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This paper discusses partial state constraint adaptive tracking control problem of switched nonlinear systems with uncertain parameters. In order to ensure boundedness of the outputs and prevent the states from violating the constraints, a barrier Lyapunov function (BLF) is employed. Based on backstepping method, an adaptive controller for the switched system is designed. Furthermore, the state-constrained asymptotic tracking under arbitrary switching is performed. The closed-loop signals keep bounded when the initial states and control parameters are given. Finally, examples and simulation results are reported to illustrate the effectiveness of the proposed controller.

There is a strong industrial background of switched system in various fields. And for exactly that reason, many researchers have discussed the theoretical and applied research of switched system, and some of them have achieved commendable results in the last decade [

Constraints are important issues in the study of physical systems, the authors in [

Moreover, parameter uncertainties are widespread in realistic systems; hereon it has already been reported that adaptive control is an effective method to deal with such uncertain. On the one hand, remarkable achievements have been obtained from the research on adaptive control of nonlinear system. In [

In the present paper, the adaptive tracking control problem of a class of switched nonlinear system with partial state constraints is solved. The progressive state tracking would not violate constraint conditions and all signals would be bounded when the parameter increases to infinitely great and approaches a certain value based on BLF. The control effectiveness of BLFs is verified by the comparative simulation results with quadratic Lyapunov functions (QLFs).

The remainder of this paper is organized as follows. Section

Consider a class of the switched nonlinear systems with uncertain parameters in the following form:

In addition,

According to (

A barrier Lyapunov function (BLF) is a scalar function

According to previous description, we can choose a BLF candidate as follows:

For function

For any positive constant

In order to make the problem more analysable and tractable, practical assumption is given for the adaptive state controller.

We can find a smooth function

In this section, an adaptive controller is designed based on backstepping method by utilizing a BLF for systems (

Define

Choose the following BLF candidate:

Consider the second component of system (

The stabilizing function and the adaptive law at this step are designed as

Using (

For the general case, we employ the BLF when

By the same token, we design stabilizing functions and adaptive laws:

Finally, we obtain the stabilizing function and the adaptive law of step

Considering system (

Choose the BLF as follows:

According to the adaptive state feedback controller (

It can be seen that all the signals are bounded. By Barbalat’s Lemma,

The design procedure of the proposed control scheme could be viewed from the block diagram in Figure

Block diagram of the adaptive control system.

In this section, we give examples and simulations to demonstrate the proposed result.

Consider the following switched nonlinear system:

The desired trajectory is

Figures

After that, we design a controller using QLF. Choose

As it can be clearly seen from Figures

State trajectory

State trajectory

Error trajectory

Error trajectory

State trajectories

Error trajectories

Considering the above system, the control objective is that the output of system

From Figure

State trajectories

Tracking error trajectories

Tracking error trajectories

In this paper, we have studied the constraint adaptive tracking control problem of switched nonlinear systems with uncertain parameters using the BLF and backstepping method. Asymptotic output tracking and states constraint have been ensured, and we guarantee that the state constraint is violated, which has been verified by the simulations. Next we will focus on studying constraint adaptive output feedback control and designing an adaptive controller and stabilization under arbitrary switching signals can be achieved by output feedback.

The authors declare that there are no conflicts of interest regarding the publication of this paper.