Prescribed-time control of stochastic high-order nonlinear systems

Hui Wang; Hui Wang

doi:10.3934/mbe.2022531

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 11399-11408. doi: 10.3934/mbe.2022531

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Research article Special Issues

Prescribed-time control of stochastic high-order nonlinear systems

Hui Wang ^,

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received: 19 July 2022 Revised: 29 July 2022 Accepted: 07 August 2022 Published: 09 August 2022

In this paper, the prescribed-time stabilization is studied for stochastic high-order nonlinear systems. Different from the previous research results on stochastic high-order nonlinear systems where only asymptotic stabilization or finite-time stabilization is considered, this paper proposes a new design to achieve stabilization in the prescribed-time. Specifically, the designed controller can ensure that the closed-loop system has an almost surely unique strong solution and the equilibrium of the closed-loop system is prescribed-time mean-square stable. The design method is verified by an example.

Keywords:

Citation: Hui Wang. Prescribed-time control of stochastic high-order nonlinear systems[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 11399-11408. doi: 10.3934/mbe.2022531

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Abstract

1. Introduction

Stochastic nonlinear control is a hot topic because of its wide application in economic and engineering fields. The pioneer work is ^{[1,2,3,4,5,6,7,8]}. Specifically, ^[1,2,3] propose designs with quadratic Lyapunov functions coupled with weighting functions and ^[4,5,6,7,8] develop designs with quartic Lyapunov function, which are further developed by ^[9,10,11,12]. Recently, a class of stochastic systems (SSs) whose Jacobian linearizations may have unstable modes, has received much attention. Such systems are also called stochastic high-order nonlinear systems (SHONSs), which include a class of stochastic benchmark mechanical systems ^[13] as a special case. In this direction, ^[14] studies the state-feedback control with stochastic inverse dynamics; ^[15] develops a stochastic homogeneous domination method, which completely relaxes the order restriction required in ^[13], and investigates the output-feedback stabilization for SHONSs with unmeasurable states; ^[16] investigates the output-feedback tracking problem; ^[17] studies the adaptive state-feedback design for state-constrained systems. It should be emphasized that ^{[13,14,15,16,17]}, only achieve stabilization in asymptotic sense (as time goes to infinity). However, many real applications appeals for prescribed-time stabilization, which permits the worker to prescribe the convergence times in advance.

For the prescribed-time control, ^[18,19] design time-varying feedback to solve the regulation problems of normal-form nonlinear systems; ^[20] considers networked multi-agent systems; ^[21,22] investigates the prescribed-time design for linear systems in controllable canonical form; ^[23] designs the output-feedback controller for uncertain nonlinear strict-feedback-like systems. It should be noted that the results in ^{[18,19,20,21,22,23]} don't consider stochastic noise. For SSs, ^[24] is the first paper to address the stochastic nonlinear inverse optimality and prescribed-time stabilization problems; The control effort is further reduced in ^[25]; Recently, ^[26] studies the prescribed-time output-feedback for SSs without/with sensor uncertainty. It should be noted that ^[24,25,26] don't consider SHONSs. From practical applications, it is important to solves the prescribed-time control problem of SHONSs since it permits the worker to set the convergence times in advance.

Motivated by the above discussions, this paper studies the prescribed-time design for SSs with high-order structure. The contributions include:

1) This paper proposes new prescribed-time design for SHONSs. Since Jacobian linearizations of such a system possibly have unstable modes, all the prescribed-time designs in ^[24,25,26] are invalid. New design and analysis tools should be developed.

2) This paper develops a more practical design than those in ^{[13,14,15,16,17]}. Different from the designs in ^{[13,14,15,16,17]} where only asymptotic stabilization can be achieved, the design in this paper can guarantee that the closed-loop system is prescribed-time mean-square stable, which is superior to those in ^{[13,14,15,16,17]} since it permits the worker to prescribe the convergence times in advance without considering the initial conditions.

