Journal of Robotics, Networking and Artificial Life, Vol. 1, No. 1 (June 2014), 39-44

Finite-time Stabilization for Nonholonomic Chained Form Systemswith Communication Delay *Hengjun Zhang, Chaoli Wang † and Gang Wang

Department of Control Science and Engineering,
University of Shanghai for Science and Technology,Shanghai, 200093, China ‡E-mail: clclwang@126.com

Abstract
In this paper, the problem of finite-time stabilization is developed for nonholonomic chained form systems withcommunication delay in the input. The finite-time control laws are presented by utilizing the switching control
strategy and the theory of finite-time stability, which can make the states of the nonholonomic chained form
systems to converge from any non-equilibrium state to the equilibrium or a given point in a finite time. Finally, thesimulation results show the effectiveness of the proposed control approach.

Keywords: Finite-time stabilization, nonholonomic chained systems, communication delay.

*This paper was partially supported by The National Natural Science Foundation (61374040); Key Discipline of Shanghai (S30501); Scientific
Innovation program (13ZZ115); Graduate Innovation program of Shanghai (54-13-302-102).

†Corresponding author.

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1. Introduction
The control and stabilization of nonholonomic systems
have received considerable attention during the last few
years. Nonholonomic systems with non-holonomic or
non-integrable constraints are frequently used todescribe the practical control systems. The
representative examples of such systems are mobilerobots, car-like vehicles, n-level trailer systems and soon. See the survey paper [1] for more details andreferences therein for more examples. So it is importantfrom the view point of applications to studynonholonomic control systems. From a theorem due toBrockett [2], the nonholonomic system cannot bestabilized at an equilibrium point by pure smooth (oreven continuous) state feedback controller.
Therefore, the stabilization problem of non-holonomic
systems is still extremely challenging which has
attracted a continuously increasing attention in thecontrol community. To overcome this problem, manymethods have been proposed, such as continuous timevarying feedback control laws [3], hybrid feedback
control laws [4] and discontinuous feedback control
laws [5] and so on. Moreover, the robust stabilization
problem has been widely researched by using thesevalid approaches.
However, most models considered are in ideal cases
without delays. From a practical point of view, delays ina control system can create significant obstacles to thestabilization problem and potentially degrade theperformance of the closed-loop system. It is well known
that guaranteeing the stability of a control system withtime delay is a challenging problem [6]. The
communication delay in the input is related toprocessing and connecting time for the packets arriving
and which also occur when actuators and controllers areconnected by networks. So it is very important to study
the delay effect on nonholonomic systems. In [7], theclassical set-point control problem is studied for rigid
robots with input-output communication delays in theclosed-loop system. The consensus problem for multiagent systems with input and communication delays is
studied based on the frequency-domain analysis in [8].
And the literature [9] presents a control strategy forstabilization of nonholonomic control systems withstrongly nonlinear uncertainties and time delay.
In order to get the better convergence speed andhigher steady state precision, many researchers havepublished literature on the finite-time stabilization. And
three kinds of global finite-time feedback stabilization
methods have been presented as follow: homogeneous
properties, backstepping constructive approach andsliding mode technology. A finite-time controller isproposed in [10] for a class of high-order nonlinear by
applying homogeneous properties of systems andbackstepping constructive approach. The literature [11]considers the finite-time stability of homogeneous
systems and gives Lyapunov stability theory of finitetime. Recently, based on backstepping constructiveapproach, finite-time stabilization is discussed in [12]for a class uncertain nonlinear systems. And the sliding
mode technology is investigated in [13]. The distributed
control laws in [14] are designed such that multiple nonholonomic mobile agents can be all in agreement within
finite time. In [15], the problem of finite-time stabilityand stabilization for retarded-type functional differentialequations is considered.
In this paper, we introduce a class of chained form
nonholonomic systems with communication delay in theinput and then study the problem of finite-timestabilization for the concerned nonholonomic systems.
A global finite-time stabilization controller is presented
by using switching control strategy, and making that the
all states converge to a given point or the equilibrium
point in a finite time.
The paper is organized as follows. In Section II, we will
give a formalization of the problem considered in thisarticle and some preliminary lemmas. Section III states
our main results including controller design and stability
analysis. In Section IV, the simulation results carried out
to validate the theoretical framework. Finally, theconclusions of the paper are given in Section V.

