# Adaptive Sliding Mode Control of Uncertain High-Order Nonholonomic Systems with Unknown Control Coefficients

Authors: Yan Zhao
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

This paper investigates the global stabilization problem for a class of high-order nonholonomic systems with unknown control coefficients and uncertain nonlinearities. An adaptive sliding mode control (SMC) law based on a constructive manipulation is proposed by adding a power integrator technique. A switching control strategy is employed in the control scheme to overcome the uncontrollability problem associated with the nonholonomic systems. The designed sliding mode controller could guarantee the attractiveness of the sliding surfaceS=0 and achieve the asymptotical convergence of the state as well as the boundedness of the estimated parameters. A simulation example is provided to demonstrate the effectiveness of the proposed scheme.


---

## Body

## 1. Introduction

Nonholonomic control system as a particular class of nonlinear systems represents a wide class of mechanical systems with nonholonomic (nonintegrable) constraints. The stabilization control for nonholonomic control systems is a formidable problem in the nonlinear control area [1], because the nonholonomic systems cannot be asymptotically stabilized by means of known linear and nonlinear control methods. The difficulty of controlling this class of systems lies in the fact that nonholonomic systems do not satisfy Brockett's necessary conditions for stability as shown in [2]. In the past two decades, many interesting approaches have been proposed to overcome the stabilization obstruction associated with nonholonomic systems, including the smooth time-varying feedback, discontinuous feedback, and nonsmooth time-varying homogeneous feedback; see [1] and references therein.Variable structure control (VSC) has been shown to be robust in the presence of system variations and external disturbances. During the past few years, the VSC strategy has been also applied to the nonholonomic system control; see [3–9]. It is noted in [5, 6]; a combined adaptive/variable structure control approach was presented for a class of uncertain nonlinear systems. The proposed method not only ensures the convergence to the sliding surface S=0, S˙=0, but also retains the asymptotical stability of the classical backstepping procedure. On the basis of the previously proposed control scheme, the so-called second order sliding mode control was exploited for nonholonomic systems in [7]. The work [8] further extended this result to a class of nonholonomic ones with unknown parameters and uncertain nonlinear drifts. In [9], the adaptive SMC problem is studied for a class of perturbed nonholonomic systems which can be transformed into the chained form with uncertain nonlinear drifts.It should be noted that the results listed previously are all focused on the nonholonomic systems with the affine control variables. High-order nonholonomic systems which include the standard chained form systems as a special case present a new challenge for nonlinear feedback control, since this class of nonholonomic control systems is neither stabilizable by time-invariant continuous state feedback nor affine in the control inputs [10]. By means of adding a power integrator developed in [11], the work [12] first investigated a class of high-order nonholonomic systems in the so-called power chained form. As the further development of this approach, [13] studied a class of uncertain nonholonomic control systems with nonlinear drifts. The recent work [14] removed the assumption of the control gain signs. Reference [15] proposed an adaptive stabilization scheme for a class of high-order chained nonholonomic systems without nonlinear drifts in the case of unknown control coefficients.Motivated by the recent progress in the feedback design for nonholonomic systems, this paper will investigate the global stabilization problem for a class of high order nonholonomic systems with unknown control coefficients and uncertain nonlinearities. A combined backstepping and adaptive SMC methodology developed in [6, 16] will be applied to the control design procedure, and adding a power integrator technique is used to deal with the nonaffine control inputs. The tuning function technique presented in [17] is also introduced. We employ a switching control strategy to overcome the stabilization burden associated with the nonholonomic systems.The paper is organized as follows. Section2 presents the problem statement. Section 3 gives the SMC design scheme for x0(t0)≠0. The switching control strategy and the simulation example are shown in Sections 4 and 5, respectively. Section 6 gives some concluding remarks.


