# Homogeneous Stabilizer by State Feedback for Switched Nonlinear Systems Using Multiple Lyapunov Functions’ Approach

Authors: Hui Ye; Bin Jiang; Hao Yang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

This paper investigates the problem of global stabilization for a class of switched nonlinear systems using multiple Lyapunov functions (MLFs). The restrictions on nonlinearities are neither linear growth condition nor Lipschitz condition with respect to system states. Based on adding a power integrator technique, we design homogeneous state feedback controllers of all subsystems and a switching law to guarantee that the closed-loop system is globally asymptotically stable. Finally, an example is given to illustrate the validity of the proposed control scheme.


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## Body

## 1. Introduction

A switched system is a hybrid system which consists of a family of subsystems, either continuous-time or discrete-time subsystems, and a switching law, which defines a specific subsystem that is active at each instant of time. In the last decade, the problem of designing switching strategies for switched systems has received a great attention [1–10] and the references therein. A switched system may be either stable or unstable, not depending on its subsystems but a particular switching signal. Because of less conservativeness, MLFs method is more preferable than other methods. It plays an important role in design for switched nonlinear systems in the control literature [4, 5]. Based on the common Lyapunov function approach, the existing designs of switched nonlinear systems were restricted to triangular form under arbitrary switchings via backstepping or forwarding method [7, 8, 10]. For switched nonlinear systems, it has not been used by homogeneous domination control approach to the best of our knowledge. The homogeneous domination approach begins with the linear domination idea proposed in [11], where a global output feedback stabilizer is constructed for uncertain nonlinear system under a linear growth condition. The work [12] deals with the stability of nonswitched nonlinear systems using the homogeneous domination approach. Using such method, how to tackle the problem of stabilization for a class of more general switched nonlinear systems is challenging. In this paper, motivated by [13], in some regular conditions, the MLFs method is utilized to completely handle the different coordinate transformations for all subsystems by constructing proper switching law. Compared with the existing literature on switched nonlinear systems, the results of this paper have some advantages. At first, the problem of global state feedback stabilization for a wider class of nonlinear systems with some special forms is studied by homogeneous domination approach, which simplifies the control design procedure. Secondly, individual coordinate transformations for subsystems are introduced into virtual controllers, which decreases the conservativeness compared with the common change of coordinate. Finally, the dual controllers and a switching law are designed by constructing the MLFs.The outline of the paper is as follows. In Section2, we provide the system formulation and preliminaries. In Section 3, we present a necessary and sufficient condition for the stabilization issue and the main result of the paper. An illustrative example is given in Section 4. Our conclusion is included in Section 5.


