# Adaptive Stabilization for Nonholonomic Systems with Unknown Time Delays

Authors: Yuanyuan Wu; Qingbo Li; Yuqiang Wu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper presents an adaptive control strategy for a class of nonholonomic systems in chained form with virtual control coefficients, nonlinear uncertainties, and unknown time delays. State scaling technique and backstepping recursive approach are applied to design a nonlinear state feedback controller, which
can guarantee the stabilization of the closed-loop systems. The simulation results are provided to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

In the last few decades, considerable efforts have been devoted to the research of nonholonomic system, which is a particular class of nonlinear systems and widespread in real world, such as mobile robots, car-like vehicle, underactuated satellites, and knife-edge. It is well known that the control of nonholonomic systems is extremely challenging, largely due to the impossibility of asymptotically stabilizing nonholonomic systems via smooth time-invariant state feedback, a well-recognized fact pointed out in [1, 2]. In order to overcome this obstruction, several approaches have been proposed, such as discontinuous feedback, time-varying feedback, and hybrid stabilization.The discontinuous feedback stabilization was first proposed in [3], and then further discussion was made in [4–7]; especially an elegant discontinuous coordinate transformation approach was presented in [5] for the stabilization problem of nonholonomic systems. Meanwhile, the smooth time-varying feedback control strategies also have drawn much attention [8–11]. To date, there have been several controller design approaches for the asymptotic stabilization or exponential regulation of nonholonomic control systems [4–14].As pointed out in [9], many nonlinear mechanical systems with nonholonomic constraints can be transformed, either locally or globally, into the nonholonomic systems in the so-called chained form. Therefore, a number of research literature resources [8–23] for such chained nonholonomic systems are provided. Recently, some new adaptive control strategies have been proposed to stabilize the nonholonomic systems. For instance, state feedback control is studied in [15–20] and output feedback control in [21–25].From a practical point of view, when modeling a mechanical system, time delay should be taken into account, and there are a few literature resources [19, 20, 25] for the nonholonomic systems with time delay. In [19, 20, 25], the problem of stabilization is studied for delayed nonholonomic systems; however, the virtual control coefficients and unknown parameter vector are not considered.In this paper, we introduce a new class of chained nonholonomic systems with unknown virtual control coefficients, uncertain nonlinearities, and unknown time delays and then study the problem of adaptive state feedback stabilization. Since the  nonholonomic system considered in this paper contains the delayed terms, it cannot be handled by existing conventional methods. The proposed constructive design method is based on a combined application of the state scaling technique, the recursive backstepping approach, and the novel Lyapunov-Krasovskii functionals. The switching control strategy for the first subsystem is employed to achieve the asymptotic stabilization.


## 2. Problem Formulation and Preliminaries

In this paper, we present an adaptive stabilization control design procedure for the following nonholonomic systems with nonlinear uncertainties and unknown time delays:(1)x˙0(t)=d0u0(t)+ϕ0(t,x0(t),θ),x˙1(t)=d1u0(t)x2(t)+φ1(y(t),y(t-τ1))+ϕ1(t,u0(t),x0(t),x(t),θ),⋮x˙i(t)=diu0(t)xi+1(t)+φi(y(t),y(t-τi))+ϕi(t,u0(t),x0(t),x(t),θ),⋮x˙n(t)=dnu1(t)+φn(y(t),y(t-τn))+ϕn(t,u0(t),x0(t),x(t),θ),y(t)=[x0(t),x1(t)]T,
where [x0(t),x(t)]T=[x0(t),x1(t),…,xn(t)]T∈Rn+1 and u(t)=[u0(t),u1(t)]T∈R2 are system states and control input, respectively. θ∈Rm is an unknown bounded parameter vector. di(0≤i≤n) are disturbed virtual control coefficients, and the individual signs are known. φid(y(t),y(t-τi))′s denote the delayed terms, which contain output delays. τi(1≤i≤n) are unknown constants; ϕi∈Rm are vectors of smooth nonlinear functions and represent unmodeled dynamic and external disturbances.Assumption 1.
For nonlinear functionsϕ0 and ϕi(1≤i≤n), there exist (known) smooth nonnegative functions  ϕ-0 and ϕ-ij(1≤j≤i≤n) such that
(2)ϕ0(t,x0(t),θ)=x0ϕ-0(x0,θ),|ϕi(t,u0,x0,xi,θ)|≤|x0|n-i+1ϕ-i0(x0,x1,θ)+∑j=1i‍|xj|ϕ-ij(u0,x0,x-i,θ),
where x-i=[x1,·,xi]T.Assumption 2.
The nonlinear functionsφi(y(t),y(t-τi))(1≤i≤n) satisfy
(3)|φi(y(t),y(t-τi))|≤|x1(t)x1(t-τi)|φ-i(y(t),y(t-τi)),
in which φ-i are (known) smooth nonnegative nonlinear functions.Remark 3.
It is clear that system (1) covers a number of important classes of uncertain nonholonomic systems that have been investigated in some existing literature resources. For instance, when di=1 and ϕi=0, ϕi=0 system (1) reduces the standard form of nonholonomic system which has been widely studied in the literature [15, 18–20]. Moreover, in Ge et al. [15], not only the virtual control coefficients di=1 and the dynamics ϕi satisfying ϕi=ϕ~iTθ are assumed but also the modeled dynamics φi do not exist. In  Liu and Zhang  [22], the virtual control coefficients and time delays have not been considered, and the expression ϕi=ϕ~iTθ is also required. While di=1 and φi and unknown parameters θ are not existent, system (1) degenerates to the one studied in Xi et al. [21]. When φi=0, together with ϕi=ϕ~iTθ, system (1) becomes the considered system in Ju et al. [23].Remark 4.
Note that here we only use the sign ofdi without any knowledge of individual virtual control coefficient di(1≤i≤n). Moreover, Assumptions 1 and 2 are imposed on the nonlinear functions ϕi and the delayed terms φi of system (1), respectively. It can be seen that some similar conditions are implied in [22].Lemma 5.
For any real-valued continuous functionf(x,y), where x∈Rn and y∈Rm, there are smooth functions a(x)≥0, b(y)≥0, c(x)≥1 and d(y)≥1 such that
(4)|f(x,y)|≤a(x)+b(y),|f(x,y)|≤c(x)d(y).By Lemma5 and Assumption 1, we know that there exist smooth functions ω0≥1, ωi0≥1, ωij≥1, ζ0≥1, ζi0≥1, and ζij≥1 such that
(5)|ϕ0(t,x0(t),θ)|≤|x0|ω0(x0)ζ0(θ),|ϕi(t,u0,x0,xi,θ)|≤|x0|n-i+1ωi0(x0,x1)ζi0(θ)+∑j=1i‍|xj|ωij(u0,x0,x-i)ζij(θ).Denote thatϑ=ζ0+∑i=1n∑j=0iζij(θ), then the above inequalities can be rewritten as follows:
(6)|ϕ0(t,x0(t),θ)|≤|x0|ω0(x0)ϑ,|ϕi(t,u0,x0,xi,θ)|≤|x0|n-i+1ωi0(x0,x1)ϑ+∑j=1i‍|xj|ωij(u0,x0,x-i)ϑ.


## 3. Adaptive Stabilization Control Design

In this section, we will design an adaptive stabilization controller for the case thatx0(t)≠0, and the case that x0(t)=0 will be considered in the next section. Now, we use two separate stages to globally asymptotically stabilize the system (1). Firstly, the control u0(t) should be designed for x0-subsystem; in the second stage, we design u1(t) to guarantee all states of the rest in system (1) converge to zero.


