Sliding Mode Control with H∞ Attenuator for Unmatched Disturbancesin a Mechanical System with Friction and a Force Constraint

Raul Rascon, Joaquin Alvarez and Luis T. Aguilar

Abstract— A method using sliding mode and non-linear H∞control is presented for an underactuated mechanical systemwith friction and a force constraint to solve a position regulationproblem. A low pass filter is suggested to be used for theobserver realization. It was proved that the nonlinear systemis globally asymptotically stable and achieves zero steady-stateposition error even in the presence of certain disturbances, oth-erwise the controller attenuates others external perturbationsand dead zone model discrepancies. The results obtained areillustrate by numerical simulations.

I. INTRODUCTION

A major problem in control engineering is a robust feed-back design that asymptotically stabilizes a nominal plantwhile also attenuating the influence of parameter variationsand external disturbances. In the last decade, this problemwas intensely studied and research efforts have resulted inthe development of a systematic design methodology fornonlinear systems. A survey of the methods for the H∞ case,fundamental in this respect, is given in [4].

Another methodology widely spread due to its effective-ness against disturbances is sliding mode control see [6].The main feature of this class of controllers is to allow thesliding mode to occur on a prescribed switching surface, sothat the system is governed by the sliding equation only, andremains insensitive to a class of disturbances and parametervariations [5]. This control method has been successfullytested for motion control of robotic manipulators, see [8]and references therein, besides, a previous work of slidingmodes control in constrained robots can be found in [7].

An important detail to consider in some mechanicalsystems is the effect of the constraint, which is a hardproblem to model, specially using H∞ control design, dueto that the system needs to be at least twice continuousdifferentiable. Efforts have been made to control constrainedmechanical systems using a wide range of methodologiessuch as sliding mode control [8], predictive control [9], andoptimal control [10] among others. Few scarce literatureexists on the control of perturbed and uncertain mechanicalsystems under constraints. For example, the problem ofunilateral constraints on the position is considered in [11].In [12] a PID-type controller for constrained robots addedto a mixed H2/H∞ control to attenuate the influence of

Work is supported by Consejo Nacional de Ciencia y Tecnologıa underGrant 127575.

R. Rascon and J. Alvarez are with CICESE Research Center, Electronicsand Telecommunication Department, P.O. BOX 434944, San Diego, CA92143-4944, (emails: rrascon{jqalvar}@cicese.mx)

L.T. Aguilar is with Instituto Politecnico Nacional, Centro de Investi-gacion y Desarrollo de Tecnologıa Digital, Avenida del parque 1310 Mesade Otay, Tijuana Mexico 22510 (email: luis.aguilar@ieee.org).

disturbances and uncertainties was constructed. In [13] isdesigned a robust motion/force tracking performance forconstrained uncertain robots. On the other hand, in [14] isproposed a dynamic position feedback/force control designedfor constrained robot actuated by DC motors.

Some previous works of sliding mode control combinedwith H∞ control methodology can be found in [15], [16].

The present paper considers that the dynamical systemsare nonlinear and, due to frictional effects and constraints,are also nonsmooth. Additionally, the motion of the system isaffected by unknown disturbances. Incomplete and imperfectstate measurements are only available.

In this study we combine the robustness properties ofsliding mode with H∞ control to design a controller whichis capable to handle the above mentioned factors and therebyyielding a good performance on real systems. The motivationfor doing so is best illustrated with an application to amechanical system.

II. PROBLEM STATEMENT

The main concern of this work is the regulation controldesign, and its stability analysis, of a mechanical systemsubject to a bilateral constraint (see Figure 1). The equations

u

k

a

m1

x1

m2

c

x2b

Fig. 1. Constrained mechanical system

of motion of the open-loop constrained mechanical systemcan be expressed in joint coordinates as

m1x1 + F1 + f = u+ w1,

m2x2 + F2 = f + w2

(1)

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

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where f is the force contact between the masses

f =

⎧⎪⎨⎪⎩−k(x2 − b− c− (x1 + a)) if x2 − b− c ≥ x1 + a,

0 if |x1 + a− x2 + b| < c,

k(x1 + a− (x2 − b)) if x1 + a ≥ x2 − b;(2)

which can be rewritten as

f = k2 (2(x1 + a+ b− x2) + c+ |x1 + a− x2 + b|−|x2 − b− c− a− x1|)

