Robust trajectory tracking for an electr

Orlov, and Edgar Yoshio Morales Teraoka Abstract—Various robust control techniques, such as integral-block, sliding-mode, and H -infinity controls, are combined to design a controller,

Robust Trajectory Tracking for an

Electrohydraulic Actuator

Alexander G Loukianov, Jorge Rivera, Yuri V Orlov, and Edgar Yoshio Morales Teraoka

Abstract—Various robust control techniques, such as

integral-block, sliding-mode, and H -infinity controls, are combined to

design a controller, forcing an electrohydraulic actuator which is

driven by a servovalve to track a chaotic reference trajectory This

approach enables one to compensate the inherent nonlinearities

of the actuator and reject matched external disturbances and

attenuate mismatched external disturbances The capabilities of

the approach are illustrated in a simulation study.

Index Terms—Electrohydraulics, H -infinity control, variable

structure systems.

I INTRODUCTION

NUMEROUS synchronization problems that come from

practice are typically considered as tracking problems In

this paper, such an interpretation of a synchronization problem

is exemplified with robust chaotic tracking for an

electrohy-draulic actuator

Nowadays, electrohydraulic actuators are very important

tools for industrial processes This is mainly due to their fast

response and great power-supply capacity with respect to the

mass or volume they occupy However, controlling

electro-hydraulic systems presents a formidable problem since their

dynamics are highly nonlinear Therefore, the investigation of

the position or force control for electrohydraulic actuators is of

great interest from both academic and industrial perspectives

This paper is motivated by coffee-harvest automation, where

electrohydraulic actuators seem very useful for shaking the

tree branches In particular, the shaking action, being viewed

a chaotic system, is very attractive for producing a broadband

spectrum

Various control techniques, including traditional

proportional-integral differential (PID) controllers [1],

recur-sive Lyapunov designs [2], adaptive neural-network-based

con-trollers [3], and partial feedbacks [4], have been used to control

Manuscript received January 15, 2008; revised November 5, 2008 First

published November 25, 2008; current version published August 12, 2009 This

work was supported by CONACYT México under Project 46069.

A G Loukianov is with the Advanced Studies and Research Center,

National Polytechnic Institute, Guadalajara, C.P 45091, México (e-mail:

louk@gdl.cinvestav.mx).

J Rivera is with the Departamento de Electrónica, Centro Universitario

de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Guadalajara,

C.P 44430, México (e-mail: jorge.rivera@cucei.udg.mx).

Y V Orlov is with the Department of Electronics and Telecomunications,

Mexican Scientific Research and Advanced Studies Center of Ensenada,

Carretera Tijuana-Ensenada, B.C 22860, México (e-mail: yorlov@cicese.mx).

E Y Morales Teraoka is with the Advanced Studies and Research Center,

National Polytechnic Institute, Guadalajara, C.P 45091, México, and also with

Tohoku University, Sendai 980-8577, Japan (e-mail: mty7810@gmail.com).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2008.2010207

either the position or force of a hydraulic actuator driven by a servovalve On the other hand, a fruitful and relatively simple approach, particularly when dealing with nonlinear plants subjected to perturbations, is based on the use of a variable structure control technique with sliding mode (SM) [5]–[10]

A sliding mode controller was proposed in [11] to control the hydraulic actuator force In this case, the relative degree turns out to be two It is, however, well known that the SM controller ensures the robustness of the closed-loop system only with respect to matched perturbations The SM concept, combined with the block control (BC) technique [12], was shown

to provide a robust stabilization of a nonlinear system in the presence of mismatched perturbations, and it was used to design

a discontinuous controller for the hydraulic actuator in [13] Since the latter controller typically provides a conservative result for improving the performance of a nonlinear perturbed

system, one can additionally apply an H ∞controller

However, a real direct implementation of such a controller

is hardly possible for high-order dynamic systems because of the numerical complexity of solving Riccati equations In order

to reduce this complexity, the integral BC/SM technique [14] is used in this paper to decompose the original system into several

subsystems of lower dimensions, and the nonlinear H ∞-control method is then separately applied to the resulting lower order subsystems