### 3.1. State Scaling

The following state scaling discontinuous transformation is introduced:(7)zi(t)=xi(t)x0n-i(t),1≤i≤n.Under the newz-coordinates, the system (1) is transformed into
(8)x˙0(t)=d0u0(t)+ϕ0(t,x0(t),θ),z˙i(t)=u0(t)x0(t)dizi+1(t)-(n-i)u0(t)x0(t)d0zi(t)+ϕ~i+φ~i,z˙n(t)=dnu1(t)+ϕ~n+φ~n,
where
(9)ϕ~i=ϕid(t,u0(t),x0(t),x(t),θ)x0n-i(t)-(n-i)ϕ0(t,u0(t),x0(t),x(t),θ)x0(t)zi(t),(10)φ~i=φi(y(t),y(t-τi))x0n-i(t).In order to obtain the estimation for the nonlinear functionsϕ~i and φ~i, the following lemmas are given.Lemma 6.
For every1≤i≤n, there exist smooth nonnegative functions ω~i0(x0,z1) and ω~ij(u0,x0,z-i) such that
(11)|ϕ~i|≤|x0(t)|ω~i0(x0(t),z1(t))ϑ+∑j=1i‍|zj(t)|ω~ij(u0(t),x0(t),z-i(t))ϑ,
where z-i=[z1,…,zi]T.Proof.
By the above inequalities (6), it can be deduced that
(12)|ϕ~i|≤|ϕi(t,uo,x0,x,θ)||x0n-i|+|(n-i)ϕ0(t,x0,θ)||x0||zi|≤|x0|ωi0(x0,x1)ϑ+∑j=1i‍|zj|hhhhhhhhhhhhhhhhhhh·|x0|i-jωij(u0,x0,x-i)ϑhhhhhhhhhhhhhhhhhhh+(n-i)|zi|ω0(x0)ϑ=|x0|ωi0(x0,z1x0n-i)ϑ+∑j=1i‍|zj|·|x0|i-jωij(u0,x0,z1x0n-i,…,zix0n-i)ϑ+(n-i)|zi|ω0(x0)ϑ.
Introduce the notation(13)ω~i0=ωi0(x0,z1x0n-i),ω~ij=|x0|i-jωij×(u0,x0,z1x0n-i,…,zix0n-i),1≤i≤j-1,ω~ii=ωii(u0,x0,z1x0n-i,…,zix0n-i)+(n-i)ω0(x0).
It is clear that ω~ij(0≤j≤i) are smooth nonnegative functions, and inequality (11) holds.Lemma 7.
For every1≤i≤n-1, the following inequality holds:
(14)|φ~i|≤|z1(t)z1(t-τi)|Γi(y(t),y(t-τi))≤|z1(t)z1(t-τi)|fi1(y(t))fi2(y(t-τi)),
where Γi, fi1, and fi2 are smooth nonnegative functions.Proof.
According to Assumption2, the nonlinear functions φ~i yield that
(15)|φ~i|=|φi(y(t),y(t-τi))||x0n-i(t)|≤|x1(t)x1(t-τi)||x0n-i(t)|φ-i(y(t),y(t-τi))=|z1(t)x0i-1(t)|·|z1(t-τi)x0n-1(t-τi)|φ-i(y(t),y(t-τi)).
LetΓi(y(t),y(t-τi))=|[1,0]y(t)|i-1|[1,0]y(t-τi)|n-1φ-i(y(t),y(t-τi)); then the above inequality can be expressed as
(16)|φ~i|≤|z1(t)z1(t-τi)|Γi(y(t),y(t-τi)).
It is seen thatΓi(y(t),y(t-τi)) are smooth functions. Then using Lemma 5, there exist smooth functions fi1≥1 and fi2≥1 such that
(17)|φ~i|≤|z1(t)z1(t-τi)|fi1(y(t))fi2(y(t-τi)).


