Further results on tracking control of nonholonomic chained systems

FURTHER RESULTS ON TRACKING CONTROL OF NONHOLONOMI ...

14th World Congress ofIFAC

C-2a-05-3

Copyright © 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China

FURTHER RESULTS ON TRACKING CONTROL OF NONHOLONOMIC CHAINED SYSTEMS Zhong-Ping

Jiang~

Henk Nijllleijern,l

* Department of Electrical Engineering, Polytechnic University,

Six Metrotech Genter, Brooklyn, NY 11201, U.S.A. ** Faculty of Mathematical Sciences, Dept of Systems, Signals

and Control, University of Twente, P. O. Box 217, 7500 AE Enschede, The Netherlands.

Abstract: In this paper we deal with the state-feedback tracking control problem for a class of nonholonomic dynamic systems in chained form. By means of the recursive technique proposed in our recent work, we first show that the semiglobal exponential tracking can be easily obtained for systems in chained form. Then, we show that, whenever the control inputs are subject to saturation constraints, a semiglobal asymptotic tracking controller can be found on the basis of Lyapunov redesign and backstcpping techniques. Copyright © 1999 IFAC Keywords: Nonlinear control; mechanical systems; tracking; saturation control.

1. INTRODUCTIOK

Recent years have witnessed a lot of research activities on the control and st.abilization of nonholonomic dynamical systems - see the survey paper (Kolmanovsky and McClamroch 1995) and references cited there for practical motivations and various novel approaches. One of the t.echnical burdens for the use of traditional control design methods to solve the st.abilization problem is that there is no (time-invariant) continuous state-feedback, because Brockett's necessary condition (Brockett 1983) is not met by such systems. The time-varying feedback appears to be a useful approach to get around the impossibility of stabilizing a nonholonomic system by means of (time-invariant) continuous state-feedback. More recently, the problem of tracking control for nonholonomic dynamical systems has also received attention - see, for instance, (Fierro and Lewis Corresponding Author. This author is also with the Faculty of Mecha.nical Engineering, Technical University 1

of Eindhoven, P.O. Box .">1.3, 5600 MB Eindhoven , The Netherlands. h. nijmeijerillmath. ut"ente .nl

1995), (Fliess et al. 1995), (Kahayama et al. 1990), (Murray et al. 1992)-(S0rdalen 1993) as well as (Jiang and Nijmeijer 1997b)-(Jiang and Nijmeijer 1999). The purpose of the paper is to continue our papers (.Jiang and Nijmeijer 1997a, .Hang and Nijmeijer 1999) and to show that the tracking methodology proposed in those papers can be extended t.o est.abilish semiglobal exponential tracking and semiglobal asymptotic tracking with input saturation. As in (Jiang and Nijmeijer 1997a, Jiang and Kijmeijer 1999), we focus on a class of nonholonomic systems in chained form, or extended chained form. As we will see below, the goals are achieved through a step-by-step control design procedure and the stability proof is rather simple as in (Jiang and Nijmeijer 1997 a, Jiang and Nijmeijer 1999) invoking advanced Lyapunov stability theory (Khalil 1996). Section 2 stat.es the problem and recalls briefly t.he control method in our recent papers (Jiang and Nijmeijer 1997 a, Jiang and Nijmeijer 1999). Section 3 gives a result on semiglobal exponential 1172

Copyright 1999 IF AC

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FURTHER RESULTS ON TRACKING CONTROL OF NONHOLONOMI ...

14th World Congress ofIFAC

stability. Section 4 proposes a modified version of the control design scheme in Section 2 in order to design saturated tracking control laws for our class of nonholonomic systems in chained form. Finally, we conclude the paper in Section 5.

It should be noted that the global tracking problem was solved in our earlier paper (Jiang and Nijmeijer 1997b) (see also (Samson and AitAbderrahim 1991» for a two-wheeled mobile robot which kinematic model can be put into a third-order system in chained form (1) via difIeomorphism and feedback.

2. PRIOR RESULTS AND PROBLEM STATEMENT

The purpose of this paper is to address the semiglobal tracking problem whenever the control inputs are subject to saturation constraints:

We consider nonholonomic mechanical systems which can be transformed into the following chained form (cf. (Murray et al. 1992))

where Ul rnax > SUPt>o IUld(t)1 and UZ rnax > IU2d(t)1 are given arbitrary constant real numbers.

