# Output Feedback Tracking Sliding Mode Control for Systems with State- and Input-Dependent Disturbances

## Abstract

**:**

## 1. Introduction

## 2. State Feedback Sliding Mode Control

**Remark**

**1.**

**Remark**

**2.**

## 3. Robust Loop Transfer Recovery Observer

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 4. Output Feedback Sliding Mode Control

**Lemma**

**2.**

**(i)**- $\parallel {\tilde{x}}_{e}\parallel \le \u03f5\left[\right(\overline{\kappa}\parallel K\parallel +\overline{\theta})\parallel {\widehat{x}}_{e}\parallel +\overline{\kappa}\rho \left({\widehat{x}}_{e}\right)+\overline{d}]/(1-\u03f5\overline{\theta})$.
**(ii)**- $\parallel K{\tilde{x}}_{e}+e\parallel \le \beta \rho \left({\widehat{x}}_{e}\right)$.
**(iii)**- $\parallel {\tilde{x}}_{e}\parallel \le \u03f5({f}_{1}\parallel {x}_{e}\parallel +{f}_{0})$ for some positive constants ${f}_{1}$ and ${f}_{0}$.
**(iv)**- $\parallel \rho \left({\widehat{x}}_{e}\right)\parallel \le {g}_{1}\parallel {x}_{e}\parallel +{g}_{0}$ for some positive constants ${g}_{1}$ and ${g}_{0}$,

**Proof.**

- (ii)
- With the upper bound in (i), an upper bound for $K{\tilde{x}}_{e}+e$ can be derived,$$\begin{array}{cc}\hfill \parallel K{\tilde{x}}_{e}& +e\parallel \le \parallel K\parallel \parallel {\tilde{x}}_{e}\parallel +\parallel e\parallel \hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \u03f5\parallel K\parallel (\overline{\kappa}\parallel u\parallel +\overline{\theta}\parallel {x}_{e}\parallel +\overline{d})+(\overline{\kappa}\parallel u\parallel +\overline{\theta}\parallel {x}_{e}\parallel +\overline{d})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =(1+\u03f5\parallel K\parallel )(\overline{\kappa}\parallel u\parallel +\overline{\theta}\parallel {x}_{e}\parallel +\overline{d})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le (1+\u03f5\parallel K\parallel )[\overline{\kappa}(\parallel K\parallel +{c}_{1})\parallel {\widehat{x}}_{e}\parallel +\overline{\theta}\parallel {x}_{e}\parallel +(\overline{\kappa}{c}_{0}+\overline{d})]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le (1+\u03f5\parallel K\parallel )\left[\right(\overline{\kappa}\parallel K\parallel +\overline{\theta})\parallel {\widehat{x}}_{e}\parallel +\overline{\theta}\parallel {\tilde{x}}_{e}\parallel +\overline{\kappa}\rho \left({\widehat{x}}_{e}\right)+\overline{d}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \frac{1+\u03f5\parallel K\parallel}{1-\u03f5\overline{\theta}}\left[\right(\overline{\theta}+\overline{\kappa}\parallel K\parallel )\parallel {\widehat{x}}_{e}\parallel +\overline{\kappa}\rho \left({\widehat{x}}_{e}\right)+\overline{d}]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \frac{1}{\gamma}\left[\right(\overline{\theta}+\overline{\kappa}\parallel K\parallel )\parallel {\widehat{x}}_{e}\parallel +\overline{\kappa}\rho \left({\widehat{x}}_{e}\right)+\overline{d}]\le \beta \rho \left({\widehat{x}}_{e}\right),\hfill \end{array}$$
- (iii)
- In this section, an alternative upper bound is presented for the estimation error ${\tilde{x}}_{e}$:$$\begin{array}{cc}\hfill \parallel {\tilde{x}}_{e}\parallel & \le \u03f5\overline{\kappa}\parallel u\parallel +\u03f5\overline{\theta}\parallel {x}_{e}\parallel +\u03f5\overline{d}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \u03f5\overline{\kappa}(\parallel K\parallel +{c}_{1})\parallel {\widehat{x}}_{e}\parallel +\u03f5\overline{\theta}\parallel {x}_{e}\parallel +\u03f5(\overline{\kappa}{c}_{0}+\overline{d})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \u03f5\overline{\kappa}(\parallel K\parallel +{c}_{1})\parallel {\tilde{x}}_{e}\parallel +(\u03f5\overline{\theta}+\u03f5\overline{\kappa}(\parallel K\parallel +{c}_{1}\left)\right)\parallel {x}_{e}\parallel +\u03f5(\overline{\kappa}{c}_{0}+\overline{d})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \u03f5({f}_{1}\parallel {x}_{e}\parallel +{f}_{0}),\hfill \end{array}$$$$\begin{array}{c}\hfill {f}_{1}=\frac{\overline{\theta}+\overline{\kappa}\parallel K\parallel +\overline{\kappa}{c}_{1}}{1-\u03f5(\overline{\kappa}\parallel K\parallel +\overline{\kappa}{c}_{1})},\phantom{\rule{1.em}{0ex}}{f}_{0}=\frac{\overline{\kappa}{c}_{0}+\overline{d}}{1-\u03f5(\overline{\kappa}\parallel K\parallel +\overline{\kappa}{c}_{1})}.\end{array}$$