## 2. System Description and Preliminaries

We consider a class of switched nonlinear systems of the form(1)x˙i=xi+1+fi,σtx¯i+gi,σtx,i=1,…,n-1x˙n=uσt+fn,σtx,where x=(x1,…,xn)T∈Rn is the system state and uσ(t) is the subsystem input. x¯i=(x1,…,xi)T,i=1,…,n. The function σ(t):[0,∞)→M={1,…,m} is a switching signal which is assumed to be a piecewise continuous (from the right) function of time. For each k∈M, uk∈R is the control input of the kth subsystem. For any k∈M, and i=1,2,…,n, the functions fi,k(·),i=1,…,n and gl,k(x),l=1,…,n-1,k∈M are C∞ with respect to their arguments and vanish at the origin; that is, fi,k(0)=0 and gl,k(0)=0. In addition, we assume that the state of system (1) does not jump at the switching instants; that is, the solution is everywhere continuous and only finite switches can occur in any finite time interval.Obviously, the structure of the switched nonlinear system (1) is much more general than the structure of the nonswitched nonlinear systems called strict-feedback form whose stabilization problem has been addressed in [12, 14] when gl,k(x)≡0 and m=1. The existence of a common Lyapunov function for all subsystems was shown to be a necessary and sufficient condition for a switched system to be asymptotically stable under arbitrary switchings [15]. If at least one of the subsystems is not asymptotically stable, we still solve the global stabilization problem for system (1) by designing controllers for all subsystems and a switching law.In the following, we collect some useful definitions and lemmas which play important roles in this paper. The innovative idea of homogeneity was introduced for the stability analysis of a nonlinear system and has led to a number of interesting results (see [16, 17]). We recall the definitions of homogeneous systems with weighted dilation.Definition 1.
For real numbersri>0,i=1,…,n and fixed coordinates (x1,x2,…,xn)∈Rn,∀ε>0.
(i) The dilationΔε(x) is defined by Δε(x)=(εr1x1,…,εrnxn),∀ε>0, with ri being called as the weights of the coordinates. For simplicity of notation, we define dilation weight Δ=(r1,…,rn).
(ii) A functionV∈C(Rn,R) is said to be homogeneous of degree τ if there is a real number τ∈R such that (2)VΔεx=ετVx1,…,xn,∀x∈Rn∖0.
(iii) A vector fieldf∈C(Rn,Rn) is said to be homogeneous of degree τ if there is a real number τ∈R such that for i=1,…,n(3)fiΔεx=ετ+rifix1,…,xn,∀x∈Rn∖0.
(iv) A homogeneousp-norm is defined as xΔ,p=∑i=1nxip/ri1/p,∀x∈Rn, for a constant p≥1. For the simplicity, we choose p=2 and write xΔ for xΔ,2.Lemma 2.
SupposeV:Rn→R is a homogeneous function of degree τ with respect to the dilation weight Δ. Then the following holds:
(i)∂V/∂xi is homogeneous of degree τ-ri with ri being the homogeneous weight of xi.
(ii) There is a constantc such that (4)Vx≤cxΔτ.
Moreover, ifV(x) is positive definite, (5)c_xΔτ≤Vx,for  a  constantc_>0.Lemma 3.
Forx∈R, y∈R, p≥1 is a constant; the following inequalities hold:(6)x+yp≤2p-1xp+yp,x+y1/p≤x1/p+y1/p≤2p-1/px+y1/p.
Ifp≥1 is an odd integer or a ratio of two odd integers,(7)x-yp≤2p-1xp-yp,xp-yp≤px-yxp-1+yp-1≤cx-yx-yp-1+yp-1,where c>0 is a constant.Lemma 4.
Supposec and d are two positive real numbers. Given any positive number γ>0, the following inequality holds:(8)xcyd≤cc+dγxc+d+dc+dγ-c/dyc+d.Lemma 5.
Let0≤δ1≤⋯≤δn be real numbers and cj≥0,j=1,…,n. Then, ∀x∈R,(9)c1xδ1+cnxδn≤∑j=1ncjxδj≤∑j=1ncjxδ1+xδn.


