Output Regulation for a Class of MIMO Uncertain Stochastic Nonlinear Systems by Active Disturbance Rejection Control Approach

Lv, Chunwan; Huang, Guoshun; Ouyang, Zhengyong; Chen, Jian; Bao, Lingxin

doi:https://doi.org/10.1155/2019/5674212

Mathematical Problems in Engineering

On this page

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

Research Article | Open Access

Volume 2019 | Article ID 5674212 | https://doi.org/10.1155/2019/5674212

Output Regulation for a Class of MIMO Uncertain Stochastic Nonlinear Systems by Active Disturbance Rejection Control Approach

Chunwan Lv,¹Guoshun Huang,¹Zhengyong Ouyang,¹Jian Chen,¹and Lingxin Bao^2,3

Academic Editor: Gen Q. Xu

Received30 Oct 2018

Accepted08 Jan 2019

Published22 Jan 2019

Abstract

In this paper, the active disturbance rejection control (ADRC) approach is applied to a class of multi-input multioutput (MIMO) uncertain stochastic nonlinear systems. An extended state observer (ESO) is first designed for estimation of both unmeasured states and stochastic total disturbance of each subsystem which represents the total effects of internal unmodeled stochastic dynamics and external stochastic disturbance with unknown statistical property. The ADRC controller based on the states of ESO is further designed to achieve the closed-loop system’s output regulation performance including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero avoiding solving any partial differential equations in the conventional output regulation theory. Some numerical simulations are presented to demonstrate the effectiveness of the proposed ADRC approach.

1. Introduction

During the past couple of years, there have been existing representative control approaches to cope with uncertainties in controlled plants such as the internal model principle [1–3], the robust control [4], and the adaptive control [5]. Nevertheless, a majority of these control methods pay attention to the worst situation so that the design of controller is comparatively conservative. The active disturbance rejection control (ADRC), as a nontraditional control strategy, was first put forward by Han in leading paper [6]. The disturbance coped with by ADRC is much more general which can be the total coupling effects of internal unmodeled system dynamics and external disturbance. The most noteworthy feature of ADRC is that an extended state observer (ESO), as its key part, is designed to estimate the disturbance in real time so that the disturbance can be cancelled in the ESO-based feedback loop. This estimation/cancellation strategy leads to less control energy consumption in the control engineering practice [7].

Many industry practitioners have been paying more attention to the ADRC approach as presented in recent survey type paper [8]. Specific applications in different fields emerge in large numbers including synchronous motors [9], DC-DC power converters [10], control system in superconducting RF cavities [11], and flight vehicles control [12]. On the other hand, there have been lots of theoretical researches like the convergence analysis of ADRC for uncertain nonlinear systems [13–16] and the stabilization and output tracking problem for distributed parameter systems by the ADRC approach [17–21].

However, tardy theoretical progress has been made about ADRC for stochastic systems. Some progress could be found such as the practical mean square convergence analysis of ESO for the open-loop of a class of MIMO stochastic nonlinear systems [22] and the practical mean square convergence analysis of ADRC for both single-input single-output stochastic nonlinear systems [23] and lower triangular stochastic nonlinear systems [24]. The practical mean square stability of MIMO stochastic nonlinear systems by the ADRC approach is addressed in [25].

Although there have been some works on ADRC for stochastic nonlinear systems like [22, 24, 25] where the considered external disturbance is the bounded stochastic noise with unknown statistical characteristics existing widely in practical systems [26–28], the output regulation problem for this class of stochastic nonlinear systems receives less attention. In [22], the authors focus on the convergence of practical mean square estimation errors by ESO for the open-loop of systems without considering the performances of the closed-loop system under the ESO-based feedback. The practical mean square stability of the ADRC’s closed-loop systems is investigated for lower triangular stochastic nonlinear systems in [24] and MIMO stochastic nonlinear ones in [25] where the output-feedback stabilization problem is just a special case of the output regulation one in this paper. As the continuous research of [22–25], the ADRC approach is applied in the output regulation problem for a class of MIMO uncertain stochastic nonlinear systems with large stochastic uncertainties including unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains, where the output regulation performance of the resulting closed-loop systems includes practical mean square reference tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero. To be specific, in this paper we consider the output regulation problem for the partial exact feedback linearizable MIMO system [29] subject to vast stochastic uncertainties as follows:where with and , , and are the state, control, and measured output of the system, respectively; denotes the state of stochastic inverse dynamics; the functions , , and are unknown; the constants are the partially unknown control coefficients with some known nominal values satisfying Assumption (A3); is a p-dimensional standard Wiener process defined on a complete probability space with being a sample space, a -field, a filtration, and the probability measure; is the external stochastic disturbance where is an unknown bounded function satisfying Assumption (A1) and is a q-dimensional standard Wiener process defined on as well and is mutually independent with ; in addition, we denote

For each , the stochastic total disturbance of each -subsystem is defined as follows:which represents the total effects of unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains. So the stochastic uncertainties in the considered system are very complex.