### 3.2. Control Design

In this section, we design the control inputsu0(t) and u1(t) subject to x0(t0)≠0. The case that the initial condition x0(t0)=0 will be treated in Section 4. The design of the control inputs here is based on the backstepping method for the transformed system (10). The recursive procedure stops once the true system inputs occur.Step 1.
For thex0-subsystem
(18)x˙0(t)=d0u0(t)+ϕ0(t,x0(t),θ),
define new variables r0=1/|d0|, r~0=r0-r^0, and ϑ~0=ϑ-ϑ^0, where r^0 and ϑ^0 are the estimates of r0 and ϑ, respectively.Consider the Lyapunov function candidate(19)V0=12x02(t)+12ϑ~02+12r0r~02.Calculating the time derivative ofV0 along the system (18)
(20)V˙0=x0(t)[d0u0(t)+ϕ0(t,x0(t),θ)]-ϑ~0ϑ^˙0-1r0r~0r^˙0.The controlleru0(t) can be chosen as
(21)u0(t)=g0(x0,ϑ^0,r^0)x0(t),
where
(22)g0(x0,ϑ^0,r^0)=sign(d0)r^0π0(x0,ϑ^0,r^0),(23)π0(x0,ϑ^0,r^0)=-1+1+(k0/r^0)22×[ω0(x0)ϑ^0+c02+(ω0(x0)ϑ^0)2].With the controlleru0(t) in (21), the time derivative of V0 satisfies
(24)V˙0=(1-r~0r0)π0(x0,ϑ^0,r^0)x02(t)+x0(t)ϕ0(t,x0(t),θ)-ϑ~0ϑ^˙0-1r0r~0r^˙0≤-c02+(ω0(x0)ϑ^0)2x02(t)-r~0r0π0(x0,ϑ^0,r^0)x02(t)+x02(t)ω0(x0)ϑ~0-ϑ~0ϑ^˙0-1r0r~0r^˙0.Choosing the following update lawsϑ^˙0 and r^˙0 as
(25)ϑ^˙0=ω0(x0)x02(t),(26)r^˙0=-π0(x0,ϑ^0,r^0)x02(t),
we have
(27)V˙0≤-c0x02(t),
where c0>0 is a positive design parameter. Therefore, it implies that x0(t), ϑ^0, and r^0 are bound. By LaSalle’s Invariant Theorem, we can further achieve that x0(t)→0, as t→∞.Remark 8.
The closed-loop dynamics ofx0-subsystem is
(28)x˙0(t)=x0(t)[d0g0(x0,ϑ^0,r^0)+ϕ-0(x0,θ)].It is seen thatψ(t)=d0g0(x0,ϑ^0,r^0)+ϕ-0(x0,θ) is bounded as x0(t), ϑ^0, r^0, and θ are bounded. On the other hand, the solution of x0-system can be computed as
(29)x0(t)=x0(t0)e∫t0t‍ψ(s)ds.Obviously, forx0(t0)≠0 and t≥t0, the solution x0(t) exists and satisfies |x0(t)|>0. That is, x0(t) does not become zero at any time instant for x0(t0)≠0. Therefore, the introduced state scaling above is effective.Under the controlleru0(t) in (25), the z-system can be rewritten as
(30)z˙i(t)=dig0(x0,ϑ^0,r^0)zi+1(t)-(n-i)d0g0(x0,ϑ^0,r^0)zi(t)+ϕ~i+φ~i,z˙n(t)=dnu1(t)+ϕ~n+φ~n.Step 2.
Forz1(t)-subsystem in (30)
(31)z˙1(t)=d1g0(x0,ϑ^0,r^0)z2(t)-(n-1)d0g0(x0,ϑ^0,r^0)z1(t)+ϕ~1+φ~1,
let r1=1/|d1|, and d~0=d0-d^0, d~1=d1-d^1, r~1=r1-r^1, ϑ~1=ϑ-ϑ^1, and υ~=ϑ2-υ^, where d^0, d^1, r^1, ϑ^1, and υ^ are the estimates of unknown parameters d0, d1, r1, ϑ, and ϑ2, respectively. Introduce the coordinate transformations ξ1(t)=z1(t) and ξ2(t)=z2(t)-α1, where α1 is regarded as the virtual control input. Construct the following Lyapunov-Krasovskii functional:
(32)V1=V0+12ξ12(t)+12d~02+12d~12+12r1r~12+12ϑ~12+12υ~2+∑l=1n‍∑k=1l‍∫t-τkt‍εlk2ξ12(σ)fk22(y(σ))dσ,
where εlk>0(1≤k≤l≤n) are scalars. Along (31), the time derivative of V1 gives
(33)V˙1=V˙0+ξ1(t)[(d^1+d~1)g0(x0,ϑ^0,r^0)ξ2(t)V˙0+ξ1(t)+d1g0(x0,ϑ^0,r^0)α1-(n-1)(d^0+d~0)g0V˙0+ξ1(t)×(x0,ϑ^0,r^0)z1(t)+φ~1+ϕ~1(d^1+d~1)g0(x0,ϑ^0,r^0)ξ2]-d~0d^˙0-d~1d^˙1-1r1r~1r^˙1-ϑ~1ϑ^˙1-υ~υ^˙+∑l=1n‍∑k=1l‍εlk2ξ12(t)fk22(y(t))-∑l=1n‍∑k=1l‍εlk2ξ12(t-τk)fk22(y(t-τk)).
By Lemmas6 and 7, the following inequalities hold:
(34)|ξ1(t)ϕ~1|≤x02(t)+ξ12(t)γ10(x0,z1)ϑ2+ξ12(t)γ11(u0,x0,z1)ϑ,|ξ1(t)φ~1|≤|ξ1(t)z1(t)z1(t-τ1)×f11(y(t))f12(y(t-τ1))|≤12ε11ξ12(t)z12(t)f112(y(t))+ε112ξ12(t-τ1)f122(y(t-τ1)),
where γ10=(1/4)ω~102(x0,z1) and γ11=ω~11(u0,x0,z1). Choose a virtual control function α1(x0,z1,ϑ^0,ϑ^1,υ^,r^0,r^1,d^0) as follows:
(35)α1=sign(d1)r^1π1,π1=-1g0(x0,ϑ^0,r^0)[∑l=1n‍∑k=1l‍c1ξ1(t)+ξ1(t)γ10(x0,z1)υ^-1g0(x0,ϑ^0,r^0)+ξ1(t)γ11(u0,x0,z1)ϑ^1-1g0(x0,ϑ^0,r^0)+12ε11ξ1(t)z12(t)f112(y(t))-1g0(x0,ϑ^0,r^0)+∑l=1n‍∑k=1l‍εlk2ξ1(t)fk22(y(t))]+(n-1)d^0z1(t),
where c1 is a positive design parameter. With the choice of the update law
(36)r^˙1=-g0(x0,ϑ^0,r^0)π1ξ1(t),
we can obtain
(37)V˙1≤-(c0-1)x02(t)-c1ξ12(t)+d^1g0(x0,ϑ^0,r^0)ξ1(t)ξ2(t)-d~0(d^˙0-τ10)-d~1(d^˙1-τ11)-υ~(υ^˙-η10)-ϑ~1(ϑ^˙1-η11)-∑l=2n‍∑k=1l‍εlk2ξ12(t-τk)fk22(y(t-τk)),
where
(38)τ10=ξ1(t)h10,h10=-(n-1)g0(x0,ϑ^0,r^0)ξ1(t),τ11=ξ1(t)h11,h11=g0(x0,ϑ^0,r^0)ξ2(t),η10=γ10(x0,z1)ξ12(t),η11=γ11(u0,x0,z1)ξ12(t).Step i (2≤i≤n-1).
Assume that, at stepi-1, a virtual control function αi-1 and a Lyapunov functional Vi-1 have been designed for the (z1,…,zi-1)-subsystem of (30) in such a way that
(39)V˙i-1≤-(c0-i+1)x02(t)-∑j=1i-1‍[cj-i+j+1]ξj2(t)+d^i-1g0(x0,ϑ^0,r^0)ξi-1(t)ξi(t)-∑j=0i-1‍(d~j+∑l=j+2i‍∂αl-1∂d^jξl(t))(d^˙j-τij)-(υ~+∑j=2i-1‍∂αj-1∂υ^ξj(t))(υ^˙-ηi0)-(ϑ~1+∑j=2i-1‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηi1)-∑l=in‍∑k=1l‍12ξ12(t-τk)fk22(y(t-τk)).
Now, we examine the(z1,…,zi)-subsystem of (30). Define ri=1/|di|, and d~i=di-d^i and r~i=ri-r^i, where d^i and r^i are the estimates of unknown parameters di and ri, respectively. Introduce the coordinate transformations ξi+1(t)=zi+1(t)-αi, where αi is regarded as a virtual control input, and construct the following Lyapunov-Krasovskii functional
(40)Vi=Vi-1+12ξi2(t)+12d~i2+12rir~i2.
Based on (39), the time derivative of  Vi along the solutions of (30) satisfies
(41)V˙i=V˙i-1+ξi(t)×{∑j=1i-1‍(d^i+d~i)g0(x0,ϑ^0,r^0)ξi+1(t)+dig0(x0,ϑ^0,r^0)αi(t)+φ~i+ϕ~i-(n-i)(d^0+d~0)g0(x0,ϑ^0,r^0)zi(t)-∂αi-1∂x0[-(d^0+d~0)g0(x0,ϑ^0,r^0)x0(t)+ϕ0]-∑j=1i-1‍∂αi-1∂zj[-(d^j+d~j)g0(x0,ϑ^0,r^0)zj+1(t)-∑j=1i-1‍∂αi-1∂zj-(n-j)(d^0+d~0)g0(x0,ϑ^0,r^0)zj(t)-∑j=1i-1‍∂αi-1∂zj+φ~j+ϕ~j]-∑j=0i-1‍∂αi-1∂r^jr^˙j-∑j=0i-2‍∂αi-1∂d^jd^˙j-∂αi-1∂ϑ^0ϑ^˙0-∂αi-1∂ϑ^1ϑ^˙1-∂αi-1∂υ^υ^˙∑j=1i-1‍}-d~id^˙i-1rir~ir^˙i.
By Lemmas5 and 7 and Young inequality, the following inequality holds:
(42)|ξi(t)[φ~i-∑j=1i-1‍∂αi-1∂zjφ~j]|≤12εiiξi2(t)z12(t)fi12(y(t))+∑j=1i-1‍12εijξi2(t)(∂αi-1∂zj)2z12(t)fj12(y(t))+∑j=1i‍εij2ξ12(t-τj)fj22(y(t-τj)).
Using Lemma6 and Young inequality, there are nonnegative smooth functions γi0(x0,z-i,ϑ^0,ϑ^1,υ^,r^0,…,r^i-1,d^0,…,d^i-2) and γi1(u0,x0,z-i) such that
(43)|ξi(t)[ϕ~i-∑j=1i-1‍∂αi-1∂zjϕ~j-∂αi-1∂x0ϕ0]|≤x02(t)+∑j=1i-1‍ξj2(t)+ξi2(t)γi0ϑ2+ξi2(t)γi1ϑ,
where γi0 and γi1 are known nonnegative functions.
Choose the following virtual control functionαi:
(44)αi=sign(di)r^iπi,πi=-d^i-1ξi-1(t)+(n-i)d^0zi(t)+∑j=1i-1‍d^j∂αi-1∂zjzj+1(t)-d^0∑j=1i-1‍(n-j)∂αi-1∂zjzj(t)+d^0∂αi-1∂x0x0(t)-1g0(x0,ϑ^0,r^0)×[∑j=2i-1‍ciξi(t)+ξi(t)γi0υ^+ξi(t)γi1ϑ^1-∂αi-1∂ϑ^0ϑ^˙0-∑j=0i-1‍∂αi-1∂r^jr^˙j-∑j=0i-2‍∂αi-1∂d^jτij-∂αi-1∂υ^ηi0-∂αi-1∂ϑ^1ηi1+12εiiξi(t)z12(t)fi12(y(t))+∑j=1i-1‍12εijξi(t)(∂αi-1∂zj)2z12(t)fj12(y(t))-∑l=2i-1‍∑j=0l-2‍ξl(t)∂αl-1∂d^jhij-∑j=2i-1‍ξj(t)∂αj-1∂υ^γi0-∑j=2i-1‍ξj(t)∂αj-1∂ϑ^1γi1],
where ci is a positive design parameter and τij(0≤j≤i-2), ηi0 and ηi1 are pending nonnegative functions to be specified in (47). Moreover, construct the following update law:
(45)r^˙i=-g0(x0,ϑ^0,r^0)πiξi(t).
Substituting inequalities (42)–(45) into (41) yields
(46)V˙i≤-(c0-i)x02(t)-∑j=1i‍[cj-i+j]ξj2(t)+d^ig0(x0,ϑ^0,r^0)ξi(t)ξi+1(t)-∑j=0i‍(d~j+∑l=j+2i‍∂αl-1∂d^jξl(t))(d^˙j-τij)-(υ~+∑j=2i‍∂αj-1∂υ^ξj(t))(υ^˙-ηi0)-(ϑ~1+∑j=2i‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηi1)-∑l=i+1n‍∑k=1l‍εlk2ξ12(t-τk)fk22(y(t-τk)),
where
(47)τi0=τi-1,0+hi0ξi(t),hi0=-(n-i)g0(x0,ϑ^0,r^0)zi(t)-∂αi-1∂x0g0(x0,ϑ^0,r^0)x0(t)+∑j=1i-1‍(n-j)∂αi-1∂zjg0(x0,ϑ^0,r^0)zj(t),τil=τi-1,l+hilξi(t),hil=-∂αi-1∂zlg0(x0,ϑ^0,r^0)zl+1(t),=-∂αi-1∂zlg(x0,ϑ^0,r^0)1≤l≤i-1,τii=hiiξi(t),hii=g0(x0,ϑ^0,r^0)ξi+1(t),ηi0=ηi-1,0+γi0ξi(t),ηi1=ηi-1,1+γi1ξi(t).Step n.
At the last step, we study the wholez-subsystem (30), and the true input u1(t) will be designed on the basis of the virtual control αi′s and the Lyapunov function Vn-1 introduced before. Here, let us consider a Lyapunov-Krasovskii function Vn as follows:
(48)Vn=Vn-1+12ξn2(t)+12rnr~n2.
Denote rn=1/|dn| and r~n=rn-r^n, where r^n is the estimate of unknown parameter rn. Recall that ξn(t)=zn(t)-αn-1, with αn-1 being a virtual control input, then
(49)V˙n-1≤-(c0-n+1)x02(t)-∑j=1n-1‍[cj-n+j+1]ξj2(t)+d^n-1g0(x0,ϑ^0,r^0)ξn-1(t)ξn(t)-∑j=0n-1‍(d~j+∑l=j+2n-1‍∂αl-1∂d^jξl(t))(d^˙j-τn-1,j)-(υ~+∑j=2n-1‍∂αj-1∂υ^ξj(t))(υ^˙-ηn-1,0)-(ϑ~1+∑j=2n-1‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηn-1,1)-∑k=1n‍εlk2ξ12(t-τk)fk22(y(t-τk)).
DifferentiatingVn along the solution of (30) gives
(50)V˙n=V˙n-1+ξn(t)×{∑j=1n-1‍dnu1(t)+φ~n+ϕ~n-∂αn-1∂x0××[(d^0+d~0)g0(x0,ϑ^0,r^0)x0(t)+ϕ0]--∑j=1n-1‍∂αn-1∂zj[(d^j+d~j)g0(x0,ϑ^0,r^0)zj+1(t)-∑j=1n-1‍∂αn-1∂zj---(n-j)(d^0+d~0)g0-∑j=1n-1‍∂αn-1∂zj×××(x0,ϑ^0,r^0)zj(t)+φ~j+ϕ~j(d^j+d~j)g0(x0,ϑ^0,r^0)zj+1(d^j+d~j)]--∑j=0n-1‍∂αn-1∂r^jr^˙j-∑j=0n-2‍∂αi-1∂d^jd^˙j--∂αn-1∂ϑ^0ϑ^˙0-∂αn-1∂ϑ^1ϑ^˙1-∂αn-1∂υ^υ^˙∑j=1n-1‍}-1rir~nr^˙n.
Similarly, by Lemmas5–7 and Young inequality, we can easily obtain that there are scalars εnj>0(1≤j≤n) and smooth nonnegative functions γn0 and γn1 such that
(51)|ξn(t)[φ~n-∑j=1n-1‍∂αn-1∂zjφ~j]|≤12εnnξn2(t)z12(t)fn12(y(t))+∑j=1n-1‍12εnjξn2(t)(∂αn-1∂zj)2z12(t)fj12(y(t))+∑j=1n‍εnj2ξ12(t-τj)fj22(y(t-τj)),|ξn(t)[ϕ~n-∑j=1n-1‍∂αn-1∂zjϕ~j-∂αn-1∂x0ϕ0]|≤x02(t)+∑j=1n-1‍ξj2(t)+ξn2(t)γn0ϑ2+ξn2(t)γn1(u0,x0,z-n)ϑ.
Next, we can design the control inputu1(t) as follows:
(52)u1(t)=sign(dn)r^nπn,(53)πn=-g0(x0,ϑ^0,r^0)×{d^n-1ξn-1(t)-∑j=1n-1‍∂αn-1∂zjaaa×[d^jzj+1(t)-(n-j)d^0zj(t)]aaa-d^0∂αn-1∂x0x0(t)∑j=1n-1‍∂αn-1∂zj}-cnξn(t)-ξn(t)γn0υ^-ξn(t)γn1ϑ^1+∂αn-1∂ϑ^0ϑ^˙0+∑j=0n-1‍∂αn-1∂r^jr^˙j+∑j=0n-2‍∂αn-1∂d^jτnj+∂αn-1∂υ^ηn0+∂αn-1∂ϑ^1ηn1-12εnnξn(t)z12(t)fn12(y(t))-∑j=1n-1‍12εnjξn(t)(∂αn-1∂zj)2z12(t)fj12(y(t))+∑l=2n-1‍∑j=0l-2‍ξl(t)∂αl-1∂d^jhnj+∑j=2n-1‍ξj(t)∂αj-1∂υ^ηn0+∑j=2n-1‍ξj(t)∂αj-1∂ϑ^1ηn1,
where cn is a positive design parameter and τnj(0≤j≤n-1), ηn0 and ηn1 are smooth nonnegative functions to be specified in (56). With the choice of the update law
(54)r^˙n=-πnξi(t),
it renders
(55)V˙n≤-(c0-n)x02(t)-∑j=1n‍[cj-n+j]ξj2(t)-∑j=0n-1‍(d~j+∑l=j+2n-1‍∂αl-1∂d^jξl(t))(d^˙j-τnj)-(υ~+∑j=2n‍∂αj-1∂υ^ξj(t))(υ^˙-ηn0)-(ϑ~1+∑j=2n‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηn1),
where
(56)τn0=τn-1,0+hn0ξn(t),hn0=-∂αn-1∂x0g0(x0,ϑ^0,r^0)x0(t)+∑j=1n-1‍(n-j)∂αn-1∂zjg0(x0,ϑ^0,r^0)zj(t),τnl=τn-1,l+hnlξi(t),hnl=-∂αn-1∂zlg0(x0,ϑ^0,r^0)zl+1(t),=-∂αn-1∂zlg(x0,ϑ^0,r^0)1≤l≤n-1,ηn0=ηn-1,0+γn0ξn(t),ηn1=ηn-1,1+γn1ξn(t).
Furthermore, employing the following update laws
(57)d^˙j=τnj,0≤j≤n-1,υ^˙=ηn0,ϑ^˙1=ηn1
eventually achieves
(58)V˙n≤-(c0-n)x02(t)-∑j=1n‍[cj-n+j]ξj2(t).This together with (48) implies that x0, ξi, υ~, ϑ~0, ϑ~1 and d~i, r~i(0≤i≤n) are bounded. Since θ and di are constant vector and constant, respectively, we know that ϑ^0, ϑ^1 and d^i, r^i(0≤i≤n) are also bounded. According to the definitions of the virtual control input αi in the above design procedure, zi are bounded as αi are bounded. It indicates that all signals of the closed loop system are bounded.LaSalle Invariant Theorem further achieves thatx0,ξi,υ~,ϑ~0,ϑ~1,d~i,r~i→0 as t→∞. The boundedness of all signals and the choice of virtual control functions αi imply that αi converge to zero, which shows that zi also tend to zero. From the transformation xi=zix0n-i, we can prove that x0→0, as t→0.The above analysis is summarized into the following theorem.Theorem 9.
For the system (1), under Assumptions 1 and 2, if the control strategies (21) and (52) are applied with an appropriate choice of the design parameters ci(0≤i≤n), the global asymptotic stabilization of the closed loop system is achieved for x0(t0)≠0.In the next section, we will deal with the stability analysis of the closed loop system with our control laws (21) and (52) as long as the initial condition x0(t0) is zero.


