# Design of Finite Time Reduced Order H∞ Controller for Linear Discrete Time Systems

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problem Definition

^{T}is the matrix transposition. Then, consider the following linear time-invariant system:

**Assumption 1.**

**Definition 1**

**Definition 2**

**Definition 3**

**Lemma**

**1.**

**Proof.**

**Remark 1.**

**Remark 2.**

## 3. Main Results

#### 3.1. FTB-H∞ Filter-Based Controller Synthesis

**Theorem 1.**

**Proof.**

**Remark 3.**

- The filter is unbiased: the error $e\left(k\right)$ does not depend explicitly on $x\left(k\right)$ and $u\left(k\right)$ if $w\left(k\right)=0$;
- The effect of disturbances on controlled output is minimized if $w\left(k\right)\ne 0$;
- The FTB-H∞ of the closed-loop system is guaranteed.

**Remark 4.**

**Remark 5.**

**Remark 6.**

- If we choose the condition (41), the Equation (40) becomes:$${L}_{f}{D}_{y}=\left({E}_{22}+Z{F}_{22}+\left({E}_{11}+Z{F}_{11}\right){E}_{33}\right){D}_{y}$$
- Now, either we use the condition (42) or the condition (43). Choosing the last condition, the Equation (40) becomes:$${L}_{f}{D}_{y}=\left({E}_{22}+Z{F}_{22}+{E}_{11}\left({E}_{33}+Z{F}_{33}\right)\right){D}_{y}$$

**Remark 7.**

#### 3.2. LMI Synthesis Conditions

**Theorem 2.**

**Proof.**

**Remark 8.**

## 4. Design Algorithm

- Set appropriate values for the parameters ${c}_{2},N,R,d,\gamma ,$ and α where $\alpha \ge 1$.
- Solve the matrices (12) and (13), and deduce $X$, $Y$, and ${K}_{u}=Y{X}^{-1}$.
- If these results are derived, then solve the matrices (60) and (61) for the given values of parameters ${c}_{2},N,R,d,\gamma ,$and $\mathsf{\alpha}$.
- Next, if the result is feasible, go then to Step 5; otherwise go back to Step 2.
- Finally, compute $\overline{Z}={Q}^{-1}Y$.
- Furthermore, the parameters ${A}_{f},G,{L}_{f},$and$K$ are computed using (34)–(39), (50)–(52), (53)–(55), (24), and (20).

**Remark 9.**

## 5. Numerical Examples

**Example 1.**

**Example 2.**

**Example 3.**

**Remark 10.**

**Remark 11.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Taoussi, M.; El Akchioui, N.; Bardane, A.; El Fezazi, N.; Farkous, R.; Tissir, E.H.; Al-Arydah, M.
Design of Finite Time Reduced Order H∞ Controller for Linear Discrete Time Systems. *Mathematics* **2023**, *11*, 31.
https://doi.org/10.3390/math11010031

**AMA Style**

Taoussi M, El Akchioui N, Bardane A, El Fezazi N, Farkous R, Tissir EH, Al-Arydah M.
Design of Finite Time Reduced Order H∞ Controller for Linear Discrete Time Systems. *Mathematics*. 2023; 11(1):31.
https://doi.org/10.3390/math11010031

**Chicago/Turabian Style**

Taoussi, Mohammed, Nabil El Akchioui, Adil Bardane, Nabil El Fezazi, Rashid Farkous, El Houssaine Tissir, and Mo’tassem Al-Arydah.
2023. "Design of Finite Time Reduced Order H∞ Controller for Linear Discrete Time Systems" *Mathematics* 11, no. 1: 31.
https://doi.org/10.3390/math11010031