# Time-Delay Luenberger Observer Design for Sliding Mode Control of Nonlinear Markovian Jump Systems via Event-Triggered Mechanism

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{∞}attenuate level γ. The challenge is to deal with the issue that transition rates may be totally unknown. Moreover, an observer-based sliding mode controller is constructed to ensure the finite-time reachability of the predefined sliding surface. Finally, a numerical example based on a robotic manipulator is given to verify the effectiveness of the proposed method.

## 1. Introduction

_{∞}control for 2-D MJSs were proposed; and systematic theory on stochastic differential equations with Markovian switching was presented in [8]; the H

_{∞}filtering problem for MJSs was studied in [9,10]; for more details, we may refer to [11,12,13,14,15] and some of their references. In another aspect, due to the existence of nonlinearity, the Takagi–Sugeno (T-S) fuzzy modeling approach has become one of the most popular and effective ways to handle the synthesis of complex nonlinear systems [16], and the investigation on T-S fuzzy model-based MJSs is also rich. In addition, the stabilization of nonlinear singular MJS with matched/unmatched uncertainties based on the T-S fuzzy model was studied in [17]; the robust H

_{∞}control involving a class of uncertain stochastic MJSs was investigated in [18], etc. [19,20,21]. As we all know, the transition rates (TRs) play a big role in the system’s performance, but getting exact TR information seems impossible due to all kinds of limits in actual systems, such as high cost, technique limitations and so on. Hence, it is necessary to study MJSs in the presence of deficient TR information. Up to now, some results can be found in dealing with this issue, but not enough. For example, some pioneer works dealt with the issues of stabilization, stochastic stability and quantized filtering for (singular) MJSs with deficiency mode information in [22,23,24,25]. However, all the results proposed in literature [22,23,24,25] need TR information, no matter partially or fully. So, what if the information of TRs for one mode to another is completely unknown? This is the new challenge we are going to deal with in this paper.

_{∞}SMC for discrete-time MJSs subject to intermittent observations was researched in [27]; in Ref. [28], the research of asynchronous SMC was investigated based on uncertain MJSs with time-varying delays. In the presence of status components unavailable, observer-based SMC arises, such as an adaptive sliding mode observer was designed for nonlinear MJSs in [29]; the research about sensor fault estimation along with fault-tolerant control for time-delay MJSs through sliding mode observer technique was studied in [30]; in [31], a reduced-order sliding mode observer was designed to realize adaptive control of T-S fuzzy modeled-based MJSs. More works can be found in [32,33,34] and references therein.

_{∞}performance in sliding mode dynamics and error dynamics with totally unknown mode transition information. The main contributions of this paper can be concluded as: (1) Compared with traditional observer, the proposed event-triggered time-delay state observer brings the benefits that error is better suppressed and better stabilization property is obtained; (2) a novel sliding surface function is proposed, based on which the observer gain matrices can be computed in the design process rather than given as in [42]; (3) a new method is proposed to give feasible strict LMI conditions for stability of MJSs with totally unknown transition information; and (4) fuzzy SMC law ensures finite-time reachability of sliding surface and keeps sliding motion of each sub-models in the presence of uncertain transition information and nonlinearities.

**Notions:**

## 2. Model Establishing and Problem Statement

**Plant Rule 1: IF**${x}_{1}\left(t\right)$ is “about 0 rad”,

**THEN**

**Plant Rule i: IF**${\vartheta}_{1}\left(t\right)$ is ${F}_{i1}$ and ${\vartheta}_{2}\left(t\right)$ is ${F}_{i2}$ and … and ${\vartheta}_{p}\left(t\right)$ is ${F}_{ip}$

**THEN**

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

_{∞}performance in sliding mode dynamics and error dynamics.