- (iv)
- According to (22),$$\begin{array}{cc}\hfill \rho \left({\widehat{x}}_{e}\right)& ={c}_{1}\parallel {\widehat{x}}_{e}\parallel +{c}_{0}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le {c}_{1}\parallel {x}_{e}\parallel +{c}_{1}\parallel {\tilde{x}}_{e}\parallel +{c}_{0}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le ({c}_{1}+\u03f5{c}_{1}{f}_{1})\parallel {x}_{e}\parallel +{c}_{0}+\u03f5{c}_{1}{f}_{0},\hfill \end{array}$$

**Theorem**

**2.**

**Proof.**

## 5. Application Example

**Remark**

**6.**

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Time history of system outputs and references (

**a**) ${y}_{1}$, ${y}_{{m}_{1}}$ and (

**b**) ${y}_{2}$, ${y}_{{m}_{2}}$.

**Figure 3.**Time history of sliding surfaces (

**a**) ${\widehat{\sigma}}_{1}$ and (

**b**) ${\widehat{\sigma}}_{2}$.

**Figure 4.**Time history of system outputs and references (

**a**) ${y}_{1}$, ${y}_{{m}_{1}}$ and (

**b**) ${y}_{2}$, ${y}_{{m}_{2}}$.

**Figure 6.**Time history of system outputs and the extended references (

**a**) ${y}_{1}$, ${y}_{{m}_{1}}$ and (

**b**) ${y}_{2}$, ${y}_{{m}_{2}}$.

**Figure 7.**Time history of the control signals (

**a**) ${u}_{1}$ and (

**b**) ${u}_{2}$ with the extended references.

**Figure 8.**Time history of the sliding variables (

**a**) ${\widehat{\sigma}}_{1}$ and (

**b**) ${\widehat{\sigma}}_{2}$ with the extended references (zoomed).

**Figure 9.**Time history of the control signals (

**a**) ${u}_{1}$ and (

**b**) ${u}_{2}$ with the extended references (zoomed).

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Yeh, Y.-L. Output Feedback Tracking Sliding Mode Control for Systems with State- and Input-Dependent Disturbances. *Actuators* **2021**, *10*, 117.
https://doi.org/10.3390/act10060117

**AMA Style**

Yeh Y-L. Output Feedback Tracking Sliding Mode Control for Systems with State- and Input-Dependent Disturbances. *Actuators*. 2021; 10(6):117.
https://doi.org/10.3390/act10060117

**Chicago/Turabian Style**

Yeh, Yi-Liang. 2021. "Output Feedback Tracking Sliding Mode Control for Systems with State- and Input-Dependent Disturbances" *Actuators* 10, no. 6: 117.
https://doi.org/10.3390/act10060117