For given, bounded, deterministic reference signals whose derivatives are assumed to be bounded, the control objective is to design an output-feedback control such that for any initial states the output converges practically to in mean square sense and at the same time converge practically to in mean square sense for all . The output-feedback stabilization problem for the class of MIMO uncertain stochastic nonlinear systems is covered by letting for all .

On the basis of a partial exact feedback linearizable MIMO system [29] widely investigated in control theory and the fact that stochastic uncertainties are ubiquitous in practical control engineering and often cause disadvantageous effects on control performance, the ADRC approach is addressed for the output regulation problem of system (1) including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero in this paper. It should be noticed that system (1) representing a partial exact feedback linearizable MIMO system [29] with vast stochastic uncertainties is quite general and has physical and engineering background. Firstly, the SISO nonlinear systems and MIMO nonlinear ones widely addressed by the ADRC approach in available literatures [6, 7, 13, 14, 21, 30] are covered as special cases of system (1) when is the function of time variable only: and . Secondly, the external stochastic disturbance is quite general in the sense that it is not required to know its statistical characteristics since the function can be unknown and the bounded stochastic noise investigated in [26–28] in many practical systems is also covered as its special case. Finally, system (1) covers some stochastic systems considered in the aforementioned literatures like SISO stochastic nonlinear systems in [31] when , , and .

The main contributions of this paper can be summarized as follows: (a) The ADRC approach is systematically proposed to solve the output regulation problem for a class of MIMO uncertain stochastic nonlinear systems without difficulty in solving any partial differential equations compared with the conventional output regulation theory. (b) The stochastic uncertainties dealt with by ADRC in this paper are very general including unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains. (c) Most available output-feedback controls for stochastic nonlinear systems are designed to guarantee the global asymptotic stability in probability provided that the noise vector field vanishes at the origin [32] or only the noise-to-state (or input-to-state) stability in probability [33] otherwise. In this paper, however, the practical mean square convergence is obtained by the ADRC approach without assuming that the noise vector field should be vanishing at the origin.

The rest of this paper is presented as follows. In Section 2, both ESO and ESO-based feedback control are designed for the -subsystem of (1), the assumptions of the main result are stated, and the output regulation performance of the closed-loop is summarized as Theorem 2. In Section 3, a rigorous proof of Theorem 2 is given. Finally, in Section 4, some numerical simulations are presented to illustrate the effectiveness of the proposed ADRC approach.

The following notations are used throughout this paper. denotes the -dimensional Euclidean space and represents the space of all real -matrices; for a vector or matrix , denotes its transpose; for a square matrix , denotes its trace; denotes the unit matrix; and denote the minimal and maximal eigenvalues of the symmetric real matrix , respectively; denotes the Euclidean norm of the vector and the corresponding induced norm when is a matrix; denotes an matrix with entries ; for a differentiable function , for ; for a twice differentiable function , for ; for a matrix valued function , .

2. Main Results

By analogy with [30], the one-parameter tuning linear ESO for the -subsystem of (1) is designed as follows:where are designed parameters such that the following matrix is Hurwitz: is the gain parameter to be tuned, and is the nominal value of satisfying Assumption (A3). ESO (4) is designed to estimate both stochastic total disturbance and unmeasured states by choosing suitable parameters and tuning the gain parameter . Specially, is the estimate of the stochastic total disturbance defined by (3). It should be noted that we only need to tune one parameter in ESO (4) based on the estimation accuracy and the variation of the stochastic total disturbance. Generally, the higher the estimation accuracy is needed and the faster the stochastic total disturbance varies, the larger the parameter needs to be set. Here and throughout the paper, we always drop for the solution of (4) by abuse of notation without confusion.

For all , letand

ESO (4) based output-feedback control is designed aswhere are defined in (13) and the feedback gain parameters are chosen such that the following matrix is Hurwitz:

To obtain practical mean square convergence of the closed-loop of the -subsystem of (1) under ESO (4) based output-feedback control (8) including ESO’s estimation of unmeasured states and stochastic total disturbance, practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability, the following assumptions are needed.

Assumption (A1) is about the unknown function defining the external stochastic disturbance.

