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Adaptive tracking control for a class of uncertain switched stochastic nonlinear systems
 Xikui Liu^{1},
 Yanxin Li^{1} and
 Yan Li^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s136620191975x
© The Author(s) 2019
 Received: 4 July 2018
 Accepted: 16 January 2019
 Published: 28 January 2019
Abstract
This paper pursues an adaptive fuzzy control scheme for a class of nonlinear systems with stochastic switching. A general controller and adaptive mechanism are designed by utilizing Lyapunov function approach and backstepping technique. It is demonstrated that the presented control method can guarantee that all the signals in the closedloop system are semiglobally uniformly ultimately bounded (SUUB) and the tracking error is convergent to a neighborhood of the origin. Finally, the simulation results verify the feasibility of the control strategy presented in this paper.
Keywords
 Adaptive fuzzy control
 Stochastic switched nonlinear systems
 Backstepping technique
1 Introduction
As is well known, a common Lyapunov function (CLF) guarantees that the switched system is stable under arbitrary switching [1]. As an impactful approach for the stability analysis, CLF has been widely employed for control synthesis of switched linear systems [2–6]. For instance, [7] used the classic quadratic Lyapunov function and solved the stabilization problem for a class of stochastic nonlinear strictfeedback systems. Based on CLF method for a class of switched nonlinear systems, [8, 9] have investigated three state feedback control methods; however, the nonlinear functions of the above control systems are known. Additionally, the backstepping technique is used for the global stabilization problem for switched nonlinear systems in strictfeedback form under arbitrary switchings [8]. The adaptive backstepping approach is a recursive design methodology for controller design. It constructs associated Lyapunov functions and feedback control laws, and its main purpose is to design the adaptive laws and virtual control functions to counteract the unknown nonlinearity of system [10]. In recent years, in view of several classes of switched nonlinear systems, some backstepping control design methods have been proposed. Nevertheless, few of them take into account the uncertainties that exist extensively in practical switched nonlinear systems [6, 11].
In the last few decades, as a typical hybrid system, switched systems have been a great concern with their increasing significance in engineering practice, such as multiagent systems, aircraftcontrol systems and circuit and power systems [12, 13]. Switched systems present switching between a set of subsystems resting with changing environmental factors. The system detects and breaks down various parameters in the changing environment, and then switches to the subsystem matching with the environment. So far, the controller design and stability analysis of switched systems have proposed remarkable results [2, 14–20]. In the actual control system, the dynamic characteristics of controlled objects such as production process, production equipment and transmission system are difficult to describe by accurate mathematical model. With the change of working environment, the components of the control system may be aged or damaged, and the characteristics of the controlled object also change. All these factors lead to some inevitable errors between the mathematical model of the controlled object and the actual object. For example, in large power systems, due to the large dimension, many systems contain unmodeled dynamic, uncertain parameters and random noise. In the actual operation, the system will also be affected by various harmonic and load disturbances. These random uncertainties of the power system bring about security risks to the normal operation of the power system. All of the above systems can be described by switching system. Since stochastic switched systems integrate the characteristics and difficulties of stochastic systems and switched systems, it is very difficult to analyze the stability and application of stochastic switched systems.
Obviously, stochastic disturbance is considered as one of the unstable sources of control systems which usually exists in many practical systems [21–23]. So, for a deterministic nonlinear system, the control for stochastic nonlinear system is much more difficult. Therefore, the research on control design and stability analysis for nonlinear stochastic systems is a significant and challenging subject, and it has been a topic of great concern in the last few years [24–28]. Specifically, some control methods based on adaptive backstepping technique for deterministic nonlinear systems [29–32] have been successfully generalized to nonlinear stochastic systems [33–39]. For instance, an outputfeedback backstepping controller was developed for a class of stochastic nonlinear systems in [40], the state feedback controller is designed for nonlinear stochastic systems with Markovian switching [41] and [42] presented the backstepping control design approaches. Nevertheless, these methods are only suitable for those nonlinear stochastic systems with known nonlinear dynamic models. Adaptive outputfeedback control methods for a class of uncertain nonlinear stochastic systems were proposed by utilizing the fuzzy logic system (FLS) and the stability of the control systems was discussed in [35]. The results of [35] were extended to a class of uncertain largescale nonlinear stochastic systems. The approaches [17, 20] decreased the adjustable parameters. The presented controller in [43] has a simple structure because the unknown virtual control signals were directly approximated via FLS. From the above, adaptive fuzzy control approach plays an important role in dealing with uncertain nonlinear systems.