## 2. Problem Statement

In this paper, we consider the following class of high-order nonholonomic systems:(1)x˙0=d0(t)u0p0,x˙1=d1(t)x2p1u0q1+ϕ1(x0,x1),⋮x˙n-1=dn-1(t)xnpn-1u0qn-1+ϕn-1(x0,x1,…,xn-1),x˙n=β(t)upn+ϕn(x0,x),
where (x0,xT)T=(x0,x1,…,xn)T∈ℝn+1 are the states, u0 and u are two control inputs, di(t)(i=0,…,n-1), β(t) are nonzero functions, and ϕi:ℝi+1→ℝ are smooth functions satisfying ϕi(0,…,0)=0(i=1,…,n). The functions di(t)(i=0,1,…,n-1), β(t) and ϕi(x0,x1,…,xi)(i=1,…,n) represent the control coefficients and the uncertain nonlinearities, respectively. It is further assumed that β(t) is bounded for t≥0, pi≥1(i=0,…,n) are odd integers, and qk(k=1,…,n-1) are positive integers.In the rest of this paper, an adaptive sliding mode controlleru will be first designed for system (1) such that, given any initial state (x0(0),x(0))∈Ω0={(x0,x)∣x0≠0}, the closed-loop system is asymptotically stabilized. Then, a switching control strategy is proposed, which guarantees that all the signals are bounded for the initial conditions (x0(0),x(0))∈ℝn+1.To this end, we make the following assumptions regarding system (1).Assumption 1.
For eachϕi(i=1,…,n-1) in (1), there exist known smooth functions φi satisfying
(2)ϕi(x0,x1,…,xi)≤(|x1|+⋯+|xi|)φi(x0,x1,…,xi).Assumption 2.
For each0≤i≤n-1, the sign of di(t) is positive, and there are unknown positive constants θ1 and θ2 such that
(3)θ1≤di(t)≤θ2,∀t≥0.Assumption 3.
There is a known smooth functionψ(x0,x) such that |ϕn(x0,x)|≤ψ(x0,x), for all (x0,x)∈ℝn+1.Next, we introduce a lemma which is crucial in establishing the main results of this paper.Lemma 4 (Young’s Inequality).
If the constantsp,q>1 and satisfy (p-1)(q-1)=1, then for all ε>0 and (x,y)∈ℝ2, there holds
(4)xy≤εpp|x|p+1qεq|y|q.