The remainder of this paper is organized as follows. In Section 2, the problem is formulated. In Section 3, the controller is designed and the stability analysis is given. Section 4 uses an example to explain the validity of the prescribed-time design. The conclusions are collected in Section 6.

2. Problem formulation

Consider the following class of SHONSs

$\begin{eqnarray} dx_{1}& = &x_{2}^{p}dt+\varphi_{1}^{T}(x)d\omega, \end{eqnarray}$

(2.1)

$\begin{eqnarray} dx_{2}& = &udt+\varphi_{2}^{T}(x)d\omega, \end{eqnarray}$

(2.2)

where $x = (x_{1}, x_{2})^{T} \in R^{2}$ and $u \in R$ are the system state and control input. $p\geq1$ is an odd integral number. The functions $\varphi_{i}: R^{2}\rightarrow R^{m}$ are smooth in $x$ with $\varphi_{i}(0) = 0$ , $i = 1, 2$ . $\omega$ is an $m$ -dimensional independent standard Wiener process.

The assumptions we need are as follows.

Assumption 1. There is a constant $c > 0$ such that

$\begin{eqnarray} |\varphi_{1}(x)|&\leq&c|x_{1}|^{\frac{p+1}{2}}, \end{eqnarray}$

(2.3)

$\begin{eqnarray} |\varphi_{2}(x)|&\leq&c(|x_{1}|^{\frac{p+1}{2}}+|x_{2}|^{\frac{p+1}{2}}). \end{eqnarray}$

(2.4)

We introduce the function:

$\begin{eqnarray} \mu(t)& = &\left(\frac{T}{t_{0}+T-t}\right)^{m}, \quad t\in[t_{0}, t_{0}+T), \end{eqnarray}$

(2.5)

where $m\geq 2$ is an integral number. Obviously, the function $\mu(t)$ is monotonically increasing on $[t_{0}, t_{0}+T)$ with $\mu(t_{0}) = 1$ and $\lim\limits_{t\rightarrow t_{0}+T}\mu(t) = +\infty$ .

Our control goal is to design a prescribed-time state-feedback controller, which guarantees that the closed-loop system has an almost surely unique strong solution and is prescribed-time mean-square stable.

3. Controller design and stability analysis

In the following, we design a time-varying controller for system (2.1)–(2.2) step by step.

Step 1. In this step, our goal is to design the virtual controller $x_{2}^{*}$ .

Define $V_{1} = \frac{1}{4}\xi_{1}^{4}$ , $\xi_{1} = x_{1}$ , from (2.1), (2.3) and (2.5) we have

$\begin{eqnarray} \mathcal{L}V_{1}(\xi_{1}) & = &\xi_{1}^{3}x_{2}^{p}+\frac{3}{2}\xi_{1}^{2}|\varphi_{1}|^{2}\\ &\leq&\xi_{1}^{3}x_{2}^{p}+\frac{3}{2}c^{2}\xi_{1}^{p+3}\\ &\leq&\xi_{1}^{3}(x_{2}^{p}-x_{2}^{*p})+\xi_{1}^{3}x_{2}^{*p}+\frac{3}{2}c^{2}\mu^{p}\xi_{1}^{p+3}. \end{eqnarray}$

(3.1)

Choosing

$\begin{eqnarray} x_{2}^{*}& = &-\mu\left(2+\frac{3}{2}c^{2}\right)^{1/p}\xi_{1}\\ &\triangleq&-\mu\alpha_{1}\xi_{1}, \end{eqnarray}$

(3.2)

which substitutes into (3.1) yields

$\begin{eqnarray} \mathcal{L}V_{1}(\xi_{1})&\leq&-2\mu^{p}\xi_{1}^{p+3}+\xi_{1}^{3}(x_{2}^{p}-x_{2}^{*p}), \end{eqnarray}$

(3.3)

where $\alpha_{1} = (2+\frac{3}{2}c^{2})^{1/p}$ .

Step 2. In this step, our goal is to design the actual controller $u$ .

