Switched adaptive stabilization of switched nonlinearly parameterized cascade systems and its application to a two inverted pendulums

Abstract

This paper studies adaptive control of switched nonlinearly parameterized cascade systems. No solvability of the adaptive control problem for subsystems is required. By exploiting the multiple Lyapunov functions method and the parameter separation technique and the tool of adding a power integrator, we develop a new switched adaptive control approach for the explicit construction of adaptive controllers of subsystems and a proper switching law that solves the adaptive stabilization problem. A key feature of the proposed adaptive controllers is its switched property, namely, each subsystem has its individual update law. A two-inverted pendulum as a practical example is also provided to demonstrate the effectiveness of the proposed design method.On the other hand, uncertainties extensively exist in real world systems. For example, such uncertainties often appear in control systems such as those in electrical, friction dynamics, dynamics of magnetic bearing, mechanical, chemical, aeronautical, and biomedical engineering [19][20][21][22]. Adaptive control is an effective methodology, which provides adaptation mechanisms to adjust controllers for systems with parametric, structural, and environmental uncertainties to achieve desired system performance [23][24][25]. For about two decades, hence, adaptive control of non-switched nonlinear systems with parametric uncertainty has been one of the most active subjects in the field of nonlinear control [26,27]. In particular, adaptive control has proven its great capability in compensating for non-switched nonlinearly parameterized systems involving inherent nonlinearity on the basic of a parameter separation technique [28].For switched systems, adaptive control was not studied until recently focusing on switched linear systems [29-32] and piecewise affine systems [33,34]. For switched nonlinear systems with parametric uncertainty, however, few results on adaptive control have appeared by now. In [35], an adaptive neural networks (NN) control scheme for strict-feedback switched nonlinear systems is proposed. By using the CLF method and backstepping, adaptive stabilization for strict-feedback switched nonlinear systems under arbitrary switchings is achieved [36]. On the basis of the MLFs technique, an adaptive disturbance rejection problem for switched nonlinear systems in strictfeedback form with unknown exosystem is studied in [37]. Tamba and Leksono and Long and Zhao [36,37] present some effective methods of the adaptive control problem for switched nonlinear systems in strict-feedback form. It should be observed that the literature mentioned earlier has focused on adaptive control of switched nonlinear systems with linear parameterization, and a common update law is used to estimate all the vector parameters in different subsystems. Most switched nonlinear systems in practice, however, do not possess linear parameterization and a common update law for all subsystems. In addition, nonlinear parameterization can be found in various practical control systems such as machine...

Quasi‐passivity‐based adaptive stabilization for switched nonlinearly parameterized systems

“…The change of the energy of the corresponding non-quasi-passive subsystem when it is active is no faster than an exponential increment rate λ 2 , as shown in (11). (10) represents the 'excess' of quasi-passivity for i ∈ I p .…”

“…3,5,10,11,14 However, in many practical applications, asymptotic stability may be difficult to obtain due to large uncertainties. First, when all the subsystems have any same relative degree, the global practical stability is achieved by combining the recursive feedback quasipassification design technique with a switched adaptive control technique.…”

“…The asymptotic stability is often the desired performance. 3,5,10,11,14 However, in many practical applications, asymptotic stability may be difficult to obtain due to large uncertainties. From an engineering point of view, the global practical stability can be often satisfactory.…”

Adaptive control under arbitrary switching for a class of switched nonlinear systems with nonlinear parameterisation

“…When the switching signal can be regarded as a component of the control input, multiple Lyapunov functions technique will be mainly adopted to design respective controller for each switched subsystem and appropriate switching laws to stabilize the whole switched systems, see, e.g. Dong, Liu, Mei & Li (2011); Wang (2013Wang ( , 2014; Chiang & Fu (2014a) ;Yang, Jiang & Cocquempot (2014); Xiong, Lam, Shu & Mao (2014); Long & Zhao (2014) and the references therein. While once the switching happening among all the switched subsystems is determined by the inherent character of system itself and cannot be achieved by the control design, which is classified as arbitrary switching signals, common Lyapunov function method is the one appropriate choice.…”

Global adaptive finite‐time stabilization of a class of switched nonlinear systems with parametric uncertainty

“…As we know, uncertainties do exist in any real world systems. Adaptive control is one of the effective ways to deal with control systems with parametric uncertainty [29][30][31][32][33][34][35][36]. Some adaptive finite-time control results have been reported, for example, Hong et al [37] have investigated the problem of the global FTS for a class of nonlinear systems in p (being odd positive integers) normal form with parametric uncertainties; by following backstepping methodology, an adaptive finite-time control law was obtained in the form of continuous time-invariant feedback.…”