Proof。 Substituting (27) into (26), a chain of interconnected track- ing error dynamics can be derived

ity using a recursive procedure from the strict feedback nonlinear

where xi "i 2 [1, n], y, and u are the state, the output and the input of i þ

system, respectively, and gi(x), "i 2 [1, n] is nonzero for all x。

Using giei+1  as the input and ei  as the output in (30) gives

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

PMSM, controller, and observer parameters。

|ifflffln{pzuffltffl} o|u{tpzu}t

|P{z0}

Parameter Value Parameter Value

Then (31) shows that the relationship between ei  and ei+1  is

strictly output passive [34] and e_ i  ¼ —ki ei       8i 2 ½1; 4] is zero-state

observable。 Therefore, each subsystem is bounded input bounded output (BIBO) stable for "i 2 [1, 4]。 Serial interconnections of BIBO stable system are also BIBO stable。 Further, the 5th tracking error dynamics becomes

k 5。651 × 105 b 0

h1 7。1693 × 109 h2 3。2569 × 1010

h3 2。1456 xn 950

f 0。5 Kv 1。3333 × 10—5

k1 3000 k2 2500

k3 2300 k4 50

e_ 5  ¼ —k5e5: ð32Þ

Consequently e5 converges exponentially to zero at the conver- gence rate k5 and ei  converges to zero "i 2 [1, 4]。 h

Remark 3。 The tracking performance of the proposed   controller

(27) requires parameters。 Therefore, it is important to know the accuracy values of parameters。 The tracking performance of the proposed controller (27) is most sensitive to the ratio of hydraulic parameters h2/h1 [38]。 If we obtain the information of full state, the parameters h1 and h2 can be easily estimated using the adaptation algorithms [38,39]。 More detailed analysis is beyond the scope of this paper, but is given in [40]。

From (23), the control law (27) can be rewritten as

u ¼ u。x; xd 。;

8             

k5 100 l1 837。7153

l2 3。5088 × 105 l3 8。5148 × 1010

l4 0。3188 l5 0。3162

KP 1500 KI 20

Kff 100

>   4  ¼ h1 pffiPffisffiffi—ffiffitffiaffiffinffiffihffiffiðffirffixffi4ffiffiÞffixffiffi3ffi