Define $\xi_{2} = x_{k}-x_{2}^{*}$ , from (3.2) we get

$\begin{eqnarray} \xi_{2} = x_{2}+\mu\alpha_{1}\xi_{1}. \end{eqnarray}$

(3.4)

By using (2.1)–(2.2) and (3.4) we get

$\begin{eqnarray} d\xi_{2}& = &\left(u+\frac{m}{T}\mu^{1+1/m}\alpha_{1}\xi_{1}+\mu\alpha_{1}x_{2}^{p}\right)dt +\left(\varphi_{2}^{T}+\mu\alpha_{1}\varphi_{1}^{T}\right)d\omega. \end{eqnarray}$

(3.5)

We choose the following Lyapunov function

$\begin{eqnarray} V_2(\xi_{1}, \xi_{2})& = &V_{1}(\xi_{1})+\frac{1}{4}\xi_{2}^{4}. \end{eqnarray}$

(3.6)

It follows from (3.3), (3.5)–(3.6) that

$\begin{eqnarray} \mathcal{L}V_2 &\leq&-2\mu^{p}\xi_{1}^{p+3}+\xi_{1}^{3}(x_{2}^{p}-x_{2}^{*p}) +\xi_{2}^{3}\left(u+\frac{m}{T}\mu^{1+1/m}\alpha_{1}\xi_{1}+\mu\alpha_{1}x_{2}^{p}\right)\\&& +\frac{3}{2}\xi_{2}^{2}|\varphi_{2}+\mu\alpha_{1}\varphi_{1}|^{2}. \end{eqnarray}$

(3.7)

By using (3.4) we have

$\begin{eqnarray} \xi_{1}^{3}(x_{2}^{p}-x_{2}^{*p})&\leq&|\xi_{1}|^{3}|\xi_{2}|(x_{2}^{p-1}+x_{2}^{*p-1})\\ &\leq&|\xi_{1}|^{3}|\xi_{2}|((\beta_{1}+1)\mu^{p-1}\alpha_{1}^{p-1}\xi_{1}^{p-1}+\beta_{1}\xi_{2}^{p-1})\\ & = &(\beta_{1}+1)\mu^{p-1}\alpha_{1}^{p-1}|\xi_{1}|^{p+2}|\xi_{2}|+\beta_{1}|\xi_{1}|^{3}|\xi_{2}|^{p}, \end{eqnarray}$

(3.8)

where

$\begin{eqnarray} \beta_{1}& = &\max\{1, 2^{p-2}\}. \end{eqnarray}$

(3.9)

By using Young's inequality we get

$\begin{eqnarray} (\beta_{1}+1)\mu^{p-1}\alpha_{1}^{p-1}|\xi_{1}|^{p+2}|\xi_{2}|\leq\frac{1}{3}\mu^{p}\xi_{1}^{p+3}+\frac{1}{3+p}\left(\frac{3+p}{3(2+p)}\right)^{-(2+p)} ((\beta_{1}+1)\alpha_{1}^{p-1})^{p+3}\mu^{-3}\xi_{2}^{p+3} \end{eqnarray}$

(3.10)

and

$\begin{eqnarray} \beta_{1}|\xi_{1}|^{3}|\xi_{2}|^{p}&\leq&\frac{1}{3}\mu^{p}\xi_{1}^{3+p}+\frac{p}{p+3}\left(\frac{3+p}{9}\right)^{-3/p} \beta_{1}^{(3+p)/p}\mu^{-3}\xi_{2}^{p+3}. \end{eqnarray}$

(3.11)

Substituting (3.10)–(3.11) into (3.8) yields

$\begin{eqnarray} \xi_{1}^{3}(x_{2}^{p}-x_{2}^{*p})&\leq&\frac{2}{3}\mu^{p}\xi_{1}^{p+3}+\Big(\frac{p}{p+3}\left(\frac{p+3}{9}\right)^{-3/p} \beta_{1}^{(p+3)/p}\\&&+\frac{1}{p+3}\left(\frac{p+3}{3(p+2)}\right)^{-(p+2)} ((\beta_{1}+1)\alpha_{1}^{p-1})^{p+3}\Big)\mu^{-3}\xi_{2}^{p+3}. \end{eqnarray}$