Assumption (A1). is continuously differentiable and twice continuously differentiable with respect to and , respectively, and there exists a known constant such that, for all ,

Remark 1. Roughly speaking, Assumption (A1) indicates that both the external stochastic disturbance and its “variation” should be bounded which is reasonable since it is as a part of the stochastic total disturbance estimated by ESO.

Assumption (A2) is a prior assumption about the unknown functions , , and in system (1).

Assumption (A2). are continuously differentiable and twice continuously differentiable with respect to and other arguments, respectively, and are locally Lipschitz continuous in uniformly in . There exist known constants and a nonnegative continuous function such that, for all , , , , ,

Assumption (A3) is about the prior estimates for the unknown control parameters in system (1).

Assumption (A3). The matrix with entries in (4) is invertible with the inverse matrix given byand the nominal values of satisfywhere are the positive definite matrix solution satisfying .

The main result on practical mean square convergence of the closed-loop of the -subsystem of (1), (4), and (8), which includes ESO’s practical mean square estimation of both unmeasured states and stochastic total disturbance and output regulation performance, is summarized in Theorem 2.

Theorem 2. Under Assumptions (A1)–(A3) and supposing that for some constant , then the closed-loop of the -subsystem of (1), (4), and (8) is practically mean square convergent in the sense that there are a constant (specified by (32) later) and a -dependent constant with such that, for any initial values and all ,andwhere is a -independent constant.

Remark 3. It should be noticed that the practical mean square convergence addressed in this paper includes ESO’s practical mean square estimation of both unmeasured states and stochastic total disturbance and output regulation performance for the closed-loop system, but the practical mean square convergence addressed in [22] only refers to ESO’s practical mean square estimation of both unmeasured states and stochastic total disturbance for the open-loop system. In addition, there does not exist an essential difference between the upper bound of the estimation error of ESO in practical mean square sense in this paper and the one in [22] since the high gain parameter in ESO is denoted by in this paper and by in [22], respectively.

3. Proof of the Main Result

Proof of Theorem 2. For all , we setBy Itô’s formula, we can obtain thatwhere we setBy Assumptions (A1)-(A2), we can easily conclude that there exist -independent positive constants such thatMoreover,Suppose that . It follows that there exist -independent positive constants such thatwhereThus, it follows that the closed-loop of the -subsystem of (1), (4), and (8) is equivalent towhere is defined as that in (18). It follows from the definition of in (18) thatThe remaining proof is arranged in the following three steps.
Step 1. We prove that the solution of system (26) is practically mean square bounded.
We first define the positive definite functions , , and as follows:where and are the positive definite matrix solutions satisfying the Lyapunov equations and , respectively.
It is easy to obtain thatApply Itô’s formula to with respect to along the solution of system (26) to obtainIt is easy to conclude that there exist and such thatwhere is given in (14). Now we suppose thatSetIt follows from (22), (24), (27), (31), and Young’s inequality thatwhere and is given in (14). It is easy to conclude that for some -independent constant . Hence for any and any , there exists such that, for all , we havefor some -independent constant .
Step 2. We prove the convergence of the estimation errors of ESO for both unmeasured states and stochastic total disturbance in practical mean square sense.
Similar to the operations used in (34), it follows from (35) that, for all , we haveSetThenWe can see from (35) that the first term of the right-hand side of (38) is bounded by multiplied by a -independent constant and the second term is bounded by multiplied by a -independent constant when ; thus there exists a -independent constant such that, for all ,and thusTherefore, for all , , we haveStep 3. We prove the convergence of the reference signals tracking errors in practical mean square.
For any and all , it follows from (34) and (40) thatwhere we setandTherefore, for any and all , we haveIt follows from (35) that the first term of the right-hand side of (45) is bounded by multiplied by a -independent constant and the second term is bounded by multiplied by a -independent constant when for all so that there exist and such that, for all , we haveThis completes the proof of Theorem 2.

4. Numerical Simulations

In this section, we aim to verify the validity of the ADRC approach by considering the following MIMO uncertain stochastic system:where are unknown parameters satisfying for a given known constant . is a bounded nonwhite noise that exists in many practical dynamical systems like the motion of oscillators [26, 27], where and denote the central frequency and strength of frequency disturbance, respectively. In this case, . In addition, the constants and the function in Assumptions (A1)-(A2) can be specified as , , , , and . We can easily check that Assumptions (A1)-(A2) hold in this case.