 1.
This paper studies the tracking control problem of switched nonlinear uncertain systems, which is different from the available methods on switched nonlinear systems. The stochastic disturbance is considered and all system functions studied in this paper are unknown completely. Therefore, compared with existing work, the controlled system is more general and the control design is more challenging.
 2.
There are two kinds of adaptive fuzzy backstepping control approaches proposed in this paper for a class of switched nonlinear uncertain systems. We propose a design approach with multiple adaptive laws in the first place. After that, another approach with only one adaptive law is presented in order to avoid too many parameters. In addition, we use the norm of the unknown weight vector of FLS basis function rather than the weight vector elements themselves as the estimated parameter at each step, which significantly reduces the number of adaptive parameters. Therefore, the presented control design approach becomes more practical to use.
The remainder of manuscript is organized as follows. The preliminaries and problem formulation are addressed in Sect. 2. A novel adaptive fuzzy control scheme is introduced in Sect. 3. A simulation example is developed in Sect. 4, Finally, conclusions are given in Sect. 5.
2 Preliminaries and problem formulation
The following notations are used in this paper. \(R_{+}\) means the set of all nonnegative real numbers, \(R^{n}\) represents the real ndimensional space, and \(R^{n\times r}\) stands for the set of all \(n\times r\) real matrices. \(\Vert X \Vert \) indicates the Euclidean norm of a vector x. \(C^{2,1}\) represents the set of all the functions \(V(x,t)\) which belong to \(C^{2}\) with respect to x and belong to \(C^{1}\) with respect to t. \(\operatorname{Tr}(A)\) means a trace of the matrix A.
2.1 Stochastic stability
Definition 1
([44])
Remark 1
The term \(\frac{1}{2}\operatorname{Tr} \{ h^{T}\frac{\partial ^{2} \mathcal{V}}{ \partial x^{2}}h \}\) is called Itô correction term, \(\frac{\partial ^{2} \mathcal{V}}{\partial x^{2}}\) will be more difficult to construct the common virtual control function and the unified adaptive mechanism for uncertain switched stochastic systems than that of deterministic system.
Lemma 1
([45])
Lemma 2
([34])
Lemma 3
(Young’s inequality [46])
2.2 Problem formulation
Assumption 1
([47])
The tracking target \(y_{v}(t)\) and its time derivatives up to the nth order are continuous and bounded.
Remark 2
When we do not consider the unknown functions and the tracking control problem, system (3) will be reduced to system (1) in [17], So, the system studied in this note is more general.
Assumption 2
([48])
Remark 3
In the existing researches on purefeedback nonlinear systems, it is usually considered that the sign of \(h_{n,r}u_{\tau (t)}\) is known. Therefore, Assumption 2 is reasonable, it is a meaningful work for the stochastic nonlinear systems.
2.3 Fuzzy logic systems
In the process of controller design and stability analysis, the FLS is adopted in order to approximate the unknown functions.
 \(R_{j}\)::

IF \(\bar{x}_{1}\) is \(\varGamma _{1}^{j}\) and … and \(\bar{x}_{n}\) is \(\varGamma _{n}^{j}\), then y is \(P^{j}\), \(j=1,2,\ldots,\aleph \),
Lemma 4
([49])
Remark 4
Lemma 4 shows that real continuous function \(f(\bar{x})\) can be expressed as a linear combination of bounded error ϵbased function vectors \(\zeta (\bar{x})\). That is, \(f(\bar{x})=\Im ^{T} \zeta (\bar{x})+\xi (\epsilon )\), \(\vert \xi (\epsilon ) \vert < \epsilon \), it plays an important role in the whole process of adaptive laws design. It is noted that \(0<\zeta ^{T}\zeta \leq 1\).