## 3. Main Results

**Remark**

**1.**

#### 3.1. Network and ZOH

#### 3.2. Luenberger State Observer Design

**Observer Rule i: IF**${\widehat{\vartheta}}_{1}\left(t\right)$ is ${M}_{i1}$ and ${\widehat{\vartheta}}_{2}\left(t\right)$ is ${M}_{i2}$ and … and ${\widehat{\vartheta}}_{n}\left(t\right)$ is ${M}_{ip}$

**THEN**

**Remark**

**2.**

#### 3.3. Sliding Surface Design

**Remark**

**3.**

- It is stochastically stable for system (17) with w(t) = 0 and the sliding mode dynamics (21).
- The measurement of H
_{∞}performance with the condition of zero-initial will be satisfied as follows:$$J=\mathbb{E}{\int}_{0}^{+\infty}\left[{y}_{e}^{T}\left(s\right){y}_{e}\left(s\right)-{\gamma}^{2}{w}^{T}\left(s\right)w\left(s\right)\right]ds<0$$

**Remark**

**4.**

#### 3.4. Stochastic Stability and H_{∞} Performance Analysis

**Remark**

**5.**

_{∞}attenuation level $\gamma $, if matrices ${P}_{m}>0$, ${Q}_{1}>0$, ${Q}_{2}>0$, ${R}_{1}>0$, ${R}_{2}>0$, ${\Omega}_{m}>0$, free weighting matrices ${S}_{km}$ ($k$ = 1, 2, 3) and ${Y}_{im}$ with appropriate dimensions exist, the following condition is satisfied for each $m\in \mathcal{S}$:

**Proof.**

_{∞}performance of overall closed-loop system will be considered. $\mathbb{E}V\left(t\right)=\mathbb{E}{\int}_{0}^{+\infty}\mathcal{L}V\left(s\right)ds\ge 0$ in the condition of zero-initial. Therefore,

_{∞}disturbance attenuation level $\gamma $. □

**Remark**

**6.**

- 1.
- $m\in {I}_{m,kn}$and${\widehat{\pi}}_{mn}$for$\forall n\in {I}_{m,kn}$are known, that is${I}_{m,kn}$are known, that is${I}_{m,kn}=\mathcal{S}$;
- 2.
- $m\in {I}_{m,kn}$and${\widehat{\pi}}_{mn}$for$\forall n\in {I}_{m,kn}$are partially known, that is${I}_{m,kn}\ne \mathcal{S}$while${I}_{m,kn}$is also not empty;
- 3.
- $m\in {I}_{m,ukn}$and${\widehat{\pi}}_{mn}$for$\forall n\in {I}_{m,kn}$are partially known, that is${I}_{m,kn}\ne \mathcal{S}$while${I}_{m,kn}$is also not empty;
- 4.
- $m\in {I}_{m,ukn}$and${\widehat{\pi}}_{mn}$for$\forall n\in {I}_{m,kn}$are all unknown, that is${I}_{m,kn}=\varphi $.

**Theorem**

**2.**

_{∞}attenuation level$\gamma $, if matrices${P}_{m}>0$,${Q}_{1}>0$,${Q}_{2}>0$,${R}_{1}>0$,${R}_{2}>0$,${U}_{mm}>0$,${W}_{mm}>0$,${\Omega}_{m}>0$,free weighting matrices ${S}_{km}$ ($k$ = 1, 2, 3) and ${Y}_{im}$ with appropriate dimensions exist, the following conditions are satisfied for each $m\in \mathcal{S}$ If $m\in {I}_{m,kn}$ and ${I}_{m,kn}=\mathcal{S}$, then

**Proof.**

**Remark**

**7.**

_{∞}performance is ensured with unknown TRs.

#### 3.5. Reachability of Sliding Surface

**Theorem**

**3.**

**Proof.**

**Remark**

**8.**

**Remark**

**9.**

## 4. Numerical Example

**Plant Rule 1: IF**${x}_{1}\left(t\right)$ is “about 0 rad”,

**THEN**

**Plant Rule 2: IF**${x}_{1}\left(t\right)$ is “about $\pi $ rad or $-\pi $ rad”,

**THEN**

_{∞}attenuation levels $\gamma $ with fixed error tolerance $\rho =0.1$ and transmission delay ${d}_{m}=0.1$, we can see the maximum allowable ${a}_{3}$ for different attenuation levels $\gamma $ in Figure 1. From these results, it is easy to obtain that the proposed scheme can reduce the average transmission frequency while maintaining the control performance.

