Finite-Time Stabilization for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Norm Stochastic Nonlinear Systems with Output Constraints

Guan, Shijun; Fang, Liandi

doi:https://doi.org/10.1155/2020/2486210

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Abstract Introduction Preliminaries Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Finite-time Control of Complex Systems and Their Applications

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Research Article | Open Access

Volume 2020 | Article ID 2486210 | https://doi.org/10.1155/2020/2486210

Finite-Time Stabilization for -Norm Stochastic Nonlinear Systems with Output Constraints

Shijun Guan¹and Liandi Fang^2,3

Guest Editor: Xiaodi Li

Received03 Jun 2020

Revised07 Jul 2020

Accepted10 Jul 2020

Published12 Aug 2020

Abstract

This paper investigates the finite-time stability problem of -norm stochastic nonlinear systems subject to output constraint. To cope with the constraint on system output, a tan-type barrier Lyapunov function (BLF) is constructed. By using the constructed BLF and the backstepping technique, a new control algorithm is proposed with a continuous state-feedback controller being designed, which guarantees not only that the requirement of output constraint is always achieved but also that the origin of the system is finite-time stable. This result is demonstrated by both the rigorous analysis and the simulation example.

1. Introduction

During the past decades, the control problem of nonlinear systems has long been a hot topic, and many control design approaches have been proposed for various kinds of nonlinear systems, such as adaptive fuzzy control [1, 2], output tracking control [3, 4], control [5, 6], and sliding mode control [7–10]. Due to their important roles in many science and industry applications, the stochastic nonlinear systems have attracted much interest in recent years. With the development of stochastic theory, various control design strategies have been developed for types of stochastic nonlinear systems by the backstepping technique, see [11–14], for examples. Especially, some works have considered -norm stochastic nonlinear systems, which are inherently nonlinear due to the fractional powers of such systems being not identically equal to one. It should be noted that the inherent nonlinearities cause the stability and control design problems, which are not very easy to be solved [15]. Luckily, the issues have been well studied for -norm stochastic nonlinear systems with different structures by the adding a power integrator technique in the existing literatures. For instance, Li et al. [16] have considered the adaptive state-feedback stabilization for -norm stochastic nonlinear systems; the output-feedback control has been addressed for -norm stochastic nonlinear systems with time-varying delays in [17]; Zhao et al. [18] have proposed a neural tracking control algorithm for -norm switched stochastic nonlinear systems. More latest studies can be found in [19–21] and the references within.

However, most of the abovementioned works about -norm stochastic nonlinear systems did not take the output constraint into consideration. As it is well known, many actual systems are subject to output constraint due to the consideration of the system performance and operation safety [22, 23]. For this reason, the constrained control issue of nonlinear systems has drawn attention from many scholars. Tee et al. [24] have first proposed the notion of the barrier Lyapunov function (BLF) and consequently have developed a control design strategy for a class of strict-feedback deterministic nonlinear systems with output constraints. After then, with the aid of BLFs, control design schemes have been presented for many deterministic nonlinear systems with different types of constraints, including stability control for nonlinear systems with time-varying or asymmetric output constraints [25, 26], adaptive control for nonlinear systems with full-state constrains [27], and sliding mode control for nonlinear systems with output constraints [28–30]. Moreover, since the finite-time control possesses some inherent advantages [31–33], techniques for the finite-time stabilization under output/state constraints have also been developed, respectively, for strict-feedback nonlinear systems [34], norm nonlinear systems [35–37], and switched nonlinear systems [38]. On the basis of these results, the constrained control schemes for some classes of stochastic nonlinear systems have also been proposed. Jin [39] has constructed an adaptive tracking controller for a class of output-constrained stochastic nonlinear systems in strict-feedback form. Later, the adaptive control problem and the finite-time control problem have been, respectively, addressed for stochastic nonlinear systems with full-state constraints in [40, 41]. Furthermore, the adaptive neural network or fuzzy constrained control problems have attracted some attention [42–46]. Nevertheless, the stochastic nonlinear systems with output constraints considered in most of the existing related works are in the strict-feedback form, rather than in -normal form. On the contrary, the existing research has mainly focused on the adaptive control problem but did not take the finite-time stabilization into account.