(3.12)

From (2.3), (2.4), (3.2) and (3.4) we have

$\begin{eqnarray} |\varphi_{2}+\mu\alpha_{1}\varphi_{1}|^{2}&\leq& 2|\varphi_{2}|^{2}+2\mu^{2}\alpha_{1}^{2}|\varphi_{1}|^{2}\\ &\leq& 4c^{2}(|x_{1}|^{1+p}+|x_{2}|^{1+p})+2c^{2}\alpha_{1}^{2}\mu^{2}|x_{1}|^{1+p}\\ &\leq& \beta_{2}\mu^{1+p}|\xi_{1}|^{p+1}+4c^{2}2^{p}|\xi_{2}|^{1+p}, \end{eqnarray}$

(3.13)

where

$\begin{eqnarray} \beta_{2}& = &4c^{2}(1+2^{p}\alpha_{1}^{p+1})+2c^{2}\alpha_{1}^{2}. \end{eqnarray}$

(3.14)

By using (3.13) we get

$\begin{array}{l} \frac{3}{2}\xi_{2}^{2}|\varphi_{2}+\mu\alpha_{1}\varphi_{1}|^{2}\leq\frac{1}{3}\mu^{p}\xi_{1}^{3+p} +\Big(\frac{2}{3+p}\left(\frac{p+3}{3(1+p)}\right)^{-(1+p)/2} \\(\frac{3}{2}\beta_{2})^{(3+p)/2}+6c^{2}2^{p}\Big)\mu^{3(1+p)/2}\xi_{2}^{p+3}. \end{array}$

(3.15)

It can be inferred from (3.7), (3.12) and (3.15) that

$\begin{eqnarray} \mathcal{L}V_2 &\leq&-\mu^{p}\xi_{1}^{p+3} +\xi_{2}^{3}\left(u+\frac{m}{T}\mu^{1+1/m}\alpha_{1}\xi_{1}+\mu\alpha_{1}x_{2}^{p}\right) +\beta_{3}\mu^{3(p+1)/2}\xi_{2}^{p+3}, \end{eqnarray}$

(3.16)

where

$\begin{eqnarray} \beta_{3}& = & \frac{p}{3+p}\left(\frac{3+p}{9}\right)^{-3/p} \beta_{1}^{(3+p)/p}+\frac{1}{3+p}\left(\frac{p+3}{3(p+2)}\right)^{-(p+2)} ((\beta_{1}+1)\alpha_{1}^{p-1})^{3+p}\\&&+\frac{2}{3+p}\left(\frac{p+3}{3(1+p)}\right)^{-(1+p)/2} (\frac{3}{2}\beta_{2})^{(3+p)/2}+6c^{2}2^{p}. \end{eqnarray}$

(3.17)

We choose the controller

$\begin{eqnarray} u& = &-\frac{m}{T}\mu^{1+1/m}\alpha_{1}\xi_{1}-\mu\alpha_{1}x_{2}^{p}-(1+\beta_{3})\mu^{3(p+1)/2}\xi_{2}^{p}, \end{eqnarray}$

(3.18)

then we get

$\begin{eqnarray} \mathcal{L}V_2 &\leq&-\mu^{p}\xi_{1}^{p+3} -\mu^{3(p+1)/2}\xi_{2}^{p+3}. \end{eqnarray}$

(3.19)

Now, we describe the main stability analysis results for system (2.1)–(2.2).

Theorem 1. For the plant (2.1)–(2.2), if Assumption 1 holds, with the controller (3.18), the following conclusions are held.

1) The plant has an almost surely unique strong solution on $[t_{0}, t_{0}+T)$ ;

2) The equilibrium at the origin of the plant is prescribed-time mean-square stable with $\lim\limits_{t\rightarrow t_{0}+T} E|x|^{2} = 0$ .