## 3. SMC Applied to Backstepping Design

In this part, we first assume thatx0(t0)≠0 for the system (1). The case of x0(t0)=0 will be discussed in the subsequent section.Forx0(t0)≠0, take u0 as follows:
(5)u0=-x0.
Choose the Lyapunov function V0(x0)=(1/2)x02, then the time derivative of V0(x0) along the first equation of (1) satisfies
(6)V˙0=-d0(t)x0p0+1≤-θ1x0p0+1≤0.
Considering p0 is an odd integer, it can be concluded that x0-subsystem is asymptotically stable. In addition, x0(t,x0(t0)) cannot converge to zero in finite time if x0(t0)≠0.In order to carry out the control design, we introduce the following input-state scaling discontinuous transformation defined by(7)z1=x1u0r1,z2=x2u0r2,…,zn=xnu0rn,
with ri=qi+piri+1, rn=0, i=1,…,n-1.Under the control law (5) and the transformation (7), the x-subsystem can be transformed into
(8)z˙1=d1(t)z2p1+d0(t)r1x0p0-1z1+ϕ-1(x0,z1),⋮z˙n-1=dn-1(t)znpn-1+d0(t)rn-1x0p0-1zn-1+ϕ-n-1(x0,z1,…,zn-1),z˙n=β(t)upn+ϕ-n(x0,z),
where ϕ-i(x0,z1,…,zi)=ϕi(x0,x1,…,xi)/u0ri,i=1,…,n.To invoke the backstepping method, we define the error variablesξ1,…,ξn as follows:
(9)ξ1=z1,ξ2=z2p1-α1p1,…,ξn=znp1⋯pn-1-αn-1p1⋯pn-1,
where α1,…,αn-1 are the virtual controllers and given by
(10)αip1⋯pi(x0,z1,…,zi,Θ^)=-ξiβi(x0,z1,…,zi,Θ^),i=1,…,n-1,
where βi(x0,z1,…,zi,Θ^)>0(i=1,…,n-1) are some smooth functions and Θ^ is the estimation of unknown parameter Θ=(1+θ22p0)max{1,θ1-1}.Lemma 5.
For every1≤i≤n-1, there are smooth nonnegative functions φ-i(x0,z1,…,zi) satisfying
(11)ϕ-i(x0,z1,…,zi)≤(|z1|+|z2|+⋯+|zi|)φ-i(x0,z1,…,zi).Proof.
According to Assumption1, it can be derived that
(12)|ϕ-i(x0,z1,…,zi)|≤u0-ri(|x1|+⋯+|xi|)φi(x0,x1,…,xi)=(|z1||u0|r1-ri+⋯+|zi-1||u0|ri-1-ri+|zi|)×φi(x0,x1,…,xi)≤(|z1|+|z2|+⋯+|zi|)(1+u0|r1-ri)×φi(x0,x1,…,xi).
In view of rk≥ri(k=1,…,i-1), one gets (11) with φ-i=(1+|u0|r1-ri)φi.In what follows, we give the SMC control design procedure using the recursive method.Step 1.
Choose the candidate Lyapunov functionV1=(n/2θ1)x02+(1/2)ξ12+(θ1/2γ1)Θ~2 for this step, where Θ~=Θ-Θ^ is the parameter estimation error, and γ1>0 is the design gain. Then, along the trajectories of system (8), we have
(13)V˙1=-nd0(t)θ1x0p0+1+ξ1(d1(t)z2p1+d0(t)r1x0p0-1z1+ϕ-1)-θ1γ1Θ~Θ^˙≤-nx0p0+1-n-1θ1ξ12-ξ12+d1(t)ξ1ξ2+d1(t)ξ1α1p1+n-1θ1ξ12+ξ12+ξ1ϕ-1+d0(t)r1x0p0-1ξ12-θ1γ1Θ~Θ^˙≤-nx0p0+1-n-1θ1ξ12-ξ12+d1(t)ξ1ξ2+d1(t)ξ1α1p1+ξ12(n-1θ1+1+φ-1+d0(t)r1x0p0-1)-θ1γ1Θ~Θ^˙≤-nx0p0+1-n-1θ1ξ12-ξ12+d1(t)ξ1ξ2+d1(t)ξ1α1p1+θ1Θξ12(n+φ-1+r1x0p0-1)-θ1γ1Θ~Θ^˙.
