# A Review of Convex Approaches for Control, Observation and Safety of Linear Parameter Varying and Takagi-Sugeno Systems

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## Abstract

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## 1. Introduction

**Notation**: The notation used in this article is quite standard. ${\mathbb{R}}^{m\times n}$ denotes the set of all matrices with m rows and n columns. If a square matrix $A\in {\mathbb{R}}^{n\times n}$ is symmetric, this fact will be denoted by $A\in {\mathbb{S}}^{n}$. Given a matrix $A\in {\mathbb{S}}^{n}$, $A\succ 0$ ($A\prec 0$) denotes positive (negative) definiteness, that is, that all its eigenvalues are positive (negative). Similarly, $A\u2ab00$ ($A\u2aaf0$) denotes positive (negative) semi-definiteness. For a matrix $A\in {\mathbb{R}}^{m\times n}$, ${A}^{T}$ and ${A}^{\u2020}$ denote its transpose and pseudo-inverse, respectively. If $A\in {\mathbb{R}}^{n\times n}$ is non-singular, ${A}^{-1}$ will denote its inverse. The symbol * denotes the transposed element in a symmetric position of a matrix. Finally, $\mathrm{He}\left\{A\right\}$ is used as a shorthand notation for $A+{A}^{T}$.

## 2. Control of Convex Systems

#### 2.1. Convex State-Feedback Control

#### 2.2. Convex Output-Feedback Control

#### 2.3. Convex Tracking Controller

#### 2.4. Model Predictive Control for Convex Systems

#### 2.5. Final Comments on Control of Convex Systems

## 3. Observation of Convex Systems

#### 3.1. Convex Observers

#### 3.2. Robust Observers

#### 3.3. Proportional-Integral Observers

#### 3.4. Descriptor Observers

## 4. Safety in Convex Systems

#### 4.1. Residual Generation for Fault Detection

#### 4.2. Unknown Input Observers-Based Fault Isolation

#### 4.3. Observer-Based Fault Estimation

#### 4.4. Multiple Model Adaptive Estimators

#### 4.5. Sliding Mode Fault Tolerant Control

#### 4.6. Fault Tolerant Control Based on Controller Reconfiguration

#### 4.7. Fault-Hiding via Virtual Actuators and Virtual Sensors

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

López-Estrada, F.-R.; Rotondo, D.; Valencia-Palomo, G.
A Review of *Convex* Approaches for Control, Observation and Safety of Linear Parameter Varying and Takagi-Sugeno Systems. *Processes* **2019**, *7*, 814.
https://doi.org/10.3390/pr7110814

**AMA Style**

López-Estrada F-R, Rotondo D, Valencia-Palomo G.
A Review of *Convex* Approaches for Control, Observation and Safety of Linear Parameter Varying and Takagi-Sugeno Systems. *Processes*. 2019; 7(11):814.
https://doi.org/10.3390/pr7110814

**Chicago/Turabian Style**

López-Estrada, Francisco-Ronay, Damiano Rotondo, and Guillermo Valencia-Palomo.
2019. "A Review of *Convex* Approaches for Control, Observation and Safety of Linear Parameter Varying and Takagi-Sugeno Systems" *Processes* 7, no. 11: 814.
https://doi.org/10.3390/pr7110814