(3)where k ∈ R is the stiffness of the spring, m1 ∈ R isthe mass, x1(t), x1(t), x1 ∈ R represent the displacement,velocity and acceleration of the mass m1, m2 ∈ R isthe second mass and x2(t), x2(t), x2 ∈ R represent thedisplacement, velocity and acceleration of the mass m2.Friction forces are involved, where Fi, with i = 1, 2 arerepresented as

Fi = σ0ixi + Fdi (4)

which are a combination of viscous friction σ0ixi and theDahl friction Fdi. The Dahl friction Fdi is governed by thefollowing dynamic model

Fdi = σ1ixi − σ1i|xi|Fdi

fci+ w2+i. (5)

where σ0i > 0, σ1i > 0, and fci > 0 are the viscousfriction coefficient, the stiffness, and the Coulomb frictionlevel, respectively; w2+i is an external disturbance which isinvolved to account for inadequacies of the friction mod-elling. Provided the mass position x1(t) and mass positionx2(t) are the only available measurements on the system,the above model (1), (3)-(5) appears to be differentiableand the unforced system possesses a multivalued set ofequilibria (x1, x2) with x1 = m where m is any constant andx2 =∈ [m+a+ b,m+a+ b+ c]. To fulfill the requirementsfor a local solution of the H∞-position regulation problem amonotonic approximation of the contact force model (3) isconsidered, this approximation ensures that the system is atleast twice continuously differentiable according to [1], [2],

f = kΔx+ kη(Δx) (6)

where

Δx = x1 + a− x2 + b + c/2 (7)

andη = −c1− e−(Δx/0.5c)

1 + e−(Δx/0.5c). (8)

(a)

f

Δ x

−c/2

c/2

(b)

f

−c/2

Δ xc/2

Fig. 2. (a) The contact force model and (b) the strictly monotonicapproximation of the contact force model

Hereinafter this approximation of f will be used, the presentcontact force approximation is inspired from [3]. Coupled tothe drive system (1) subject to position measurements of themasses, it constitutes a twice continuous differentiable ap-proximation of the underlying mechanical system, operatingunder uncertaintiesw1(t), w2(t) to be attenuated. As a matterof fact, these uncertainties involve discrepancies between thephysical contact force model (3) and its approximation (6)-(8).

III. DESIGN PROCEDURE

The objective of the sliding mode controller using an H∞criterion is to regulate the position of mass x2(t) to a desiredreference also attenuate unmatched disturbances, the slidingmode control stage is going to eliminate the effect of w1(t)and the H∞ control stage is going to attenuate the effects ofw2(t), w3(t), w4(t), w5(t) and w6(t) which are consideredas unmatched disturbances. For this purpose the nonlineardrive system (1) with friction (4)-(5) and force contact (6)-(8)is thus to design first of all a sliding mode controller so as toobtain the closed-loop system in which all these trajectoriesare bounded and the output x2(t) asymptotically decays toa desired position xd2 via xd1 as t → ∞ in the presenceof a bounded disturbance supt |w1(t)| ≤ M otherwise thenonlinear H∞ controller is going to attenuate the influenceof the external disturbances w = [w1, w2, w3, w4, w5, w6]

T ,once we have designed the sliding mode controller whichinvolve the H∞ control in its own design, the H∞ controldesign takes place. We can shift the equilibrium point of (1)to the origin by introducing the transformation based on theposition error also the integral value of the position error isintroduce for control purposes.

q1 =∫ t

0q2dt q2 = x1 − xd1 q3 = x1

q4 = Fd1 q5 =∫ t

0 q6dt q6 = x2 − xd2

q7 = x2 q8 = Fd2

(9)

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 435

where xd2 = xd1 + a + b + c/2, and the system dynamicsincluding Dahl friction model is as follows

d

dt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

q3

q4

q5

q6

q7

q8

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q2

q3

−σ01q3−q4m1

− fm1

+ um1

+ w1

m1

σ11q3 − σ11|q3|q4fc1

+ w3

q6

q7

−σ02q7−q8m2

+ fm2

+ w2

m2

σ12q7 − σ12|q7|q8fc2

+ w4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(10)

where

f = kΔq + kη(Δq) (11)

with Δq = q2 + xd1 + a− q6 − xd2 + b+ c/2 and replacingxd2 into Δq is as follows

Δq = q2 − q6

andη = −c1− e−(Δq/0.5c)

1 + e−(Δq/0.5c). (12)

Coupled to the drive system (10) subject to motor positionmeasurements, it is subsequently shown to constitute a twicecontinuous differentiable approximation of the underlyingmechanical system (Figure 1) operating under uncertaintiesto be attenuated.

