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

Authors: Enchang Cui; Yuanwei Jing; Xiaoting Gao
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

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.


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## Body

## 1. 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 [1–3]. Since nonlinearity is the nature of the universe, more and more attention of switched nonlinear system has been drawn by control field [4–6]. 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 [10–13] 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 [15–17], 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 [18–20] 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. Section2 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.


## 2. Problem Statements and Preliminaries

### 2.1. Problem Formulation

Consider a class of the switched nonlinear systems with uncertain parameters in the following form:(1)x˙i=Fiσtxi+θbσtgixixi+1x˙n=Fnσtxn+θbσtgnxnuy=x1,where θbi∈Rn are the unknown piecewise constant parameters. σ(t)∈P={1,2,…,ps} is a non-Zeno switching signal which is right continuous. u∈R is the control input. Fki and gk are smooth vector fields with Fki(0)=0 and gk(x)≠0,∀x∈Rn.In addition,Fij(xi),∀j∈P is unknown switched nonlinear function which can be linearly parameterized as(2)Fiσtxi=∑k=1pfkixiθfkiσt,where fki(xi) is smooth function and θfi∈Rn is a vector of uncertain parameters satisfying θ∈Ωθ with known compact set Ωθ. The parameters θbi and θfi,∀i∈P, the switching time instants Tk,  k=1,2,…, and the switching index σ(t)∈P=1,2,…,ps are all unknown. There exist positive constants Pi such that fkixi≤Pi for all xi≤b,i=1,2,…,n due to smoothness property.According to (1) and (2), system (1) can be transformed in to the following form:(3)x˙i=Fiσtxi+θbσtgixixi+1=∑k=1pfkixiθfkiσt+θbσtgixixi+1,i=1,2,…,n.The control target is to design an adaptive controller for system (1) such that y tracks a desired trajectory yd asymptotically; that is, limt→∞⁡y-yd=limt→∞⁡z1=0 and xi≤b, b 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]).
θ
b
i for all i’s have the same sign and whose common lower bound is known, that is, 0<θ-b<θbi.


### 2.2. Preliminaries

Definition 2 (see [12]).
A barrier Lyapunov function (BLF) is a scalar functionV(x), defined with respect to the system x˙=f(x) on an open region D containing the origin, that is, continuous, positive definite, has continuous first-order partial derivatives at every point of D, has the property V(x)→∞ as x approaches the boundary of D, and satisfies Vxt≤b∀t≥0 along the solution of x˙=f(x) for x0∈D and some positive constant b.
According to previous description, we can choose a BLF candidate as follows:(4)V1=12log⁡kb12kb12-z12,where log⁡· denotes the natural logarithm. kb1 is the constraint on z1 and satisfies the condition z1<kb1. Thus it can be seen that V1 is positive definite, which grows to infinity when its argument approaches to its finite limit kb1.Lemma 3 (see [24]).
For function∑k=1pf-k1(x1)θfk1σ(t), a smooth function h1(x1) and an unknown positive constant l1 can be found:(5)∑k=1pf-k1x1θfk1σt≤l1h1x1.Lemma 4 (see [10]).
For any positive constantkb1, any z1 satisfying z1<kb1, we have(6)log⁡kb12kb12-z12<z1kb12-z12.In order to make the problem more analysable and tractable, practical assumption is given for the adaptive state controller.Assumption 5 (see [24]).
We can find a smooth functionf-k1(x1) such that(7)fk1x1=f-k1x1z1.


## 2.1. Problem Formulation

Consider a class of the switched nonlinear systems with uncertain parameters in the following form:(1)x˙i=Fiσtxi+θbσtgixixi+1x˙n=Fnσtxn+θbσtgnxnuy=x1,where θbi∈Rn are the unknown piecewise constant parameters. σ(t)∈P={1,2,…,ps} is a non-Zeno switching signal which is right continuous. u∈R is the control input. Fki and gk are smooth vector fields with Fki(0)=0 and gk(x)≠0,∀x∈Rn.In addition,Fij(xi),∀j∈P is unknown switched nonlinear function which can be linearly parameterized as(2)Fiσtxi=∑k=1pfkixiθfkiσt,where fki(xi) is smooth function and θfi∈Rn is a vector of uncertain parameters satisfying θ∈Ωθ with known compact set Ωθ. The parameters θbi and θfi,∀i∈P, the switching time instants Tk,  k=1,2,…, and the switching index σ(t)∈P=1,2,…,ps are all unknown. There exist positive constants Pi such that fkixi≤Pi for all xi≤b,i=1,2,…,n due to smoothness property.According to (1) and (2), system (1) can be transformed in to the following form:(3)x˙i=Fiσtxi+θbσtgixixi+1=∑k=1pfkixiθfkiσt+θbσtgixixi+1,i=1,2,…,n.The control target is to design an adaptive controller for system (1) such that y tracks a desired trajectory yd asymptotically; that is, limt→∞⁡y-yd=limt→∞⁡z1=0 and xi≤b, b 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]).
θ
b
i for all i’s have the same sign and whose common lower bound is known, that is, 0<θ-b<θbi.