3 Main results
In this section, the adaptive fuzzy control scheme of system (3) is proposed by combining the FLS with adaptive backstepping technique and CLF approach. In Sect. 3.1, a specific design process will be given. In each step, we will design a virtual control function \(\sigma _{i}\) via using a proper CLF \(V_{i}\), and the control law \(u_{k}\) will finally be designed. In Sect. 3.2, in order to avoid repetition, a final CLF will be only adopted to prove the design procedure.
3.1 Adaptive control design under multiple adaptive laws
Define: \(\bar{y}_{v}^{(t)}= [ y_{v},y_{v}^{(1)},\ldots,y_{v} ^{(j)} ] ^{T}\), \(j=1,2,\ldots, n\), with \(y_{v}^{(j)}\) denoting the jth derivative of \(y_{v}\).
At step j of the design process, the unknown function \(\hat{f}_{j,r} \) is approximated by a FLS \(\Im _{j,r}(x_{j})\). For this purpose, define a constant \(\varsigma _{j}=\frac{ \Vert \Im _{j,r} \Vert ^{2}}{b _{k}}\), \(j=1,2,\ldots,n\), denote \(\hat{\varsigma }_{j}\) as the estimation of \(\varsigma _{j}\), and the estimation error is \(\tilde{\varsigma } _{j}=\varsigma _{j}\hat{\varsigma }_{j}\).
Now, we give detailed backstepping design process in the following steps.
Remark 5
Note that the FLS is directly used to approximate unknown nonlinear function \(\hat{f}_{1,r}\) rather than only the unknown function \(f_{1,r}\). This method will be used in the remaining design steps.
Theorem 1
Consider a class switched stochastic nonlinear system (3), under Assumptions 1 and 2, for bounded initial conditions, parameter adaptive laws (35), the control law (46), and the intermediate control signals (47), guarantee that all the signals in the closedloop system are SUUB and the tracking error is convergent to a neighborhood of the origin.
Remark 6
In [43], the adaptive tracking problem for a class of switched nonlinear systems was investigated. By combining the backstepping technique with the approximation scheme of FLS, a design approach with multiple adaptive laws was developed. In this paper, Theorem 1 generalizes the result of Theorem 1 in [43]. Considering the stochastic disturbances, the systems in this paper are more common.
3.2 Adaptive control design under one adaptive law
Theorem 2
Consider a class switched stochastic nonlinear system (3), under Assumptions 1 and 2, for bounded initial conditions, the control law (60), and the intermediate control signals (61), guarantee that all the signals in the closedloop system are SUUB and the tracking error converging to a neighborhood of the origin.
Remark 7
In [43], the adaptive tracking problem for a class of switched nonlinear systems was investigated. By combining the backstepping technique with the approximation approach of FLS, a design scheme with only one adaptive laws was developed. In this paper, it is noted that Theorem 2 generalizes the result of Theorem 2 in [43].
4 Simulation example
In this section, a simulation example is proposed in order to certify the control performance and the feasibility of the presented method in the previous sections.
Example 1
Remark 8
Example 2
5 Conclusion
This paper studied the adaptive tracking control problem for a class of stochastic nonlinear systems under arbitrary switchings. It was noted that the nonlinear functions and stochastic disturbances of the system were completely unknown. For the sake of releasing the computational burden, the unknown nonlinear function of the system was estimated by employing the approximation property of FLS, then the adaptive backstepping technique was used to construct a class of adaptive fuzzy control. Under arbitrary switching conditions, the presented controller could ensure that all the signals in the closedloop system remained bounded in probability and the system output converged to a small neighborhood of the reference signal. Finally, simulation results further showed the effectiveness of the proposed approaches.
Declarations
Acknowledgements
We are thankful to the reviewers for their useful corrections and suggestions, which improved the quality of this paper.
Funding
This research work has been supported financially by the National Natural Science Foundation of China (Grant no. 61402265, 61573227), Shandong Provincial Natural Science Foundation (No. ZR2018MF013, ZR2016FM48), the Research Fund for the Taishan Scholar Project of Shandong Province of China, SDUST Research Fund (No. 2015TDJH105) and the Fund for Postdoctoral Application Research Project of Qingdao (01020120607).