Proof. By using Young's inequality we get

$\begin{eqnarray} \frac{1}{4}\mu \xi_{1}^{4} &\leq&\mu^{p}\xi_{1}^{p+3}+\beta_{4}\mu^{-3}, \end{eqnarray}$

(3.20)

$\begin{eqnarray} \frac{1}{4}\mu \xi_{2}^{4} &\leq&\mu^{3(p+1)/2}\xi_{2}^{p+3}+\beta_{4}\mu^{-3}, \end{eqnarray}$

(3.21)

where

$\begin{eqnarray} \beta_{4}& = &\frac{p-1}{p+3}(\frac{3+p}{4})^{-4/(p-1)}(\frac{1}{4})^{(3+p)/(p-1)}. \end{eqnarray}$

(3.22)

It can be inferred from (3.19)–(3.21) that

$\begin{eqnarray} \mathcal{L}V_2 &\leq&-\frac{1}{4}\mu \xi_{1}^{4}-\frac{1}{4}\mu \xi_{2}^{4}+2\beta_{4}\mu^{-3}\\ & = &-\mu V_{2}+2\beta_{4}\mu^{-3}. \end{eqnarray}$

(3.23)

From (2.1), (2.2) and (3.18), the local Lipschitz condition is satisfied by the plant. By (3.23) and using Lemma 1 in ^[24], the plant has an almost surely unique strong solution on $[t_{0}, t_{0}+T)$ , which shows that conclusion 1) is true.

Next, we verify conclusion 2).

For each positive integer $k$ , the first exit time is defined as

$\begin{eqnarray} \rho_k = \inf\{t: t\geq t_0, \; |x(t)|\geq k\}. \end{eqnarray}$

(3.24)

Choosing

$\begin{eqnarray} V = e^{\int^{t}_{t _{0}}\mu(s)ds}V_{2}. \end{eqnarray}$

(3.25)

From (3.23) and (3.25) we have

$\begin{eqnarray} \mathcal{L}V& = &e^{\int^{t}_{t _{0}}\mu(s)ds}(\mathcal{L}V_{2}+\mu V_{2})\leq 2\beta_{4}\mu^{-3}e^{\int^{t}_{t _{0}}\mu(s)ds}. \end{eqnarray}$

(3.26)

By (3.26) and using Dynkin's formula we get

$\begin{eqnarray} EV(\rho_k\wedge t, x(\rho_k\wedge t))& = &V_{n}(t_{0}, x_{0})+E\left\{\int^{\rho_k\wedge t}_{t_{0}}\mathcal{L}V(x(\tau), \tau)d\tau\right\} \\ &\leq&V_{n}(t_{0}, x_{0})+2\beta_{4}\int^{ t}_{t_{0}}\mu^{-3}e^{\int^{\tau}_{t _{0}}\mu(s)ds}d\tau. \end{eqnarray}$

(3.27)

Using Fatou Lemma, from (3.27) we have

$\begin{eqnarray} EV(t, x)&\leq&V_{n}(t_{0}, x_{0})+2\beta_{4}\int^{ t}_{t_{0}}\mu^{-3}e^{\int^{\tau}_{t _{0}}\mu(s)ds}d\tau, \quad t\in[t_{0}, t_{0}+T). \end{eqnarray}$

(3.28)

By using (3.25) and (3.28) we get

$\begin{eqnarray} EV_{2}&\leq&e^{-\int^{t}_{t _{0}}\mu(s)ds}\left(V_{n}(t_{0}, x_{0})+2\beta_{4}\int^{ t}_{t_{0}}\mu^{-3}e^{\int^{\tau}_{t _{0}}\mu(s)ds}d\tau\right), \quad t\in[t_{0}, t_{0}+T). \end{eqnarray}$

(3.29)

By using (3.6) and (3.29) we obtain

$\begin{eqnarray} \lim\limits_{t\rightarrow t_{0}+T} E|x|^{2}& = &0. \end{eqnarray}$