The stochastic total disturbance is defined byAn ESO (49) is designed for system (47) as follows:In this case the corresponding matrices in (5) becomewhere all eigenvalues are -1 and thus these two matrices are Hurwitz. Choose in (7) as follows:with the corresponding matrices in (9)being Hurwitz. ESO (49) based feedback control is designed aswhere in this case in (8) are specified as . Both system (47) and system (49) are discretized by the Milstein approximation method proposed in [34]. In Figures 1 and 2, the initial values arethe time discrete step is taken asand the gain in (49) is taken asThe uncertain parameters and reference signals in Figures 1 and 2 are taken asandrespectively.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

It is observed from both Figures 1 and 2 that ESO (49) is very valid to estimate both the state and the stochastic total disturbance defined in (48), despite varying system parameters and reference signals between (57) and (58). In addition, it is seen from Figures 1 and 2 that the tracking effects of to and to are very satisfactory. Figure 2 shows that the estimation effects of ESO and the performance of reference signals tracking are still very effective regardless of the larger parameters than the one given in (57), which validates the good robust performance of ADRC.

5. Concluding Remarks

In this paper, the active disturbance rejection control (ADRC) approach is applied to the output regulation problem for a class of multi-input multioutput (MIMO) uncertain stochastic nonlinear systems subject to vast stochastic uncertainties. The stochastic uncertainties of each subsystem including unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains are first regarded as the stochastic total disturbance of each subsystem. The stochastic total disturbance is then estimated by a linear ESO in real time and cancelled in the ESO-based output feedback loop. The output regulation performance of the resulting ADRC’s closed-loop system is obtained with rigorous theoretical proof including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero. Finally, the validity of the estimation performance of ESO, efficient reference tracking of ADRC, and good robust performance of ADRC are shown by some numerical simulations.

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 Natural Science Foundation of China (Grant Nos. 11801077, 11501108, 11771151, and 11501106), the Natural Science Foundation of Guangdong Province (Grant Nos. 2015A030313636, 2016A030313835, and 2018A030310357), the Project of the Department of Education of Guangdong Province (Grant No. 2017KTSCX191), the Research Fund for Distinguished Young Scholars of Fujian Agriculture and Forestry University (Grant No. XJQ201611), and the Training Programme Foundation for Excellent Young Scholars of Guangdong Province (Grant No. YQ2015164).