Authors’ contributions
The main idea of this paper was proposed by the first and last authors, while the second and last authors reviewed and modified the paper. Furthermore, all authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Vu, L., Liberzon, D.: Common Lyapunov functions for families of commuting nonlinear systems. Syst. Control Lett. 54, 405–416 (2005) MathSciNetView ArticleGoogle Scholar
 Huang, L.T., Li, Y.M., Tong, S.C.: Fuzzy adaptive output feedback control for a class of switched nontriangular structure nonlinear systems with timevarying delays. Int. J. Syst. Sci. 49(1), 132–146 (2018) MathSciNetView ArticleGoogle Scholar
 Briat, C., Seuret, A.: Convex dwelltime characterizations for uncertain linear impulsive systems. IEEE Trans. Autom. Control 57(12), 3241–3246 (2012) MathSciNetView ArticleGoogle Scholar
 Briat, C., Seuret, A.: Affine characterizations of minimal and modedependent dwelltimes for uncertain linear switched systems. IEEE Trans. Autom. Control 58(5), 1304–1310 (2013) MathSciNetView ArticleGoogle Scholar
 Xiang, W., Xiao, J.: Stabilization of switched continuoustime systems with all modes unstable via dwelltime switching. Automatica 50(2), 940–945 (2014) MathSciNetView ArticleGoogle Scholar
 Liu, Y.J., Lu, S., Tong, S., Chen, X.: Adaptive controlbased Barrier Lyapunov functions for a class of stochastic nonlinear systems with full state constraints. Automatica 87, 83–93 (2018) MathSciNetView ArticleGoogle Scholar
 Pan, Z.G., Basar, T.: Adaptive controller design for tracking and disturbance attenuation in parametric strictfeedback nonlinear systems. IEEE Trans. Autom. Control 43(8), 1066–1083 (1998) MathSciNetView ArticleGoogle Scholar
 Ma, R.C., Zhao, J.: Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings. Automatica 46(11), 1819–1823 (2010) MathSciNetView ArticleGoogle Scholar
 Niu, B., Zhao, J.: Tracking control for outputconstrained nonlinear switched systems with a barrier Lyapunov function. Int. J. Syst. Sci. 44(5), 978–985 (2013) MathSciNetView ArticleGoogle Scholar
 Krstic, M., Kanellakopoulos, I., Kokotovic, P.V.: Nonlinear and Adaptive Control Design. Wiley, New York (1995) MATHGoogle Scholar
 Long, L.J., Zhao, J.: \(H_{\infty }\) control of switched nonlinear systems in pnormal form using multiple Lyapunov functions. IEEE Trans. Autom. Control 57, 1285–1291 (2012) MathSciNetView ArticleGoogle Scholar
 Liberzon, D., Morse, A.S.: Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19(5), 59–70 (1999) View ArticleGoogle Scholar
 Homaee, O., Zakariazadeh, A., Jadid, S.: Realtime voltage control algorithm with switched capacitors in smart distribution system in presence of renewable generations. Int. J. Electr. Power Energy Syst. 54(1), 187–197 (2014) View ArticleGoogle Scholar
 Yang, H., Jiang, B., Cocquempot, V.: A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes. Nonlinear Anal. Hybrid Syst. 13, 45–60 (2014) MathSciNetView ArticleGoogle Scholar
 Wang, M., Zhao, J.: Quadratic stabilization of a class of switched nonlinear systems via single Lyapunov function. Nonlinear Anal. Hybrid Syst. 4(1), 44–53 (2010) MathSciNetView ArticleGoogle Scholar
 Xi, C.J., Zhai, D., Dong, J.X., Zhang, Q.L.: Approximationbased adaptive fuzzy tracking control for a class of switched nonlinear purefeedback systems. Int. J. Syst. Sci. 48(12), 2463–2472 (2017) MathSciNetView ArticleGoogle Scholar