(3.30)

4. A simulation example

Consider the prescribed-time stabilization for the following system

$\begin{eqnarray} &&dx_1 = x_2^{3}dt+0.1x_1^{2}d\omega, \end{eqnarray}$

(4.1)

$\begin{eqnarray} &&dx_2 = udt+ 0.2x_1x_2d\omega, \end{eqnarray}$

(4.2)

where $p_{1} = 3$ , $p_{2} = 1$ . Noting $0.2x_1x_2\leq 0.1(x_{1}^{2}+x_{2}^{2})$ , Assumption 1 is satisfied.

Choosing

$\begin{eqnarray} \mu(t)& = &\left(\frac{1}{1-t}\right)^{2}, \quad t\in[0, 1), \end{eqnarray}$

(4.3)

According to the design method in Section 3, we have

$\begin{eqnarray} u& = & -3\mu^{1.5}x_{1}-1.5\mu x_{2}^{3}-57\mu^{6}(x_{2}+1.5\mu x_{1})^{3} \end{eqnarray}$

(4.4)

In the practical simulation, we select the initial conditions as $x_{1}(0) = -6$ , $x_{2}(0) = 5$ . shows the responses of (4.1)–(4.4), from which we can obtain that $\lim\limits_{t\rightarrow t_{0}+T} E|x_{1}|^{2} = \lim\limits_{t\rightarrow t_{0}+T} E|x_{2}|^{2} = 0$ , which means that the controller we designed is effective.

Figure 1. The responses of closed-loop system (4.1)–(4.4).

DownLoad: Full-Size Img PowerPoint

5. Concluding remarks

This paper proposes a new design method of prescribed-time state-feedback for SHONSs. the controller we designed can guarantee that the closed-loop system has an almost surely unique strong solution and the equilibrium at the origin of the closed-loop system is prescribed-time mean-square stable. The results in this paper are more practical than those in ^{[13,14,15,16,17]} since the design in this paper permits the worker to prescribe the convergence times in advance without considering the initial conditions.

There are some related problems to investigate, e.g., how to extend the results to multi-agent systems ^[27], impulsive systems ^[28,29,30] or more general high-order systems ^{[31,32,33,34]}.

Acknowledgments

This work is funded by Shandong Provincial Natural Science Foundation for Distinguished Young Scholars, China (No. ZR2019JQ22), and Shandong Province Higher Educational Excellent Youth Innovation team, China (No. 2019KJN017).