Denote ρ1(x0,z1)=n+φ-1+r1x0p0-1, τ1=γ1ξ12ρ1(x0,z1), and a direct substitution leads to
(14)V˙1≤-nx0p0+1-n-1θ1ξ12-ξ12+d1(t)ξ1ξ2+d1(t)ξ1α1p1+θ1Θ^ξ12ρ1-θ1γ1Θ~(Θ^˙-τ1).
Choose the virtual control law
(15)α1p1(x0,z1,Θ^)=-ξ1ρ1(x0,z1)1+Θ^2≜-ξ1β1(x0,z1,Θ^),
and one gets
(16)V˙1≤-nx0p0+1-n-1θ1ξ12-ξ12+d1(t)ξ1ξ2-θ1γ1Θ~(Θ^˙-τ1).Step i
(
2
≤
i
<
n
)
. Suppose that at Stepi-1, there is a positive definite and proper Lyapunov function Vi-1 satisfying
(17)V˙i-1≤-(n-i+2)x0p0+1-n-i+1θ1∑k=1i-1‍ξk2-∑k=1i-1‍ξk2+di-1(t)ξi-12-(1/p1⋯pi-2)(zipi-1-αi-1pi-1)+(τi-1-Θ^˙)(θ1γ1Θ~-∑k=2i-1‍∂Wk∂Θ^).
Let
(18)Vi=Vi-1+Wi,Wi=∫αi-1zi‍(sp1⋯pi-1-αi-1p1⋯pi-1)2-(1/p1⋯pi-1)ds.
Then Vi is a C2, positive definite and proper function such that
(19)V˙i=V˙i-1+∂Wi∂x0x˙0+∑k=1i-1‍∂Wi∂zkz˙k+∂Wi∂ziz˙i+∂Wi∂Θ^Θ^˙≤-(n-i+2)x0p0+1-n-i+1θ1∑k=1i-1‍ξk2-∑k=1i-1‍ξk2+di-1(t)ξi-12-(1/p1⋯pi-2)(zipi-1-αi-1pi-1)+∂Wi∂x0x˙0+∑k=1i-1‍∂Wi∂zkz˙k+ξi2-(1/p1⋯pi-1)(ϕ-i+d0(t)rix0p0-1zi)+di(t)ξi2-(1/p1⋯pi-1)zi+1pi+(τi-1-Θ^˙)(θ1γ1Θ~-∑k=2i-1‍∂Wk∂Θ^)+∂Wi∂Θ^Θ^˙.
With the help of Young's Inequality, we can obtain
(20)di-1(t)ξi-12-(1/p1⋯pi-2)(zipi-1-αi-1pi-1)≤21-(1/p1⋯pi-2)θ2|ξi-1|2-(1/p1⋯pi-2)|ξi|1/p1⋯pi-2≤14θ1ξi-12+ξi222p1⋯pi-2-3p1⋯pi-2×(4p1⋯pi-2-2p1⋯pi-2)2p1⋯pi-2-1θ12p1⋯pi-2-1θ22p1⋯pi-2≤14θ1ξi-12+θ1Θξi222p1⋯pi-2-3p1⋯pi-2(4p1⋯pi-2-2p1⋯pi-2)2p1⋯pi-2-1≜14θ1ξi-12+θ1Θξi2ϱi1(x0,z1,…,zi,Θ^),
where ϱi1 and ϱij(i=2,…,n;j=1,…,5) given later are smooth nonnegative functions.From (18), we get
(21)∂Wi∂x0x˙0=(2-1p1⋯pi-1)d0(t)x0p0∂αi-1p1⋯pi-1∂x0×∫αi-1zi‍(sp1⋯pi-1-αi-1p1⋯pi-1)1-(1/p1⋯pi-1)ds≤(4-2p1⋯pi-1)θ2|x0|p0|ξi||∂αi-1p1⋯pi-1∂x0|≤x0p0+1+ξi2((4-2p1⋯pi-1)θ2|∂αi-1p1⋯pi-1∂x0|)p0+1ξip0-1≤x0p0+1+θ1Θξi2((4-2p1⋯pi-1)|∂αi-1p1⋯pi-1∂x0|)p0+1ξip0-1≜x0p0+1+θ1Θξi2ϱi2(x0,z1,…,zi,Θ^).
In view of
(22)∂Wi∂zk2∂Wi∂zk=-(2-1p1⋯pi-1)∂αi-1p1⋯pi-1∂zk∂Wi∂zk2∂Wi∂zk=×∫αi-1zi‍(sp1⋯pi-1-αi-1p1⋯pi-1)1-(1/p1⋯pi-1)ds∂Wi∂zk∂Wi∂zk2≤(4-2p1⋯pi-1)|ξi||∂αi-1p1⋯pi-1∂zk|,∂αi-1p1⋯pi-1∂zk=-∂ξk∂zkβk-∑l=ki-1‍ξl∂βl∂zk∂αi-1p1⋯pi-1∂zk=-p1⋯pk-1βkzkp1⋯pk-1-1-∑l=ki-1‍ξl∂βl∂zk,
it can be derived that
