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 电液系统的输出反馈非线性控制英文文献和中文翻译(8):http://www.youerw.com/fanyi/lunwen_96540.html