## 5. Conclusions

_{∞}performance of the sliding mode dynamics and the error dynamics were ensured in terms of LMI conditions. In addition, a fuzzy sliding mode controller was constructed to guarantee the finite-time reachability of the predefined sliding surface. Finally, numerical examples based on robotics were presented to verify the effectiveness of the proposed method.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Krasovskii, N.M.; Lidskii, E.A. Analytical design of controllers in systems with random attributes. Autom. Remote Control
**1961**, 22, 1021–2025. [Google Scholar] - Ugrinovskii, V.; Pota, H.R. Decentralized control of power systems via robust control of uncertain Markov jump parameter systems. Int. J. Control
**2005**, 78, 662–677. [Google Scholar] [CrossRef] - Andrieu, C.; Davy, M.; Doucet, A. Efficient particle filtering for jump Markov systems. Application to time-varying autoregressions. IEEE Trans. Signal Process.
**2003**, 51, 1762–1770. [Google Scholar] [CrossRef][Green Version] - Wallace, V.L.; Rosenberg, R.S. Markovian models and numerical analysis of computer system behavior. In Proceedings of the Spring Joint Computer Conference, Boston, MA, USA, 26–28 April 1966; pp. 141–148. [Google Scholar] [CrossRef]
- Ellis, R.S. Large deviations for the empirical measure of a Markov chain with an application to the multivariate empirical measure. Ann. Probab.
**1988**, 16, 1496–1508. [Google Scholar] [CrossRef] - Souza, C.E.D. Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems. IEEE Trans. Autom. Control
**2006**, 51, 836–841. [Google Scholar] [CrossRef] - Gao, H.; Lam, J.; Xu, S.; Wang, C. Stabilization and H∞ control of two-dimensional Markovian jump systems. IMA J. Math. Control Inf.
**2004**, 21, 377–392. [Google Scholar] [CrossRef] - Mao, X.; Yuan, C. Stochastic Differential Equations with Markovian Switching; Imperial College Press: London, UK, 2006. [Google Scholar] [CrossRef][Green Version]
- Wu, L.; Shi, P.; Gao, H.; Wang, C. H∞ filtering for 2D Markovian jump systems. Automatica
**2008**, 44, 1849–1858. [Google Scholar] [CrossRef] - Xu, S.; Lam, J.; Mao, X. Delay-dependent H∞ control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans. Circuits Syst. I Regul. Pap.
**2007**, 54, 2070–2077. [Google Scholar] [CrossRef] - Fei, Z.; Gao, H.; Shi, P. New results on stabilization of Markovian jump systems with time delay. Automatica
**2009**, 45, 2300–2306. [Google Scholar] [CrossRef] - Zhang, L.; Huang, B.; Lam, J. H∞ model reduction of Markovian jump linear systems. Syst. Control Lett.
**2003**, 50, 103–118. [Google Scholar] [CrossRef] - Jilkov, V.P.; Li, X.R. Online Bayesian estimation of transition probabilities for Markovian jump systems. IEEE Trans. Signal Process.
**2004**, 52, 1620–1630. [Google Scholar] [CrossRef] - Wu, Z.; Su, H.; Chu, J. H∞ filtering for singular Markovian jump systems with time delay. Int. J. Robust Nonlinear Control
**2010**, 20, 939–957. [Google Scholar] [CrossRef] - Costa, O.L.V.; Guerra, S. Stationary filter for linear minimum mean square error estimator of discrete-time Markovian jump systems. IEEE Trans. Autom. Control
**2002**, 47, 1351–1356. [Google Scholar] [CrossRef] - Takagi, T.; Sugeno, M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern.
**1985**, SMC-15, 116–132. [Google Scholar] [CrossRef] - Wang, Y.; Xia, Y.; Shen, H.; Zhou, P. SMC design for robust stabilization of nonlinear Markovian jump singular systems. IEEE Trans. Autom. Control
**2018**, 63, 219–224. [Google Scholar] [CrossRef] - Saravanakumar, R.; Ali, M.S.; Karimi, H.R. Robust H∞ control of uncertain stochastic Markovian jump systems with mixed time-varying delays. Int. J. Syst. Sci.
**2017**, 48, 862–872. [Google Scholar] [CrossRef] - Park, I.S.; Kwon, N.K.; Park, P. H∞ control for Markovian jump fuzzy systems with partly unknown transition rates and input saturation. J. Frankl. Inst.
**2018**, 355, 2498–2514. [Google Scholar] [CrossRef] - Shen, H.; Li, F.; Yan, H.; Karimi, H.R.; Lam, H. Finite-time event-triggered H∞ control for T–S fuzzy Markov jump systems. IEEE Trans. Fuzzy Syst.
**2018**, 26, 3122–3135. [Google Scholar] [CrossRef][Green Version] - Dong, S.; Wu, Z.; Pan, Y.; Su, H.; Liu, Y. Hidden-Markov-model-based asynchronous filter design of nonlinear Markov jump systems in continuous-time domain. IEEE Trans. Cybern.
**2019**, 49, 2294–2304. [Google Scholar] [CrossRef] - Zhang, L.; Boukas, E.-K. Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica
**2009**, 45, 463–468. [Google Scholar] [CrossRef] - Xiong, L.; Tian, J.; Liu, X. Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities. J. Frankl. Inst.