(23)∑k=1i-1∂Wi∂zkz˙k‍≤(4-2p1⋯pi-1)|ξi|×∑k=1i-1‍(p1⋯pk-1βkzkp1⋯pk-1-1+∑l=ki-1‍|ξl||∂βl∂zk|)|z˙k|≤(4-2p1⋯pi-1)|ξi|×∑k=1i-1p1⋯pk-1βkzkp1⋯pk-1-1×(θ2|zk+1|pk+θ2rkx0p0-1|zk|+(|z1|+⋯+|zk|)φ-kθ2|zk+1|pk+θ2rkx0p0-1|zk|)+(4-2p1⋯pi-1)|ξi|∑k=1i-1‍∑l=ki-1‍|ξl||∂βl∂zk||z˙k|≤(4-2p1⋯pi-1)|ξi|×∑k=1i-1‍p1⋯pk-1βk×(θ2zk+1p1⋯pk+θ2(1+rkx0p0-1)zkp1⋯pk-1+(|z1|p1⋯pk-1+⋯+|zk-1|p1⋯pk-1+k|zk|p1⋯pk-1)φ-kθ2zk+1p1⋯pk+θ2(1+rkx0p0-1)zkp1⋯pk-1)+(4-2p1⋯pi-1)|ξi|∑l=1i-1‍|ξl|∑k=1l‍|∂βl∂zk||z˙k|≤(4-2p1⋯pi-1)|ξi|×∑k=1i-1p1⋯pk-1βk‍×(θ2|ξk+1|+θ2(βk+1+rkx0p0-1)|ξk|+θ2(1+rkx0p0-1)βk-1|ξk-1|+∑l=1k-1‍2pl⋯pk-1-1(|ξl|pl⋯pk-1+|ξl-1βl-1|pl⋯pk-1)×φ-l+k(|ξk|+βk-1|ξk-1|)φ-k(βk+1+rkx0p0-1)|ξk|)+(4-2p1⋯pi-1)|ξi|∑l=1i-1‍|ξl|∑k=1l‍|∂βl∂zk||z˙k|≤14θ1∑k=1i-1‍ξk2+ξi28θ1(4-2p1⋯pi-1)2×∑k=1i-1‍(p1⋯pk-1βk)2·(θ22((2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2)+2∑l=k+1i-1‍(22pk⋯pl-1-2φ-k2·ξk2pk+1⋯pl-1-2+22pk+1⋯pl-1-2×βk2φ-k+12ξk2pk+1⋯pl-1-2)+2k2(φ-k2+βk2φ-k+12)θ22((2+2βk+2rkx0p0-1)2)+ξi2(4-2p1⋯pi-1)2∑k=1i-1∑l=1k‍(∂βk∂zl)2z˙l2≤14θ1∑k=1i-1ξk2+θ1Θξi2(4-2p1⋯pi-1)2×(∑k=1i-1‍8(p1⋯pk-1βk)2·((2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2+2∑l=k+1i-1(22pk⋯pl-1-2φ-k2·ξk2pk+1⋯pl-1-2‍+22pk+1⋯pl-1-2βk2φ-k+12ξk2pk+1⋯pl-1-2)+2k2(φ-k2+βk2φ-k+12)(2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2)+∑k=1i-1‍∑l=1k‍(∂βk∂zl)2z˙l2∑k=1i-1‍8(p1⋯pk-1βk)2)≜14θ1∑k=1i-1‍ξk2+θ1Θξi2ϱi3(x0,z1,…,zi,Θ^).
By Lemma 5, we can obtain
(24)ξi2-(1/p1⋯pi-1)(ϕ-i+d0(t)rix0p0-1zi)≤|ξi|2-(1/p1⋯pi-1)×(∑k=1i|zk|φ-i+θ2rix0p0-1×(|ξi|1/p1⋯pi-1+|ξi-1βi-1|1/p1⋯pi-1)∑k=1i|zk|φ-i+θ2rix0p0-1)≤|ξi|2-(1/p1⋯pi-1)×(∑k=1i‍(|ξk|1/p1⋯pk-1+|ξk-1βk-1|1/p1⋯pk-1)×φ-i+θ2rix0p0-1·(|ξi|1/p1⋯pi-1+|ξi-1βi-1|1/p1⋯pi-1)∑k=1i‍(|ξk|1/p1⋯pk-1+|ξk-1βk-1|1/p1⋯pk-1))=|ξi|2-(1/p1⋯pi-1)×∑k=1i-1|ξk|1/p1⋯pi-1|ξk|(pk⋯pi-1-1)/p1⋯pi-1‍×φ-i+|ξi|2-(1/p1⋯pi-1)×∑k=1i-1‍|ξk|1/p1⋯pi-1·|ξk|(pk+1⋯pi-1-1)/p1⋯pi-1βk1/p1⋯pkφ-i+θ2rix0p0-1|ξi|2-(1/p1⋯pi-1)|ξi-1βi-1|1/p1⋯pi-1+ξi2(φ-i+θ2rix0p0-1)≤14θ1∑k=1i-1ξk2+ξi2×(2p1⋯pi-1-12p1⋯pi-1(4θ1p1⋯pi-1)1/(2p1⋯pi-1-1)×(∑k=1i-1(|ξk|(2pk⋯pi-1-1)/(2p1⋯pi-1-1)+|ξk|(2pk+1⋯pi-1-1)/(2p1⋯pi-1-1)×βk2pk+1⋯pi-1/(2p1⋯pi-1-1))×φ-i2p1⋯pi-1/(2p1⋯pi-1-1)+(θ2rix0p0-1)2p1⋯pi-1/(2p1⋯pi-1-1)×βi-12/(2p1⋯pi-1-1)∑k=1i-1(|ξk|(2pk⋯pi-1-1)/(2p1⋯pi-1-1))+φ-i+θ2rix0p0-12p1⋯pi-1-12p1⋯pi-1(4θ1p1⋯pi-1)1/(2p1⋯pi-1-1))≤14θ1×∑k=1i-1ξk2+θ1Θξi2×(×βi-12/(2p1⋯pi-1-1)∑k=1i-1(|ξk|(2pk+1⋯pi-1-1)/(2p1⋯pi-1-1))+φ-i+rix0p0-1(4p1⋯pi-1)1/(2p1⋯pi-1-1)×(∑k=1i-1(|ξk|(2pk+1⋯pi-1-1)/(2p1⋯pi-1-1)×βk2pk+1⋯pi-1/(2p1⋯pi-1-1)+|ξk|(2pk⋯pi-1-1)/(2p1⋯pi-1-1))×φ-i2p1⋯pi-1/(2p1⋯pi-1-1)+(rix0p0-1)2p1⋯pi-1/(2p1⋯pi-1-1)×βi-12/(2p1⋯pi-1-1)∑k=1i-1(|ξk|(2pk+1⋯pi-1-1)/(2p1⋯pi-1-1))+φ-i+rix0p0-1)≜14θ1∑k=1i-1ξk2+θ1Θξi2ϱi4(x0,z1,…,zi,Θ^).
Substituting (20)–(24) into (19), we have
