origin of the tracking error dynamics (36) in the backstepping controller is exponentially stable since there a Pback  exists that

backðx3; x4ÞPback  þ PbackAbackðx3; x4 Þ¼ —Qback; ð37Þ

It was reported that the passivity-based control is simpler and more straightforward  than  the backstepping algorithm in  [22]。  In

is defined。 Notice that the difference is the addition of ‘gi—1ei—1’ to i ; 8i 2 ½2; 4] for the backstepping control law (35)。 With the backstepping control law (35), the tracking error dynamics becomes

[22], the controller gains are chosen to be ki = jgij " i 2 [1, 4] and k5 > 0。 These gain selections guarantee that the origin of the error dynamics (34) is asymptotically stable。 Generally, when the high

W。 Kim et al。 / Mechatronics 22 (2012) 766–777 771

gain observer is implemented in the feedback controller,   the If the controller gains ki  of the passivity-based controller (33) are

closed-loop system is in the form of singular perturbation sys-

designed to satisfy c < 1

, then the origin of the tracking error

tem。 According to Theorem 11。4 [34], the origins of the re- duced-order system and the boundary-layer system should be exponentially stable for the closed-loop stability of the singular perturbation system。 Since the passivity-based controller guaran- tees that the tracking error dynamics (34) is asymptotically stable unlike the backstepping controller, it is difficult to design a high gain observer with a passivity-based controller。 However, in EHS (9), since g1, g2, g4, and g5  are constant and g3(x3, x4) is

bounded,   a   positive   constant   c  exists   such   that   kBe(e, x3,

x4)k2 6 ckek2。

2kmax ðPe Þ

dynamics (34) is exponentially  stable。

Proof。 Let us define the quadratic Lyapunov function Ve(e)= eTPee。 The derivative of Ve gives us

V_ e  ¼ —kek2  þ 2eT PeBeðe; x  ; x  Þ 6 —ke  2  þ 2k ðP  Þcke  2

2 3     4 k2 max     e k2

6 —ð1 — 2kmaxðPeÞcÞkek2 : ð39Þ

Pe  depends on the design of controller gain ki。 If the controller  gains

k  are designed to satisfy   < 1

2kmaxðPe Þ

, then the origin of the tracking

Lemma 1。 Consider the EHS (9)。 Let us define the positive definite matrix, Pe as

PeAe þ AT Pe ¼ —I: ð38Þ

error dynamics (34) is exponentially stable。 h

In Lemma 1, the controller law should be designed to   satisfy

<       1 。 Ideally, it is better as the controller gains are higher。

2kmax ðPe Þ

However,  the  higher  control  gain  may  amplify  the  noise  in the

Fig。 3。  EHS model used in the   simulations。

772 W。 Kim et al。 / Mechatronics 22 (2012)   766–777