Model reference adaptive control for nonlinear switched systems under asynchronous switching

Abstract

In this paper, the problem of model reference adaptive control for nonlinear switched systems with parametric uncertainties is investigated. Asynchronous switching between subsystems and adaptive controllers is also considered. Firstly, a state feedback adaptive controller is designed. Then, sufficient conditions ensuring the global practical stability of the error switched system with average dwell time are proposed. The boundedness of all signals in the closed-loop system is guaranteed by the proposed adaptive controller. Finally, a practical example is given to demonstrate the validity of the main results.(MRAC) philosophy to piecewise smooth systems. For MIMO dynamical systems with partially known dynamics, [15] proposed the composite model reference adaptive control tracking design technique to improve tracking performance.Uncertainties also exist extensively in switched systems, which brings considerable difficulties for researchers. The main problem is that we need to compare the values of the adjacent Lyapunov functions at the switching instants. However, for switched systems with parametric uncertainties, as parametric estimation errors exist in Lyapunov functions, it is impossible to exactly know the values of the Lyapunov functions at a switching instant, which makes the existing stability conditions inapplicable. Some attempt has been made to overcome this obstacle. By the parameter projection adaptive law, a small tracking error is achieved based on the dwell time method [14,16]. Inspired by Sang and Tao [14], Wu et al.[6] developed a novel method that the conditions of the dwell time method are satisfied outside a ball centered at the origin with a given radius, then the global practical stability is obtained. However, for nonlinear switched systems with parametric uncertainties, the problem becomes much more complicated because of the existence of nonlinear terms. Up to now, few results on MRAC for nonlinear switched systems have been reported, which motivates us on this note.What is more, in practice, the switching signal available to the controllers is often later than the switching of subsystems, which results in asynchronous switching in the closed-loop system. It is worth noting that, asynchronous switching causes unavailability of matching conditions for MRAC, which may lead to instability. In recent years, asynchronous switching has become a hot research direction, and a lot of results have been developed [17][18][19]. However, in practice, uncertainties and asynchronous switching can coexist in a switched system, which brings out a challenge. In [19], by the dwell time method, the MRAC was studied under asynchronous switching for a class of linear switched systems. However, for nonlinear switched systems, there is a lack of study. This is another motivation of the present study.In this paper, we present an MRAC scheme for nonlinear switched systems based on the average dwell time method under asynchronous switching. Compared with the existing literature on MRAC for switched systems, the res...

Model reference resilient control for the helicopter with time‐varying disturbance

“…Proof. Based on the above analysis, the derivative of the Lyapunov function (18) is less than zero under conditions (30) and (31). Hence, according to Definition 1, we confirm that system (16) is asymptotic stability with H ∞ performance, and the gains of the state observer, the disturbance observer, and the resilient controller are obtained from the Step (i) to Step (iii).…”

“…Using Schur complement Lemma for (31), we obtain − 1 I + X 1 X 1 < 0, which implies X 1 X 1 < 1 I. Hence, using the LMI Toolbox of the MATLAB for (30) and (31), the gain of the resilient controller K 01 = Y 2 X −1 1 . Remark 6.…”

“…For Assumption 2, it is a common assumption for model reference control problem and always be presented in many excellent articles before discussing this issue, such as the works of Guo et al 29 and Xie and Zhao. 30 The idea of model reference control method is as follows: firstly, design a reference model to describe the expected performance of the controlled system; then, construct a controller to guarantee that the reference model system could be tracked by the controlled system. In fact, the helicopter system is a controllable system, and the poles can be assigned arbitrarily.…”

“…Then, choose the appropriate matrices P 01 , P 02 , P 03 , and Y 1 to guarantee that inequation (27) holds, which implies that (24) holds. Note that matrix in (24) is a principal minor of the matrix in (29), and (29) would be true under (30) holds. Finally, the resilient controller gain is obtained if linear matrix inequation (30) is feasible.…”

“…Step (iii) There exist matrices X 1 > 0 and Y 2 , constants 1 > 0, 2 > 0, and 1 > 0, such that (30) and (31) hold. The gain of the resilient controller is given as K 01 = Y 2 X −1 1 .…”