Motivated by the above discussions, we will investigate the problem of the finite-time stabilization for a class of -norm stochastic nonlinear systems with output constraints and unknown time-varying parameters. First of all, a BLF-based control strategy will be developed by the backstepping approach. Secondly, applying stochastic Lyapunov theorems and ItÔ’s formula, the constructed state-feedback controller is rigorously proved to be able to ensure the achievement of the output constraint and the finite-time stability of the considered systems simultaneously. Finally, the main result of this paper will be further demonstrated by a simulation example.

2. Problem and Preliminaries

2.1. Problem Statement

The following class of stochastic nonlinear systems are considered:where is a -dimension standard Wiener process; are system state, control input, and output, respectively; is the time-varying parameter; the nonlinear functions and are continuous and satisfy ; and the fractional powers ’s meet the requirement . The output is required to satisfywhere is a known positive constant.

This paper aims to design a continuous state-feedback controller for system (1), which can ensure that the origin of the closed-loop system is finite-time stable in probability and the requirement of the output constraint is achieved.

2.2. Preliminaries

Notations 1. For , let and . For any and , denote .
Consider the following stochastic system:where and are continuous satisfying and .

Definition 1 (see [13]). For any given , associated with system (1), the second-order differential operator is defined as follows:

Definition 2 (see [24]). Suppose that is an open set containing the origin and is positive definite and continuously differentiable. Then, for system is called a BLF if for each solution starting from , as , for all and for some .

Assumption 1 (see [36]). For , there exist known positive constants and such that .

Assumption 2. For , there are a constant and known nonnegative smooth functions , such thatfor all , where .

Remark 1. Note that condition (4) is borrowed from [34]. However, the systems considered in [34] are -norm deterministic nonlinear systems, while we consider -norm stochastic nonlinear systems with drift terms ’s and diffusion terms ’s in this paper. In light of , the value of is generally taken as for simplicity, where and represent even and odd integers, respectively. Then, the value of each can be obtained by applying and . It can also be observed that both the denominator and numerator of each are odd.

Lemma 1 (see [13]). Suppose that there exists a positive Lyapunov function , which satisfies . If is with respect to (3) and satisfies , then system (3) has a solution for any initial value.

Lemma 2 (see [13]). Suppose that system (3) admits a solution for each initial value. If there are class functions ϱ and ϱ, a positive Lyapunov function , real numbers , and , such that

For all , then the origin of system (3) is finite-time stable in probability.

Lemma 3 (see [37]). Let be positive real numbers. For any , we have

Lemma 4 (see [8]). Let ; for any , one haswhere if and if .

Lemma 5 (see [10]). Let. For any , we have(i)(ii)(iii)

3. Main Results

3.1. A Tan-Type BLF

Before carrying out the control design for system (1), we should handle the output constraint issue.

Firstly, we denote and . Let be a constant parameter satisfying , where the value of is chosen as below:(i)If for all , , then (ii)If for all , , then

Consequently, it is clear that .

Then, a tan-type BLF can be constructed on as follows:where is given by Assumption 2 and is defined as above.

It is not hard to obtain from the expression of thatwhere .

Remark 2. It should be noted that the BLF is modified from [34], which is constructed by fully taking the advantage of the given nonlinear growth conditions. As stated in [34], the control strategy based on is a universal method, which can handle stochastic systems with or without output constraints.

3.2. Controller Design and Stability Analysis

In what follows, a continuous state-feedback controller will be constructed, and the stability of system (1) under the designed controller will be rigorously analysed. To this end, a theorem is presented to describe the main result.

Theorem 1. Suppose Assumptions 1-2 hold for system (1). For any constant , there is a continuous state-feedback controller such that(i)The output of system (1) is kept in a given constrained set in the sense of probability, i.e., (ii)The origin of the closed-loop system is finite-time stable in probability.

Proof. The proof contains three parts. First of all, the design procedure of the controller is explicitly displayed. Then, the system output is proved to be kept in the given constrained set with probability one. In the last part, the finite-time stability of system (1) is rigorously analysed.