IV. SLIDING MODE CONTROL USING H∞ ATTENUATOR

The control objective is to determine a feedback controllerto solve the following regulation problem, such that that theclosed-loop response satisfies

limt→∞ ‖ q6(t) ‖ = 0. (13)

A sliding-mode control law is now designed for system(10), where u∞ is a nonlinear H∞ controller as in [1], [2].We look for a controller that directs the trajectories to thedynamical sliding surface

s = q2 + q3 + q5 + q6 −∫ t

0

u∞dt. (14)

The control law which ensures that is given by

u = σ01q3 + q4 + f

+m1 (−q3 − q6 − q7 + u∞ − λs− βsign(s)) .(15)

The proposed control law will be acting at all times, i.e.,when the system is in free or in constrained motion (incontact with the constraint). The values λ and β are positivegain parameters which will be tuned to ensure the motion ofthe trajectories be directed toward the sliding surface.

Since the sliding surface equation (14) is a dynamicalvariable, we will add s as another state in (10). This leadsto the extended system

d

dt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

q3

q4

q5

q6

q7

q8

s

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q2

q3

−σ01q3−q4m1

− fm1

+ um1

+ w1

m1

σ11q3 − σ11|q3|q4fc1

+w3

q6

q7

−σ02q7−q8m2

+ fm2

+ w2

m2

σ12q7 − σ12|q7|q8fc2

+w4

q3 +−σ01q3−q4−f+u+w1

m1

+ q6 + q7 − u∞

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

V. STABILITY ANALYSIS

We analize in this section the stability of the closed-loopsystem (16), using the controller (15), and conclude aboutthe overall stability.

By substituting (15) into (16), the closed-loop system takesthe form

d

dt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

q3

q4

q5

q6

q7

q8

s

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q2

q3

−q3 − q6 − q7 + u∞ − λs− βsign(s) + w1

m1

σ11q3 − σ11|q3|q4fc1

+w3

q6

q7

−σ02q7−q8m2

+ fm2

+ w2

m2

σ12q7 − σ12|q7|q8fc2

+w4

−λs− βsign(s) + w1

m1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(17)

Now, we ensure the existence of sliding modes by verify-ing ss < 0. To this end, note that supt |w1(t)| ≤ M for allt and some constant M > 0

ss = s(−λs− βsign(s) + w1

m1

)≤ −λs2 −

(β − M

m1

)|s|.

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 436

We conclude the existence of sliding modes on the surfaces = q2 + q3 + q5 + q6 −

∫ t

0 u∞dt while the condition β −Mm1

> 0 remains valid. Also, we can demonstrate finite timeconvergence of the trajectories to the surface s = 0 usingthe quadratic function

V (s) = s2, (18)

and compute its time derivative along the solutions of (17),

V (s(t)) ≤ −2λs2 − 2(β − M

m1

)|s|

≤ −2(β − M

m1

)|s| = −2

(β − M

m1

)√V (s(t)).

(19)From (19) it follows that

V (t) = 0 for t ≥ t0 +

√V (t0)(

β− Mm1

) = tf .