 Si, W.J., Dong, X.D.: Adaptive neural DSC for stochastic nonlinear constrained systems under arbitrary switchings. Nonlinear Dyn. 12, 1–14 (2017) MathSciNetMATHGoogle Scholar
 Zhao, X.D., Liu, X.W., Yin, S., Li, H.Y.: Improved results on stability of continuoustime switched positive linear systems. Automatica 50, 614–621 (2014) MathSciNetView ArticleGoogle Scholar
 Zhao, X.D., Yin, S., Li, H.Y., Niu, B.: Switching stabilization for a class of slowly switched systems. IEEE Trans. Autom. Control 60(1), 221–226 (2015) MathSciNetView ArticleGoogle Scholar
 Zhao, X.D., Zhang, L.X., Shi, P., Liu, M.: Stability and stabilization of switched linear systems with modedependent average dwell time. IEEE Trans. Autom. Control 57, 1809–1815 (2012) MathSciNetView ArticleGoogle Scholar
 Liu, X.K., Li, Y., Zhang, W.H.: Stochastic linear quadratic optimal control with constraint for discretetime systems. Appl. Math. Comput. 228, 264–270 (2004) MathSciNetMATHGoogle Scholar
 Li, Y., Liu, X.K.: \(H_{}\) index for nonlinear stochastic systems with state and input dependent noises. Int. J. Fuzzy Syst. 20, 759–768 (2018) MathSciNetView ArticleGoogle Scholar
 Li, Y., Zhang, W.H., Liu, X.K.: \(H_{}\) index for discretetime stochastic systems with Markovian jump and multiplicative noise. Automatica 90, 268–293 (2018) MathSciNetGoogle Scholar
 Liu, Y.L., Ma, H.J.: Adaptive fuzzy tracking control of nonlinear switched stochastic systems with prescribed performance and unknown control directions. IEEE Trans. Syst. Man Cybern. 99, 1–10 (2017) Google Scholar
 Yin, Y., Shi, P., Liu, F.: Gainscheduled robust fault detection on timedelay stochastic nonlinear systems. IEEE Trans. Ind. Electron. 58(10), 4908–4916 (2011) View ArticleGoogle Scholar
 Li, W., Jing, Y., Zhang, S.: Adaptive statefeedback stabilization for a large class of highorder stochastic nonlinear systems. Automatica 47(4), 819–828 (2011) MathSciNetView ArticleGoogle Scholar
 Zhou, Q., Shi, P.: A new approach to networkbased \(H_{\infty }\) control for stochastic systems. Int. J. Robust Nonlinear Control 22(9), 1036–1059 (2012) MathSciNetView ArticleGoogle Scholar
 Wang, H., Liu, P.X., Niu, B.: Robust fuzzy adaptive tracking control for nonaffine stochastic nonlinear switching systems. IEEE Trans. Cybern. 99, 1–10 (2018) Google Scholar
 Liu, Z., Wang, F., Zhang, Y., Chen, X., Philip Chen, C.L.: Adaptive tracking control for a class of nonlinear systems with fuzzy deadzone input. IEEE Trans. Syst. Man Cybern. 23(1), 193–204 (2015) Google Scholar
 Wang, F., Liu, Z., Lai, G.Y.: Fuzzy adaptive control of nonlinear uncertain plants with unknown dead zone output. Fuzzy Sets Syst. 263(1), 27–48 (2015) MathSciNetView ArticleGoogle Scholar
 Liu, Y.J., Tong, S.C.: Adaptive neural network tracking control of uncertain nonlinear discretetime systems with nonaffine deadzone input. IEEE Trans. Syst. Man Cybern. 45(3), 497–505 (2015) Google Scholar
 Liu, Y.J., Tong, S.C.: Adaptive fuzzy control for a class of nonlinear discretetime systems with backlash. IEEE Trans. Fuzzy Syst. 22(5), 1359–1365 (2014) View ArticleGoogle Scholar
 Zhang, H., Xia, Y.: Adaptive tracking control for a class of stochastic switched systems. Int. J. Control 91(6), 1–16 (2018) MathSciNetMATHGoogle Scholar
 Tong, S.C., Li, Y., Li, Y.M., Liu, Y.J.: Observerbased adaptive fuzzy backstepping control for a class of stochastic nonlinear strictfeedback systems. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 41(6), 1693–1704 (2011) View ArticleGoogle Scholar