The position tracking problem, where a prescribed trajectory

to follow is chaotic, is under study The plant model is a non-linear system that presents the dynamics of an external cylinder load (a spring and a damper in parallel), a friction model, and

an approximation of the servovalve dynamics Although such

a plant model is greatly simplified compared with the actual system dynamics, it captures all essential features of the real dynamics The present model is of the relative degree of four The proposed control approach is as follows The plant model

is first represented in the nonlinear block-controllable form [12] In order to design a nonlinear sliding surface, the BC technique, combined with an integral element [14], is then used

to linearize the nominal unperturbed part of the plant dynam-ics and reject an unknown constant part of the mismatched

perturbation in each block of the system After that, the H ∞ -control approach [15] is separately applied to each block to attenuate the mismatched disturbances in the SM dynamics A discontinuous control strategy, ensuring the stability of the SM,

is finally proposed under an a priori known upper bound on

the control signal Due to the nature of the proposed combined

BC/SM/H ∞ approach, the resulting controller is expected to yield the desired robustness properties against both matched

and mismatched unknown disturbances with a priori known

0278-0046/$26.00 © 2009 IEEE

Fig 1 Piston with load.

norm bounds Such a combination results in solving a finite

number of reduced-order Riccati equations rather than solving

a full-order Riccati equation The state measurements are

as-sumed to be available, throughout

II MATHEMATICALMODEL

The complete mathematical model used to describe the

be-havior of the electrohydraulic actuator consists of the dynamics

of a hydraulic actuator disturbed by an external load and the

dynamics of a servovalve This model is naturally separated into

three parts, namely, the mechanical, hydraulic, and servovalve

subsystems In what follows, these parts are briefly described

A Mechanical Subsystem

The piston disturbed by an external load, being modeled as a

spring and a damper in parallel attached to the piston, is shown

in Fig 1

By applying Newton’s law, the system dynamics are derived

in the form

f i=−k s x p − b d v p+ Λa P L − F r (v) (1)

where x p is the piston position, v p = dx p /dt is the piston

velocity, a is the acceleration of the piston, 

f i represents

the acting forces, P L is the load pressure, F r is the internal

friction of the cylinder, m is the actuator mass, k s is the load

spring stiffness, b dis the load viscous damping, and Λais the

piston area Introducing the state variables x1= x p , x2= v p,

and x3= P L and adding to (1) an unknown force M (t) as a

perturbation, the first two state space equations of the plant are

thus given by

˙x1= x2

˙x2= 1

m(−k s x1− b d x2+ Λa x3− F r (x2)− M(t)) (2)

B Hydraulic Subsystem

The dynamics of the cylinder are derived in [16] for a

symmetric actuator Defining the load pressure P L to be the

pressure across the actuator piston, the derivative of the load

pressure is given by the total load flow through the actuator

divided by the fluid capacitance

V t

4β e P˙L=−Λ a ˙x p − C tm P L + Q L (3)

where P L is the load pressure, V tis the total actuator volume,

β e is the effective bulk modulus, C tmis the coefficient of total

leakage due to pressure, and Q Lis the turbulent hydraulic fluid flow through an orifice The relationship between spool-valve

displacement x v and the load flow Q Lis given by

Q L = C d wx v



P s − sgn(x v )P L

where C d is the valve discharge coefficient, w is the spool-valve area gradient, P s is the supply pressure, and ρ is the hydraulic

fluid density The spool area gradient for a cylindrical spool can

be approximated simply as the circumference of the valve at each port Combining (3) and (4) results in the load pressure state equation

˙

P L=4β e

V t (−Λ a v − C tm P L) +4β e C d wx v

V t



P s−sgn(x v )P L

By setting x4= x v, the latter equation takes the form

˙x3=−αx2− βx3+ γx4



P s − sgn(x4)x3 (5) with the following constant parameters:

α = (4Λ a β e /V t)

β = (4C tm β e /V t)

γ = (4C d wβ e /V t)

1/ρ.