SUPt>o

(1)

where x = (Xl' ... ' Xn) E IRn is the state, U (Ul, U2) E lR? is the control input.

=

We assume, like in (Jiang and Nijmeijer 1997b, Jiang and Nijmeijer 1997 a, Jiang and Nijmeijer 1999), that the reference trajectory Xd = (Xld, ... , Xnd) is generated by a system in chained form:

Before approaching this practically important saturation issue, we first summarize how the semiglobal tracking problem without input saturation was solved in (Jiang and Nijmeijer 1997 a, Jiang and Nijmeijer 1999) by means of a recursive technique. Step a : Let Xe = X - Xd and introduce the global change of state coordinates Yi

Yn-l Yn

(2)

Based on this model reference control, several solutions were given in (Jiang and Nijmeijer 1997 a, Jiang and Nijmeijer 1999) to the following semiglobal tracking problem for a class of nonlinear systems in chained form (1).

Ul

=

PI (t, X),

U2

=

PZ(t, x)

(3)

such that, for any initial tracking error x e (0) = X(O) - Xd(O) in 5, all the signals of the closed loop system (1)-(3) are uniformly bounded over [0,00). Furthermore, the tracking errors xe(t) := x(t) Xd (t) satisfy lim Ix(t) - xd(t)1

t--++oo

O.

(4)

-

1::; i ::; n - 2

Xn-iXle,

(6)

so that the tracking error x.-dynamics is transformed into:

where Ud = (Uld, U2d) is the time-varying reference control input.

Definition 1. The tracking control problem is said to be semi-globally solvable for the system (1) if, given any compact set 5 E IRn containing the origin, we can design appropriate Lipschitz continuous time-varying state-feedback controllers of the form

= X(n-i+l)e = X2e = Xl. ,

Yn-3 = UldYn-Z

X2 (Ul

Yn-2 = UldYn-l

UZYn

Yn-l = U2 Yn =

Ul

-

Uld)Yn

(7)

UZd Uld

Clearly, for any vector (X2d, .•• , X(n-l)d), the tracking error x. is zero if and only if Y = (Yl, ... , Yn) is equal to zero. With this in mind, the state feedback semiglobal tracking problem for the system (1) has been translated into a state feedback semiglobal stabilization problem for the transformed time-varying system (7). Step b : Using the currently popular integrator backstepping methodology (cf. CM. Krstic and Kokotovic 1995», we find a new change of state coordinates

1h

= Yl,

rh

= Yi

fin =

Yn,

2::; i ::; n - 1

(8)

- eti-l (Yl, ... , Yi-l, 4>1, ... ,4>i-d

where the functions ,pi'S and eti'S are defined by

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FURTHER RESULTS ON TRACKING CONTROL OF NONHOLONOMI ...

A. ( ) 'f'1 U1d

= U U+1 1d

Proposition 1. (Jiang and Nijrneijer 1997a) Assume that Xid(t) (2 ~ i ~ n - 1), Ud(t) and its derivatives u~~ (t) up to order n - 2 are bounded on [0, =). If Uld(t) does not go to zero as t -t +00, given any compact neighborhood S of the origin in lRn, we set Ci > 0 for alII ~ i ~ n and can find a sufficiently large>. > 0 so that, for any initial tracking conditions Xe (0) in S, the solutions of the resulting closed-loop system (7), (14) and (15) and the tracking errors Xe (t) are uniformly bounded. In particular,

(9)

,

1 d ( (i-2») cPi -_ -Uld -d cPi-1 U1d,···, u 1d t

(10)

2
i-I

oai-1 -/J--Yk+I k=l Yk

+L

14th World Congress ofIFAC

(12) lim

t-.>+oo

Ix(t) - xd(t)1

o

(16)

e

with 2: n - 3 a nonnegative integer and the c/s positive real numbers.

It was shown in (Jiang and Nijmeijer 1999) that the conditions of Proposition 1 on the reference states and inputs can be weakened to guarantee semiglobal asymptotic tracking. It was also shown in (Jiang and Nijmeijer 1997a, Jiang and Nijmeijer 1999) that exponential tracking is achieved under the extra condition that liminft-.>+oo IUld(t)1 > ·0. Indeed, we show next that a semiglobal exponential tracking result can be ensured under the same condition.

