# An LMI Approach to Nonlinear State-Feedback Stability of Uncertain Time-Delay Systems in the Presence of Lipschitzian Nonlinearities

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## Abstract

**:**

## 1. Introduction

_{∞}stabilization control for the time-delay Takagi–Sugeno fuzzy systems under nonlinear perturbations and sampled-data input was investigated. By using output feedback, the authors of [15] studied the robust stabilization problem of a class of time-varying time-delay dynamical systems, which were not perfectly known, where the system output was modeled through a nonlinear function depending on the delayed inputs and states. In [16], the state-feedback stability control of switched discrete-time singular systems in the presence of time-varying state delays was presented. In [17], the stabilization problem of nonlinear cascade time-delay systems using the converse Lyapunov stabilization and invariant set theories was presented.

_{∞}control [24] and H

_{2}control [25]. Dealing with difficult problems for which there is no analytical solution is another significant feature of the LMI approach that attracts the attention of many researchers since it offers numerically tractable means [26,27]. Furthermore, powerful algorithms exist, such as interior-point ones, to provide a way of dealing with LMI problems. Two robust H

_{∞}state-feedback controllers based on LMIs for time-delay discrete-time systems and uncertain switched impulsive linear systems were proposed in [28] and [29], respectively. In [30], a robust H

_{∞}fuzzy-logic controller for Takagi–Sugeno time-delay bilinear discrete-time systems in the presence of disturbances was presented in which the stabilization conditions were formulated as LMI. A combination of Lyapunov parameter-dependent function and LMI was also used in [31,32] to develop a control scheme for uncertain systems subject to time delay. Moreover, based on the LMI approach, the problem of stabilization of a uniform Euler–Bernoulli beam and a two-dimensional Burgers’ equation have been investigated in [33] and [34], respectively. To the best of the authors’ knowledge, little consideration has been drawn to the problem of nonlinear state-feedback stability for nonlinear time-delay systems with Lipschitz nonlinearities via LMIs.

- -
- Design of a nonlinear state-feedback stabilizing controller for nonlinear systems in the presence of time delays, Lipschitz nonlinearity, and parametric uncertainties.
- -
- Achievement of asymptotic stabilization based on the Lyapunov–Krasovskii stabilization theory and the LMI approach.
- -
- The suggested control scheme is rather straightforward; there is no difficulty in the employment of this technique.
- -
- Application of the offered method on a nonlinear unstable system and a rotational inverted pendulum to prove the efficiency of the method.

## 2. Problem Description

**Lemma**

**1.**

**Lemma 2**

**(Schur Complement) [36].**

**Assumption**

**1.**

**Remark**

**1.**

## 3. Nonlinear State-Feedback Stabilization

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

## 4. Simulation and Experimental Results

#### 4.1. Example A: Unstable Nonlinear System

^{®}YALMIP

^{®}solver as:

#### 4.2. Example B: A Practical Rotational Inverted Pendulum

^{®}solver, matrices $H$, $P$, ${P}_{1}$, $F$ are obtained as:

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**State trajectories of the unstable nonlinear system (showing the stabilization efficiency of the proposed method compared to the method of [1]).

**Figure 2.**Time responses of the system outputs y

_{1}(t) and y

_{2}(t) (which show the better stabilization of the proposed method rather than the method of [1]).

**Figure 3.**System states for the second scenario with different initial conditions and larger uncertainties (showing the better stabilization performance of the proposed method).

**Figure 4.**System outputs for the second scenario with different initial conditions and larger uncertainties (displaying the better stabilization performance of the proposed method compared to the method of [1]).

**Figure 5.**Schematic diagram of the RIP system (which illustrates that the connected motor to the arm causes the balancing control of the inverted pendulum).

**Figure 6.**States of the RIP system (which shows that the angular position and velocity of the pendulum and arm are stabilized).

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**MDPI and ACS Style**

Golestani, M.; Mobayen, S.; HosseinNia, S.H.; Shamaghdari, S.
An LMI Approach to Nonlinear State-Feedback Stability of Uncertain Time-Delay Systems in the Presence of Lipschitzian Nonlinearities. *Symmetry* **2020**, *12*, 1883.
https://doi.org/10.3390/sym12111883

**AMA Style**

Golestani M, Mobayen S, HosseinNia SH, Shamaghdari S.
An LMI Approach to Nonlinear State-Feedback Stability of Uncertain Time-Delay Systems in the Presence of Lipschitzian Nonlinearities. *Symmetry*. 2020; 12(11):1883.
https://doi.org/10.3390/sym12111883

**Chicago/Turabian Style**

Golestani, Mehdi, Saleh Mobayen, S. Hassan HosseinNia, and Saeed Shamaghdari.
2020. "An LMI Approach to Nonlinear State-Feedback Stability of Uncertain Time-Delay Systems in the Presence of Lipschitzian Nonlinearities" *Symmetry* 12, no. 11: 1883.
https://doi.org/10.3390/sym12111883