## 3. Homogenous Stabilizer by State Feedback

In this section, we first construct a set of Lyapunov functions for all subsystems of the following switched nonlinear systems:(10)x˙i=xi+1+fi,σtx¯i,i=1,…,n-1x˙n=uσt+fn,σtxwhich is exactly in (1) with gi,σ(t)(x)=0. Then we present the homogeneous controllers of subsystems and a switching law for system (10) under some regular conditions.To the best of our knowledge, no results on switched nonlinear systems (10) in the use of the homogeneous domination approach have appeared. For system (10), a common Lyapunov function is difficult to find because a common coordinate transformation of all subsystems is difficult to be exploited. On the other hand, if system (1) has only one subsystem, the design problem reduces to the classical stabilization design problem for nonswitched nonlinear systems. According to the structure of system (1), however, system (1) cannot constructively be stabilized by the existing methods such as the adding a power integrator technique proposed in [18, 19]. To obtain the homogenous state feedback controllers of subsystems and a switching law, which globally stabilize system (1), some assumptions are made as follows.Assumption 6.
Fori=1,…,n, and k∈M, there are constants τk≥0 and ρk≥0, such that(11)fi,kx¯i≤ρkx1ri+1,k/r1,k+x2ri+1,k/r2,k+⋯+xiri+1,k/ri,kwith the constants ri,k’s defined as (12)r1,k=1,ri+1,k=ri,k+τk.Assumption 7.
There are some constantsal,l=1,…,k such that(13)gi,kx≤a1ξ1,1n-1τ1/r1,k+⋯+ξi,1n-1τ1/r1,k+⋯+akξ1,kn-1τk/ri,k+⋯+ξi,kn-1τk/ri,k,where ξi,k will be defined later.Assumption 8.
In the above formula, the variables must meet(14)τi≤min⁡τ1,…,τi-1.Remark 9.
WhenM={1} and gi,k(x)=0, Assumption 6 is reduced to that in [12, 14]. In presence of gi,k(x), system (1) is more general compared with previous lower-triangular or upper-triangular system. If τk=0, Assumptions 6 and 7 become linear growth conditions. Assumption 8 plays an important role in the later control design. For simplicity, in this paper we assume τi=qi/pi with qi an even integer and pi an odd integer. Based on this, ri,j=1+j-1τi,j=1,…,n+1 will be odd in both denominator and numerator.Theorem 10.
Under Assumptions6–8, there are constants δi,i=1,…,n and continuous functions χk,l(x)>0,k,l∈M, such that for ∀x≠0,k∈M(15)a∑i=1nδiξi,k2rn,k/ri,k+∑l=1mχk,lxVlξ¯n,l-Vkξ¯n,k<0.Then, there exist homogeneous state feedback controllers and a switching law solving the global stabilization problem of system (1).Proof.
The proof is carried out by using an inductive argument, which enables us to simultaneously construct a Lyapunov function which is positive definite and proper, as well as homogeneous stabilizers. First construct a set of Lyapunov functions for all subsystems of system (10) using adding a power integrator technique for each k.