2. Problem Statement
A class of chained form nonholonomic systems with
communication delay in the input to be considered inthis paper is described by

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 x = x u (t − τ ),
2 11
 1
 
 x = x u (t − τ ),n 11
 n −1 xn = u2 (t − τ 2 ),=where xky=( t ) x ( t ) + ∑ Lτ(iA, Bi )ut ,(1)( x1 ,, xn ) ∈ R n , ui (i = 1, 2) are theui (t − τ i ) with both τ 1 andτ 2 being positive constants.

=Lτ(iA, Bi )ut

mfor every0∫τ e

A ( −τ i − s )− i=y ( t ) Ay ( t ) + Bu ( t ) ,k(5)∑e τ B .
−A iii =0

Lemma 2.2 If the system (5) is finite-time stable bya feedback control law

sig ( x ) = x sign ( x ) ,αu (t ) = k (t ) f ( y (t )) ,where α > 0 , x denotes the absolute value of realwith k (t ) bounded and f : R

number x and sign ( ⋅) denotes a sign function.
Next, we will introduce the following lemmas thatare needed for our controllers later.
Lemma 2.1[11] Consider the following nonlinearsystemy ( t ) = f ( y ( t ) )Bi u (t + s )ds,

then from (3) and (4), we can havewith B =For convenience, some symbols are given in thefollowing at first:
αwhere ut ( s ) =u (t + s ) :[ −τ i , 0] → Rs ∈ [−τ i , 0] andTui
inputs of systems, and=(4)i =0(2)where define that V : D → R is continuous on ann

open neighborhood of origin and f ( 0 ) = 0 . The[(6)→ R continuous suchnthat f (0) = 0 and there exists a function α of classf ( x) m ≤ α ( x n ) , then the system (3)κ such that‖‖‖‖is finite-time stable by the feedback controllerku ( t ) k ( t ) f  x ( t ) + ∑ Lτ(iA, Bi )ut  .
=i =0 x1 ( t ) = x2 ( t ) ,  xn −1 ( t ) = xn ( t ) , x (= n t ) u (t −τ ) ,(8)which can be denoted in short byx (t ) = Ax(t ) + Bu (t − τ ),with

0 1 0  0Moreover, if D
= N= R , the origin is globally    finite-time stable equilibrium.
=A =, B
 1 
We consider the systemk0   0 0
x ( t ) =Ax ( t ) + ∑ Bi u ( t − τ i ), t ≥ 0,(3)ni =0where x ∈ R , u ∈ R , A ∈ Rnmn×n, Bi ∈ Rn×mandτ i are positive constants. According to the paper [15],we know that if let(7)

Now we consider the finite-time stabilization of theintegrator systems with delay in the input described byequilibrium x = 0 of the systemis finite-time stable if:(i) it is asymptotically stable, in N , an open
neighborhood of origin ,with N ⊆ D ;(ii) it is finite-time convergent in N , that is , for
any initial condition x0 ∈ N \ {0} , there is a settling
time T > 0 such that every solution x ( t , x0 ) ofsystem (2) is defined with x ( t , x0 ) ∈ N \ {0} fort ∈ 0, T ) and satisfies lim x ( t , x0 ) = 0 , andx →Tx ( t , x0 ) = 0 , if t ≥ T .

n0
   .
0
 
1

According to the Lemma 2.2 and the literature [15],it is not difficult for us to get the following lemma.
Lemma=2.3 Let k ( k1 ,  , kn ) ∈ R such thatn

the polynomial s + kn sn

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there exists ε ∈ (0,1) such that for all α ∈ (1 − ε ,1) ,the system (8) is globally finite-time stable under thecontinuous feedback control law

uα =−k1sig α1 ( y1 ) −  − kn sig α n ( yn ) , (9)where− Aτ

B , α1 ,  , α n satisfy

α iα i +1
, 2 ≤ i ≤n=
α i −12α i +1 − α i(10), α n +1 1.
=
α n α=3.

with the matrix A and B are same as in the Lemma
2.3.
When t ≥ T1 + τ 2 , we also have(y=( t ) x ( t ) + Lτ( A, B )ut ,and B = eyi ( t ) =c i −1 xi ( t ) + Lτ( 2A, B ) c n −1u2 , B =e − Aτ 2 B.

the n -order subsystem (12) can be written as x1 = cx2 ,  x = cx ,n n −1= xn u2 ( t − τ 2 ) .