(20)Hence, V (t) converges to zero in finite time and, in con-sequence, a motion along the manifold s = 0 occursin the discontinuous system (17). Thus, in the followingdevelopments, we will assume that system (17) is in slidingmode, so s = s = 0 for t ≥ tf . From (14) it is shownthat the dynamics of system (17) once on sliding mode, aredescribed by

d

dt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q1

q2

q3

q4

q5

q6

q7

q8

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q2

q3

−q3 − q6 − q7

σ11q3 − σ11|q3|q4fc1

+ w3

q6

q7

−σ02q7−q8m2

+ fm2

+ w2

m2

σ12q7 − σ12|q7|q8fc2

+ w4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

0

1

0

0

0

0

0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

u∞ (21)

let us analyze trajectories (q3, q6) in (21) considering theinput u∞ = 0, through the following quadratic function

V (q3, q6) = (q3 + q6)2 (22)

where V (q3, q6) is a positive definite and also radiallyunbounded function. The time derivative of V (q3, q6) alongthe trajectories of (21) with the input u∞ = 0 is given by

V (q3, q6) = 2qT3 q3 + 2qT3 q6 + 2qT6 q3 + 2qT6 q6

= −2 (q3 + q6)2< 0.

(23)

By this way, while the system remains in s = 0, thetrajectories (q3, q6) of (21) with the input u∞ = 0 convergeto zero as t→ ∞.

Thus, the regulation problem for q6 in the deviation system(21) can formally be stated as a nonlinear H∞-controlproblem.

In the sequel, the investigation will be confine to the H∞position regulation problem, where

1) The output to be controlled is given by

z = ρ

[0

q6

]+

[1

0

]u∞ (24)

with a positive weight coefficient ρ,2) The positions q2, q6 are the available measurements

and are corrupted by the error vectors w5(t), w6(t) ∈R.

y =

[q2 + xd1 + w5

q6 + xd2 + w6

], (25)

The H∞ control problem in question is thus stated asfollows. Given the system representation (21)-(25) and a realnumber γ > 0, it is required to find (if any) a causal dynamicfeedback controller

u∞ = K(ξ) (26)

with internal state ξ ∈ R8 such that the undisturbed closed-loop state q6 is uniformly asymptotically stable around theorigin and its L2 gain is locally less than γ, i.e., inequality

∫ T

0

‖z(t)‖2dt < γ2∫ T

0

‖w(t)‖2dt (27)

is satisfied for all T > 0 and all piecewise continuousfunctions w(t) = [w1(t), w2(t), w3(t), w4(t), w5(t), w6(t)]

T

for which the corresponding state trajectory of the closed-loop system, initialized at the origin, remains in someneighbourhood of this point.

VI. H∞ SYNTHESIS

The above H∞ control problem is nothing else than anonlinear H∞ control problem for the nonsmooth system(21) modelled by equations of the form as in [1], [2]

q = f(q) + g1(q)w + g2(q)u (28)

z = h1(x) + k12(q)u (29)

y = h2(q) + k21(q) (30)

12th IEEE Workshop on Variable Structure Systems,VSS’12, January 12-14, Mumbai, 2012

978-1-4577-2067-3/12/$26.00 c©2011 IEEE 437

when equations (28)-(30) are specified with

f1(q) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q2q3

−q3 − q6 − q7σ11q3q6q7

−σ02

m2q7 − q8

m2+ k

m2q2 − k

m2q6 +

kηm2

σ12q7

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (31)

f2(q) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

000

−σ11|q3|q4fc1000

−σ12|q7|q8fc2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (32)

g1(q) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 00 0 0 0 0 01

m10 0 0 0 0

0 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 1

m2

0 0 0 00 0 0 1 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, (33)

g2(q) =

⎡⎣02×1

105×1

⎤⎦ , h1(q) = ρ

[0q6

], (34)

h2(q) =

[q2 + xd1q6 + xd2

], k12(q) =

[10

], (35)

k21(q) =

[0 0 0 0 1 00 0 0 0 0 1

]. (36)

Then the output feedback is as in [1], [2]

ξ =2∑

i=1

fi(ξ)

+

[1

γ2g1(ξ)g

T1 (ξ)− g2(ξ)g

T2 (ξ)

]Pεξ

+ZεCT2 [y − h2(ξ)], (37)

u∞ = −gT2 (ξ)Pεξ, (38)

subject to (31)-(36) is a local solution of the H∞-positionregulation problem for the system under constraint forces(21).

VII. SIMULATION RESULTS

Performance issues and robustness properties of the pro-posed controller (15) using the disturbance attenuator u∞ in(38) are tested in numerical simulations and compared witha classical sliding mode controller design. Since only thestates q2 and q6 measurements are available, the H∞ filter(37) was applied to have access to the remaining states.

