#
Discrete Output Regulator Design for the Linearized **Saint–Venant–Exner** Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Assumption**

**1.**

**Remark**

**1.**

- The closed-loop system is exponentially stable;
- For the closed-loop system, the tracking error $e(t\to \infty )=0,\forall x\left(0\right)\in X,z\left(0\right)\in {\Re}^{e}$;

#### 2.1. System Properties

#### 2.1.1. Linearized System Stability

**Lemma**

**1.**

**Proof.**

#### 2.1.2. Resolvent and Transfer Function

**Lemma**

**2.**

**Proof.**

## 3. Continuous Time Regulator Design

#### 3.1. System Stabilization

**Remark**

**2.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

#### 3.2. Output Regulation

**Lemma**

**5.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**6.**

**Proof.**

**Remark**

**5.**

#### 3.3. System Observer Design

**Lemma**

**7.**

**Proof.**

#### 3.4. Exosystem Observer

#### Exosystem Finite-Time Observer

## 4. Discrete Time Regulator Design

- The closed-loop system is stable;
- For the closed-loop system, the tracking error ${e}_{k}(k\to \infty )=0,\forall {x}_{k=0}\in X,{z}_{k=0}\in \mathbb{R}$;

#### 4.1. Discrete Representation

**Assumption**

**2.**

**Lemma**

**8.**

**Proof.**

#### 4.2. Discrete System Stabilization

**Lemma**

**9.**

**Proof.**

#### 4.3. Discrete Output Regulation

#### 4.3.1. Discrete Regulator Equations

**Lemma**

**10.**

**Proof.**

#### 4.4. Discrete System Observer Design

**Lemma**

**11.**

**Proof.**

#### 4.5. Discrete Exosystem Observer Design

**Lemma**

**12.**

**Proof.**

#### 4.6. Finite-Time Discrete Exosystem Observer Design

## 5. Results

#### 5.1. Continuous Time Regulation

#### 5.2. Discrete Time Regulation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. System Stabilization: Backstepping

## Appendix B. Observer Design: Backstepping

## Appendix C. Discrete System Observer Stability

## Appendix D. Discrete Exosystem Observer Stability

**Lemma**

**A1.**

**Proof.**

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**Figure 1.**PDE system (Equation (2)) representation.

**Figure 2.**System eigenvalue distribution for the parameters given in Table 1.

**Figure 5.**System stabilization with the control law given in Equation (22).

**Figure 6.**System stabilization and output regulation with the control law given in Equation (10).

**Figure 7.**System observer error, using the observer developed in Section 3.3.

**Figure 8.**System closed-loop response, using the observer developed in Section 3.3 and the control law given in Equation (10).

**Figure 9.**System closed-loop response (on the left), using the system and exosystem observers and the control law given in Equation (10). On the right, the exosystem states and its observer states.

**Figure 10.**System closed-loop response (on the left), using the system observer, finite-time exosystem observer and the control law given in Equation (10). On the right, the exosystem states and its observer states.

**Figure 11.**Discrete system stabilization with the control law given in Equation (53), considering ${z}_{k}=0$.

**Figure 12.**Discrete system stabilization and output regulation with the control law given in Equation (53), assuming that ${z}_{k}$ are known states.

**Figure 13.**Discrete system closed-loop response, using the observer developed in Section 4.4 and the control law given in Equation (53).

**Figure 14.**Discrete system closed-loop response (on the left), using the system and exosystem observers and the control law given in Equation (53). On the right, the discrete exosystem states and its observer states.

**Figure 15.**Discrete system closed-loop response (on the left), using the observability matrix to reconstruct the exosystem states. On the right, the discrete exosystem states and the observer states.

Parameter | Value | Parameter | Value |
---|---|---|---|

${q}_{1}={q}_{2}$ | 1 | ${\rho}_{1}={\rho}_{2}$ | $0.5$ |

${\sigma}_{11}={\sigma}_{21}={\beta}_{1}$ | $0.2$ | ${\sigma}_{12}={\sigma}_{22}={\beta}_{2}$ | $0.05$ |

${\alpha}_{1}$ | $0.1$ | $\mu $ | 2 |

${\gamma}_{1}$ | $0.5$ | ${\gamma}_{2}$ | 1 |

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**MDPI and ACS Style**

Cassol, G.O.; Dubljevic, S. Discrete Output Regulator Design for the Linearized * Saint–Venant–Exner* Model.

*Processes*

**2020**,

*8*, 915. https://doi.org/10.3390/pr8080915

**AMA Style**

Cassol GO, Dubljevic S. Discrete Output Regulator Design for the Linearized * Saint–Venant–Exner* Model.

*Processes*. 2020; 8(8):915. https://doi.org/10.3390/pr8080915

**Chicago/Turabian Style**

Cassol, Guilherme Ozorio, and Stevan Dubljevic. 2020. "Discrete Output Regulator Design for the Linearized * Saint–Venant–Exner* Model"

*Processes*8, no. 8: 915. https://doi.org/10.3390/pr8080915