3.2.1. Part I: Design Procedure

Step 1. Let , and choose the Lyapunov function . Then, we can directly get from Definition 1 thatwhere is a nonnegative function and is a virtual controller required to be design after later.
Then, we designSubstituting (12) into (13) yields

Step 2. We denote and define the positive Lyapunov function on as withSince is valid, one can obtainUsing Definition 1 again, we havewhere is the virtual controller required to be designed later.
In the following, each term in the right hand of (16) will be estimated by its upper bound.
Firstly, applying and Lemma 5, it is easily obtained thatIt can be deduced from (19) and Lemma 3 thatwhere is a function.
Secondly, one can obtain from Assumption 1 and Lemma 4 thatwhere and are nonnegative smooth functions.
Additionally, note that . Then, as stated in [36], there exist functions , and , such thatThen, using (18), (21), (25), Assumptions 1-2, and Lemma 3, we can inferwhereis a nonnegative function.
Moreover, from (19), Assumption 2, and Lemma 3, it can be deduced thatwhere is a function.
On the contrary, it is noted thatThen, applying (22), (26), Assumption 2, and Lemma 3, there clearly exist nonnegative functions , and such thatSubstituting equations (31)–(33) into (30), one obtainswhere is a function.
Let . Design the virtual controller asSubstituting (20), (27), (29), (34), and (35) into (16), one can obtainInductive Step. Suppose at step , there exist a Lyapunov function , and a range of continuous virtual controllers defined aswith , such thatThen, the following property can be inferred.

Proposition 1. Choose the th Lyapunov function as with

Then, is on and there exists a virtual controller such thatwhere

The proof of above Proposition 1 is provided in the Appendix.

Step 3. In light of the inductive step, when and , Proposition 1 holds. Thus, we choose the overall Lyapunov function as withand define the virtual controller asThen, is clearly a function on , and it is easy to obtain thatTherefore, we can designwhich results in

3.2.2. Part II: Verification of Keeping the Output Constraint

For any , by Ito’s formula and (46), we can deduce

From and (47), it can be further verified thatwhich indicates

Hence, . Then, Part I of Theorem 1 is proved.

3.2.3. Part III: Stability Analysis

Based on the definition of , we easily obtain the fact that is radially unbounded. Combining the fact with (40) and Lemma 1 directly infers that system (1) has a solution for any .

On the contrary, by a simple calculation, one obtains

In addition, since , it is easy to get that for all . Then, applying the characteristics of tangent functions, it is not difficult to infer that

Now, let and . According to (44), (45), and Lemma 4, we obtain

So, one further obtains

In other words, . Consequently, it is directly deduced from Lemma 2 that the origin of system (1) under controller (39) is finite-time stable in probability.

Remark 3. Comparing with the existing results in most of the literatures, the paper mainly focuses on the finite-time stabilization, instead of the boundness of tracking error. Moreover, the proposed approach can be extend to the tracking control by introducing a coordinate transformation before constructing the BLF.

4. Simulation

In this section, we will provide the simulation results of the following example to illustrate the validity of the proposed strategy:where , , and . Then, , , and . Since , Assumption 1 is satisfied. And Assumption 2 is also satisfied with , , and . Furthermore, note that . Thus, one can select .

Now, let and . In view of design procedure in Part I of the proof, we can design the virtual controller as

Next, according to the design procedure, we denote and further obtain that

Let and ; then, the controller can be designed as

Finally, we suppose and select some different initial states ’s with each satisfying . The simulation results of system (48) are shown in Figures 1 and 2. Figure 1 curves trajectories of under different initial values, which illustrate that the output constraint is always not violated. Meanwhile, the trajectories of is given in Figure 2. It can be observed from the two figures that system (48) under controller (51) is finite-time stable.

5. Conclusion

In this paper, the stability issue is addressed for a class of -norm stochastic systems with output constraints and unknown time-varying parameters. Using a tan-type BLF, the finite-time control strategy is proposed by the adding a power integrator technique. On this basis, the designed controller has been proved to ensure that the origin of the closed-loop system is finite-time stable in probability and the system output is kept in a pre-given set. This conclusion has also been verified by the simulation results. It should be pointed out that the proposed approach is not applicable to the case of asymmetrical output constraints. In the future, we will try to modify the proposed method to be suitable for stochastic systems with asymmetrical output constraints or multi-input multi-output stochastic systems.

Appendix

Proof of Proposition 1. For , one can get from the definition of thatThen, we haveIn the following, we introduce some sub-propositions to simplify the proof.

Proposition A.1. For , there exist smooth nonnegative functions and such that

Proof. For ,, one haswhich meanswhere are nonnegative smooth functions.

Proposition A.2. For , there exist nonnegative functions such that

Proof. Since , it can be gotten from Lemma 5 thatFrom (A.7) and Lemma 3, one can verify thatwhere is a function.