Fig. 3. Experimental Platform ECP-210 Modified to Have a ForceConstraint

In the simulation, performed in MatLab, the dynamicalmodel as in (16) is studied under the following parame-ters: desired position of the second mass xd2 = 22.69cm,viscous friction coefficients (σ01 = 7.695, σ02 = 2.1141),stiffness coefficients (σ11 = 1, σ12 = 1), Coulomb frictionlevels (fc1 = 1, fc2 = 1), values of the masses in kilograms(m1 = 1.06238kg.,m2 = 1.111kg.), values of the distancesin centimetres (a = 14.647cm, b = 5.151cm, c = 1.26cm),the spring stiffness k = 375.42Nm. The controller feedbackgains are set λ = 2 and β = 1. The parameters γ = 1.6,ρ = 1 and ε = 500 of the nonsmooth H∞ controller.The parameters of the mechanical system are real valuestaken from the control laboratory, the experimental resultsare under development using the platform ECP-210 modifiedto have a force constraint as in Figure 3. In the disturbedcase were additionally applied unknowns but bounded torquedisturbances governed by

w1 = 1.5 cos(0.5t) w2 = 0.2 sin(3t)

w3 = 0.4 sin(t) exp(−0.2t)(39)

The initial values were set to q1(0) = 0, q2(0) = −0.015m,q3(0) = 0, q4(0) = 0, q5(0) = 0, q6(0) = −0.005m,q7(0) = 0, q8(0) = 0 and s(0) = −1, respectively.Initial conditions for H∞ filter (37) were set to ξ =[0, 0, 0, 0, 0, 0, 0, 0]. Good performance and desired robust-ness properties of the system (16) under sliding-mode controllaw with H∞ attenuator (15), (38), using H∞ filter (37) areconcluded from Figure 4. A comparison of the sliding mode-H∞ control as (15), (38) is made against a classical slidingmode control, considering full state feedback and using thesame feedback gains for both controllers, the undisturbedcase is shown in Figure 5 and the disturbed case (particularcase when the condition β− M

m1

> 0 for existence of slidingmodes on the surface s is not fullfilled) is shown in Figure6.

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978-1-4577-2067-3/12/$26.00 c©2011 IEEE 438

0 5 10 15 20 25−12

−10

−8

−6

−4

−2

0

2

4

6x 10

−3

time (sec)

q 6 pos

ition

err

or (

m)

0 5 10 15 20 25−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

time (sec)

q 7 vel

ocity

(m

/s)

0 5 10 15 20 25−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

time (sec)

s sl

idin

g m

otio

n

0 5 10 15 20 25−4

−2

0

2

4

6

8

10

12

14

time (sec)

Con

trol

(N

.m)

Fig. 4. closed loop system using H∞ filter for output feedback (undis-turbed case).

0 5 10 15 20 25−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

time (sec)

q 6 pos

ition

err

or (

m)

SMC−H

∞

SMC

0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

q 7 vel

ocity

(m

/s)

SMC−H

∞

SMC

0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

time (sec)

s sl

idin

g m

otio

n

SMC−H

∞

SMC

0 5 10 15 20 25−6

−4

−2

0

2

4

6

8

10

12

14

time (sec)

Con

trol

(N

.m)

SMC−H

∞

SMC

Fig. 5. Comparison of sliding mode-H∞ control against sliding modecontrol using state feedback (undisturbed case).

0 5 10 15 20 25−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time (sec)

q 6 pos

ition

err

or (

m)

SMC−H

∞

SMC

0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (sec)

q 7 vel

ocity

(m

/s)

SMC−H

∞

SMC

0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

time (sec)

s sl

idin

g m

otio

n

SMC−H

∞

SMC

0 5 10 15 20 25−10

−5

0

5

10

15

20

time (sec)

Con

trol

(N

.m)

SMC−H

∞

SMC

Fig. 6. Comparison of sliding mode-H∞ control against sliding modecontrol using state feedback (disturbed case).