References

B. A. Francis and W. M. Wonham, “The internal model principle of control theory,” Automatica, vol. 12, no. 5, pp. 457–465, 1976.
View at: Publisher Site | Google Scholar | MathSciNet
J. Huang, Nonlinear Output Regulation: Theory and Applications, vol. 8 of Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2004.
View at: Publisher Site | MathSciNet
L. Liu, Z. Chen, and J. Huang, “Parameter convergence and minimal internal model with an adaptive output regulation problem,” Automatica, vol. 45, no. 5, pp. 1306–1311, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
Z. H. Qu, Robust Control of Nonlinear Uncertain Systems, John Wiley & Sons, Inc., New York, NY, USA, 1998.
View at: MathSciNet
M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc, New York, NY, USA, 1995.
J. Q. Han, “From PID to active disturbance rejection control,” IEEE Transactions on Industrial Electronics, vol. 56, no. 3, pp. 900–906, 2009.
View at: Publisher Site | Google Scholar
Q. Zheng, L. Q. Gao, and Z. Gao, “On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics,” in Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), pp. 3501–3506, New Orleans, La, USA, December 2007.
View at: Publisher Site | Google Scholar
Z. Gao, “On the centrality of disturbance rejection in automatic control,” ISA Transactions®, vol. 53, no. 4, pp. 850–857, 2014.
View at: Publisher Site | Google Scholar
H. Sira-Ramírez, J. Linares-Flores, C. García-Rodríguez, and M. A. Contreras-Ordaz, “On the control of the permanent magnet synchronous motor: an active disturbance rejection control approach,” IEEE Transactions on Control Systems Technology, vol. 22, no. 5, pp. 2056–2063, 2014.
View at: Publisher Site | Google Scholar
B. Sun and Z. Gao, “A DSP-based active disturbance rejection control design for a 1-kW H-bridge DC-DC power converter,” IEEE Transactions on Industrial Electronics, vol. 52, no. 5, pp. 1271–1277, 2005.
View at: Publisher Site | Google Scholar
J. Vincent, D. Morris, N. Usher et al., “On active disturbance rejection based control design for superconducting RF cavities,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 643, no. 1, pp. 11–16, 2011.
View at: Publisher Site | Google Scholar
Y. Xia and M. Fu, Compound Control Methodology for Flight Vehicles, Springer-Verlag, Berlin, Germany, 2013.
B.-Z. Guo and Z.-L. Zhao, “On the convergence of an extended state observer for nonlinear systems with uncertainty,” Systems & Control Letters, vol. 60, no. 6, pp. 420–430, 2011.
View at: Publisher Site | Google Scholar
B. Guo and Z. Zhao, “On convergence of the nonlinear active disturbance rejection control for MIMO systems,” SIAM Journal on Control and Optimization, vol. 51, no. 2, pp. 1727–1757, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-L. Zhao and B.-Z. Guo, “A novel extended state observer for output tracking of {MIMO} systems with mismatched uncertainty,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 63, no. 1, pp. 211–218, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-H. Wu and B.-Z. Guo, “Approximate decoupling and output tracking for {MIMO} nonlinear systems with mismatched uncertainties via {ADRC} approach,” Journal of The Franklin Institute, vol. 355, no. 9, pp. 3873–3894, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
B.-Z. Guo and H.-C. Zhou, “The active disturbance rejection control to stabilization for multi-dimensional wave equation with boundary control matched disturbance,” IEEE Transactions on Automatic Control, vol. 60, no. 1, pp. 143–157, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
H. Zhou, “Output-based disturbance rejection control for 1-D anti-stable Schr\"Odinger equation with boundary input matched unknown disturbance,” International Journal of Robust and Nonlinear Control, vol. 27, no. 18, pp. 4686–4705, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H.-C. Zhou and H. Feng, “Disturbance estimator based output feedback exponential stabilization for Euler-Bernoulli beam equation with boundary control,” Automatica, vol. 91, pp. 79–88, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
H. Feng and B.-Z. Guo, “A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 62, no. 8, pp. 3774–3787, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
H. Feng and B.-Z. Guo, “Active disturbance rejection control: Old and new results,” Annual Reviews in Control, vol. 44, pp. 238–248, 2017.
View at: Publisher Site | Google Scholar
Z. H. Wu and B. Z. Guo, “Extended state observer for MIMO nonlinear systems with stochastic uncertainties,” International Journal of Control, Article ID 1475750, In press.
View at: Google Scholar
Z. H. Wu and B. Z. Guo, “On convergence of active disturbance rejection control for a class of uncertain stochastic nonlinear systems,” International Journal of Control, in Press.
View at: Publisher Site | Google Scholar
B.-Z. Guo and Z.-H. Wu, “Active disturbance rejection control approach to output-feedback stabilization of lower triangular nonlinear systems with stochastic uncertainty,” International Journal of Robust and Nonlinear Control, vol. 27, no. 16, pp. 2773–2797, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
Z. H. Wu and B. Z. Guo, “Active disturbance rejection control to MIMO nonlinear systems with stochastic uncertainties: approximate decoupling and output-feedback stabilisation,” International Journal of Control, in Press.
View at: Google Scholar
Z. L. Huang, W. Q. Zhu, Y. Q. Ni, and J. M. Ko, “Stochastic averaging of strongly non-linear oscillators under bounded noise excitation,” Journal of Sound and Vibration, vol. 254, no. 2, pp. 245–267, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
F. Hu, L. C. Chen, and W. Q. Zhu, “Stationary response of strongly non-linear oscillator with fractional derivative damping under bounded noise excitation,” International Journal of Non-Linear Mechanics, vol. 47, no. 10, pp. 1081–1087, 2012.
View at: Publisher Site | Google Scholar
Z. L. Huang and W. Q. Zhu, “Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations,” Probabilistic Engineering Mechanics, vol. 19, no. 3, pp. 219–228, 2004.
View at: Publisher Site | Google Scholar
A. Isidori, Nonlinear Control Systems, Springer-Verlag, London, 3rd edition, 1995.
Z. Gao, “Scaling and bandwidth-parameterization based controller tuning,” in Proceedings of the American Control Conference, pp. 4989–4996, Denver, Colo, USA, June 2003.
View at: Google Scholar
B.-Z. Guo, Z.-H. Wu, and H.-C. Zhou, “Active disturbance rejection control approach to output-feedback stabilization of a class of uncertain nonlinear systems subject to stochastic disturbance,” IEEE Transactions on Automatic Control, vol. 61, no. 6, pp. 1613–1618, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
H. Deng and M. Krstić, “Output-feedback stochastic nonlinear stabilization,” IEEE Transactions on Automatic Control, vol. 44, no. 2, pp. 328–333, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
H. Deng and M. Krstić, “Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” Systems & Control Letters, vol. 39, no. 3, pp. 173–182, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2019 Chunwan Lv et al. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

774

Downloads

867

Citations