## 3.1. State Scaling

The following state scaling discontinuous transformation is introduced:(7)zi(t)=xi(t)x0n-i(t),1≤i≤n.Under the newz-coordinates, the system (1) is transformed into
(8)x˙0(t)=d0u0(t)+ϕ0(t,x0(t),θ),z˙i(t)=u0(t)x0(t)dizi+1(t)-(n-i)u0(t)x0(t)d0zi(t)+ϕ~i+φ~i,z˙n(t)=dnu1(t)+ϕ~n+φ~n,
where
(9)ϕ~i=ϕid(t,u0(t),x0(t),x(t),θ)x0n-i(t)-(n-i)ϕ0(t,u0(t),x0(t),x(t),θ)x0(t)zi(t),(10)φ~i=φi(y(t),y(t-τi))x0n-i(t).In order to obtain the estimation for the nonlinear functionsϕ~i and φ~i, the following lemmas are given.Lemma 6.
For every1≤i≤n, there exist smooth nonnegative functions ω~i0(x0,z1) and ω~ij(u0,x0,z-i) such that
(11)|ϕ~i|≤|x0(t)|ω~i0(x0(t),z1(t))ϑ+∑j=1i‍|zj(t)|ω~ij(u0(t),x0(t),z-i(t))ϑ,
where z-i=[z1,…,zi]T.Proof.
By the above inequalities (6), it can be deduced that
(12)|ϕ~i|≤|ϕi(t,uo,x0,x,θ)||x0n-i|+|(n-i)ϕ0(t,x0,θ)||x0||zi|≤|x0|ωi0(x0,x1)ϑ+∑j=1i‍|zj|hhhhhhhhhhhhhhhhhhh·|x0|i-jωij(u0,x0,x-i)ϑhhhhhhhhhhhhhhhhhhh+(n-i)|zi|ω0(x0)ϑ=|x0|ωi0(x0,z1x0n-i)ϑ+∑j=1i‍|zj|·|x0|i-jωij(u0,x0,z1x0n-i,…,zix0n-i)ϑ+(n-i)|zi|ω0(x0)ϑ.
Introduce the notation(13)ω~i0=ωi0(x0,z1x0n-i),ω~ij=|x0|i-jωij×(u0,x0,z1x0n-i,…,zix0n-i),1≤i≤j-1,ω~ii=ωii(u0,x0,z1x0n-i,…,zix0n-i)+(n-i)ω0(x0).
It is clear that ω~ij(0≤j≤i) are smooth nonnegative functions, and inequality (11) holds.Lemma 7.
For every1≤i≤n-1, the following inequality holds:
(14)|φ~i|≤|z1(t)z1(t-τi)|Γi(y(t),y(t-τi))≤|z1(t)z1(t-τi)|fi1(y(t))fi2(y(t-τi)),
where Γi, fi1, and fi2 are smooth nonnegative functions.Proof.
According to Assumption2, the nonlinear functions φ~i yield that
(15)|φ~i|=|φi(y(t),y(t-τi))||x0n-i(t)|≤|x1(t)x1(t-τi)||x0n-i(t)|φ-i(y(t),y(t-τi))=|z1(t)x0i-1(t)|·|z1(t-τi)x0n-1(t-τi)|φ-i(y(t),y(t-τi)).
LetΓi(y(t),y(t-τi))=|[1,0]y(t)|i-1|[1,0]y(t-τi)|n-1φ-i(y(t),y(t-τi)); then the above inequality can be expressed as
(16)|φ~i|≤|z1(t)z1(t-τi)|Γi(y(t),y(t-τi)).
It is seen thatΓi(y(t),y(t-τi)) are smooth functions. Then using Lemma 5, there exist smooth functions fi1≥1 and fi2≥1 such that
(17)|φ~i|≤|z1(t)z1(t-τi)|fi1(y(t))fi2(y(t-τi)).


