intranet.ceautomatica.esintranet.ceautomatica.es/sites/default/files/upload/13/files/04... ·...

Control Fraccionario de Sistemas Híbridos y en Red. Aplicaciones

Blas M. Vinagre Jara

IX Simposio CEA de Ingeniería de ControlMadrid, 11-12 Abril 2011

lunes 11 de abril de 2011

Introducción

Control Fraccionario de Sistemas Híbridos y en Red. Aplicaciones - IX Simposio CEA de Ingeniería de Control - Madrid, 11-12 Abril 2011

• Proyectos: “MODELADO, ANÁLISIS Y GUIADO DE UN SISTEMA DE VEHÍCULOS EN

RED” (ref. DPI2005-07980-C03-03), Ministerio de Educación y Ciencia, Diciembre 2005-2008.

“CONTROL Y COORDINACIÓN DE UN SISTEMA DE VEHÍCULOS AUTÓNOMOS EN RED EN EL MARCO DE LOS SISTEMAS HÍBRIDOS” (ref. TRA2008-06602-C03-02/AUT), Ministerio de Ciencia e Innovación, Diciembre 2008-2011.

• Equipo:• Universidad de Extremadura:

• Blas M. Vinagre Jara• Inés Tejado Balsera• S. Hassan HosseinNia

• Universidad Nacional de Educación a Distancia:• Ángel Pérez de Madrid y Pablo• Carolina Mañoso Hierro• Miguel Romero Hortelano

• Universidad de Sevilla:• Ángel Rodríguez Castaño


CONTENIDO


1. Control Fraccionario

2. Sistemas de Control en Red

2.1 Modelado

2.2 Estimación en redes con pérdidas

2.3 Estrategias de Control

3. Sistemas Híbridos. Casos de Aplicación

4. Trabajo futuro


Fundamentos - Operadores


1. Control Fraccionario 3. Sistemas Híbridos 2. Sistemas de Control en Red 4. Trabajo futuro

4 Blas M. Vinagre and Concepcion A. Monje

I!c f (t) !

1

"(!)

! t

c(t! #)!!1 f (#)d#, t > c, ! " R+. (1)

When we deal with dynamic systems, it is usual that f (t) is a causal function oft, and so, in what follows, the definition of the fractional-order integral to be used

will consider 0 as the lower limit of the integral.

The definition in (1) cannot be used for the fractional-order derivative by direct

substitution of ! by!! because we have to proceed carefully in order to guarantee

the convergence of the integrals involved in the definition and to preserve the prop-

erties of the ordinary derivative of integer order. After some subtle mathematical

considerations, and introducing the positive integer m so that m! 1 < ! < m,

Riemann–Liouville’s definition of the fractional-order derivative of order ! " R+

has the following form:

RD! f (t)! D

mI

m!! f (t) =dm

dtm

"

1

"(m!!)

! t

0

f (#)

(t! #)!!m+1d#

#

, (2)

where m! 1< ! < m, m " N.

An alternative definition of the fractional-order derivative was introduced by

Caputo as

CD! f (t)! I

m!!Dm f (t) =

1

"(m!!)

! t

0

f (m)(#)

(t! #)!!m+1d#, (3)

where m! 1< ! < m, m " N.

There are the following relations between these two definitions in (2) and (3):

RD! f (t) =C D

! f (t)+m!1

$k=0

tk!!

"(k!!+ 1)f (k)

$

0+%

, (4)

RD!

&

f (t)!m!1

$k=0

f (k)$

0+% tk

k!

'

=C D! f (t). (5)

Due to its importance in applications, we will consider hereGrunwald–Letnikov’s

definition, based on the generalization of the backward difference. This definition

has the form

D! f (t)|t=kh = lim

h#0

1

h!

k

$j=0

(!1) j(

!

j

)

f (kh! jh). (6)

Laplace integral transform is a fundamental tool in systems and control engi-

neering. For this reason, we will give the equivalents of the defined fractional-order

operators in the Laplace domain. These equivalents are:

L [I ! f (t)] = s!!F(s), (7)


I!c f (t) !

1

"(!)

! t

c(t! #)!!1 f (#)d#, t > c, ! " R+. (1)










RD! f (t)! D

mI

m!! f (t) =dm

dtm

"

1

"(m!!)

! t

0

f (#)

(t! #)!!m+1d#

#

, (2)

where m! 1< ! < m, m " N.


Caputo as

CD! f (t)! I

m!!Dm f (t) =

1

"(m!!)

! t

0

f (m)(#)

(t! #)!!m+1d#, (3)

where m! 1< ! < m, m " N.


RD! f (t) =C D

! f (t)+m!1

$k=0

tk!!

"(k!!+ 1)f (k)

$

0+%

, (4)

RD!

&

f (t)!m!1

$k=0

f (k)$

0+% tk

k!

'

=C D! f (t). (5)



has the form

D! f (t)|t=kh = lim

h#0

1

h!

k

$j=0

(!1) j(

!

j

)

f (kh! jh). (6)




L [I ! f (t)] = s!!F(s), (7)


I!c f (t) !