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	Z. G. Pan, T. Basar, Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems, IEEE Trans. Autom. Control, 43 (1998), 1066–1083. https://doi.org/10.1109/9.704978 doi: 10.1109/9.704978
[2]	Z. G. Pan, T. Basar, Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion, SIAM J. Control Optim., 37 (1999), 957–995. https://doi.org/10.1137/S0363012996307059 doi: 10.1137/S0363012996307059
[3]	Z. G. Pan, Y. Liu, S. Shi, Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics, Sci. China, 44 (2001), 292–308. https://doi.org/10.1007/BF02714717 doi: 10.1007/BF02714717
[4]	H. Deng, M. Krsti $\acute{c}$ , Stochastic nonlinear stabilization, part i: a backstepping design, Syst. Control Lett., 32 (1997), 143–150. https://doi.org/10.1016/S0167-6911(97)00068-6 doi: 10.1016/S0167-6911(97)00068-6
[5]	H. Deng, M. Krsti $\acute{c}$ , Output-feedback stochastic nonlinear stabilization, IEEE Trans. Autom. Control, 44 (1999), 328–333. https://doi.org/10.1109/9.746260 doi: 10.1109/9.746260
[6]	H. Deng, M. Krsti $\acute{c}$ , Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance, Syst. Control Lett., 39 (2000), 173–182. https://doi.org/10.1016/S0167-6911(99)00084-5 doi: 10.1016/S0167-6911(99)00084-5
[7]	H. Deng, M. Krsti $\acute{c}$ , R. J. Williams, Stabilization of stochastic nonlinear driven by noise of unknown covariance, IEEE Trans. Autom. Control, 46 (2001), 1237–1253. https://doi.org/10.1109/9.940927 doi: 10.1109/9.940927
[8]	M. Krsti $\acute{c}$ , H. Deng, Stabilization of Uncertain Nonlinear Systems, Springer, New York, 1998.
[9]	W. Q. Li, L. Liu, G. Feng, Cooperative control of multiple nonlinear benchmark systems perturbed by second-order moment processes, IEEE Trans. Cybern., 50 (2020), 902–910. https://doi.org/10.1109/TCYB.2018.2869385 doi: 10.1109/TCYB.2018.2869385
[10]	W. Q. Li, M. Krsti $\acute{c}$ , Stochastic adaptive nonlinear control with filterless least-squares, IEEE Trans. Autom. Control, 66 (2021), 3893–3905. https://doi.org/10.1109/TAC.2020.3027650 doi: 10.1109/TAC.2020.3027650
[11]	W. Q. Li, X. X. Yao, M. Krsti $\acute{c}$ , Adaptive-gain observer-based stabilization of stochastic strict-feedback systems with sensor uncertainty, Automatica, 120 (2020), 109112. https://doi.org/10.1016/j.automatica.2020.109112 doi: 10.1016/j.automatica.2020.109112
[12]	W. Q. Li, M. Krsti $\acute{c}$ , Mean-nonovershooting control of stochastic nonlinear systems, IEEE Trans. Autom. Control, 66 (2021), 5756–5771. https://doi.org/10.1109/TAC.2020.3042454 doi: 10.1109/TAC.2020.3042454
[13]	X. J. Xie, N. Duan, Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system, IEEE Trans. Autom. Control, 55 (2010), 1197–1202. https://doi.org/10.1109/TAC.2010.2043004 doi: 10.1109/TAC.2010.2043004
[14]	X. J. Xie, J. Tian, State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics, Int. J. Robust Nonlinear, 17 (2007), 1343–1362. https://doi.org/10.1002/rnc.1177 doi: 10.1002/rnc.1177
[15]	W. Q. Li, X. J. Xie, S. Y. Zhang, Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions, SIAM J. Control Optim., 49 (2011), 1262–1282. https://doi.org/10.1137/100798259 doi: 10.1137/100798259
[16]	W. Q. Li, L. Liu, G. Feng, Output tracking of stochastic nonlinear systems with unstable linearization, Int. J. Robust Nonlinear, 28 (2018), 466–477. https://doi.org/10.1002/rnc.3877 doi: 10.1002/rnc.3877
[17]	R. H. Cui, X. J. Xie, Adaptive state-feedback stabilization of state-constrained stochastic high-order nonlinear systems, Sci. China Inf. Sci., 64 (2021), 200203. https://linkspringer.53yu.com/article/10.1007/s11432-021-3293-0
[18]	Y. D. Song, Y. J. Wang, J. C. Holloway, M. Krsti $\acute{c}$ , Time-varying feedback for robust regulation of normal-form nonlinear systems in prescribed finite time, Automatica, 83 (2017), 243–251. https://doi.org/10.1016/j.automatica.2017.06.008 doi: 10.1016/j.automatica.2017.06.008