(25)V˙i≤-(n-i+1)x0p0+1-n-i+1θ1∑k=1i‍ξk2-∑k=1i‍ξk2+34θ1∑k=1i-1‍ξk2+θ1Θξi2(n-i+2+ϱi1+ϱi2+ϱi3+ϱi4)+di(t)ξi2-(1/p1⋯pi-1)zi+1pi+(τi-1-Θ^˙)(θ1γ1Θ~-∑k=2i-1‍∂Wk∂Θ^)+∂Wi∂Θ^Θ^˙.
Denote
(26)ρi(x0,z1,…,zi,Θ^)=n-i+2+ϱi1+ϱi2+ϱi3+ϱi4,τi=τi-1+γ1ξi2ρi(x0,z1,…,zi,Θ^).
Substituting (26) into (25), we have
(27)V˙i≤-(n-i+1)x0p0+1-n-i+1θ1∑k=1i‍ξk2-∑k=1i‍ξk2+34θ1∑k=1i-1‍ξk2+di(t)ξi2-(1/p1⋯pi-1)zi+1pi+θ1Θ^ξi2ρi+(τi-Θ^˙)(θ1γ1Θ~-∑k=2i‍∂Wk∂Θ^)+∑k=2i-1‍∂Wk∂Θ^(τi-τi-1)+∂Wi∂Θ^τi.
According to (26), there holds
(28)∑k=2i-1‍∂Wk∂Θ^(τi-τi-1)+∂Wi∂Θ^τi≤∑k=2i-1‍γ1(4-2p1⋯pk-1)|∂αk-1p1⋯pk-1∂Θ^||ξk|ξi2ρi+γ1(4-2p1⋯pi-1)|∂αi-1p1⋯pi-1∂Θ^||ξi|∑k=1i‍ξk2ρk≤14θ1∑k=1i-1‍ξk2+1θ1ξi2+θ1ξi2(2γ1∑k=2i-1‍(4-2p1⋯pk-1)2|∂αk-1p1⋯pk-1∂Θ^|2ξi2ρi2+2γ1(4-2p1⋯pi-1)2|∂αi-1p1⋯pi-1∂Θ^|2∑k=1i‍ξk2ρk2+γ14(4-2p1⋯pi-1)2|∂αi-1p1⋯pi-1∂Θ^|2ξi2ρi2(2γ1∑k=2i-1‍(4-2p1⋯pk-1)2|∂αk-1p1⋯pk-1∂Θ^|2)≜14θ1∑k=1i-1‍ξk2+1θ1ξi2+θ1ξi2ϱi5(x0,z1,…,zi,Θ^).
As a result, (27) can be further transformed into
(29)V˙i≤-(n-i+1)x0p0+1-n-iθ1∑k=1i‍ξk2-∑k=1i‍ξk2+di(t)ξi2-(1/p1⋯pi-1)zi+1pi+θ1ξi2(ρiΘ^+ϱi5)+(τi-Θ^˙)(θ1γ1Θ~-∑k=2i‍∂Wk∂Θ^).
Choose the ith virtual control law
(30)αip1⋯pi(x0,z1,…,zi,Θ^)=-ξi(ρi(x0,z1,…,zi,Θ^)1+Θ^2+ϱi5(x0,z1,…,zi,Θ^)ρi(x0,z1,…,zi,Θ^)1+Θ^2)p1⋯pi-1,
and a direct substitution into (29) results in
(31)V˙i≤-(n-i+1)x0p0+1-n-iθ1∑k=1i‍ξk2-∑k=1i‍ξk2+di(t)ξi2-(1/p1⋯pi-1)(zi+1pi-αipi)+(τi-Θ^˙)(θ1γ1Θ~-∑k=2i‍∂Wk∂Θ^).Step n. Let
(32)Vn=Vn-1+∫αn-1zn‍(sp1⋯pn-1-αn-1p1⋯pn-1)2-(1/p1⋯pn-1)ds+12γ2Λ~2,
where
(33)V˙n-1≤-2x0p0+1-1θ1∑k=1n-1‍ξk2-∑k=1n-1‍ξk2+dn-1(t)ξn-12-(1/p1⋯pn-2)(znpn-1-αn-1pn-1)+(τn-1-Θ^˙)(θ1γ1Θ~-∑k=2n-1‍∂Wk∂Θ^),γ2>0 is an arbitrary constant, Λ~=Λ-Λ^ with Λ^ is the estimate of the unknown constant Λ=max{θ2,θ22p0p1⋯pn-2}.Choose the parameter update law as(34)Θ^˙=τn-1=∑k=1n-1‍γ1ξk2ρk,
and then
(35)V˙n=V˙n-1+∂Wn∂x0x˙0+∑k=1n-1‍∂Wn∂zkz˙k+∂Wn∂znz˙n+∂Wn∂Θ^Θ^˙+1γ2Λ~Λ~˙≤-2x0p0+1-1θ1∑k=1n-1‍ξk2-∑k=1n-1‍ξk2+dn-1(t)ξn-12-(1/p1⋯pn-2)(znpn-1-αn-1pn-1)+∂Wn∂x0x˙0+∑k=1n-1‍∂Wn∂zkz˙k+ξn2-(1/p1⋯pn-1)ϕ-n+dn(t)ξn2-(1/p1⋯pn-1)upn+∂Wn∂Θ^Θ^˙+1γ2Λ~Λ~˙.
According to Lemma 4, (9), and (10), we get
(36)dn-1(t)ξn-12-(1/p1⋯pn-2)(znpn-1-αn-1pn-1)≤14ξn-12+ξn222p1⋯pn-2-3p1⋯pn-2×(4p1⋯pn-2-2p1⋯pn-2)2p1⋯pn-2-1θ22p1⋯pn-2≤14ξn-12+Λξn222p1⋯pn-2-3p1⋯pn-2(4p1⋯pn-2-2p1⋯pn-2)2p1⋯pn-2-1≜14ξn-12+Λξn2ϱn1(x0,z,Θ^).
By Lemma 4, it can be concluded that
(37)∂Wn∂x0x˙0≤x0p0+1+ξn2((4-2p1⋯pn-1)θ2|∂αn-1p1⋯pn-1∂x0|)p0+1ξnp0-1≤x0p0+1+Λξn2×((4-2p1⋯pn-1)|∂αn-1p1⋯pn-1∂x0|)p0+1ξnp0-1≜x0p0+1+Λξn2ϱn2(x0,z,Θ^).
With the same strategy as previously, it follows mentioned that