## 2.2. Preliminaries

Definition 2 (see [12]).
A barrier Lyapunov function (BLF) is a scalar functionV(x), defined with respect to the system x˙=f(x) on an open region D containing the origin, that is, continuous, positive definite, has continuous first-order partial derivatives at every point of D, has the property V(x)→∞ as x approaches the boundary of D, and satisfies Vxt≤b∀t≥0 along the solution of x˙=f(x) for x0∈D and some positive constant b.
According to previous description, we can choose a BLF candidate as follows:(4)V1=12log⁡kb12kb12-z12,where log⁡· denotes the natural logarithm. kb1 is the constraint on z1 and satisfies the condition z1<kb1. Thus it can be seen that V1 is positive definite, which grows to infinity when its argument approaches to its finite limit kb1.Lemma 3 (see [24]).
For function∑k=1pf-k1(x1)θfk1σ(t), a smooth function h1(x1) and an unknown positive constant l1 can be found:(5)∑k=1pf-k1x1θfk1σt≤l1h1x1.Lemma 4 (see [10]).
For any positive constantkb1, any z1 satisfying z1<kb1, we have(6)log⁡kb12kb12-z12<z1kb12-z12.In order to make the problem more analysable and tractable, practical assumption is given for the adaptive state controller.Assumption 5 (see [24]).
We can find a smooth functionf-k1(x1) such that(7)fk1x1=f-k1x1z1.


## 3. Main Results

In this section, an adaptive controller is designed based on backstepping method by utilizing a BLF for systems (3).Step 1.
Definez1 for x1 as the tracking error which is z1=x1-yd. Consider the first component of the system (3) and we have the first-order partial derivative as(8)z˙1=x˙1-y˙d=∑k=1pfk1x1θfk1σt+θbσtg1x1x2-y˙d,
Choose the following BLF candidate:(9)V1=12log⁡kb12kb12-z12+l~12,where l~1=l1-l^1, l^1 denotes the estimate of l1, and l~˙1=-l^˙1. We design a stabilizing function and an adaptive law as(10)α1=1θ-bg1-l^1h1z1-λ1z1+y˙d,l^˙1=z12h1kb12-z12,where λ1 is a positive gain. Besides, let z2=x2-α1. On the basis of (10) and Assumption 1, we know that the time derivate of V1 satisfies(11)V˙1=z1z˙1kb12-z12+l~1l~˙1=z1∑k=1pfk1x1θfk1σt+θbσtg1z2+α1y˙dkb12-z12+l~1l~˙1≤z12l1h1+θbσtg1z1z2+1/θ-bg1-l^1h1z1-λ1z1+y-˙dz1-y-˙dz12kb12-z12+l~1-z12h1kb12-z12≤-λ1z12kb12-z12,where y-˙d can be found that y˙d=y-˙dz1.Step 2.
Consider the second component of system (3). Since z˙2=x˙2-α˙1, we can find a α-˙1 satisfying α˙1=α-˙1z2 and choose the following BLF candidate:(12)V2=∑i=1212log⁡kbi2kbi2-zi2+l~i2.
The stabilizing function and the adaptive law at this step are designed as(13)α2=1θ-bg2-l^2h2z2-λ2z2-α˙1-kb22-z22kb12-z12θ-bg1z1,l^˙1=z22h2kb22-z22.
Using (13) and Assumption 1, we know that the time derivate of V2 satisfies (14)V˙2=z1z˙1kb12-z12+l~1l~˙1+z2z˙2kb22-z22+l~2l~˙2≤-λ1z12kb12-z12+z1z2θbσtg1kb12-z12+z2z˙2kb22-z22+l~2l~˙2≤-λ1z12kb12-z12+z1z2θbσtg1kb12-z12+z2∑k=1pfk2x2θfk2σt+θbσtg2x2z3+α2-α˙1kb22-z22+l~2l~˙2≤-λ2z22kb22-z22-λ1z12kb12-z12≤0.
For the general case, we employ the BLF whenn>2:(15)Vi=∑i=1j12log⁡kbi2kbi2-zi2+l~i2,2<j<n.
By the same token, we design stabilizing functions and adaptive laws:(16)αi=1θ-bgi-l^ihizi-λizi+α˙i-1-kbi2-zi2kbi-12-zi-12θ-bgi-1zi-1,l^˙1=zi2hikbi2-zi2.
Finally, we obtain the stabilizing function and the adaptive law of stepn:(17)αn=1θ-bgn-l^nhnzn-λnzn+α˙n-1-θ-bgn-1zn-1kbn-12-zn-12,(18)u=αn,(19)l^˙n=zn2hn.Theorem 6.
Considering system (3), the adaptive state feedback controller (18) and the adaptive law (19) can guarantee limt→∞⁡x1-yd=0 and xi≤b under arbitrary switching signal and b is a positive constant.Proof.
Choose the BLF as follows:(20)V=∑i=1n-112log⁡kbi2kbi2-zi2+l~i2+12zn2+l~n2.
According to the adaptive state feedback controller (18) and the adaptive law (19), we obtain the time derivative of the BLF:(21)V˙≤-∑i=1n-1λizi2kbi2-zi2+zn-1znθbσtgn-1kbn-12-zn-12+znz˙n+l~nl~˙n≤-∑i=1nλizi2kbi2-zi2≤0.
It can be seen that all the signals are bounded. By Barbalat’s Lemma,zi will converge to zero. According to the nature of BLF, we can guarantee limt→∞⁡x1-yd=0 and xi≤b, among them b being a positive constant. The output of the system tracks the desired trajectory yd asymptotically and certain states are constrained.
The design procedure of the proposed control scheme could be viewed from the block diagram in Figure1.Figure 1
Block diagram of the adaptive control system.