[19]	Y. D. Song, Y. J. Wang, M. Krsti $\acute{c}$ , Time-varying feedback for stabilization in prescribed finite time, Int. J. Robust Nonlinear, 29 (2019), 618–633. https://doi.org/10.1002/rnc.4084 doi: 10.1002/rnc.4084
[20]	Y. J. Wang, Y. D. Song, D. J. Hill, M. Krsti $\acute{c}$ , Prescribed finite time consensus and containment control of networked multi-agent systems, IEEE Trans. Cybern., 49 (2019), 1138–1147. https://doi.org/10.1109/TCYB.2017.2788874 doi: 10.1109/TCYB.2017.2788874
[21]	J. Holloway, M. Krsti $\acute{c}$ , Prescribed-time observers for linear systems in observer canonical form, IEEE Trans. Autom. Control, 64 (2019), 3905–3912. https://doi.org/10.1109/TAC.2018.2890751 doi: 10.1109/TAC.2018.2890751
[22]	J. Holloway, M. Krsti $\acute{c}$ , Prescribed-time output feedback for linear systems in controllable canonical form, Automatica, 107 (2019), 77–85. https://doi.org/10.1016/j.automatica.2019.05.027 doi: 10.1016/j.automatica.2019.05.027
[23]	P. Krishnamurthy, F. Khorrami, M. Krsti $\acute{c}$ , Robust adaptive prescribed-time stabilization via output feedback for uncertain nonlinear strict-feedback-like systems, Eur. J. Control, 55 (2020), 14–23. https://doi.org/10.1016/j.ejcon.2019.09.005 doi: 10.1016/j.ejcon.2019.09.005
[24]	W. Q. Li, M. Krsti $\acute{c}$ , Stochastic nonlinear prescribed-time stabilization and inverse optimality, IEEE Trans. Autom. Contr., 67 (2022), 1179–1193. https://doi.org/10.1109/TAC.2021.3061646 doi: 10.1109/TAC.2021.3061646
[25]	W. Q. Li, M. Krsti $\acute{c}$ , Prescribed-time control of stochastic nonlinear systems with reduced control effort, J. Syst. Sci. Complex., 34 (2021), 1782–1800. https://doi.org/10.1007/s11424-021-1217-7 doi: 10.1007/s11424-021-1217-7
[26]	W. Q. Li, M. Krsti $\acute{c}$ , Prescribed-time output-feedback control of stochastic nonlinear systems, IEEE Trans. Autom. Control, 68 (2023). https://doi.org/10.1109/TAC.2022.3151587 doi: 10.1109/TAC.2022.3151587
[27]	W. Q. Li, L. Liu, G. Feng, Distributed output-feedback tracking of multiple nonlinear systems with unmeasurable states, IEEE Trans. Syst. Man Cybern., 51 (2021), 477–486. https://doi.org/10.1109/TSMC.2018.2875453 doi: 10.1109/TSMC.2018.2875453
[28]	X. D. Li, D. W. Ho, J. D. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 99 (2019), 361–368. https://doi.org/10.1016/j.automatica.2018.10.024 doi: 10.1016/j.automatica.2018.10.024
[29]	X. D. Li, S. J. Song, J. H. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Autom. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
[30]	X. D. Li, D. X. Peng, J. D. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
[31]	R. H. Cui, X. J. Xie, Finite-time stabilization of output-constrained stochastic high-order nonlinear systems with high-order and low-order nonlinearities, Automatica, 136 (2022), 110085. https://doi.org/10.1016/j.automatica.2021.110085 doi: 10.1016/j.automatica.2021.110085
[32]	R. H. Cui, X. J. Xie, Finite-time stabilization of stochastic low-order nonlinear systems with time-varying orders and FT-SISS inverse dynamics, Automatica, 125 (2021), 109418. https://doi.org/10.1016/j.automatica.2020.109418 doi: 10.1016/j.automatica.2020.109418
[33]	R. H. Cui, X. J. Xie, Output feedback stabilization of stochastic planar nonlinear systems with output constraint, Automatica, 143 (2022), 110471. https://doi.org/10.1016/j.automatica.2022.110471 doi: 10.1016/j.automatica.2022.110471
[34]	R. H. Cui, X. J. Xie, Adaptive state-feedback stabilization of state-constrained stochastic high-order nonlinear systems, Sci. China Inf. Sci., 64 (2021), 1–11. https://doi.org/10.1007/s11432-021-3293-0 doi: 10.1007/s11432-021-3293-0

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Mathematical Biosciences and Engineering

Prescribed-time control of stochastic high-order nonlinear systems

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Prescribed-time control of stochastic high-order nonlinear systems

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