C Servovalve Subsystem

By applying frequency response analysis and an HP Digital Dynamic Analyzer [17] to servovalve dynamics, it is estab-lished that the following second-order linear model:

2.4315 × 105

s2+ 6.2529 × 102s + 2.5676 × 105 (6) matches the measured frequency response

x4(s) u(s) = K a

1/τ

s + (1/τ ) where u is the control input, τ = 1/573 s −1is the time constant,

and K a > 0 Then, the dynamics of the servovalve subsystem

can be approximated as

˙x4=−1

τ x4+

K a

Then, using (2), (5), and (7), we obtain the plant model as

˙x1= x2

˙x2= 1

m(−k s x1− b d x2+ Λa x3− w(x2, t))

˙x3= − αx2− βx3+



γ

P s − sgn(x4)x3



x4

˙x4= −1

τ x4+

K a

where x = (x1, , x4)T, w(x2, t) = F r (x2) + M (t), and the

control input is bounded by

with u0as a positive constant

III CONTROLLERDESIGN

Assuming that all the state variables are available for

mea-surement, the control problem to be addressed is as follows

The position of the actuator x1is to track a reference trajectory

r(t) generated by the Chen chaotic attractor [18]

˙r(t) = a[y(t) − r(t)]

˙

y(t) = (c − a)r(t) − r(t)z(t) + cy(t)

where a, b, and c are constant parameters In the sequel, we

present a control algorithm, solving the problem

A Nonlinear Sliding Surface

The sliding surface design procedure consists of three steps

Step 1: The position tracking error

z1= x1− r(t)

is first introduced as a new variable, with z2 as its derivative,

i.e.,

z2= ˙z1= x2− ˙r(t).

Next, the integral block

z01=

t



−∞

is designed After that, the first two blocks of (8), with the

integrator (11), can be represented in the new variables z01, z1,

and z2as

˙z01= z1

˙z1= z2

˙z2= − a21z1− a22z2+ ¯b2x3+ ¯d2(t) + g2w(z2, t)

(12) where a21= k s /m; a22= b d /m; ¯b2= Λa /m; d¯2(t) =

(−k s /m)r(t)−(b d /m) ˙r(t)−¨r(t); g2=−1/m; and w(z2, t) =

w(x2, t)| x2=z2 + ˙r(t)

Following the BC technique [12], the variable x3, considered

in (12) as a virtual control, is selected so as to compensate

the known perturbation term ¯d2(t) and introduce the new term

−m01z01with a design parameter m01> 0

x3= ¯b −12

− ¯ d2(t) − m01z01+ v1 (13)

where v1is an auxiliary virtual control (or the second part of the

virtual control x3) which is to be designed to reject the unknown

perturbation w(z2, t) Thus, substituting (13) in the last block

(12) yields

˙z2=−a21z1− a22z2− m01z01+ v1+ g2w(z2, t) (14)

and the subsystem (12), (14), with the state vector ξ1=

(z01, z1, z2)T, is presented in the form

˙

ξ1= A1ξ1+ B11w(z2, t) + B12v1 (15) where

A1=

⎡

−m01 −a21 −a22

⎤

⎦

B12=

⎡

⎣00 1

⎤

⎦

B11=

⎡

⎣00

g2

⎤

⎦

C1T=

⎡

⎣11 1

⎤

⎦

To achieve the robustness property of the subsystem (15)

against the disturbance w(z2, t), the optimal robust control technique [15] is applied The desired value v 1d of the virtual

control v1in (15) is thus chosen to be in the form

v 1d=−BT

12P ε,1 ξ1=−k01z01− k1z1− k2z2 (16)

where P ε,1 is a positive definite solution of the following Riccati equation:

0 = P ε,1 A1+ AT1P ε,1 + C1TC1+ ε1I + P ε,1

 1

γ2B11B11T − B12B12T



P ε,1 (17)

with some positive ε1and γ1 The parameters k01, k1, and k2

are thus determined by the solution of (17) Note that virtual control law attenuates the influence of the external disturbances

w on the output z1of the closed loop (15), (16) in the sense that the inequality

∞



0

z21dt ≤ γ2

1

∞



0

w2dt

holds for all square integrable disturbances w and some atten-uation level γ1> 0 The design parameter m01is additionally utilized to alleviate solving the Riccati equation (17)

If the new variable

is introduced, then substituting v1= ¯b2x3+ ¯d2(t) + m01z01

(13) and (16) in (18) results in the following nonlinear

trans-formation:

z3= ¯b2x3+ ¯d2(t) + (m01+ k01)z01+ k1z1+ k2z2 (19)

which is equivalent to

¯

b2x3=− ¯ d2(t) − (m01+ k01)z01− k1z1− k2z2+ z3. (20)

By substituting (20) in (12), one derives

˙z01= z1

˙z1= z2

˙z2=− (m01+ k01)z01− (a21+ k1)z1− (a22+ k2)z2

+ z3+ g2w(z2, t).

Step 2: Using the nonlinear transformation (19), a

straight-forward algebra reveals

˙z3= ¯f3(ξ1, z3) + ¯b3(x)x4+ ¯d3(t) + k2g2w(z2, t) (21)

where

¯

3(ξ1, z3) =− ((k2− β)(m01+ k0)) z01

− (−βk1− k01− m01+ k2(a21+ k1)) z1

− ¯b2α − βk2− k1+ k2(a22+ k2) z2

− (β − k2)z3

¯b

3(x) = ¯ b2γ

P s − sgn(x4)x3

¯

d3(t) = − a21βr − (¯b2α + a22β + a21) ˙r

− (β + a22)¨r − ¨r˙.

Let us now choose the virtual control x4 in (21) to cancel

the dynamic term ¯f3(ξ1, z3) + ¯d3(t) and introduce the new

dynamics−m03z03− m3z3, i.e.,

x4=− ¯b −1

3 (x)¯

3(ξ1, z3) + ¯d3(t) + ¯b −13 (x)( −m03z03− m3z3+ v3) (22)

where v3 is an auxiliary virtual control (or the second part of

the virtual control x4) and m03and m3are design parameters

Introducing the integral

z03=

t



−∞

z3dt

and substituting (22) in (21) results in

˙z3= − m03z03− m3z3+ v3+ k2g2w(z2, t). (24)

Being represented in matrix form, the latter equations are as

follows:

˙

ξ3= A3ξ3+ B31w(z2, t) + B32v3 (25)

where ξ3= (z03, z3)T; A3=



−m03 −m3



; B31= 0

k2g2



;

B32=0

1



; and CT

3 =

 1 1



Using the solution P ε,3 of the Riccati equation [this equation is similar to (17)]

0 = P ε,3 A3+ AT3P ε,3 + C3TC3+ ε3I + P ε,3

 1

γ2B31B31T − B32B32T



P ε,3 (26)

the desired value v 3d of the virtual control v3is defined by

v 3d=−BT

32P ε,3 ξ3=−k03z03− k3z3 (27)

where k03and k3are positive parameters Introducing the new variable

and substituting v3= ¯b3(x)x4+ ¯f3(ξ1, z3)+ ¯d3(t)+m03z03+

m3z3(22) and (27) in (28) yield

z4= ¯b3(x)x4+ ¯f3(ξ1, z3) + ¯d3(t) + (k03+ m03)z03+ (k3+ m3)z3.