In the new coordinates, the system (7) reads as

fh = -Ui~+21h + ulifh - X n -2(Ul -

-u U+2Yi ld

U1dYi-l

-

[ Xn-i-I

-

X (UI -

U1d)'iln

i-I L

oai-l ] -/J--Xn-k-I k=l Yk

-

Uld)'fln

2£+2-U 1d Yn-2 -

'iln-2

+ UldYi+l -

+Uld'iln-l -

Yn-I Yn

= =

U2Y n _

L

- O - - X n - k - l (Ul k=l Yk

U2

U2d

Ul

Uld

As mentioned, the semiglobal asymptotic tracking and local exponential tracking results were based on the systematic control design scheme which we recalled in Section 2. We show that the same design procedure implies a semiglobal exponential tracking result if we choose the design parameters Ci'S appropriately. More specifically,

UldY n _3

n-3 0an _ 3 +

Uld)Yn

.2 n -2

-

3. SEMIGLOBAL EXPONENTIAL TRACKING

(13)

Proposition 2. Under the conditions of Proposition 1, if the reference input Uld satisfies the property that liminft-.>oo IU1d(t)1 > 0, then given any compact neighborhood S of the origin in lRn , we can pick a sufficiently large>. > 0 such that, for any initial tracking conditions xe(tO) in S, the tracking errors Xe (t) satisfy

It has been shown in (Jiang and Nijmeijer 1997a, Jiang and Nijmeijer 1999) that the tracking problem is semi globally solved for the system (1) if the following tracking controllers are chosen:

IXe(t)1 ~ kle-k2(t-to)lxe(to)1 ,

n - 2 oa n -2

+ L -a-.-UldYi+l i=l

(17)

"It 2: to 2: 0, where kl and k2 are positive real numbers (which may depend on S).

(14)

Y.

Proof. Consider the Lyapunov function candidate

+

2 ) ( _Y n -2 - Y_n - l aa 0Yn-2 n -

] U2

(15)

where Cn-l, Cn , ). > 0 and 6. 1 , a smooth function M ore preCIse . 1y, o f ( Y1, . . . ,Yn-I, UId, . . . ,u (n-3» . 1d the following result can be established.

where

Y :=

(Yl ' ...

,'iln-l, Un)·

It is easy to verify that the time derivative of V along the solutions of (13), (14) and (15) satisfies

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Copyright 1999 IF AC

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FURTHER RESULTS ON TRACKING CONTROL OF NONHOLONOMI ...

Choose

n-2

V·

< -2 2£+2 _ - '"" w CiYi u ld

-

-2 Cn-lYn-l -

-2 cnY n

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(19)

n-l

i=1

U2

By assumptions, there exist a constant lu > 0 and a finite time instant t* ;::: 0 such that IUld(t)1 ;::: 1'/.1 for all t ;::: t*. Define a positive real number u by

=

U2d

+ Uld L

(26)

aiYi

;=1 -1

n-ln-2 U1

=

( U1d -

,\ -

~~

)

x

bijYiXj

Then, (19) implies

Vet, y)

(27)

(28)

~ -2uV(y) ,

V t ;::: t*

(20) with

Cn

> 0, bij

E JR. Differentiating the function

Hence, it follows from Gronwall's lemma (Khalil 1996) together with (19) that, for any pair of time instants t ;::: to ;::: 0,

V(y(t»

::; e 2at + e- 2a (t-t O)V(y(to »

(21)

with respect to the solutions of (7) yields

vir = -Uldl(Yl, ... ,Yn-lW -

In other words, there exist two positive constants k3 and k4 such that, V t ;::: to ;::: 0

It is important to note that this inequality is true for all solutions xe(t) initiated from any given compact set S, which are bounded because of Proposition 1 in (Jiang and Nijmeijer 1997 a).

By the definition of y and Y in (8) to (12) and (6) respectively, it follows using a recursive argument that there exist two constants kl and k2 (which may depend on S) such that for every initial tracking error xe(tO) in S, the corresponding solution xe(t) satisfies (t ;::: to ;::: 0)

(23)

In this case, we assume that the sign of Uld(t) is constant and known for all t ;::: O. Without loss of generality, we assume that Uld(t) ;::: 0 and lim inft--+oo Uld(t) > o.