Proposition A.3. For , there exist nonnegative functions such that

Proof. For , there are nonnegative functions such thatThen, from (A.10), we can getwhere and are nonnegative functions. Let , which directly verify (A.9).

Proposition A.4. For , there exist nonnegative functions such that

Proof. For , according to Proposition A.1 and Lemma 3, there exist nonnegative functions such that

Proposition A.5. For , there exist nonnegative functions such that

Proof. Note thatFirst of all, one gets from (A.10) thatwhere is a function.
From equation (A.10) and Proposition A.1, it can be inferred thatwhere is a function.
Then, one getswhere .
Besides, if , there exist functions such thatMeanwhile, if , we haveHence, for , combining (A.19) with (A.20) yieldsThen, for , it can be deduced from (A.21) and Lemma 3 thatwhere are functions. Hence, one haswhere . Similarly, applying (A.10), Proposition A.1 and Lemma 3, we getwhere and are nonnegative functions. Substituting (A.18), (A.23)–(A.25) into (A.15) directly infers (A.14). Till now, the proof of Proposition A.5 is finished.
Combining Propositions A.1–A.5 with (A.2) yieldswhere is a functions.
DesignThen, substituting the value of into (A.27) yields that (40) holds.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Project of Anhui Province Outstanding Young Talent Support Program with Grant no. gxyq2018089.