Then, combining (23) and (24) with (27) and (28), one derives

˙z03= z3

˙z3= − (m03+k03)z03− (k3+m3)z3+k2g2w(z2, t)+z4 Step 3: Since the function ¯ b3(x) = ¯ b2γ

P s − sgn(x4)x3

is discontinuous, we select the sliding variable s to depend on the signum of x4

s =



z+4, for x4> 0

z −4, for x4< 0

with

z4+= ¯f3(ξ1, z3) + (k03+ m03)z03+ (k3+ m3)z3

z4−= ¯f3(ξ1, z3) + (k03+ m03)z03+ (k3+ m3)z3

where ¯b+3(x) = ¯ b2γ √

P s − x3 and ¯b −3(x) = ¯ b2γ √

P s + x3 Thus, the closed-loop system

˙s =

 ¯

4(ξ, z4) + ¯b+4(x)u + ¯ d4(t), for x4> 0

¯

4(ξ, z4) + ¯b −4(x)u + ¯ d4(t), for x4< 0 (31)

with

¯b+

4(x) = γ¯ b2Ka

τ



P s − x3, ¯b −4(x) = γ¯ b2Ka

τ



P s + x3 where ξ = (ξ1, ξ3)T, ¯f4(ξ, z4), and ¯d4(t) = d( ¯ d3(t))/dt are

continuous functions

B Discontinuous Control

Taking into account (9), the discontinuous control law is

defined as

u =



−u0

¯

b+4(x) −1sign

z4+ , for x4> 0

−u0

¯

b −4(x) −1sign

z4− , for x4< 0. (32)

In the rest of the section, the stability analysis is presented To

prove the stability of the closed-loop subsystem (31) with (32),

we first consider x4> 0 and a Lyapunov function candidate

such as

V+= 1 2

z4+ 2.

Then

˙

V+ = z+4 ¯

4(ξ, z4)− u0sign

z+4 + ¯d4(t)

≤ −z+

4 u0− ¯f4(ξ, z4) + ¯d4(t) .

For x4< 0, we have

V −=1

2

z −4 2

˙

V − = z4−¯

4(ξ, z4)− u0sign

z4− + ¯d4(t)

≤ −z −

4 u0− ¯f4(ξ, z4) + ¯d4(t) . Hence, under the following assumption:

u0> ¯f4(ξ, z4) + ¯d4(t)+ δ

where δ > 0, both functions V+ and V − meet the condition

˙

V ≤ −δ √ 2V , which is well known [5], to ensure that the state

vector of the closed-loop system reaches the sliding surface

s = 0 in a finite time.

C SM Dynamics

Once achieved, the sliding motion on s = 0 is governed by

the SM equation [5]

˙z01= z1

˙z1= z2

˙z2= − (m01+ k01)z01− (a21+ k1)z1− (a22+ k2)z2

+ g2w(z2, t) + z3

˙z03= z3

˙z3= − (m03+ k03)z03− (k3+ m3)z3+ k2g2w(z2, t)

(33)

where the parameters k0, k1, k2, k03, and k3are defined by the

solutions P ε,1 and P ε,3to the appropriate Riccati equations To

analyze stability, (33) is represented as

˙

ξ1=

A1− B12BT12P ε,1 ξ1+ B11w(z2, t) + A13ξ3 (34)

˙

ξ3=

A3− B32BT32P ε,3 ξ3+ B31w(z2, t) (35)

where

A1− B12BT12P ε,1

=

⎡

−(m01+ k01) −(a21+ k1) −(a22+ k2)

⎤

⎦

B11=

⎡

⎣00

g2

⎤

⎦

B12=

⎡

⎣00 1

⎤

⎦

A13=

⎡

⎣00 00

⎤

⎦

A3− B32BT32P ε,3

=



−(m03+ k03) −(m3+ k3)



B31=

 0

k2g2



.

Using the Lyapunov function

V s= 

i=1,3

ξTi P ε,i ξ i

and calculating its derivative along solutions to (34) and (35) yield

˙

V s= 

i=1,3



ξ iT

A i − B i2 B i2TP ε,i

T

P ε,i ξ i + ξ iTP ε,i

× A i − B i2 BTi2 P ε,i ξ i + 2ξ iTP ε,i B i1 w(z2, t)



From the structure of (34) and (35), it follows that

B i2 = q i B i1 , i = 1, 3

with q1= 1

g2

and q3= 1

k2g2

. (37) Using (37), we rewrite (17) and (26) as

A i − B i2 B i2TP ε,i TP ε,i + P ε,i

A i − B i2 B i2TP ε,i

=−CT

i C i − ε i I − 1

γ2

i

P ε,i B i1 B i1TP ε,i − P ε,i B i2 B i2TP ε,i

=−CT

i C i − ε i I −

 1

γ2

i

+ q2i



P ε,i B i1 BTi1 P ε,i ,

i = 1, 3.