**2012**, 349, 2193–2214. [Google Scholar] [CrossRef] - Kao, Y.; Xie, J.; Wang, C. Stabilization of singular Markovian jump systems with generally uncertain transition rates. IEEE Trans. Autom. Control
**2014**, 59, 2604–2610. [Google Scholar] [CrossRef] - Wei, Y.; Qiu, J.; Karimi, H.R. Quantized filtering for continuous-time Markovian jump systems with deficient mode information. Asian J. Control
**2015**, 17, 1914–1923. [Google Scholar] [CrossRef] - Liu, M.; Zhang, L.; Shi, P.; Zhao, Y. Sliding mode control of continuous-time Markovian jump systems with digital data transmission. Automatica
**2017**, 80, 200–209. [Google Scholar] [CrossRef] - Zhang, H.; Wang, J.; Shi, Y. Robust H∞ sliding-mode control for Markovian jump systems subject to intermittent observations and partially known transition probabilities. Syst. Control Lett.
**2013**, 62, 1114–1124. [Google Scholar] [CrossRef] - Song, J.; Niu, Y.; Zou, Y. Asynchronous sliding mode control of Markovian jump systems with time-varying delays and partly accessible mode detection probabilities. Automatica
**2018**, 93, 33–41. [Google Scholar] [CrossRef] - Li, H.; Shi, P.; Yao, D.; Wu, L. Observer-based adaptive sliding mode control for nonlinear Markovian jump systems. Automatica
**2016**, 64, 133–142. [Google Scholar] [CrossRef] - Liu, M.; Shi, P.; Zhang, L.; Zhao, X. Fault-tolerant control for nonlinear Markovian jump systems via proportional and derivative sliding mode observer technique. IEEE Trans. Circuits Syst. I Regul. Pap.
**2011**, 58, 2755–2764. [Google Scholar] [CrossRef] - Jiang, B.; Karimi, H.R.; Yang, S.; Gao, C.; Kao, Y. Observer-based adaptive sliding mode control for nonlinear stochastic Markov jump systems via T–S fuzzy modeling: Applications to robot arm model. IEEE Trans. Ind. Electron.
**2021**, 68, 466–477. [Google Scholar] [CrossRef] - Liu, Z.; Yu, J. Non-fragile observer-based adaptive control of uncertain nonlinear stochastic Markovian jump systems via sliding mode technique. Nonlinear Anal. Hybrid Syst.
**2020**, 38, 100931. [Google Scholar] [CrossRef] - Karimi, H.R. A sliding mode approach to H∞ synchronization of master-slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties. J. Frankl. Inst.
**2012**, 349, 1480–1496. [Google Scholar] [CrossRef][Green Version] - Zohrabi, N.; Reza Momeni, H.; Hossein Abolmasoumi, A. Sliding mode control of Markovian jump systems with partly unknown transition probabilities. IFAC Proc. Vol.
**2013**, 46, 947–952. [Google Scholar] [CrossRef] - Yue, D.; Tian, E.; Han, Q. A delay system method for designing event-triggered controllers of networked control systems. IEEE Trans. Autom. Control
**2013**, 58, 475–481. [Google Scholar] [CrossRef] - Su, X.; Liu, X.; Shi, P.; Song, Y.-D. Sliding mode control of hybrid switched systems via an event-triggered mechanism. Automatica
**2018**, 90, 294–303. [Google Scholar] [CrossRef] - Peng, C.; Han, Q.; Yue, D. To transmit or not to transmit: A discrete event-triggered communication scheme for networked Takagi–Sugeno fuzzy systems. IEEE Trans. Fuzzy Syst.
**2013**, 21, 164–170. [Google Scholar] [CrossRef] - Behera, A.K.; Bandyopadhyay, B.; Yu, X. Periodic event-triggered sliding mode control. Automatica
**2018**, 96, 61–72. [Google Scholar] [CrossRef] - Dimarogonas, D.V.; Frazzoli, E.; Johansson, K.H. Distributed event-triggered control for multi-agent systems. IEEE Trans. Autom. Control
**2012**, 57, 1291–1297. [Google Scholar] [CrossRef] - Cheng, J.; Park, J.H.; Zhang, L.; Zhu, Y. An asynchronous operation approach to event-triggered control for fuzzy Markovian jump systems with general switching policies. IEEE Trans. Fuzzy Syst.
**2018**, 26, 6–18. [Google Scholar] [CrossRef] - Wang, L.; Wang, Z.; Wei, G.; Alsaadi, F.E. Finite-time state estimation for recurrent delayed neural networks with component-based event-triggering protocol. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 1046–1057. [Google Scholar] [CrossRef] [PubMed] - Wu, L.; Gao, Y.; Liu, J.; Li, H. Event-triggered sliding mode control of stochastic systems via output feedback. Automatica
**2017**, 82, 79–92. [Google Scholar] [CrossRef] - Huai-Ning, W.; Kai-Yuan, C. Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control. IEEE Trans. Syst. Man Cybern. Part B
**2006**, 36, 509–519. [Google Scholar] [CrossRef] [PubMed] - Tanaka, K.; Kosaki, T. Design of a stable fuzzy controller for an articulated vehicle. IEEE Trans. Syst. Man Cybern. Part B
**1997**, 27, 552–558. [Google Scholar] [CrossRef] [PubMed] - Meyn, S.P.; Tweedie, R.L. Markov Chains and Stochastic Stability; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Boukas, K. Stabilization of stochastic nonlinear hybrid systems. Int. J. Innov. Comput. Inf. Control
**2005**, 1, 131–141. [Google Scholar] [CrossRef] - Xiong, J.; Lam, J. Robust H
_{2}control of Markovian jump systems with uncertain switching probabilities. Int. J. Syst. Sci.**2009**, 40, 255–265. [Google Scholar] [CrossRef][Green Version] - Jiang, B.; Gao, C. Decentralized adaptive sliding mode control of large-scale semi-Markovian jump interconnected systems with dead-zone input. IEEE Trans. Autom. Control
**2021**. [Google Scholar] [CrossRef]