1

"(!)

! t

c(t! #)!!1 f (#)d#, t > c, ! " R+. (1)










RD! f (t)! D

mI

m!! f (t) =dm

dtm

"

1

"(m!!)

! t

0

f (#)

(t! #)!!m+1d#

#

, (2)

where m! 1< ! < m, m " N.


Caputo as

CD! f (t)! I

m!!Dm f (t) =

1

"(m!!)

! t

0

f (m)(#)

(t! #)!!m+1d#, (3)

where m! 1< ! < m, m " N.


RD! f (t) =C D

! f (t)+m!1

$k=0

tk!!

"(k!!+ 1)f (k)

$

0+%

, (4)

RD!

&

f (t)!m!1

$k=0

f (k)$

0+% tk

k!

'

=C D! f (t). (5)



has the form

D! f (t)|t=kh = lim

h#0

1

h!

k

$j=0

(!1) j(

!

j

)

f (kh! jh). (6)




L [I ! f (t)] = s!!F(s), (7)


I!c f (t) !

1

"(!)

! t

c(t! #)!!1 f (#)d#, t > c, ! " R+. (1)










RD! f (t)! D

mI

m!! f (t) =dm

dtm

"

1

"(m!!)

! t

0

f (#)

(t! #)!!m+1d#

#

, (2)

where m! 1< ! < m, m " N.


Caputo as

CD! f (t)! I

m!!Dm f (t) =

1

"(m!!)

! t

0

f (m)(#)

(t! #)!!m+1d#, (3)

where m! 1< ! < m, m " N.


RD! f (t) =C D

! f (t)+m!1

$k=0

tk!!

"(k!!+ 1)f (k)

$

0+%

, (4)

RD!

&

f (t)!m!1

$k=0

f (k)$

0+% tk

k!

'

=C D! f (t). (5)



has the form

D! f (t)|t=kh = lim

h#0

1

h!

k

$j=0

(!1) j(

!

j

)

f (kh! jh). (6)




L [I ! f (t)] = s!!F(s), (7)

Fractional-order PID 5

L [RD! f (t)] = s!F(s)!

m!1

"k=0

sk!

RD!!k!1 f (t)

"

t=0, (8)

L [CD! f (t)] = s!F(s)!

m!1

"k=0

s!!k!1 f (k) (0) . (9)

2.3 Fractional-order systems

2.3.1 Models

Based on the operators introduced previously, the equations for a continuous-time

dynamic system of fractional order can be written as follows:

H (D!0!1!2···!m)(y1,y2, · · · ,yl) = G#

D#0#1#2···#n)(u1,u2, · · · ,uk

$

, (10)

where yi,ui are functions of time and H(·),G(·) are the combination laws of thefractional-order derivative operator. For the linear time-invariant single-variable

case, the following equation would be obtained:

anD!ny(t)+ an!1D

!n!1y(t)+ · · ·+ a0D!0y(t)

= bmD#mu(t)+ bm!1D

#m!1u(t)+ · · ·+ b0D#0u(t).

(11)

If in the previous equation all the orders of derivation are integer multiples of a

base order ! , that is, !k,#k = k!, ! " R+, the system will be of commensurate

order, and (11) becomes

n

"k=0

akDk!y(t) =

m

"k=0

bkDk!u(t). (12)

If in (12) ! = 1/q, q " Z+, the system will be of rational order.

In the case of discrete-time systems (or discrete equivalents of continuous-time

systems) we can obtain models of the form

an$!nh y(t)+ an!1$

!n!1h y(t)+ · · ·+ a0$

!0h y(t)

= bm$#mh u(t)+ bm!1$

#m!1h u(t)+ · · ·+ b0$

#0h u(t),

(13)

where $%hdenotes the difference operator with step size h and order % .

Applying the Laplace transform to (11) with zero initial conditions, or the Z

transform to (13), the input-output representations of the fractional-order systems

can be obtained. In the case of continuous models, a fractional-order system will be

given by a transfer function of the form

G(s) =Y (s)

U(s)=bms

#m + bm!1s#m!1+ · · ·+ b0s

#0

ans!n + an!1s!n!1+ · · ·+ a0s!0

. (14)


Fundamentos - Sistemas




L [RD! f (t)] = s!F(s)!

m!1

"k=0

sk!

RD!!k!1 f (t)

"

t=0, (8)

L [CD! f (t)] = s!F(s)!

m!1

"k=0

s!!k!1 f (k) (0) . (9)


2.3.1 Models



H (D!0!1!2···!m)(y1,y2, · · · ,yl) = G#

D#0#1#2···#n)(u1,u2, · · · ,uk

$

, (10)



anD!ny(t)+ an!1D

!n!1y(t)+ · · ·+ a0D!0y(t)

= bmD#mu(t)+ bm!1D

#m!1u(t)+ · · ·+ b0D#0u(t).