## 4. Simulations

In this section, we give examples and simulations to demonstrate the proposed result.Example 1.
Consider the following switched nonlinear system:(22)x˙1=f11f12f13θ1σ+θbσg1x1x2x˙2=f21f22f23θ2σ+θbσg2x2u,σt:0,∞⟶1,2,where f11f12f13=x12x11, f21f22f23=x22x2cos(x2), g1x1=2x12+2, g2x2=x2+3. Parameters θ1σ, θ2σ, and θbσ are unknown. According to previous discussions, we can find smooth functions and satisfy the conditions of Theorem 6 that (23)h1x1=2x12+5,h2x2=x22+2.
The desired trajectory isyd=sin⁡x, and the tracking error constraint is z1<2, z2<10.
Figures2 and 3 present the simulation results of the outputs of the system tracking the desired trajectory yd 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. ChooseV1=1/2z12+l~12, and we can show that ϕ1 is a common virtual control for the first component of the system (22). Then, define z2 for x2 as the tracking error which z2=x2-ϕ1, V2=∑i=121/2z22+l~22 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 Figures6 and 7, we know that the asymptotic tracking performance is achieved and x1 tracks the desired trajectory yd 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.Figure 2
State trajectoryx1 tracking converges the desired trajectory yd=sin⁡x.Figure 3
State trajectoryx2 tracking converges the desired trajectory α1.Figure 4
Error trajectoryz1 converges to zero.Figure 5
Error trajectoryz2 converges to zero.Figure 6
State trajectoriesx1 tracking converge the desired trajectory (BLF-QLF).Figure 7
Error trajectoriesz1 converge to zero (BLF-QLF).Example 2.
Considering the above system, the control objective is that the output of systemx1 tracks the desired trajectory yd=0.3. Two different tracking error constraints are given that z1<0.2, z1<0.15.
From Figure8, we know that asymptotic tracking performance is achieved when the BLF is used with the initial value x0=0.25,-1T. 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 z1 satisfy constraints using the BLF. The errors z1 do not transgress boundary that z1<0.2 and z1<0.15, and the objective is achieved.Figure 8
State trajectoriesx1 tracking converges the desired trajectory (z1<0.2).Figure 9
Tracking error trajectoriesz1 satisfies error constraint z1<0.2.Figure 10
Tracking error trajectoriesz1 satisfies error constraint z1<0.15.


## 5. 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 designing an adaptive controller and stabilization under arbitrary switching signals can be achieved by output feedback.



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