(38)∑k=1n-1‍∂Wn∂zkz˙k≤14∑k=1n-1‍ξk2+ξn28(4-2p1⋯pn-1)2∑k=1n-1‍(p1⋯pk-1βk)2·(θ22((2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2)+2∑l=k+1n-1‍(22pk+1⋯pl-1-2βk2φ-k+12ξk2pk+1⋯pl-1-222pk⋯pl-1-2φ-k2·ξk2pk+1⋯pl-1-2+22pk+1⋯pl-1-2βk2φ-k+12ξk2pk+1⋯pl-1-2)+2k2(φ-k2+βk2φ-k+12)θ22((2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2)θ22((2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2))+ξn2(4-2p1⋯pn-1)2∑k=1n-1‍∑l=1k‍(∂βk∂zl)2z˙l2≤14∑k=1n-1‍ξk2+Λξn2(4-2p1⋯pn-1)2×(∑k=1n-1‍∑l=1k‍(∂βk∂zl)2z˙l2∑k=1n-1‍8(p1⋯pk-1βk)2·((2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2+2∑l=k+1n-1‍(22pk⋯pl-1-2φ-k2·ξk2pk+1⋯pl-1-2+22pk+1⋯pl-1-2βk2φ-k+12ξk2pk+1⋯pl-1-2)+2k2(φ-k2+βk2φ-k+12)(2+2βk+2rkx0p0-1)2+(βk+rk+1x0p0-1βk)2)+∑k=1n-1‍∑l=1k‍(∂βk∂zl)2z˙l2)≜14∑k=1n-1‍ξk2+Λξn2ϱn3(x0,z,Θ^).
In terms of (34), it can be seen that
(39)∂Wn∂Θ^Θ^˙≤γ1(4-2p1⋯pn-1)|∂αn-1p1⋯pn-1∂Θ^||ξn|∑k=1n-1‍ξk2ρk≤14∑k=1n-1‍ξk2+ξn2γ12(4-2p1⋯pn-1)2|∂αn-1p1⋯pn-1∂Θ^|2∑k=1n-1‍ξk2ρk2≜14∑k=1n-1‍ξk2+ξn2ϱn5(x0,z,Θ^).
Substituting (36)–(39) into (35), we have
(40)V˙n≤-x0p0+1-1θ1∑k=1n-1‍ξk2+Λξn2(ϱn1+ϱn2+ϱn3+ϱn4)+ξn2ϱn5+|ξn|2-(1/p1⋯pn-1)∥ψ∥+β(t)ξn2-(1/p1⋯pn-1)upn+1γ2Λ~Λ~˙,
with ϱn4(x0,z,Θ^)=0. Denote
(41)ρn(x0,z,Θ^)=ϱn1+ϱn2+ϱn3+ϱn4,ω=γ2ξn2ρn(x0,z,Θ^).
As a result, (40) can be further transformed into
(42)V˙n≤-x0p0+1-1θ1∑k=1n-1‍ξk2+β(t)ξn2-(1/p1⋯pn-1)upn+ξn2(ρnΛ^+ϱn5)+|ξn|2-(1/p1⋯pn-1)∥ψ∥+1γ2Λ~(ω-Λ^˙).
Inspired by [6, 16], the error variable ξn=znp1⋯pn-1-αn-1p1⋯pn-1 can be viewed as an adaptive sliding surface in the x-coordinate. Denote the sliding manifold by S=ξn, then the adaptive sliding mode control and the parameter update law can be chosen as
(43)upn=-1β((ρn(x0,z,Θ^)1+Λ^2+ϱn5(x0,z,Θ^))ξn1/p1⋯pn-1×(ρn(x0,z,Θ^)1+Λ^2+ϱn5(x0,z,Θ^))-sgn(S)(k+∥ψ∥)(ρn(x0,z,Θ^)1+Λ^2+ϱn5(x0,z,Θ^))),(44)Λ^˙=ω,
where k>0 is a constant and sgn(·) is the sign function. Accordingly, we obtain
(45)V˙n≤-x0p0+1-1θ1∑k=1n-1‍ξk2-k|S|2-(1/p1⋯pn-1).SinceVn is a positive definite and proper Lyapunov function, (45) guarantees the convergence of the system trajectory to the origin of the sliding surface S=ξn in finite time. In addition, the convergence properties of the states and the parameter estimate remain unchanged with respect to those of the standard backstepping design procedure. That is, the states (x0(t),x(t)) of system (1) converge to zero asymptotically and the parameter estimates Θ^(t), Λ^(t) are bounded.