## 3.2. Control Design

In this section, we design the control inputsu0(t) and u1(t) subject to x0(t0)≠0. The case that the initial condition x0(t0)=0 will be treated in Section 4. The design of the control inputs here is based on the backstepping method for the transformed system (10). The recursive procedure stops once the true system inputs occur.Step 1.
For thex0-subsystem
(18)x˙0(t)=d0u0(t)+ϕ0(t,x0(t),θ),
define new variables r0=1/|d0|, r~0=r0-r^0, and ϑ~0=ϑ-ϑ^0, where r^0 and ϑ^0 are the estimates of r0 and ϑ, respectively.Consider the Lyapunov function candidate(19)V0=12x02(t)+12ϑ~02+12r0r~02.Calculating the time derivative ofV0 along the system (18)
(20)V˙0=x0(t)[d0u0(t)+ϕ0(t,x0(t),θ)]-ϑ~0ϑ^˙0-1r0r~0r^˙0.The controlleru0(t) can be chosen as
(21)u0(t)=g0(x0,ϑ^0,r^0)x0(t),
where
(22)g0(x0,ϑ^0,r^0)=sign(d0)r^0π0(x0,ϑ^0,r^0),(23)π0(x0,ϑ^0,r^0)=-1+1+(k0/r^0)22×[ω0(x0)ϑ^0+c02+(ω0(x0)ϑ^0)2].With the controlleru0(t) in (21), the time derivative of V0 satisfies
(24)V˙0=(1-r~0r0)π0(x0,ϑ^0,r^0)x02(t)+x0(t)ϕ0(t,x0(t),θ)-ϑ~0ϑ^˙0-1r0r~0r^˙0≤-c02+(ω0(x0)ϑ^0)2x02(t)-r~0r0π0(x0,ϑ^0,r^0)x02(t)+x02(t)ω0(x0)ϑ~0-ϑ~0ϑ^˙0-1r0r~0r^˙0.Choosing the following update lawsϑ^˙0 and r^˙0 as
(25)ϑ^˙0=ω0(x0)x02(t),(26)r^˙0=-π0(x0,ϑ^0,r^0)x02(t),
we have
(27)V˙0≤-c0x02(t),
where c0>0 is a positive design parameter. Therefore, it implies that x0(t), ϑ^0, and r^0 are bound. By LaSalle’s Invariant Theorem, we can further achieve that x0(t)→0, as t→∞.Remark 8.
The closed-loop dynamics ofx0-subsystem is
(28)x˙0(t)=x0(t)[d0g0(x0,ϑ^0,r^0)+ϕ-0(x0,θ)].It is seen thatψ(t)=d0g0(x0,ϑ^0,r^0)+ϕ-0(x0,θ) is bounded as x0(t), ϑ^0, r^0, and θ are bounded. On the other hand, the solution of x0-system can be computed as
(29)x0(t)=x0(t0)e∫t0t‍ψ(s)ds.Obviously, forx0(t0)≠0 and t≥t0, the solution x0(t) exists and satisfies |x0(t)|>0. That is, x0(t) does not become zero at any time instant for x0(t0)≠0. Therefore, the introduced state scaling above is effective.Under the controlleru0(t) in (25), the z-system can be rewritten as
(30)z˙i(t)=dig0(x0,ϑ^0,r^0)zi+1(t)-(n-i)d0g0(x0,ϑ^0,r^0)zi(t)+ϕ~i+φ~i,z˙n(t)=dnu1(t)+ϕ~n+φ~n.Step 2.
Forz1(t)-subsystem in (30)
(31)z˙1(t)=d1g0(x0,ϑ^0,r^0)z2(t)-(n-1)d0g0(x0,ϑ^0,r^0)z1(t)+ϕ~1+φ~1,
let r1=1/|d1|, and d~0=d0-d^0, d~1=d1-d^1, r~1=r1-r^1, ϑ~1=ϑ-ϑ^1, and υ~=ϑ2-υ^, where d^0, d^1, r^1, ϑ^1, and υ^ are the estimates of unknown parameters d0, d1, r1, ϑ, and ϑ2, respectively. Introduce the coordinate transformations ξ1(t)=z1(t) and ξ2(t)=z2(t)-α1, where α1 is regarded as the virtual control input. Construct the following Lyapunov-Krasovskii functional:
(32)V1=V0+12ξ12(t)+12d~02+12d~12+12r1r~12+12ϑ~12+12υ~2+∑l=1n‍∑k=1l‍∫t-τkt‍εlk2ξ12(σ)fk22(y(σ))dσ,
where εlk>0(1≤k≤l≤n) are scalars. Along (31), the time derivative of V1 gives
(33)V˙1=V˙0+ξ1(t)[(d^1+d~1)g0(x0,ϑ^0,r^0)ξ2(t)V˙0+ξ1(t)+d1g0(x0,ϑ^0,r^0)α1-(n-1)(d^0+d~0)g0V˙0+ξ1(t)×(x0,ϑ^0,r^0)z1(t)+φ~1+ϕ~1(d^1+d~1)g0(x0,ϑ^0,r^0)ξ2]-d~0d^˙0-d~1d^˙1-1r1r~1r^˙1-ϑ~1ϑ^˙1-υ~υ^˙+∑l=1n‍∑k=1l‍εlk2ξ12(t)fk22(y(t))-∑l=1n‍∑k=1l‍εlk2ξ12(t-τk)fk22(y(t-τk)).
By Lemmas6 and 7, the following inequalities hold:
(34)|ξ1(t)ϕ~1|≤x02(t)+ξ12(t)γ10(x0,z1)ϑ2+ξ12(t)γ11(u0,x0,z1)ϑ,|ξ1(t)φ~1|≤|ξ1(t)z1(t)z1(t-τ1)×f11(y(t))f12(y(t-τ1))|≤12ε11ξ12(t)z12(t)f112(y(t))+ε112ξ12(t-τ1)f122(y(t-τ1)),
where γ10=(1/4)ω~102(x0,z1) and γ11=ω~11(u0,x0,z1). Choose a virtual control function α1(x0,z1,ϑ^0,ϑ^1,υ^,r^0,r^1,d^0) as follows:
(35)α1=sign(d1)r^1π1,π1=-1g0(x0,ϑ^0,r^0)[∑l=1n‍∑k=1l‍c1ξ1(t)+ξ1(t)γ10(x0,z1)υ^-1g0(x0,ϑ^0,r^0)+ξ1(t)γ11(u0,x0,z1)ϑ^1-1g0(x0,ϑ^0,r^0)+12ε11ξ1(t)z12(t)f112(y(t))-1g0(x0,ϑ^0,r^0)+∑l=1n‍∑k=1l‍εlk2ξ1(t)fk22(y(t))]+(n-1)d^0z1(t),
where c1 is a positive design parameter. With the choice of the update law
(36)r^˙1=-g0(x0,ϑ^0,r^0)π1ξ1(t),
we can obtain
(37)V˙1≤-(c0-1)x02(t)-c1ξ12(t)+d^1g0(x0,ϑ^0,r^0)ξ1(t)ξ2(t)-d~0(d^˙0-τ10)-d~1(d^˙1-τ11)-υ~(υ^˙-η10)-ϑ~1(ϑ^˙1-η11)-∑l=2n‍∑k=1l‍εlk2ξ12(t-τk)fk22(y(t-τk)),
where
(38)τ10=ξ1(t)h10,h10=-(n-1)g0(x0,ϑ^0,r^0)ξ1(t),τ11=ξ1(t)h11,h11=g0(x0,ϑ^0,r^0)ξ2(t),η10=γ10(x0,z1)ξ12(t),η11=γ11(u0,x0,z1)ξ12(t).Step i (2≤i≤n-1).
Assume that, at stepi-1, a virtual control function αi-1 and a Lyapunov functional Vi-1 have been designed for the (z1,…,zi-1)-subsystem of (30) in such a way that
(39)V˙i-1≤-(c0-i+1)x02(t)-∑j=1i-1‍[cj-i+j+1]ξj2(t)+d^i-1g0(x0,ϑ^0,r^0)ξi-1(t)ξi(t)-∑j=0i-1‍(d~j+∑l=j+2i‍∂αl-1∂d^jξl(t))(d^˙j-τij)-(υ~+∑j=2i-1‍∂αj-1∂υ^ξj(t))(υ^˙-ηi0)-(ϑ~1+∑j=2i-1‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηi1)-∑l=in‍∑k=1l‍12ξ12(t-τk)fk22(y(t-τk)).
Now, we examine the(z1,…,zi)-subsystem of (30). Define ri=1/|di|, and d~i=di-d^i and r~i=ri-r^i, where d^i and r^i are the estimates of unknown parameters di and ri, respectively. Introduce the coordinate transformations ξi+1(t)=zi+1(t)-αi, where αi is regarded as a virtual control input, and construct the following Lyapunov-Krasovskii functional
(40)Vi=Vi-1+12ξi2(t)+12d~i2+12rir~i2.
Based on (39), the time derivative of  Vi along the solutions of (30) satisfies
(41)V˙i=V˙i-1+ξi(t)×{∑j=1i-1‍(d^i+d~i)g0(x0,ϑ^0,r^0)ξi+1(t)+dig0(x0,ϑ^0,r^0)αi(t)+φ~i+ϕ~i-(n-i)(d^0+d~0)g0(x0,ϑ^0,r^0)zi(t)-∂αi-1∂x0[-(d^0+d~0)g0(x0,ϑ^0,r^0)x0(t)+ϕ0]-∑j=1i-1‍∂αi-1∂zj[-(d^j+d~j)g0(x0,ϑ^0,r^0)zj+1(t)-∑j=1i-1‍∂αi-1∂zj-(n-j)(d^0+d~0)g0(x0,ϑ^0,r^0)zj(t)-∑j=1i-1‍∂αi-1∂zj+φ~j+ϕ~j]-∑j=0i-1‍∂αi-1∂r^jr^˙j-∑j=0i-2‍∂αi-1∂d^jd^˙j-∂αi-1∂ϑ^0ϑ^˙0-∂αi-1∂ϑ^1ϑ^˙1-∂αi-1∂υ^υ^˙∑j=1i-1‍}-d~id^˙i-1rir~ir^˙i.
By Lemmas5 and 7 and Young inequality, the following inequality holds:
(42)|ξi(t)[φ~i-∑j=1i-1‍∂αi-1∂zjφ~j]|≤12εiiξi2(t)z12(t)fi12(y(t))+∑j=1i-1‍12εijξi2(t)(∂αi-1∂zj)2z12(t)fj12(y(t))+∑j=1i‍εij2ξ12(t-τj)fj22(y(t-τj)).