References

S. C. Tong, X. Min, and Y. X. Li, “Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions,” IEEE Transactions on Cybernetics, 2020.
View at: Publisher Site | Google Scholar
S. Tong and Y. Li, “Robust adaptive fuzzy backstepping output feedback tracking control for nonlinear system with dynamic uncertainties,” Science China Information Sciences, vol. 53, no. 2, pp. 307–324, 2010.
View at: Publisher Site | Google Scholar
D. Yang, X. Li, and J. Qiu, “Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback,” Nonlinear Analysis: Hybrid Systems, vol. 32, pp. 294–305, 2019.
View at: Publisher Site | Google Scholar
X. Li, X. Yang, and T. Huang, “Persistence of delayed cooperative models: impulsive control method,” Applied Mathematics and Computation, vol. 342, pp. 130–146, 2019.
View at: Publisher Site | Google Scholar
H. Shen, F. Li, H. Yan, H. R. Karimi, and H.-K. Lam, “Finite-time event-triggered $\mathcal{H}_{\infty }$ control for T-S fuzzy markov jump systems,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 5, pp. 3122–3135, 2018.
View at: Publisher Site | Google Scholar
H. Shen, Z. G. Huang, J. D. Cao, and J. H. Park, “Exponential filtering for continuous-time switched neural networks under persistent Dwell-Time switching regularity,” IEEE Transactions on Cybernetics, vol. 50, 2020.
View at: Publisher Site | Google Scholar
S. Ding, A. Levant, and S. Li, “Simple homogeneous sliding-mode controller,” Automatica, vol. 67, no. 5, pp. 22–32, 2016.
View at: Publisher Site | Google Scholar
K. Q. Mei and S. H. Ding, “Second-order sliding mode controller design subject to an upper-triangular structure,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019.
View at: Publisher Site | Google Scholar
X. Liu, X. Su, P. Shi, and C. Shen, “Observer-based sliding mode control for uncertain fuzzy systems via event-triggered strategy,” IEEE Transactions on Fuzzy Systems, vol. 27, no. 11, pp. 2190–2201, 2019.
View at: Publisher Site | Google Scholar
S. Ding and S. Li, “Second-order sliding mode controller design subject to mismatched term,” Automatica, vol. 77, pp. 388–392, 2017.
View at: Publisher Site | Google Scholar
Z. F. Li, T. S. Li, G. Feng, R. Zhao, and Q. H. Shan, “Neural network-based adaptive control for pure-feedback stochastic nonlinear systems with time-varying delays and dead-zone input,” Systems, Man, and Cybernetics: Systems, 2018.
View at: Publisher Site | Google Scholar
T. S. Li, Z. F. Li, D. Wang, and C. L. P. Chen, “Output-feedback adaptive neural control for stochastic nonlinear time-varying delays systems with unknown control directions,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 6, pp. 1188–1201, 2015.
View at: Google Scholar
J. Yin and S. Khoo, “Continuous finite-time state feedback stabilizers for some nonlinear stochastic systems,” International Journal of Robust and Nonlinear Control, vol. 25, no. 11, pp. 1581–1600, 2015.
View at: Publisher Site | Google Scholar
Y. Li, K. Li, and S. Tong, “Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO nonstrict feedback systems,” IEEE Transactions on Fuzzy Systems, vol. 27, no. 1, pp. 96–110, 2019.
View at: Publisher Site | Google Scholar
L. Fang, L. Ma, S. Ding, and D. Zhao, “Finite-time stabilization for a class of high-order stochastic nonlinear systems with an output constraint,” Applied Mathematics and Computation, vol. 358, pp. 63–79, 2019.
View at: Publisher Site | Google Scholar
W. Li, X. Liu, and S. Zhang, “Further results on adaptive state-feedback stabilization for stochastic high-order nonlinear systems,” Automatica, vol. 48, no. 8, pp. 1667–1675, 2012.
View at: Publisher Site | Google Scholar
W. T. Zha, J. Y. Zhai, and S. M. Fei, “Output feedback control for a class of stochastic high-order nonlinear systems with time-varying delays,” International Journal of Robust and Nonlinear Control, vol. 24, no. 16, pp. 2243–2260, 2014.
View at: Publisher Site | Google Scholar
X. Zhao, X. Wang, G. Zong, and X. Zheng, “Adaptive neural tracking control for switched high-order stochastic nonlinear systems,” IEEE Transactions on Cybernetics, vol. 47, no. 10, pp. 3088–3099, 2017.
View at: Publisher Site | Google Scholar
Z. Song and J. Zhai, “Decentralized output feedback stabilization for switched stochastic high-order nonlinear systems with time-varying state/input delays,” ISA Transactions, vol. 90, pp. 64–73, 2019.
View at: Publisher Site | Google Scholar
L. Liu, S. Xu, X.-J. Xie, and B. Xiao, “Observer-based decentralized control of large-scale stochastic high-order feedforward systems with multi time delays,” Journal of the Franklin Institute, vol. 356, no. 16, pp. 9627–9645, 2019.
View at: Publisher Site | Google Scholar
B. Niu, M. Liu, and A. Li, “Global adaptive stabilization of stochastic high-order switched nonlinear non-lower triangular systems,” Systems & Control Letters, vol. 136, p. 104596, 2020.
View at: Publisher Site | Google Scholar
J. H. Park, H. Shen, X. H. Chang, and T. H. Lee, Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals, Springer, Berlin, Germany, 2019.
L. Fang, L. Ma, S. Ding, and D. Zhao, “Robust finite‐time stabilization of a class of high‐order stochastic nonlinear systems subject to output constraint and disturbances,” International Journal of Robust and Nonlinear Control, vol. 29, no. 16, pp. 5550–5573, 2019.
View at: Publisher Site | Google Scholar
K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov Functions for the control of output-constrained nonlinear systems,” Automatica, vol. 45, no. 4, pp. 918–927, 2009.