(38)

To estimate the crossing term 2ξ1TP ε,1 A13ξ3in (36), we use the following relation:

XTY + YTX ≤ XTRX + YTR −1 Y, R > 0.

Set X = P ε,1 ξ1; Y = A13ξ3; and R = I2, and then

2ξ1TP ε,1 A13ξ3≤ ξT

1P ε,1 P ε,1 ξ1+ ξ3TAT13A13ξ3. (39) Define, now, the region ξ1 < r1, r1> 0 such that the

perturbation term w(z2, t) is bounded by

w(z2, t)2≤ b1ξ1 + b2,

with b1> 0 and b2> 0∀ξ1 < r1. (40)

Setting ω i = BT

i1 P ε,i ξ i , i = 1, 3 and then substituting (38),

(39), and (40) in (36) result in

˙

V s ≤ 

i=1,3



−ξT

i C iTC i ξ i − ε i ξ iTξ i −

 1

γ2

i

+ q 2(R13) i



× ξT

i P ε,i B i1 BTi1 P ε,i ξ i+ 2ξT

i P ε,i B i1  w(z2, t)

+ 2ξ1TP ε,1 A13ξ3

i=1,3



−c i ξ i 2− ε i ξ i 2−

 1

γ2

i

+ q i 2(R13)



ω i 2

+ b1ω i 2+ 2b2ω i 

+ 2b1ξ12+ d1ξ12+ d3ξ32

i=1,3



for

2b1+ d1≤ c1and γ i ≤

 1

b1− q2

i , i = 1, 3 (43)

where c i = λmin(C iTC i ); i = 1, 3; d1= λmax(P ε,1 P ε,1); and

d3= λmax(AT13A13) The condition (42) is satisfied [see the

parameters of (34) and (25)], while the condition (43) for given

d1, c1, and q1 can be achieved by appropriately choosing (or

adjusting) the regionξ1 < r1in (40) and the parameters γ1

and γ3in the Riccati equations (17) and (26) Further, from the

definition of ω i, it follows that

ω i  ≤ α i ξ i , with α i =

λ i max

P ε,i B i1 BT

i1 P ε,i

(44) Taking into account (44), (41) yields

˙

V s ≤ 

i=1,3



−ε i ξ i 2+ 2b2α i ξ i  − ε i θ i ξ i 2+ ε i θ i ξ i 2

i=1,3



− ε i(1− θ i)ξ i 2− (ε i θ i ξ i  − 2b2α i)ξ i 

i=1,3



− ε i(1− θ i)ξ i 2

(45)

provided that

ε i θ i ξ i  − 2b2α i > 0, 0 < θ i < 1, i = 1, 3.

Fig 2 Block diagram of the closed-loop system.

Therefore, the derivative (45) is negative definite for all

ξ i  > 2b2α i

ε i θ i

= μ i

Hence, a solutionξ i (t)  is ultimately bounded by [22]

ξ i (t)  ≤ ¯b i , ¯ b i = μ i



λmax(P ε,i)

λmin(P ε,i), i = 1, 3.

On the other hand, from the H ∞-optimal-control methodol-ogy [15], it follows that the feedback, which is designed based

on proper solutions of the Riccati equations (17) and (26),

attenuates the influence of the external disturbances w on the output error z1, z1= x1− r(t).