Using Lemma6 and Young inequality, there are nonnegative smooth functions γi0(x0,z-i,ϑ^0,ϑ^1,υ^,r^0,…,r^i-1,d^0,…,d^i-2) and γi1(u0,x0,z-i) such that
(43)|ξi(t)[ϕ~i-∑j=1i-1‍∂αi-1∂zjϕ~j-∂αi-1∂x0ϕ0]|≤x02(t)+∑j=1i-1‍ξj2(t)+ξi2(t)γi0ϑ2+ξi2(t)γi1ϑ,
where γi0 and γi1 are known nonnegative functions.
Choose the following virtual control functionαi:
(44)αi=sign(di)r^iπi,πi=-d^i-1ξi-1(t)+(n-i)d^0zi(t)+∑j=1i-1‍d^j∂αi-1∂zjzj+1(t)-d^0∑j=1i-1‍(n-j)∂αi-1∂zjzj(t)+d^0∂αi-1∂x0x0(t)-1g0(x0,ϑ^0,r^0)×[∑j=2i-1‍ciξi(t)+ξi(t)γi0υ^+ξi(t)γi1ϑ^1-∂αi-1∂ϑ^0ϑ^˙0-∑j=0i-1‍∂αi-1∂r^jr^˙j-∑j=0i-2‍∂αi-1∂d^jτij-∂αi-1∂υ^ηi0-∂αi-1∂ϑ^1ηi1+12εiiξi(t)z12(t)fi12(y(t))+∑j=1i-1‍12εijξi(t)(∂αi-1∂zj)2z12(t)fj12(y(t))-∑l=2i-1‍∑j=0l-2‍ξl(t)∂αl-1∂d^jhij-∑j=2i-1‍ξj(t)∂αj-1∂υ^γi0-∑j=2i-1‍ξj(t)∂αj-1∂ϑ^1γi1],
where ci is a positive design parameter and τij(0≤j≤i-2), ηi0 and ηi1 are pending nonnegative functions to be specified in (47). Moreover, construct the following update law:
(45)r^˙i=-g0(x0,ϑ^0,r^0)πiξi(t).
Substituting inequalities (42)–(45) into (41) yields
(46)V˙i≤-(c0-i)x02(t)-∑j=1i‍[cj-i+j]ξj2(t)+d^ig0(x0,ϑ^0,r^0)ξi(t)ξi+1(t)-∑j=0i‍(d~j+∑l=j+2i‍∂αl-1∂d^jξl(t))(d^˙j-τij)-(υ~+∑j=2i‍∂αj-1∂υ^ξj(t))(υ^˙-ηi0)-(ϑ~1+∑j=2i‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηi1)-∑l=i+1n‍∑k=1l‍εlk2ξ12(t-τk)fk22(y(t-τk)),
where
(47)τi0=τi-1,0+hi0ξi(t),hi0=-(n-i)g0(x0,ϑ^0,r^0)zi(t)-∂αi-1∂x0g0(x0,ϑ^0,r^0)x0(t)+∑j=1i-1‍(n-j)∂αi-1∂zjg0(x0,ϑ^0,r^0)zj(t),τil=τi-1,l+hilξi(t),hil=-∂αi-1∂zlg0(x0,ϑ^0,r^0)zl+1(t),=-∂αi-1∂zlg(x0,ϑ^0,r^0)1≤l≤i-1,τii=hiiξi(t),hii=g0(x0,ϑ^0,r^0)ξi+1(t),ηi0=ηi-1,0+γi0ξi(t),ηi1=ηi-1,1+γi1ξi(t).Step n.
At the last step, we study the wholez-subsystem (30), and the true input u1(t) will be designed on the basis of the virtual control αi′s and the Lyapunov function Vn-1 introduced before. Here, let us consider a Lyapunov-Krasovskii function Vn as follows:
(48)Vn=Vn-1+12ξn2(t)+12rnr~n2.
Denote rn=1/|dn| and r~n=rn-r^n, where r^n is the estimate of unknown parameter rn. Recall that ξn(t)=zn(t)-αn-1, with αn-1 being a virtual control input, then
(49)V˙n-1≤-(c0-n+1)x02(t)-∑j=1n-1‍[cj-n+j+1]ξj2(t)+d^n-1g0(x0,ϑ^0,r^0)ξn-1(t)ξn(t)-∑j=0n-1‍(d~j+∑l=j+2n-1‍∂αl-1∂d^jξl(t))(d^˙j-τn-1,j)-(υ~+∑j=2n-1‍∂αj-1∂υ^ξj(t))(υ^˙-ηn-1,0)-(ϑ~1+∑j=2n-1‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηn-1,1)-∑k=1n‍εlk2ξ12(t-τk)fk22(y(t-τk)).
DifferentiatingVn along the solution of (30) gives
(50)V˙n=V˙n-1+ξn(t)×{∑j=1n-1‍dnu1(t)+φ~n+ϕ~n-∂αn-1∂x0××[(d^0+d~0)g0(x0,ϑ^0,r^0)x0(t)+ϕ0]--∑j=1n-1‍∂αn-1∂zj[(d^j+d~j)g0(x0,ϑ^0,r^0)zj+1(t)-∑j=1n-1‍∂αn-1∂zj---(n-j)(d^0+d~0)g0-∑j=1n-1‍∂αn-1∂zj×××(x0,ϑ^0,r^0)zj(t)+φ~j+ϕ~j(d^j+d~j)g0(x0,ϑ^0,r^0)zj+1(d^j+d~j)]--∑j=0n-1‍∂αn-1∂r^jr^˙j-∑j=0n-2‍∂αi-1∂d^jd^˙j--∂αn-1∂ϑ^0ϑ^˙0-∂αn-1∂ϑ^1ϑ^˙1-∂αn-1∂υ^υ^˙∑j=1n-1‍}-1rir~nr^˙n.
Similarly, by Lemmas5–7 and Young inequality, we can easily obtain that there are scalars εnj>0(1≤j≤n) and smooth nonnegative functions γn0 and γn1 such that
(51)|ξn(t)[φ~n-∑j=1n-1‍∂αn-1∂zjφ~j]|≤12εnnξn2(t)z12(t)fn12(y(t))+∑j=1n-1‍12εnjξn2(t)(∂αn-1∂zj)2z12(t)fj12(y(t))+∑j=1n‍εnj2ξ12(t-τj)fj22(y(t-τj)),|ξn(t)[ϕ~n-∑j=1n-1‍∂αn-1∂zjϕ~j-∂αn-1∂x0ϕ0]|≤x02(t)+∑j=1n-1‍ξj2(t)+ξn2(t)γn0ϑ2+ξn2(t)γn1(u0,x0,z-n)ϑ.
Next, we can design the control inputu1(t) as follows:
(52)u1(t)=sign(dn)r^nπn,(53)πn=-g0(x0,ϑ^0,r^0)×{d^n-1ξn-1(t)-∑j=1n-1‍∂αn-1∂zjaaa×[d^jzj+1(t)-(n-j)d^0zj(t)]aaa-d^0∂αn-1∂x0x0(t)∑j=1n-1‍∂αn-1∂zj}-cnξn(t)-ξn(t)γn0υ^-ξn(t)γn1ϑ^1+∂αn-1∂ϑ^0ϑ^˙0+∑j=0n-1‍∂αn-1∂r^jr^˙j+∑j=0n-2‍∂αn-1∂d^jτnj+∂αn-1∂υ^ηn0+∂αn-1∂ϑ^1ηn1-12εnnξn(t)z12(t)fn12(y(t))-∑j=1n-1‍12εnjξn(t)(∂αn-1∂zj)2z12(t)fj12(y(t))+∑l=2n-1‍∑j=0l-2‍ξl(t)∂αl-1∂d^jhnj+∑j=2n-1‍ξj(t)∂αj-1∂υ^ηn0+∑j=2n-1‍ξj(t)∂αj-1∂ϑ^1ηn1,
where cn is a positive design parameter and τnj(0≤j≤n-1), ηn0 and ηn1 are smooth nonnegative functions to be specified in (56). With the choice of the update law
(54)r^˙n=-πnξi(t),
it renders
(55)V˙n≤-(c0-n)x02(t)-∑j=1n‍[cj-n+j]ξj2(t)-∑j=0n-1‍(d~j+∑l=j+2n-1‍∂αl-1∂d^jξl(t))(d^˙j-τnj)-(υ~+∑j=2n‍∂αj-1∂υ^ξj(t))(υ^˙-ηn0)-(ϑ~1+∑j=2n‍∂αj-1∂ϑ^1ξj(t))(ϑ^˙1-ηn1),
where
(56)τn0=τn-1,0+hn0ξn(t),hn0=-∂αn-1∂x0g0(x0,ϑ^0,r^0)x0(t)+∑j=1n-1‍(n-j)∂αn-1∂zjg0(x0,ϑ^0,r^0)zj(t),τnl=τn-1,l+hnlξi(t),hnl=-∂αn-1∂zlg0(x0,ϑ^0,r^0)zl+1(t),=-∂αn-1∂zlg(x0,ϑ^0,r^0)1≤l≤n-1,ηn0=ηn-1,0+γn0ξn(t),ηn1=ηn-1,1+γn1ξn(t).
Furthermore, employing the following update laws
(57)d^˙j=τnj,0≤j≤n-1,υ^˙=ηn0,ϑ^˙1=ηn1
eventually achieves
(58)V˙n≤-(c0-n)x02(t)-∑j=1n‍[cj-n+j]ξj2(t).This together with (48) implies that x0, ξi, υ~, ϑ~0, ϑ~1 and d~i, r~i(0≤i≤n) are bounded. Since θ and di are constant vector and constant, respectively, we know that ϑ^0, ϑ^1 and d^i, r^i(0≤i≤n) are also bounded. According to the definitions of the virtual control input αi in the above design procedure, zi are bounded as αi are bounded. It indicates that all signals of the closed loop system are bounded.LaSalle Invariant Theorem further achieves thatx0,ξi,υ~,ϑ~0,ϑ~1,d~i,r~i→0 as t→∞. The boundedness of all signals and the choice of virtual control functions αi imply that αi converge to zero, which shows that zi also tend to zero. From the transformation xi=zix0n-i, we can prove that x0→0, as t→0.The above analysis is summarized into the following theorem.Theorem 9.
For the system (1), under Assumptions 1 and 2, if the control strategies (21) and (52) are applied with an appropriate choice of the design parameters ci(0≤i≤n), the global asymptotic stabilization of the closed loop system is achieved for x0(t0)≠0.In the next section, we will deal with the stability analysis of the closed loop system with our control laws (21) and (52) as long as the initial condition x0(t0) is zero.