## 4. Switching Control Strategy

In this section, we consider the case ofx0(t0)=0. Without loss of generality, we assume that t0=0. As to the case of x0(0)≠0, we have given the controller (5) and (43) for u0 and u of system (1), respectively. Now, we turn to how to select control laws while x0(0)=0.We consider the constant control if|x0| is small enough. When x0(0)=0, we choose u0 as follows:
(46)u0(0)=u0*,u0*>0.
In accordance with the first subsystem of (1), we know that
(47)x˙0(0)=d0(0)u0p0(0)=d0(0)u0*p0>0.
So, for any small enough constant c0*, there exists an instant ts*>0 such that x0(ts*)=c0*. During the finite time interval [0,ts*), a new adaptive sliding mode control u=u*(x0,x) using u0 as defined in (46) and two new parameter update laws can be obtained following the procedure described in the previous section. Combining with x0(ts*)=c0*≠0, we can switch the control inputs u0 and u into (5) and (43), respectively.Now we state the main result of this paper.Theorem 6.
Suppose that Assumptions1–3 are satisfied. If the proposed control design procedure together with the previous switching control strategy is applied to system (1), then, for any initial conditions in the state space Ω={(x0,x)∈ℝn+1}, system (1) will be asymptotically stabilized at the equilibrium, and specifically, the states are regulated to the sliding surface S=0 in finite time while keeping the estimated parameters bounded.


## 5. Simulation Example

Consider the following example:(48)x˙0=d0(t)u03,x˙1=d1(t)x23u02+14x1,x˙0=u3,
where x=(x0,x1,x2)T is the state, u0 and u are the control inputs of the system, 0<θ1≤d1(t), d2(t)≤θ2 are unknown control coefficients with θ1 and θ2 being unknown positive constants. It can be seen that system (48) is in the form of (1) with p0=p1=p2=3, q1=2, ϕ2(x0,x1,x2)=(1/4)x2. Denote Θ^ as the estimation of Θ=(1+θ26)max{1,θ1-1} and Λ=max{θ2,θ26}.Here, we just consider the case ofx0(0)≠0. In the first subsystem of (48), we choose the control law
(49)u0=-x0.
Using the developed design procedure in Section 3, we can obtain that
(50)u3=-ξ21/3(ρ2(x0,x1,x2,Θ^)1+Λ^2-ξ21/3+ϱ25(x0,x1,x2,Θ^)ρ2(x0,x1,x2,Θ^)1+Λ^2)==-ksgn(S),Θ^˙=γ1ξ12ρ1,Λ^˙=γ2ξ22ρ2,
where ξ1=x1/u02, ρ1=9/4+2x02, β1=ρ1(1+Θ^2)1/2, ξ2=x23+ξ1β1, S=ξ2, ρ2=1+(40/3)4x04ξ14ξ22(1+Θ^2)2+100/3β12(β12+4x04+1/16)+10/3β1, ϱ25=(100/9)γ12ξ14ρ14, k, and γ1 and γ2 are positive design parameters.Let the unknown functions bed0(t)=1-(1/2)cost, d1(t)=1+(1/4)sint. The unknown constants can be chosen as θ1=1/2 and θ2=3/2. Take k=γ1=γ2=1. Simulation results of the closed-loop system are shown in Figure 1 with the initial condition (x0(0),x1(0),x2(0))=(1,-0.1,0.25).Figure 1
States of the closed-loop system.


## 6. Conclusions

In this paper, the global stabilization problem is considered for a class of uncertain high-order nonholonomic systems with unknown control coefficients. By adding a power integrator technique, we propose a combined adaptive/sliding mode control scheme. To get around the stabilization burden associated with the nonholonomic control systems, a switching control strategy is exploited in this procedure. The designed adaptive sliding mode controller guarantees the global asymptotical stabilization of the closed-loop system as well as the boundedness of the parameter estimates. Moreover, the attractiveness of the sliding surfaceS=0 in finite time is also obtained with the help of the proposed control approach.



---

*Source: 427137-2013-07-11.xml*