View at: Publisher Site | Google Scholar
K. P. Tee, B. Ren, and S. S. Ge, “Control of nonlinear systems with time-varying output constraints,” Automatica, vol. 47, no. 11, pp. 2511–2516, 2011.
View at: Publisher Site | Google Scholar
L. D. Fang, L. Ma, S. H. Ding, and J. H. Park, “Finite-time stabilization of high-order stochastic nonlinear systems with asymmetric output constraints,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020.
View at: Publisher Site | Google Scholar
Y.-J. Liu and S. Tong, “Barrier Lyapunov Functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints,” Automatica, vol. 64, pp. 70–75, 2016.
View at: Publisher Site | Google Scholar
S. Ding, W.-H. Chen, K. Mei, and D. J. Murray-Smith, “Disturbance observer design for nonlinear systems represented by input-output models,” IEEE Transactions on Industrial Electronics, vol. 67, no. 2, pp. 1222–1232, 2020.
View at: Publisher Site | Google Scholar
S. H. Ding, J. H. Park, and C. C. Chen, “Second-order sliding mode controller design with output constraint,” Automatica, vol. 112, Article ID 108704, 2020.
View at: Publisher Site | Google Scholar
S. Ding, K. Mei, and S. Li, “A new second-order sliding mode and its application to nonlinear constrained systems,” IEEE Transactions on Automatic Control, vol. 64, no. 6, pp. 2545–2552, 2019.
View at: Publisher Site | Google Scholar
X. Yang, X. D. Li, X. Li, and Q. Xi, “Review of stability and stabilization for impulsive delayed systems,” Mathematical Biosciences & Engineering, vol. 15, no. 6, pp. 1495–1515, 2018.
View at: Publisher Site | Google Scholar
Q. K. Duan, S. H. Ding, and X. H. Yu, “Composite super-twisting sliding mode control design for PMSM speed regulation problem based on a novel disturbance observer,” IEEE Transactions on Energy Conversion, 2020.
View at: Publisher Site | Google Scholar
L. Liu, W. X. Zheng, and S. H. Ding, “An adaptive SOSM controller design by using a sliding-mode-based filter and its application to buck converter,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 67, no. 7, pp. 2409–2418, 2020.
View at: Google Scholar
R. Ma, Y. Liu, S. Zhao, and J. Fu, “Finite-time stabilization of a class of output-constrained nonlinear systems,” Journal of the Franklin Institute, vol. 352, no. 12, pp. 5968–5984, 2015.
View at: Publisher Site | Google Scholar
C. C. Chen, “A unified approach to finite‐time stabilization of high‐order nonlinear systems with and without an output constraint,” International Journal of Robust and Nonlinear Control, vol. 29, no. 2, pp. 393–407, 2019.
View at: Publisher Site | Google Scholar
C. C. Chen and G. S. Chen, “A new approach to stabilization of high-order nonlinear systems with an asymmetric output constraint,” International Journal of Robust and Nonlinear Control, vol. 30, no. 2, pp. 756–775, 2020.
View at: Publisher Site | Google Scholar
C. C. Chen and Z. Y. Sun, “A unified approach to finite-time stabilization of high-order nonlinear systems with an asymmetric output constraint,” Automatica, vol. 111, Article ID 108581, 2020.
View at: Publisher Site | Google Scholar
S. Huang and Z. Xiang, “Finite-time stabilisation of a class of switched nonlinear systems with state constraints,” International Journal of Control, vol. 91, no. 6, pp. 1300–1313, 2018.
View at: Publisher Site | Google Scholar
X. Jin, “Adaptive fault tolerant tracking control for a class of stochastic nonlinear systems with output constraint and actuator faults,” Systems & Control Letters, vol. 107, pp. 100–109, 2017.
View at: Publisher Site | Google Scholar
Y.-J. Liu, S. Lu, S. Tong, X. Chen, and C. L. P. Chen, “Adaptive control-based barrier Lyapunov functions for a class of stochastic nonlinear systems with full state constraints,” Automatica, vol. 87, pp. 83–93, 2018.
View at: Publisher Site | Google Scholar
J. Li, J. Xia, W. Sun, G. Zhuang, and Z. Wang, “Finite-time tracking control for stochastic nonlinear systems with full state constraints,” Applied Mathematics and Computation, vol. 338, pp. 207–220, 2018.
View at: Publisher Site | Google Scholar
L. D. Fang, H. S. Ding, J. H. Park, and L. Ma, “Adaptive fuzzy control for nontriangular stochastic high-order nonlinear systems subject to asymmetric output constraints,” IEEE Transactions on Cybernetics, 2020.
View at: Publisher Site | Google Scholar
Y. Liu, H. Ma, and H. Ma, “Adaptive fuzzy fault-tolerant control for uncertain nonlinear switched stochastic systems with time-varying output constraints,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 5, pp. 2487–2498, 2018.
View at: Publisher Site | Google Scholar
B. Niu, W. Ding, H. Li, and X. Xie, “A novel neural-network-based adaptive control scheme for output-constrained stochastic switched nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 2, pp. 418–432, 2019.
View at: Publisher Site | Google Scholar
L. D. Fang, S. H. Ding, J. H. Park, and L. Ma, “Adaptive fuzzy control for stochastic high-order nonlinear systems with output constraints,” IEEE Transactions on Fuzzy Systems, 2020.
View at: Publisher Site | Google Scholar
Y. Li and S. Tong, “Adaptive fuzzy output constrained control design for multi-input multioutput stochastic nonstrict-feedback nonlinear systems,” IEEE Transactions on Cybernetics, vol. 47, no. 12, pp. 4086–4095, 2017.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Shijun Guan and Liandi Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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