## 4. Switching Controller

Several switching controllers have been proposed in some existing literature resources. As well known, the choice of a constant feedback foru0(t) may lead to a finite escape. That is, the solution x0(t) issued from the origin may blow up before the switch. Usually, the phenomenon occurs for systems with non-Lipschitz nonlinearities. In this paper, the term ϕ0(t,x0(t),θ) in x0(t)-subsystem does not satisfy the Lipschitz conditions; then we should apply a novel switching control design. When the initial state x0(t0)=0, choose controller u0(t) as
(59)u0(t)=g0(x0,ϑ^0,r^0)x0(t)+sign(d0)λ,t∈[t0,ts],
where λ>0 is a constant, g0(x0,ϑ^0,r^0) is defined in (22), and update laws of the parameters ϑ^0 and r^0 are chosen as in (25) and (26), respectively.With controlleru0(t) in (59), the derivative of the Lyapunov function V0 in (19) along x0-subsystem gives that
(60)V˙0≤-c0x02(t)+λ|d0|x0(t)=-c0[x0(t)-λ|d0|2c0]2+λ2|d0|24c02.The above inequality indicates thatx0(t), ϑ^0, and r^0 are bounded; then state x0(t) cannot blow up during the time period [t0,ts).Forx0(t)=0 and Assumption 2, we have
(61)x˙0(t)=d0u0(t)+ϕ0(t,x0(t),θ)≥-[-d0g0(x0,ϑ^0,r^0)+ω0(x0)ϑ]x0(t)+λ|d0|=-ϖ0(t)+λ|d0|,
where ϖ=-d0g0(x0,ϑ^0,r^0)+ω0(x0)ϑ. Obviously, ϖ is bounded.The above inequality indicates that(62)x0(t)≥e-∫t0t‍ϖ(s)ds∫t0t‍λ|d0|e-∫t0s‍ϖ(τ)dτds+x0(t0)e-∫t0t‍ϖ(s)ds=e-∫t0t‍ϖ(s)ds∫t0t‍λ|d0|e-∫t0s‍ϖ(τ)dτds.It is clear that when0<t≤ts, |x0(t)|>0. Therefore, the state scaling coordinate transformation in (8) is effective, and we can use the following switching control strategy for x0(t0)=0.During the time period[t0,ts], using the controller u0(t) in (59), when t=t0 and u0(t) is a constant, the controller u1(t) can be designed implying the simple nonlinear backstepping method. When t0<t≤ts, choose the controller u1(t) as the iterative procedure in Section 3. Since x0(ts)≠0 at ts, we switch the control law u0(t) and u1(t) into (21) and (52), respectively.Theorem 10.
For the system (1), under Assumptions 1 and 2, if the above switching control strategy is applied with an appropriate choice of the design parameters ci(0≤i≤n), then the closed-loop system is globally asymptotically regulated at the origin for x0(t0)=0.