Step  1. Choose a Lyapunov function V1,k=(r1,k/(2rn,k-τk))x1(2rn,k-τk)/r1,k. For each subsystem k of switched system (10), the time derivative of V1,k along the trajectory of (10) is(16)V˙1,k=x12rn,k-τk/r1,k-1x2+f1,kx1≤x12rn,k-τk/r1,k-1x2-x2,k∗+x2,k∗+x1r2,k/r1,kρk;the last inequality is deduced by Assumption 6. Then choosing the virtual controller x2,k∗ as (17)x2,k∗=-x1r2,k/r1,kn+ρk≔-x1r2,k/r1,kβ1,kwe can obtain(18)V˙1,kx1≤x12rn,k-τk/r1,k-1x2-x2,k∗-nx12rn,k/r1,k.
Inductive Step. Suppose, at step i-1, there exists a Lyapunov function Vi-1,k, which is positive define and homogeneous with respect to (12) and a set of virtual controllers x1,k∗,…,xi,k∗ defined by(19)x1,k∗=0,x2,k∗=-ξ1,kr2,k/r1,kβ1,k,⋮xj,k∗=-ξj-1,krj,k/rj-1,kβj-1,k,ξ1,k=x1-x1,k∗ξ2,k=x2-x2,k∗⋮ξj,k=xj-xj,k∗,where j=2,…,i and β1,k>0,…,βi-1,k>0 such that (20)V˙i-1,k≤ξi-1,k2rn,k-τk/ri-1,k-1xi-xi,k∗-n-i+2∑j=1i-1ξj,k2rn,k/rj,k.
Next, we claim that (20) also holds at step i. To prove this claim, we consider the Lyapunov function (21)Vi,k=Vi-1,k+ri,k2rn,k-τkξi,k2rn,k-τk/ri,kwhose derivative is(22)V˙i,k≤-n-i+2∑j=1i-1ξj,k2rn,k/rj,k+ξi-1,k2rn,k-τk/ri-1,k-1ξi,k+ξi,k2rn,k-τk/ri,k-1xi+1+fi,k·-∑l=1i-1∂xi,k∗∂xlx˙l.
Next, we estimate the last term on the right-hand side of (22). By Lemma 4, there are constants ci,k>0 and di,k>0 such that(23)ξi-1,k2rn,k-τk/ri-1,k-1ξi,k=ξi-1,k2rn,k-τk-ri-1,k/ri-1,kξi,k1/ri,kri,k≤ci,kξi-1,k2rn,k/ri-1,k+di,kξi,k2rn,k/ri,k.
By Lemma3, there is a constant ρ~k such that Assumption 6 can be rewritten as(24)fi,kx¯i≤ρkx1ri+1,k/r1,k+x2ri+1,k/r2,k+⋯+xiri+1,k/ri,k=ρk∑j=1iξj,k-ξj-1,krj,k/rj-1,kβj-1,kri,k+τk/rj,k≤ρ~k∑j=1iξj,kri,k+τk/rj,k.
On the other hand, one has(25)∑l=1i-1∂xi,k∗∂xlx˙l=∑l=1i-1∂-ξi-1,kτk/ri-1,kβi-1,k∂xlxl+1+fl,k·.
According to definition (19), one obtains (26)∂xi,k∗∂xl=∂-ξi-1,kτk/ri-1,kβi-1,k∂xl≤mξi-1,kτk/ri-1,k⋯ξl,kτk/rl,k.
Combining (26) and (24) into (25), we have(27)∑l=1i-1∂xi,k∗∂xlx˙l≤∑l=1i-1mξi-1,kτk/ri-1,k⋯ξl,kτk/rl,kξl+1,k-ξl,krl+1,k/rl,kβl,k+ρ~k∑j=1iξj,kri,k+τk/rj,k.
By Lemma5 and the fact that (i-l)τk+rl,k=ri,k, (27) becomes(28)∑l=1i-1∂xi,k∗∂xlx˙l≤∑l=1i-1m~∑j=1l+1ξj,kτk+ri,k/rj,k≤m¯ξ1,kτk+ri,k/r1,k+⋯+ξi,kτk+ri,k/ri,k,where m~ and m¯ are positive constants.
Combining (23), (24), and (28) into (22) yields(29)V˙i,k≤n-i+2∑j=1i-1ξj,k2rn,k/rj,k+ci,kξi-1,k2rn,k/ri-1,k+di,kξi,k2rn,k/ri,k+ξi,k2rn,k-τk/ri,k-1xi+1+ξi,k2rn,k-τk/ri,k-1fi,k·-∑l=1i-1∂xi,k∗∂xlx˙l≤ξi,k2rn,k-τk/ri,k-1ρ~k∑j=1iξj,kri,k+τk/rj,k-m¯ξ1,kτk+ri,k/r1,k+⋯+ξi,kτk+ri,k/ri,k-n-i+2∑j=1i-1ξj,k2rn,k/rj,k+ci,kξi-1,k2rn,k/ri-1,k+di,kξi,k2rn,k/ri,k+ξi,k2rn,k-τk/ri,k-1xi+1≤-n-i+2∑j=1i-1ξj,k2rn,k/rj,k+ci,kξi-1,k2rn,k/ri-1,k+di,kξi,k2rn,k/ri,k+ξi,k2rn,k-τk/ri,k-1ρ~k+m¯∑j=1iξj,kri,k+τ/rj,k+ξi,k2rn,k-τk/ri,k-1xi+1.
Choosing appropriate constantsci,k, di,k, mi,k, and ni,k, (29) becomes(30)V˙i,k≤-n-i+1∑j=1iξj,k2rn,k/rj,k+ξi,k2rn,k-τk/ri,k-1×xi+1+di,k+ni,kξi,kri+1,k/ri,k.