Finally, a closed-loop block diagram is shown in Fig 2

IV SIMULATIONS

Although for the control law design we used a fourth-order model (8), in order to simulate the closed-loop system, we consider a servovalve model as a second-order transfer function (6) Hence, the plant state space is given by

˙x1= x2

˙x2= 1

m(−k s x1− b d x2+ Λa x3− w(x2, t))

˙x3= − αx2− βx3+



γ

P s − sgn(x4)x3



x4

˙x4= x5

˙x5= − a n x4− b n x5+ c n u where a n , b n , and c n are suitable constants computable with (6) The unmodeled servovalve dynamics can cause switching oscillations Such oscillations are named “chattering” [5] The chattering comes out as low control accuracy, vibrations in mechanical parts, and undesirable heat losses in electric power circuits To avoid the chattering, the following observer is designed:

˙ˆx4=−1

τ xˆ4+

K a

τ u + l1(x4− ˆx4)

where the positive parameter l1is chosen such that the estimate

error system

˙¯

x4=−

 1

τ + l1



¯

x4

is stable Thus, the resulting controller is

u =



−u0

¯

b+4(x) −1 sign

ˆ4+ , for x4> 0

−u0

¯

b −4(x) −1 sign

ˆ4− , for x4< 0

where ˆz4+= η+(ξ1, z03, z3, ˆ x4) (29) and ˆz4− = η − (ξ1, z03, z3,

ˆ

x4) (30)

The performance of the controller constructed in the previous

section is now observed in simulations Robustness features

of the closed-loop system are studied through disturbing the

electrohydraulic actuator by the signal

w(t) = 1500 + 240 sin(12t) + F r (x2)

where the used static friction model F r[2] includes Karnopp’s

stick–slip friction [19], [20] and the Stribeck effect [21]

F r (x2) = F r,1 (x2) + F r,2 (x2) + F r,3 (x2) + F r,4 (x2) (46)

with

F r,1 (x2) = if x2<−0.01, then −125e( 1

0.013 (x2+0.01) ) −100

else, it is zero

F r,2 (x2) = if x2>−0.01 and x2< 0.01 and Δx2(δ) < 0

then 225; else, it is zero

F r,3 (x2) = if x2> −0.01 and x2< 0.01 and Δx2(δ) > 0

then−225; else, it is zero

F r,4 (x2) = if x2> −0.01 then + 125e( − 1

0.013 (x2−0.01)) + 100

else, it is zero, where Δx2(δ) = x2(δk) − x2(δ(k − 1)), i.e.,

the increment of x2in discrete time

A typical velocity–friction plot of such model is shown in

Fig 3 Regarding Karnopp’s friction, there are two key points,

namely, a stick phase occurs when the velocity is within a

small critical velocity range, and there is a maximum value for

friction when the mass under consideration sticks

The parameter values were taken from [13]:

Parameter value

P s 1.03 × 107Pa;

Λa 3.26 × 10 −4m2;

α 1.51 × 1010N/m3;

kg/(ms2√

m)];

a n 2.5676 × 105;

b n 6.2529 × 102;

c n 2.4315 × 105

Fig 3 Friction model used in the system model, including Karnopp and Stribeck models.

Fig 4 Output tracking with control law designed in [13].

Using these parameters, the value of ¯b −14 (x) is

esti-mated as ¯b4(x) = (γ¯ b2Ka/τ ) √

P s ± x3= (γΛ a Ka/

mτ ) √

P s ± x3= (7.28 · 108· 3.2 · 10 −4 · 0.947)/(24 · 0.0017)

√ 1.03 · 107± x3= 5.4 · 106√

1.03 · 107± x3. From this,

we can conclude that (¯b4(x(t))) −1 < 1, which guarantees

|u(t)| < u0in (32) The parameters for the proposed controller are as follows:

(m01+ k01) 133.33;

(m03+ k03) 0.423;

(k3+ m3) 420.91;

Fig 5 Output tracking with proposed control.

Fig 6 Control action.