VIII. CONCLUSIONS

It was developed a fully practical framework for slidingmode control involving H∞ control methodology in itsown design. The afore mentioned design procedure has

been shown to be eminently suited to solving a positionregulation problem for a mechanical system with frictionand a force constraint. To facilitate exposition, the frictionmodel chosen for treatment has been confined to the Dahlmodel augmented with viscous friction. In the mechanicalsystem a strictly monotonic approximation of well-knowndead zone model is used to adequate the system accordingto the nonlinearH∞ control methodology design. The slidingmode-H∞ output regulation synthesis proposed is suited toglobally solve the regulation problem when the inequalityβ− M

m1

> 0 is satisfied, even in the presence of disturbancessupt |w1(t)| ≤ M , whenever this inequality is not satisfiedthe controller will attenuate the disturbances and dead zonemodel discrepancies (as in Figure 6), but it is enough toincrease the gain parameter β in order to fulfill β− M

m1

> 0and turn it into an asymptotically stable system. Effectivenessof the design procedure has been supported by simulations.Experimental verification of the sliding mode-H∞ control isin progress and will be published elsewhere.

REFERENCES

[1] Aguilar, L., Orlov, Y., and Acho, L., 2002, “Nonlinear H∞-Controlon Nonsmooth Time Varying Systems With Application to FrictionMechanical Manipulators,” Automatica, 39, pp. 1531-1542.

[2] Isidori, A., and Astolfi, A., 1992, “Disturbance Attenuation and H∞-Control via Measurement Feedback in Nonlinear Systems,” IEEETrans. Autom. Control, 37(9), pp. 1283-1293.

[3] Merzouki, R., Cadiou, J.C., and M’Sirdi, N.K., 2004, “Compensationof Friction and Backlash Effects in an Electrical Actuator,” J. Systemsand Control Engineering, 218, pp. 75-84.

[4] Isidori, A. “A tool for semiglobal stabilization of uncertain nonminimum-phase nonlinear systems via output feedback,” IEEE Trans-actions on Automatic Control, Vol. 48, No. 10, pp. 1817–1827 (2000).

[5] V. Utkin, Sliding Modes and Their Applications. Moscow: Mir, 1978.[6] V. Utkin, Sliding modes in control optimization. Berlin: Springer-

Verlag, 1992.[7] K.-Y. Lian and C.-R. Lin, “Sliding-mode motion/force control of

constrained robots,” IEEE Transactions on Automatic Control, vol.43, no. 8, pp. 11011103, (1998).

[8] Sabanovic, A., M. Elitas, and K. Ohnishi, “Sliding Modes in Con-strained Systems Control”, IEEE Transactions on Industrial Electron-ics, Vol. 55, pp. 3332–3339 (2008).

[9] Adetola V., D. DeHaan, and M. Guay, “Adaptive model predictivecontrol for constrained nonlinear systems”, Systems & Control Letters,Vol. 58, No. 5, pp. 320–326 (2009).

[10] Christophersen F.J. “Optimal control of constrained piecewise affinesystems”, Lectures notes in control and information sciences, Springer,(2007).

[11] Brogliato, B., S-I. Niculescu, and P. Orhant, “On the control of finite-dimensional mechanical systems with unilateral constraints,” IEEETransactions on Automatic Control, Vol. 42, No. 2, pp. 200–215(1997).

[12] Tseng C-S. “Mixed H2/H∞ adaptive tracking control design foruncertain constrained robots,” Asian Journal of Control, Vol. 7, No.3, pp. 296–309 (2005).

[13] Chiu C-S., K-Y. Lian, and T-C. Wu “Robust adaptive motion/forcetracking control design for uncertain constrained robot manipulators,”Automatica, Vol. 40, No. 12, pp. 2111–2119 (2004).

[14] Chang Y.-C. and C.-H. Lee “Robust tracking control for constrainedrobots actuated by DC motors without velocity measurements,” IEEProc. Control Theory and Applications, Vol. 146, No. 2, pp. 147–156(1999).

[15] A.R. Ghafari-Kashani, J. Faiz and M.J. Yazdanpanah“Integration ofnon-linear H∞ and sliding mode control techniques for motioncontrol of a permanent magnet synchronous motor,” IET Electr. PowerAppl., Vol. 4, Iss. 4, pp. 267280 (2010).

[16] F. Castanos and L. Fridman, “Dynamic switching surfaces for outputsliding mode control: An approach”, Automatica, Vol. 47, Issue 9,Pages 1957-1961, (September 2011).

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