# Adaptive State-Quantized Control of Uncertain Lower-Triangular Nonlinear Systems with Input Delay

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Assumption**

**1.**

## 3. Adaptive State-Quantized Tracking Control in the Presence of Input Delay

**Remark**

**1.**

**Step 1:**Using (1) and (4), the dynamics of ${s}_{1}$ is defined as ${\dot{s}}_{1}={x}_{2}+{g}_{1}+{\delta}_{1}-{\dot{y}}_{r}$. By considering a Lyapunov function ${V}_{1}=(1/2){s}_{1}^{2}$ and using (3), we have

**Step i $(i=2,\dots ,n-1)$:**Consider the Lyapunov function ${V}_{i}=(1/2){s}_{i}^{2}$. Using (3)–(5), the time derivative of ${V}_{i}$ is represented by

**Step n:**A Lyapunov function is defined as ${V}_{n}=(1/2){s}_{n}^{2}$. Using (3)–(5), the time derivative of ${V}_{n}$ is represented by

**Remark**

**2.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

## 4. Simulation

#### 4.1. Example 1

#### 4.2. Example 2

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Estimation results of the proposed control system for Example 1 (

**a**) ${\widehat{\gamma}}_{j}$, $j=2,3$; and (

**b**) $\parallel {\widehat{\Theta}}_{j}\parallel $, $j=2,3$.

**Figure 3.**Input delay compensator and control input of the proposed control system for Example 1: (

**a**) $\rho $; and (

**b**) u.

**Figure 5.**Estimation results of the proposed control system for Example 2: (

**a**) ${\widehat{\gamma}}_{j}$, $j=1,2,3$; and (

**b**) $\parallel {\widehat{\Theta}}_{j}\parallel $, $j=1,2,3$.

**Figure 6.**Input delay compensator and control input of the proposed control system for Example 2: (

**a**) $\rho $; and (

**b**) u.

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Yoo, S.J.
Adaptive State-Quantized Control of Uncertain Lower-Triangular Nonlinear Systems with Input Delay. *Mathematics* **2021**, *9*, 763.
https://doi.org/10.3390/math9070763

**AMA Style**

Yoo SJ.
Adaptive State-Quantized Control of Uncertain Lower-Triangular Nonlinear Systems with Input Delay. *Mathematics*. 2021; 9(7):763.
https://doi.org/10.3390/math9070763

**Chicago/Turabian Style**

Yoo, Sung Jin.
2021. "Adaptive State-Quantized Control of Uncertain Lower-Triangular Nonlinear Systems with Input Delay" *Mathematics* 9, no. 7: 763.
https://doi.org/10.3390/math9070763