4. BOUNDED STATE-FEEDBACK TRACKING The main purpose of this section is to show how to modify the control design procedure in Section 2 so that a bounded state-feedback law can be designed to achieve semiglobal asymptotic tracking. As in (Jiang and Nijmeijer 1997a, Jiang and Nijmeijer 1999), such a semiglobal asymptotic tracking controller with saturation constraint (5) is found using a stepwise scheme based on the transformed system (7), which we formulate below.

(24)

is an (n - 1) x (n - 1) stable matrix. Let P > 0 be the solution to the Lyapunov matrix equation

ATp

=

-J

Remark 1. Note that U2 in (26) is a linear feedback control law. It is also of interest to note that the semiglobal exponential convergence of the tracking errors Xe (t) can be even guaranteed by application of a linear feedback law of the type

(31)

Let (al' ... ,an-d be a vector so that

+

(30)

V t 2: to ;::: 0, where ks and k6 are positive real numbers (which may depend on S).

We close this section by another result on semiglobal exponential tracking which is obtained via a simpler approach and is of independent interest.

PA

Proposition 3. Assume that Xid(t) (2 ::; i ::; n 1), Ud(t) and its derivative Uld(t) are bounded on [0,00). If Uld(t) 2: 0 for all t ;::: 0 and lim inft--+oo Uld(t) > 0, then given any compact neighborhood S of the origin in IRn, we can pick a sufficiently large ,\ > 0 such that, for any initial tracking conditions Xe (to) in S, the tracking errors Xe (t) satisfy

IXe(t)1 ~ k5e-ks(t-to)lxe(to)1 ,

The proof of Proposition 2 is therefore completed.

A

(29)

Finally, the same semiglobal exponential tracking result can be similarly proved as in Proposition 2.

(22)

IXe(t)1 ::; k 1 e- k2 (t-t O) IXe(to)1

cnY;

(25)

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Step 1 : We begin with the Yrsubsystem of (7) with Y2 viewed as the virtual control input. Let Zl = YI and VI := ~zr The time derivative of VI along the solutions of (7) is:

VI = ZlY2Uld -

) 0 0'n_2 - n-2 L -0-. -UldYi+l Y. -,0,.2(Z2, X)(Ul - Uld)Yn

ZIXn-2(UI - Uld)Yn (32)

This leads us to choose a virtual control function like 0'1 = a and introduce a new variable Z2 = Y20'1. Consider the function

(36)

i=l

00'n-2 ( Zn-l--- OYn-2 1

+ where

x=

cn-2Zn-2)

+ Vn - 2 (Zn-2)

U2Yn

(X2, .. . , X n -2).

Therefore, by choosing the control law U2 as

(33) where

Cl

>

a is a

design parameter.

n-2 0 0'n_2 -O-.-UldYi+l i=1 Y.

+L

Then, (32) gives

(37)

where It : lR --+ lR is any bounded, "first-sector third-sector" function (Le. !tea) = and zh(z) > a for all z "# 0.), (36) yields:

°

Step 2 : Consider the (Yl, Y2)-subsystem of (7) with Y3 viewed as the virtual control input. Let V 2 := W 1 + ~z~. Differentiating V 2 with respect to the time yields

V2 =

(Y3 X

+ 1 +c~(Zl)) Z2 U ld -

( CIZIXn-2 1 + VI (zt}

. Vn - I

=

-zn-I!t(zn-d

+

(

00'n-2 Zn-l-I'l-UYn-2

Yn(Ul - Uld)

+ Y2 X n-3)

This leads us to choose a virtual control function 0'2 (YI) and a new variable as

Step n : Now, consider the Lyapunov function for the whole system (7)

Y3 =

where.>.. >

a is a

design parameter.

