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。 电液系统的输出反馈非线性控制英文文献和中文翻译(5):http://www.youerw.com/fanyi/lunwen_96540.html