Next, we construct controllers for each subsystem and a switching law which meet the requirements of the MLFs framework for system (1). We can choose the Lyapunov function Vk=(1/2)ξ1,k2+∑i=2n(ri,k/(2rn,k-τk))ξi,k(2rn,k-τk)/ri,k. From (16)–(30), it is not hard to arrive at(31)V˙k=x1x2+f1,kx1+g1,kx+∑i=2n-1ξi,k2rn,k-τk/ri,k-1×xi+1+fi,kx¯i+gi,kx-∑l=1i-1∂xi,k∗∂xlx˙l+ξn,k2rn,k-τk/ri,k-1uk+fn,kx-∑l=1n-1∂xn,k∗∂xlx˙l.
Using inequalities (26), (27), (28), and (30), one has(32)V˙k≤-2∑j=1n-1ξj,k2rn,k/rj,k+ξn-1,k2rn,k-τk/rn-1,k-1xn+pξn-1,krn,k/rn-1,k+x1g1,kx+∑i=2n-1ξi,k2rn,k-τk/ri,k-1gi,kx-∑l=1i-1∂xi,k∗∂xlgl,kx+ξn,k2rn,k-τk/rn,k-1uk+fn,kx-∑l=1n-1m1ξn-1,kτk/rn-1,k⋯ξl,kτk/rl,kξl+1,k-ξl,krl+1,k/rl,kβl,k+ρ~k∑j=1lξj,krl,k+τk/rj,k+gl,kx,where p,m1, and ρ~k are positive constants.
According to Lemma4 and (19), there exist positive constants p, p1, and p2 such that(33)ξn-1,k2rn,k-τk/rn-1,k-1xn+pξn-1,krn,k/rn-1,k≤pξn-1,k2rn,k/rn-1,k+ξn-1,krn,k/rn-1,kξn,k-βn-1,kξn-1,krn,k/rn-1,k≤p1ξn-1,k2rn,k/rn-1,k+p2ξn,k2rn,k/rn,k.
By Lemma4 and (24) and (28), there exists a constant αi such that(34)ξn,k2rn,k-τk/rn,k-1fn,kx¯-∑l=1n-1m1ξn-1,kτk/rn-1,k⋯ξl,kτk/rl,kξl+1,k-ξl,krl+1,k/rl,kβl,k+ρ~k∑j=1lξj,krn,k+τ/rj,k≤ξn,k2rn,k-τk/rn,k-1ρn∑j=1nξj,krn,k+τ/rj,k-m2ξ1,kτk+rn,k/r1,k+⋯+ξn,kτk+rn,k/rn,k≤αi∑i=1nξi,k2rn,k/ri,k.
Combining (33) and (34) into (32), we obtain(35)V˙k≤-2∑j=1n-1ξj,k2rn,k/rj,k+p1ξn-1,k2rn,k/rn-1,k+p2ξn,k2rn,k/rn,k+ξn-1,k2rn,k-τk/rn-1,k-1x1g1,kx+∑i=2n-1ξi,k2rn,k-τk/ri,k-1gi,kx-∑l=1i-1∂xi,k∗∂xlgl,kx+ξn,k2rn,k-τk/rn,k-1uk+∑i=1nαiξi,k2rn,k/ri,k-ξn,k2rn,k-τk/rn,k-1∑l=1n-1m1ξn-1,kτk/rn-1,k⋯ξl,kτk/rl,kgl,kx.
After the merger, we have(36)V˙k≤-2∑j=1n-1ξj,k2rn,k/rj,k+p1ξn-1,k2rn,k/rn-1,k+p2ξn,k2rn,k/rn,k+ξn,k2rn,k-τk/rn,k-1uk+αi∑i=1nξi,k2rn,k/ri,k+ξn-1,k2rn,k-τk/rn-1,k-1∑i=2n-1ξi,k2rn,k-τk/ri,k-1gi,kx+m2∑i=2n∑l=1i-1ξi,k2rn,k-τk/ri,k-1ξi-1,kτk/ri-1,k⋯ξl,kτk/rl,kgl,kx.
According to Assumptions7 and 8, one has(37)ξn-1,k2rn,k-τk/rn-1,k-1∑i=2n-1ξi,k2rn,k-τk/ri,k-1gi,kx+m2∑i=2n∑l=1i-1ξi,k2rn,k-τk/ri,k-1ξi-1,kτk/ri-1,k⋯ξl,kτk/rl,kgl,kx≤ξn-1,k2rn,k-τk/rn-1,k-1∑i=2n-1ξi,k2rn,k-τk/ri,k-1a1ξ1,1n-1τ1/r1,k+⋯+ξi,1n-1τ1/r1,k+⋯+akξ1,kn-1τk/ri,k+⋯+ξi,kn-1τk/ri,k+m2∑i=2n∑l=1i-1ξi,k2rn,k-τk/ri,k-1ξi-1,kτk/ri-1,k⋯ξl,kτk/rl,ka1ξ1,1n-1τ1/r1,k+⋯+ξl,1n-1τ1/r1,k+⋯+akξ1,kn-1τk/ri,k+⋯+ξl,kn-1τk/ri,k≤di∑j=1nξj,k2rn,k/rj,k.
Choose the state feedback controllersuk,k∈M and a reasonable function qk(x) as (38)uk=-ξn,krn,k+τk/rn,kqkx.
By appropriate choice, we have(39)V˙k≤∑i=1nδiξi,k2rn,k/ri,k≤-∑l=1mχklxVlξ¯nl-Vkξ¯nk,∀x≠0,where Vk is a positive definite and proper Lyapunov function. Now we design the switching law as follows: (40)σt=arg⁡mink∈MVkξ1,k,…,ξn,k.
By inequalities (15), (39), and (40), when σ(t)=k, one has V˙k<0,∀x≠0. As a result, according to the MLFs method, the closed-loop system (1) and (38) under the switching law (40) is globally asymptotically stable.


