-1} \boldsymbol{B}\_2. \end{cases}$$

*In this case, a feedback gain in the controller (4) is given as in (30).*