## 5. Simulation Example

In this section, a numerical example will be given to illustrate that the proposed systematic control law design method is effective.Example 1.
Consider the following system:(63)x˙0(t)=d0u0(t)+(x0(t))θ0,x˙1(t)=d1u0(t)x2(t)+12ln(1+x12(t))ex0(t)x12(t-0.3)+x1(t)θ1x1(t),x˙2(t)=d2u1(t)+x1(t)ex0(t-0.2)x13/2(t-0.2)+ln(1+(θ2x2(t))2),
where d0, d1, and d2 are unknown virtual control coefficients, and θ0, θ1, and θ2 are unknown bounded parameters. Our purpose is to design controllers u0(t) and u1(t) such that the states of the closed-loop system tend to zero when t→∞.To apply the proposed design method, we make the estimation of nonlinear functions in system (62) as follows:
(64)(x0(t))θ0≤|x0(t)|e(θ0-1)ln|x0(t)|≤|x0(t)|e(1/2)(θ0-1)2+(1/2)ln2|x0(t)|≤|x0(t)|e(1/8)ln2(1+x02(t))·e(1/2)(θ0-1),x1(t)θ1x1(t)≤|x1(t)|ex1(t)lnθ1≤|x1(t)|e(1/2)x12(t)e(1/2)ln2θ1,ln(1+(θ2x2(t))2)≤|x2(t)|·|θ2|.Letϑ=e(1/2)(θ0-1)+e(1/2)(θ0-1)+|θ2|; then the above inequalities can be deduced as
(65)(x0(t))θ0≤|x0(t)|e(1/8)ln2(1+x02(t))ϑ,x1(t)θ1x1(t)≤|x1(t)|e(1/2)x12(t)ϑ,ln(1+(θ2x2(t))2)≤|x2(t)|ϑ.Introduce the following coordinate transformation:(66)z1(t)=x1(t)x0(t),z2(t)=x2(t);
then system (62) can be rewritten as
(67)x˙0(t)=d0u0(t)+(x0(t))θ0,z˙1(t)=u0(t)x0(t)d1z2(t)-u0(t)x0(t)d0z1(t)+x1(t)θ1x1(t)x0(t)-(x0(t))θ0x0(t)z1(t)×(1/2)ln(1+x12(t))ex0(t)x12(t-0.3)x0(t),z˙2(t)=d2u1(t)+x1(t)ex0(t-0.2)x13(t-0.2)+ln(1+(θ2x2(t))2).Forx0(t)-subsystem, design the following controller u0(t):
(68)u0(t)=g0(x0,ϑ^0,r^0)x0(t),
with
(69)g0=sign(d0)r^0π0,π0=-1+1+(k0/r^0)22[ω0ϑ^0+c02+(ω0ϑ^0)2],
where ω0=e(1/8)ln2(1+x02(t)) and k0, and c0 are design constants. Moreover, the adaptation laws of r^0 and ϑ^0 are chosen as
(70)ϑ^˙0=ω0x02(t),r^˙0=-π0x02(t).Defineξ1(t)=z1(t) and  ξ2(t)=z2(t)-α1;  according to the design procedure in Section 3, the following controller u1(t) can be given:
(71)u1(t)=sign(d2)r^2π2,
with
(72)π2=-c2ξ2(t)-d^1g0ξ1(t)+∂α1x0(t)d^0g0x0(t)+∂α1z1(t)g0[d^1z2(t)-d^0z1(t)]+∂α1ϑ^0ϑ^˙0-ξ2(t)γ20υ^-ξ2(t)γ21ϑ^1-12ε21ξ2(t)(∂α1z1(t))2ξ12(t)e2x0(t)-12ε22ξ2(t)ξ12(t)x02(t)+∂α1r^0r^˙0+∂α1r^1r^˙1+∂α1d^0[-g0ξ12(t)-∂α1x0(t)g0x0(t)ξ2(t)+∂α1d^0++∂α1z1(t)g0ξ1(t)ξ2(t)]+∂α1ϑ^1[ξ12(t)ω~11+ξ22(t)γ21],
where the virtual control function α1 can be expressed as follows:
(73)α1=sign(d1)r^1π1,π1=d^0ξ1(t)-1g0[12ε11c1ξ1(t)+ξ1(t)ω~11ϑ^1d^0ξ1(t)5∓1g0+12ε11ξ13(t)e2x0(t)+ε118ξ13(t)x04(t)].It can be seen thatε11, ε21, and ε22 are positive constants; γ20=1; and ω~11 and γ21 represent, respectively,
(74)ω~11=e(1/8)ln2(1+x02(t))+e(1/2)z12(t)x02(t),(75)γ21=12{sign(d1)r^1[d^0-1g0(12ε11c1+ω~11ϑ^1hhhhhhhhhhhhhlhhhhhhl+12ε11ξ12(t)e2x0(t)hhhhhhhhhhhlhhhhhhhhh+ε118ξ12(t)x04(t)12ε11)1g0]1g0}2+14(∂α1x0(t))2ω02+12(∂α1z1(t))2ω~112.The adaptation laws of the variablesr^1, r^2, d^0, and d^1 in the controller u1(t) and υ^, ϑ^1 are chosen as
(76)r^˙1=-g0π1ξ1(t),r^˙2=-π2ξ2(t),d^˙0=-g0ξ12(t)-∂α1x0(t)g0x0(t)ξ2(t)+∂α1z1(t)g0ξ1(t)ξ2(t),d^˙1=g0ξ1(t)ξ2(t)-∂α1z1(t)g0ξ2(t)z2(t),υ^˙=ξ22(t)γ20,ϑ^˙1=ξ12(t)ω~11+ξ22(t)γ21.For simulation use, we pickd0=3.5, d1=2.5, d2=3, θ0=1.5, θ1=0.5, θ2=0.5, and the initial state condition [1.5-0.51]T, and the initial values of parameter estimates d^0, d^1, r^0, r^1, ϑ^0, υ^, and ϑ^1 are 0.5, 1.5, 0.1, 0.5, 0.3, 0.2, 0.1, and 0.5, respectively (Figures 3, 4, and 5). In addition, we take the other parameters as k0=5, c0=3, c1=2, c2=5, ε11=2, and ε21=ε22=1. Simulation results are shown in Figures 1 and 2. It is obvious that the states x0(t), x1(t), and x2(t) of the system (62) converge to zero, and the control laws u0(t) and u1(t) also tend to zero.Statesx0(t), x1(t), and x2(t).
(a)
(b)
(c)Controllersu0(t) and u1(t).
(a)
(b)Parametersr^0, r^1, and r^2.
(a)
(b)
(c)Parametersϑ^0, υ^1, and ϑ^1.
(a)
(b)
(c)Parametersd^0 and d^1.
(a)
(b)


## 6. Conclusion

The state feedback adaptive stabilization was investigated for a new class of nonholonomic systems with unknown virtual control coefficients, nonlinear uncertainties, and unknown time delays. It should be mentioned that the stabilization approaches in some existing literature resources may fail to be applied for the concerned systems. In order to overcome the difficulties from the time delay, we introduce novel Lyapunov-Krasovskii functionals, and a recursive technique is proposed to design the adaptive controller. To make the state scaling transformation effective, the switching control strategy is employed to achieve the asymptotic stabilization.



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*Source: 520390-2013-12-19.xml*