Symbol | Meaning |
---|---|

$\vartheta \left(t\right)$ | angle position of the robot arm |

$u\left(t\right)$ | the input of control |

$D\left(t\right)$ | coefficient of viscous friction |

$L$ | length of the arm |

$J$ | moment of inertia |

$M$ | mass of the pay load |

$g$ | the acceleration of gravity |

Mode m | Parameter M | Parameter J |
---|---|---|

1 | 1 | 1 |

2 | 1.5 | 2 |

3 | 2 | 2.5 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cheng, M.; Zhang, C.; Qiu, J.; Wu, Z.; Gao, Q.
Time-Delay Luenberger Observer Design for Sliding Mode Control of Nonlinear Markovian Jump Systems via Event-Triggered Mechanism. *Machines* **2021**, *9*, 259.
https://doi.org/10.3390/machines9110259

**AMA Style**

Cheng M, Zhang C, Qiu J, Wu Z, Gao Q.
Time-Delay Luenberger Observer Design for Sliding Mode Control of Nonlinear Markovian Jump Systems via Event-Triggered Mechanism. *Machines*. 2021; 9(11):259.
https://doi.org/10.3390/machines9110259

**Chicago/Turabian Style**

Cheng, Min, Chunyang Zhang, Jin Qiu, Zhengtian Wu, and Qing Gao.
2021. "Time-Delay Luenberger Observer Design for Sliding Mode Control of Nonlinear Markovian Jump Systems via Event-Triggered Mechanism" *Machines* 9, no. 11: 259.
https://doi.org/10.3390/machines9110259