As a consequence, given any C2 > 0, the time derivative of the positive definite and proper function

As in (Jiang and Nijmeijer 1997a, Jiang and Nijmeijer 1999), whenever choosing the tracking controller Ul as

(34)

(40)

satisfies

-

(

00'n-2 Zn-l OYn-2 - 1

cn-2zn-2)]

+ V n - 2 (Zn-2)

with h arbitrary bounded, "first-sector thirdsector" function, the time derivative of Vn along the solutions of (7) gives Vn = -zn-l!t(zn-d -

Step n - 1 : As a direct application of backstepping (cf. (Jiang and Nijmeijer 1999), we obtain a function

which satisfies

znh(zn)

(41)

Finally, we can prove the following result using the same type of arguments as in (Jiang and Nijmeijer 1999, Proof of Prop. 1). Proposition 4. Assume that Xid(t) (2 ::; i ::; n-l), Ud(t) and Uld(t) are bounded for all t E [0, (0). If Uld(t) does not go to zero as t --+ +00, the semiglobal asymptotic tracking problem is solvable for system (7) subject to input saturation (5). More precisely, given any compact neighborhood S of the origin in lR n , we can find suitable design

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FURTHER RESULTS ON TRACKING CONTROL OF NONHOLONOMI ...

functions h, h and tune the design parameters A, El to En-2 such that, for any initial tracking conditions xe(O) in S, the input constraint (5) holds while the solutions of the resulting closedloop system (7), (37) and (40) satisfy (5) and the tracking errors Xe (t) converge to zero as t -t
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Jiang, Z. P. and H. Nijmeijer (1999). A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Trans. Automat. Control. Kanayama, Y., Y. Kimura, F. Miyazaki and T. Noguchi (1990). A stable tracking control scheme for an autonomous mobile robot. Proc. IEEE 1990 Int. Conf. on Robotics and Automation pp. 384-389. Khalil, H. K. (1996). Nonlinear Systems. 2nd Ed., Prentice Hall. Upper Saddle River. Kolmanovsky, 1. and N. H. McClamroch (1995). Developments in nonholonomic control systems. IEEE Control Systems Magazine 15,20-36. M. Krstic, 1. Kanellakopoulos and P. V. Kokotovic (1995). Nonlinear and Adaptive Control Design. John Wiley & Sons. New York. Murray, R. M., G. Walsh and S. Sastry (1992). Stabilization and tracking for nonholonomic control systems using time-varying state feedback. IFAC Nonlinear Control Systems Design pp. 109-114. Samson, C. and K. Ait-Abderrahim (1991). Feedback control of a nonholonomic wheeled cart in cartesian space. Proc. 1991 IEEE Int. Conf. Robotics and Automation pp. 11361141. S0rdalen, O. J. (1993). Feedback Control of Nonholonomic Mobile Robots. Ph.D. thesis, Norwegian Institute of Technology.

Remark 2. As in (Jiang and Nijmeijer 1999), exponential stability of the closed-loop system can be guaranteed provided the reference input Ud satisfies the conditions of (Jiang and Nijmeijer 1999, Corollary 2). It is also worth noting that extension to a simplified dynamic model as introduced in (Jiang and Nijmeijer 1999) is rather straightforward.

5. CONCLUSION In this paper we have shown that recent results in our work (Jiang and Nijmeijer 1997a, Jiang and Nijmeijer 1999) can be easily extended to the case of semiglobal exponential tracking and tracking under input constraints. As before, the desired tracking control laws have been designed on the basis of a recursive control design procedure. The semiglobal results presented in this paper are meaningful from a practical point of view because chain ability is in general not global for nonlinear mechanical systems with nonholonomic constraints. Nevertheless, our results have shown the boundedness property for the solutions in any (bounded) domain where we have chain ability. Last but not the least it is more or less direct to extend our results in this paper to systems in extended chained form, or dynamic model.

6. REFERENCES Brockett, R. W. (1983). Asymptotic stability and feedback stabilization. Differential Geometric Control Theory, R.W. Brockett, R.S. Millman and H.J. Sussmann, eds. pp. 181-191. Fierro, R. and F. L. Lewis (1995). Control of a nonholonomic mobile robot: backstepping kinematics into dynamics. Froc. 34th IEEE Conf. Dec. Control pp. 3805-3810. Fliess, M., J. Levine, P. Martin and P. Rouchon (1995). Design of trajectory stabilizing feedback for driftless flat systems. Froc. 3rd ECC pp. 1882-1887. Jiang, Z. P. and H. Nijmeijer (1997a). Backstepping-based tracking control of nonholonomic chained systems. ECC'97, Paper No. 672. Jiang, Z. P. and H. Nijmeijer (1997b). Tracking control of mobile robots: a case study in backstepping. A utomatica 33, 1393-1399.

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