## 4. An Illustrative Example

In this section, we give an example to demonstrate the validity of the proposed method. Consider the switched nonlinear system as shown in [13](41)x˙1=x2x˙2=x3+gσtxx˙3=x4x˙4=uσt+m2grJ2-kr24J2sin⁡x3+kr24J2sin⁡x2,where σ(t)={1,2}, m2 = 2.5 kg, g = 9.81 m/s2, r = 0.5 m, and J2 = 0.625 kg. g1(x)=-(1/32)x4-(1/64)x32-(1/16)x2x32-(5/64)x1x32-(1/4)x3-(13/16)x2-(27/32)x1 and g2(x)=-(1/36)x4-(1/36)x32-(1/9)x2x32-(5/36)x1x32-(7/36)x3-(11/18)x2-(11/18)x1. Following the design in Section 3, choosing the MLFs functions Vk=(1/2)ξ12+(1/2)ξ22+(1/2)ξ32+(1/2)ξ4k2, where ξ1=x1, ξ2=x2+2x1, ξ3=x3+4x2+5x1, ξ4,1=x4+5x3+14x2+12x1+(2+x32)ξ3, and ξ4,1=x4+5x3+14x2+12x1+(3+0.5x32)ξ3, we design the controllers u1=-ξ3-2ξ41-9.6sin⁡x3-10sin⁡x2-5x4-14(x3+g1(x))-12x2-2x3x4ξ3(2+x32)(x4+4x3+5x2+4g1(x)) and u2=-ξ3-2ξ42-9.6sin⁡x3-10sin⁡x2-5x4-14(x3+g2(x))-12x2-2x3x4ξ3(3+0.5x32)(x4+4x3+5x2+4g2(x)).In the simulation, Figures1–3 illustrate the effectiveness of control design under the switching signal σ(t)=arg⁡mink=1,2⁡{Vk} with the initial condition x10,x20,x30,x40T=-2,2,0.3,-1T.Figure 1
The curves of statesx1, x2, x3, and x4.Figure 2
The curve of the control inputu.Figure 3
The switching signalσ(t).


## 5. Conclusion

This paper has discussed the problem of global state feedback stabilization for a class of switched nonlinear systems. We design homogeneous controllers based on adding a power integrator technique and a switching law to guarantee asymptotical stability of the closed-loop switched nonlinear system.



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