Adaptive Control of a Class of Switched Nonlinear System with Partial State Constraints Using a Barrier Lyapunov Function

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.


Introduction
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 [1][2][3].Since nonlinearity is the nature of the universe, more and more attention of switched nonlinear system has been drawn by control field [4][5][6].It is also worth mentioning that a class of switched nonlinear system in lower triangular form is considered in [7], of which the backstepping method is used and a common virtual control is constructed to achieve the aim.Nevertheless, relatively accurate parameters are required by most control strategy for switched nonlinear system.Apparently, these desirable demands could not be met in practical terms.
Constraints are important issues in the study of physical systems, the authors in [8,9] emphasized this problem in nonlinear saturation and performance and safety specifications, respectively.Taking a practical example for illustrative purposes, the attitude-control mass expulsion system of electrostatic microactuators is confronted with constraints, owing to the necessity of both the position and the speed of the movable electrode to be controlled.However, the development of constrained control is restrained by meeting the practical requirement.In this scenario, the extensive attention of the design and the analysis of constrained control has been increased.Recently, barrier Lyapunov functions (BLFs) have been proposed to solve the control problem of nonlinear systems.The state of the system will be not contradictory to the constraints [10][11][12][13] utilizing BLFs.Although the BLF is proven to settle constrained control problems [14], the parameters of switched nonlinear systems are expected to be deterministic.
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 [15][16][17], for instance, the control method is proposed and the preferable control performances are provided aiming at nonlinear time-delay systems.On the other hand, many researchers study the adaptive control problem of switched systems with uncertain parameters [18][19][20] with the development of adaptive control, and preliminary results have been obtained [21,22].In [23], an adaptive tracking controller is designed for switched stochastic nonlinear systems with unknown actuator dead-zone; the satisfying control performance is obtained as well.Among the aforementioned works, few authors have addressed the important issue that considering the uncertainty and the constraint together.Hence, the approach to design constraint adaptive controllers for switched nonlinear systems has been not reported to the best of our knowledge.
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 2 formulates the control problem.The partial state constraint problems of barrier Lyapunov functions are elucidated.Section 3 develops the adaptive controllers with state constraints; the stability analysis and the proof process are presented as well.Section 4 corroborates the expected effectiveness of the proposed controller by means of selected simulation results.Finally, Section 5 concludes this paper.
According to (1) and (2), system (1) can be transformed in to the following form: The control target is to design an adaptive controller for system (1) such that  tracks a desired trajectory   asymptotically; that is, lim →∞ (−  ) = lim →∞  1 = 0 and |  | ≤ ,  is a positive constant.Moreover, in order to make the problem here more tractable, we give the following assumption, which is common but practical.
Assumption 1 (see [24]).   for all 's have the same sign and whose common lower bound is known, that is, 0 <   < |   |.

Preliminaries
Definition 2 (see [12]).A barrier Lyapunov function (BLF) is a scalar function (), defined with respect to the system ẋ = () on an open region  containing the origin, that is, continuous, positive definite, has continuous firstorder partial derivatives at every point of , has the property () → ∞ as  approaches the boundary of , and satisfies (()) ≤  ∀ ≥ 0 along the solution of ẋ = () for (0) ∈  and some positive constant .
According to previous description, we can choose a BLF candidate as follows: where log (⋅) denotes the natural logarithm. 1 is the constraint on  1 and satisfies the condition | 1 | <  1 .Thus it can be seen that  1 is positive definite, which grows to infinity when its argument approaches to its finite limit  1 .
Lemma 3 (see [24]).For function 1 , a smooth function ℎ 1 ( 1 ) and an unknown positive constant  1 can be found: Lemma 4 (see [10]).For any positive constant  1 , any In order to make the problem more analysable and tractable, practical assumption is given for the adaptive state controller.

Main Results
In this section, an adaptive controller is designed based on backstepping method by utilizing a BLF for systems (3).
Step 1. Define  1 for  1 as the tracking error which is  1 =  1 −   .Consider the first component of the system (3) and we have the first-order partial derivative as Choose the following BLF candidate: where l1 =  1 − l1 , l1 denotes the estimate of  1 , and l 1 = − l 1 .We design a stabilizing function and an adaptive law as where  1 is a positive gain.Besides, let  2 =  2 −  1 .On the basis of (10) and Assumption 1, we know that the time derivate of  1 satisfies where ẏ  can be found that ẏ  = ẏ   1 .
Step 2. Consider the second component of system (3).Since ż 2 = ẋ 2 − α 1 , we can find a α 1 satisfying α 1 = α 1  2 and choose the following BLF candidate: The stabilizing function and the adaptive law at this step are designed as Using (13) and Assumption 1, we know that the time derivate of  2 satisfies For the general case, we employ the BLF when  > 2: By the same token, we design stabilizing functions and adaptive laws: Finally, we obtain the stabilizing function and the adaptive law of step : Theorem 6. Considering system (3), the adaptive state feedback controller (18) and the adaptive law (19) can guarantee lim →∞ ( 1 −   ) = 0 and |  | ≤  under arbitrary switching signal and  is a positive constant.Proof.Choose the BLF as follows: According to the adaptive state feedback controller (18) and the adaptive law (19), we obtain the time derivative of the BLF: It can be seen that all the signals are bounded.By Barbalat's Lemma,   will converge to zero.According to the nature of BLF, we can guarantee lim →∞ ( 1 −   ) = 0 and |  | ≤ , among them  being a positive constant.The output of the system tracks the desired trajectory   asymptotically and certain states are constrained.
The design procedure of the proposed control scheme could be viewed from the block diagram in Figure 1.

Simulations
In this section, we give examples and simulations to demonstrate the proposed result.
Example 1.Consider the following switched nonlinear system: Mathematical Problems in Engineering The desired trajectory is   = sin , and the tracking error constraint is | 1 | < 2, | 2 | < 10.Figures 2 and 3 present the simulation results of the outputs of the system tracking the desired trajectory   asymptotically, and the tracking errors constraints are never violated.And it is evident from Figures 4 and 5 that tracking error trajectories converge to zero.
After that, we design a controller using QLF.Choose  1 = (1/2)( 2  1 + l2 1 ), and we can show that  1 is a common virtual control for the first component of the system (22).Then, define  2 for  2 as the tracking error which 2 ) is a common Lyapunov for the system (22).We will follow the same design procedure for the design of the controller based on the QLF method and compare the simulation results between BLF and QLF.
As it can be clearly seen from Figures 6 and 7, we know that the asymptotic tracking performance is achieved and  1 tracks the desired trajectory   asymptotically based on BLF method.However, when the QLF is utilized with the same parameters and design processes, the output is more volatile than the BLF one, and the convergence rate is slower than the BLF method.
Example 2. Considering the above system, the control objective is that the output of system  1 tracks the desired trajectory   = 0.3.Two different tracking error constraints are given that | 1 | < 0.2, | 1 | < 0.15.
From Figure 8, we know that asymptotic tracking performance is achieved when the BLF is used with the initial value (0) = (0.25, −1)  .However, when the QLF is used with the same initial value, the state constraint is violated.The simulation results in Figures 9 and 10 show that we can know that tracking errors  1 satisfy constraints using the BLF.The errors  1 do not transgress boundary that | 1 | < 0.2 and | 1 | < 0.15, and the objective is achieved.

Conclusions
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

2. 1 .
Problem Formulation.Consider a class of the switched nonlinear systems with uncertain parameters in the following form:

Figure 1 :
Figure 1: Block diagram of the adaptive control system.