 Li, J., Chen, W.S., Li, J.M.: Adaptive NN outputfeedback decentralized stabilization for a class of largescale stochastic nonlinear strictfeedback systems. Int. J. Robust Nonlinear Control 21(4), 452–472 (2011) MathSciNetView ArticleGoogle Scholar
 Shen, Y., Han, Y., Shahnazi, R., Haghani, A.: Fuzzy adaptive tracking control of constrained nonlinear switched stochastic purefeedback systems. IEEE Trans. Cybern. 47(3), 579–588 (2017) View ArticleGoogle Scholar
 Si, W.J., Dong, X.D.: Adaptive neural control for MIMO stochastic nonlinear purefeedback systems with input saturation and fullstate constraints. Neurocomputing 275, 298–307 (2018) View ArticleGoogle Scholar
 Ma, H., Liu, Y., Ye, D.: Adaptive output feedback tracking control for nonlinear switched stochastic systems with unknown control directions. IET Control Theory Appl. 12(4), 484–494 (2018) MathSciNetView ArticleGoogle Scholar
 Han, J., Zhang, H., Wang, Y., Zhang, K.: Fault estimation and faulttolerant control for switched fuzzy stochastic fystems. IEEE Trans. Fuzzy Syst. 99, 1 (2018) Google Scholar
 Deng, H., Krstic, M.: Outputfeedback stochastic nonlinear stabilization. IEEE Trans. Autom. Control 44(2), 328–333 (1999) MathSciNetView ArticleGoogle Scholar
 Wu, Z., Xie, X., Shi, P.: Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching. Automatica 45(4), 997–1004 (2009) MathSciNetView ArticleGoogle Scholar
 Xia, Y., Fu, M., Shi, P.: Adaptive backstepping controller design for stochastic jump systems. IEEE Trans. Autom. Control 54(12), 2853–2859 (2009) MathSciNetView ArticleGoogle Scholar
 Zhao, X., Zheng, X., Niu, B., Liu, L.: Adaptive tracking control for a class of uncertain switched nonlinear systems. Automatica 52, 185–191 (2015) MathSciNetView ArticleGoogle Scholar
 Liu, S.J., Zhang, J.F., Jiang, Z.P.: Decentralized adaptive outputfeedback stabilization for largescale stochastic nonlinear systems. Automatica 43(2), 238–251 (2007) MathSciNetView ArticleGoogle Scholar
 Wang, M., Zhang, S.Y., Chen, B., Luo, F.: Direct adaptive neural control for stabilization of nonlinear timedelay systems. Sci. China Inf. Sci. 53(4), 800–812 (2010) MathSciNetView ArticleGoogle Scholar
 Wang, H.Q., Chen, B., Liu, X.P., Liu, K.F., Lin, C.: Robust adaptive fuzzy tracking control for purefeedback stochastic nonlinear systems with input constraints. IEEE Trans. Cybern. 43(6), 2093–2104 (2013) View ArticleGoogle Scholar
 Wang, F., Liu, Z., Zhang, Y., Philip Chen, C.L.: Adaptive fuzzy control for a class of stochastic purefeedback nonlinear systems with unknown hysteresis. IEEE Trans. Fuzzy Syst. 24(1), 140–152 (2016) View ArticleGoogle Scholar
 Yu, Z.X., Li, S.G., Du, H.B.: Razumikhin–Nussbaumlemmabased adaptive neural control for uncertain stochastic purefeedback nonlinear systems with timevarying delays. Int. J. Robust Nonlinear Control 23(11), 1214–1239 (2013) MathSciNetView ArticleGoogle Scholar
 Chen, B., Liu, X., Liu, K., Lin, C.: Direct adaptive fuzzy control of nonlinear strictfeedback systems. Automatica 45(6), 1530–1535 (2009) MathSciNetView ArticleGoogle Scholar
 Wang, H.Q., Chen, B., Liu, K.F., Liu, X.P., Lin, C.: Adaptive neural tracking control for a class of nonstrictfeedback stochastic nonlinear systems with unknown backlashlike hysteresis. IEEE Trans. Neural Netw. 25(5), 947–958 (2014) View ArticleGoogle Scholar
 Ma, R.C., Zhao, J.: Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings. Automatica 46, 1819–1823 (2010) MathSciNetView ArticleGoogle Scholar