> xd d

: 5 ¼ x5 þ x_ 4 — k4 e4 :

4。2。Gain selection for exponential  stability

From the control law (33), the tracking error dynamics becomes

e_  ¼ Aee þ Beðe; x3 ; x4Þ; ð34Þ

where e ¼ ½ e1 ;   e2;   e3;   e4;   e5 ]T ,

Fig。 2。  Block diagrams of the two methods。

e_  ¼ Abackðx3 ; x4 Þe; ð36Þ

where

2 —k1 0 0 0 0  3

6

6

Ae ¼ 6

0 —k2 0 0 0  7

0 0 —k3 0 0  7

2 —k1 g1 0 0 0  3

6 —g1  —k2 g2 0 0  7

6 7

6   0 0 0 —k4 0  7

Abackðx3 ; x4 Þ¼ 6

6

0 —g2 —k3 g3 ðx3 ; x4 Þ 0   7

7

0 0 0 0 —k5

T

6   0 0 —g3ðx3 ; x4Þ —k4 g4    7

0 0 0 —g4 —k5

Beðe; x3; x4 Þ¼ ½ g1 e2;    g2e3 ;    g3 ðx3 ; x4Þe4;    g4 e5 ;    0 ]  :

The passivity-based controller is designed to create passive interconnected subsystems passive。 On the other hand, the back- stepping controller is designed to guarantee the passivity of the en- tire tracking error dynamics for strict feedback nonlinear systems [41]。 Now in order to compare the tracking error dynamics (34) in a passivity-based controller to the tracking error dynamics in the general backstepping controller, the backstepping control    law

Remark 4。 In  backstepping  control  design,  the  additional  term gi   1e     in  xd     makes  A     ðx  ; x  Þ ¼ A     þ A      ðx  ; x  Þ where Aback0 < 0 and Aback1 ðx3; x4Þ ¼ —Aback1 ðx3; x4Þ。 Therefore, the tracking error dynamics (36) becomes strict passive。 Unlike the origin of  the

tracking error dynamics (34) in the passivity-based controller, the