6 7

the current input [mA], and h1 :¼ 4be Cdw ; h2 :¼ 4be AP ; h3 :¼ 4be Ctl 。 6 7

Vt pffiqffi Vt Vt

4 5

/5 ðxÞ— /5ð^xÞ

Assumption 1。  In an actual EHS, all states are physically bounded。

Therefore, the initial values of all states are and remain in the compact set。

Let us define the scaled estimation error   as

T 。 。T

g ¼ ½ g1     g2     g3     g4     g5 ]   ¼ x1 x2 x3 x4      x5     ;

Under Assumption 1, the nonlinear term, h1      ffiPffiffiffiffiffiffiffiffiffiffitffiaffiffiffinffiffiffihffiffiffiffiffirffiffixffiffiffiffiffiffixffiffiffiffix4

W。 Kim et al。 / Mechatronics 22 (2012) 766–777 769

where e is a small positive constant。 Using the newly defined vari- able g, the estimation error dynamics (13) is transformed into a fast dynamics in a singularly perturbed form  as

Remark 2。 In practice, g3 is seldom zero, since jx3j is seldom close to Ps。 In the rare case that g3 = 0, g3 is set to a small positive number to avoid the problem of piding by zero [38]。 Therefore we   assume

In (9), we see that the EHS (9) can be substituted into the strict feedback  form  (22)  as follows

Theorem 1。 [32] Consider the estimation error dynamics (13)。 If e is designed to satisfy the  condition

where Pg is a positive definite matrix and cD is a positive constant, then

^x goes to x by the high gain observer, and kmax(A) and kmin(A) are the maximum and minimum eigenvalues of the matrix A, respectively。

Proof。 The estimation error dynamics is given by (15)。 To study the convergence of the equilibrium points g =0 of (15), we define a Lyapunov candidate function

Vg ¼ egT Pgg: ð17Þ

In (23), f1 = 0, f4 = 0, and gi(x) – 0 "i 2 [1,5]。

We define xd as the desired position。 Then the tracking errors can be written as

ei  ¼ xi  — xd ;    8i 2 ½1; 5]: ð24Þ

We differentiate each tracking error to create the tracking error dynamics  as follows

e_ i  ¼ x_ i  — x_ d  ¼ fi  þ g xi   1  — x_ d ;    8i 2 ½1; 4]

Since Ag is Hurwitz, a positive definite matrix Pg exists such that

V_ g  ¼ —gT g þ 2gT PgDðe; x; ^xÞ: ð19Þ

Since cD  exists satisfying

Theorem 2。  [22,32]Consider  the EHS  (9)。 Given  an  arbitrary desired

we see that

1, if the desired states x2; 。。。 ; x5  and the control input u are

defined as

4。Controller design and gain selection for exponential stability

4。1。Passivity-based controller design

Passivity-based control [21,22] is designed to guarantee   passiv-

where control gain ki > 0, "i 2 [1, 5] , then the origin of the tracking error dynamics is asymptotically stable。