4 Þ 3 —  PsÞx4

Vt   _

/ðxÞ¼ 6 —h2x2  — h3 x3  þ h1ðpffiPffiffiffiffiffiffiffiffiffiffitffiaffiffiffinffiffiffihffiffiffiffiffirffixffiffiffiffiffiffiffixffiffiffiffi

ffiffiffiffiffi 7

QLAP x_ p þ Ctl PL þ 4b PL ; ð6Þ 6 0 7

2

where Ctl ¼ Cil þ Cel is the total leakage coefficient [m5/Ns], Cil is the

—xnx4  — 2fxnx5

T

internal leakage coefficient [m5/Ns],

Cel is the external leakage coef-

¼ ½ 0    /2ðxÞ    /3ðxÞ    0    /5ðxÞ ]  ;

ficient [m5/Ns], Vt is the total actuator volume [m3], and be is the effective bulk modulus of the system [N/m2]。

B ¼ 。

0   0   0   0   x2Kv 。 ;

Remark 1。 To smooth the function sgn(xv), sgn(xv) is replaced with tanh(rxv) where r is a sufficiently large positive constant。 This modification is reasonable due to the leakage of the valve spool

[35–37]。

Combining the control flow rate Eq。 (4) and the load flow rate continuity Eq。 (6), the fluid dynamic equation of the actuator is gi- ven by

and

C ¼ ½ 1   0   0   0   0 ]:

To estimate x, we design a high gain observer such    that

^x_  ¼ A^x þ /ð^xÞþ Bu^ þ Lðy — y^Þ; ð11Þ

where ^x is the estimation of states and L is the observer gain。 We analyze the convergence of the estimation error and stability   of

4beAP

4beCtl

4beCdw qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

the closed-loop system。 Let us define the estimation error   as

P_ L  ¼— 

t

x_  p —

t

PL þ

Vt pffiqffiffiffi

ðPs — tanhðrxv ÞPL Þxv : ð7Þ

2 ~x1 3

2 x1  — ^x1 3

Finally, by applying Newton’s second law, the actuator’s force

where m is the mass of the piston [kg], k is the load spring  constant

[N/m], and b is the viscous damping coefficient [N/(m/s)]。

Combining 3–8, the dynamics of the EHS can be formulated as a state space representation:

x_   ¼ f ðx; uÞ;

Differentiating the estimation error gives  us

~x_  ¼ Ao~x þ dðx; ^xÞ; ð13Þ

where x1  is the position of the piston [m], x2  is the velocity of the

piston [m/s], x3 is the load pressure [N/m2], x4 is the spool position

^

6 /2 ðxÞ— /2ð^xÞ 7

6 7

^ 6 ^ 7 T

of the servo valve [m], x5 is the velocity of the servo valve [m/s], u is

dðx; xÞ¼ /ðxÞ— /ðxÞ¼ 6 /3 ðxÞ— /3ðxÞ 7 ¼ ½ 0    d2      d3      0    d5 ]  ;