The proposed control law is compared with that

ob-tained in [13], which was designed to force the

electrohy-draulic actuator to follow a chaotic reference signal r(t) in

the presence of a constant disturbance, by means of high

gains

The reference trajectoryx(t) is generated by the Chen chaotic

attractor (10) with a = 35, b = 3, and c = 28 The states r,

y, and z of (10) are multiplied for a suitable constant (0.002)

in order to meet the actuator amplitude restrictions Fig 4

shows the trajectory of the electrohydraulic actuator driven

by the controller from [13] Fig 5 shows the actuator

posi-tion under the control scheme proposed in this paper One

can observe that the proposed controller is more robust to

external disturbances and parameter variations compared with

that of [13] because the tracking is more accurate under the

presence of the same disturbance Finally, Fig 6 shows the

control action corresponding to the proposed discontinuous

control law

V CONCLUSION

A robust controller design is proposed for forcing the elec-trohydraulic actuator to approximately track desired chaotic position reference trajectories in the presence of both matched and mismatched disturbances The controller is developed on

the basis of the BC SM approach and H ∞-control methodology Good performance of the closed-loop system is obtained in spite of strong disturbances affecting the system

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Alexander G Loukianov was born in Moscow,

Russia, in 1946 He received the Dipl Eng degree from Polytechnic Institute, Moscow, Russia, in 1975 and the Ph.D degree in automatic control from the Institute of Control Sciences, Russian Academy of Sciences, Moscow, in 1985.

He was with the Institute of Control Sciences

in 1978, and was the Head of the Discontinuous Control Systems Laboratory from 1994 to 1995.

In 1995–1997, he held a visiting position with the University of East London, London, U.K Since April 1997, he has been with the Advanced Studies and Research Center,

National Polytechnic Institute, Guadalajara, México, as a Professor of electrical

engineering graduate programs In 1992–1995, he was in charge of an industrial

project between his institute and the largest Russian car plant and also of

several international projects supported by INTAS and INCO-COPERNICUS,

Brussels He has published more than 90 technical papers in international

jour-nals and conferences and has served as a Reviewer for different international

journals and conferences His research interests include nonlinear systems

control and variable structure systems with sliding mode as applied to electric

drives and power systems control, robotics, space, and automotive control.

Jorge Rivera was born in El Rosario, Sinaloa,

México, in 1975 He received the B.Sc degree from the Technological Institute of the Sea, Mazatlán, México, in 1999 and the M.Sc and Ph.D degrees

in electrical engineering from the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, México, in 2001 and 2005, respectively.

Since 2006, he has been with the Universidad de Guadalajara, Guadalajara, México, as a full-time Professor with the Departamento de Electrónica, Centro Universitario de Ciencias Exactas e Ingenierías, Guadalajara, Mexico His research interests include regulator

theory, sliding-mode control, discrete-time nonlinear control systems, and their

applications to electrical machines.

Yuri V Orlov received the M.S degree from the

Mechanical–Mathematical Faculty, Moscow State University, Moscow, Russia, in 1979, the Ph.D de-gree in physics and mathematics from the Institute

of Control Science, Moscow, in 1984, and the Dr.Sc degree in physics and mathematics from Moscow Aviation Institute, Moscow, in 1990.

From 1979 to 1992, he was with the Institute

of Control Sciences, Russian Academy of Sciences, Moscow, Russia He was also a part-time Professor with Moscow Aviation Institute from 1989 to 1992.

He has been a Full Professor with the Department of Electronics and Telecom-munications, Mexican Scientific Research and Advanced Studies Center of Ensenada, Ensenada, México, since 1993 His research interests include math-ematical methods in control, analysis, and synthesis of nonlinear nonsmooth discontinuous time-delay distributed parameter systems and their applications

to electromechanical systems.

Edgar Yoshio Morales Teraoka was born in

Guadalajara, México, in 1978 He received the B.S degree in electromechanical engineering from Uni-versidad Panamericana, Guadalajara, in 2001 and the M.S degree in electrical engineering from the Advanced Studies and Research Center, National Polytechnic Institute, Guadalajara, Mexico He is currently working toward the Ph.D degree in electric and communications engineering at Tohoku Univer-sity, Sendai, Japan.