Controller Design

In this section, the main results will be presented. Basedon the switching control strategy, the finite-time x1 = c 0 x1 ,
 
n −1
 xn = c xn ,u = c n −1u .
2
 2states of the systems (1) can converge to the equilibrium
point in a finite time. To solve the finite-timestabilization problem, we divide the systems (1) into afirst-order subsystem(11)

and a n -order subsystem= x1 x2u1 ( t − τ 1 ) , = xn −1 xnu1 ( t − τ 1 ) ,= xn u2 ( t − τ 2 ) .

stabilization
if let k ( k1 , , kn ) ∈ R such that the=npolynomial s n + kn s n −1 +  + k2 s + k1 is Hurwitz.
And choosing the proper parameters c > 0 and

α 0 , α1 , , α n ∈ ( 0,1) , then take the following
switching control laws:
When t < T1 + τ 2 , we haveu1 = c,(15)
−k1sig α1 ( y1 ( t ) ) −  − kn sig α n ( yn ( t ) ) ,u2 =(16)With the help of transformation(16), the subsystem(15) can be expressed as x1 = x2 ,  xn −1 = xn ,= xn u2 ( t − τ 2 ) .

(12)Theorem 3.1The systems (1) can be finite-time(15)

Consider the following transformation

controllers u1 and u2 are designed to make that all=x0 u ( t − τ 1 ) ,)2u =−k0 sig α0 ( x0 ) 1 + x0 ( t − τ 1 ) ,1(14)u2 = 0.
Proof: When t < T1 + τ 2 , because of u1 = c , so(17)

Based on Lemma 2.3, the control lawu2 =−k1sig α1 ( y1 ( t ) ) −  − kn sig α n ( y n ( t ) ) , (18)whereyi ( t ) =xi ( t ) + Lτ( 2A, B )u2 , B =e − Aτ 2 B,
the A and B are the same as in the Lemma 2.3, andα1 ,, α n satisfy the conditions (10). It can render thesubsystem (17) globally finite-time stable. Substituting
the transformation (16) into the control law (18), thestates x1 , x2 ,  , xn and u2 can converge to the origin
in a finite time T1 + τ 2 via the control law (13).

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In this section, we will give an illustrative exampleof Theorem 3.1. Consider the following nonholonomic
chained form systems with communication delayx0 u1 ( t − τ 1 ) ,== x1 x2u1 ( t − τ 1 ) ,x2 u2 ( t − τ 2 ) .
=Choosing τ=1(21)τ=
0.5 and the initial values of2system (21)are ( x0 , x1 , x2 ) =( −2, −1, 2) . Selecting theparameters k1 = 2 ,k2 = 3 , c = 1 and α 0 = 0.5 ,α 2 = 0.4 , by the formula (10), we have α1 = 0.25 .

Fig. 1 The state response of system (21)

From Fig. 1, it is easy to see that the state variablesof systems can be driven to the equilibrium point in afinite time. As shown in Fig. 2, the control inputs u1
and u2 can also converge to the origin.
5. Conclusions

Fig. 2 The response of controllers

When t ≥ T1 + τ 2 , the states x1 , x2 ,  , xn havereached on the equilibrium point, thus, we only consider
the first order subsystem (11).
Construct a Lyapunov function

V ( x0 ( t ) ) =1
2x0 ( t ) .
2Taking the time derivative of V ( x0 ) along(1+α2− x0 0 1 + x0 ( t − τ 1 )V =x0u1 =≤ − x01+α 0= −2(1+α 0 ) 2V(1+α 0 ) 2(19)) (20).

So the subsystem (11) can be stabilized within the1−αsettling time T2 ≤ | x0 (t ) | 0 (1 − α 0 ) via theswitching control law (14). Hence, the statevariables
x0 , x1 , , xn , and the control inputs u1 , u2 canconverge to the equilibrium point after thetime
T = T1 + T2 + τ 1 + τ 2 . This completes the proof ofTheorem 3.1.
4. Simulation

In this paper, a globally finite-time stabilization
problem is presented for nonholonomic chained form
systems with communication delays in the input. Byutilizing finite-time stability theory and switching
control strategy, the nonholonomic chained systems canbe divided into different subsystems, and then thecontrollers of subsystems are given with different
methods such that each state of the systems canconverge to the equilibrium point in a finite time.
Finally, a simulation example is provided to verify thatthe designed controllers are effective.

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