(11)




n

"k=0

akDk!y(t) =

m

"k=0

bkDk!u(t). (12)




an$!nh y(t)+ an!1$

!n!1h y(t)+ · · ·+ a0$

!0h y(t)


#m!1h u(t)+ · · ·+ b0$

#0h u(t),

(13)






G(s) =Y (s)

U(s)=bms

#m + bm!1s#m!1+ · · ·+ b0s

#0

ans!n + an!1s!n!1+ · · ·+ a0s!0

. (14)


L [RD! f (t)] = s!F(s)!

m!1

"k=0

sk!

RD!!k!1 f (t)

"

t=0, (8)

L [CD! f (t)] = s!F(s)!

m!1

"k=0

s!!k!1 f (k) (0) . (9)


2.3.1 Models



H (D!0!1!2···!m)(y1,y2, · · · ,yl) = G#

D#0#1#2···#n)(u1,u2, · · · ,uk

$

, (10)



anD!ny(t)+ an!1D

!n!1y(t)+ · · ·+ a0D!0y(t)

= bmD#mu(t)+ bm!1D

#m!1u(t)+ · · ·+ b0D#0u(t).

(11)




n

"k=0

akDk!y(t) =

m

"k=0

bkDk!u(t). (12)




an$!nh y(t)+ an!1$

!n!1h y(t)+ · · ·+ a0$

!0h y(t)


#m!1h u(t)+ · · ·+ b0$

#0h u(t),

(13)






G(s) =Y (s)

U(s)=bms

#m + bm!1s#m!1+ · · ·+ b0s

#0

ans!n + an!1s!n!1+ · · ·+ a0s!0

. (14)


L [RD! f (t)] = s!F(s)!

m!1

"k=0

sk!

RD!!k!1 f (t)

"

t=0, (8)

L [CD! f (t)] = s!F(s)!

m!1

"k=0

s!!k!1 f (k) (0) . (9)


2.3.1 Models



H (D!0!1!2···!m)(y1,y2, · · · ,yl) = G#

D#0#1#2···#n)(u1,u2, · · · ,uk

$

, (10)



anD!ny(t)+ an!1D

!n!1y(t)+ · · ·+ a0D!0y(t)

= bmD#mu(t)+ bm!1D

#m!1u(t)+ · · ·+ b0D#0u(t).

(11)




n

"k=0

akDk!y(t) =

m

"k=0

bkDk!u(t). (12)




an$!nh y(t)+ an!1$

!n!1h y(t)+ · · ·+ a0$

!0h y(t)


#m!1h u(t)+ · · ·+ b0$

#0h u(t),

(13)






G(s) =Y (s)

U(s)=bms

#m + bm!1s#m!1+ · · ·+ b0s

#0

ans!n + an!1s!n!1+ · · ·+ a0s!0

. (14)


In the case of discrete-time systems, the discrete-time transfer function will be of

the form

G(z) =bm

!

!!

z!1"""m + bm!1

!

!!

z!1"""m!1+ · · ·+ b0

!

!!

z!1"""0

an (! (z!1))#n + an!1 (! (z!1))#n!1 + · · ·+ a0 (! (z!1))#0, (15)

where!

!!

z!1""

is the Z transform of the operator $1h , or, in other words, thediscrete equivalent of the Laplace operator, s.

As can be seen in the previous equations, a fractional-order system has an

irrational-order transfer function in the Laplace domain or a discrete transfer func-

tion of unlimited order in the Z domain, since only in the case of #k " Z, there

will be a limited number of coefficients (!1)l!#kl

"

different from zero. Because of

this, it can be said that a fractional-order system has an unlimited memory or is

infinite-dimensional, and obviously the systems of integer order are just particular

cases.

In the case of a commensurate-order system, the continuous-time transfer func-

tion is given by

G(s) =

m

%k=0

bk(s# )k

n

%k=0

ak(s# )k. (16)

2.3.2 Dynamic behaviour and stability

In a general way, the study of the stability of fractional-order systems can be carried

out by studying the solutions of the differential equations that characterize them. An

alternative way is the study of the transfer function of the system (14). To carry out

this study it is necessary to remember that a function of the type

ans#n + an!1s

#n!1 + ....+ a0s#0 , (17)

with #i " R+, is a multi-valued function of the complex variable s whose domain

can be seen as a Riemann surface [24,82] of a number of sheets which is finite only

in the case of #i, #i " Q+, the principal sheet being defined by !& < arg(s) < & .In the case of #i " Q+, that is, # = 1/q, q being a positive integer, the q sheets ofthe Riemann surface are determined by

s= |s|ej' , (2k+ 1)& < ' < (2k+ 3)& , k =!1,0, · · · ,q! 2. (18)

Correspondingly, the case of k = !1 is the principal sheet. For the mappingw= s# , these sheets become the regions of the plane w defined by

w= |w|ej( , #(2k+ 1)& < ( < #(2k+ 3)& . (19)


Fundamentos - Dinámica




This mapping is illustrated in Figure 1 and Figure 2 for the case of w = s1/3.

Figure 1 represents the Riemann surface that corresponds to the transformation

introduced above, and Figure 2 represents the regions of the complex plane w that

correspond to each sheet of the Riemann surface. These three sheets correspond to

k=

!

"

"

#

"

"

$

!1, !! < arg(s)< ! , (the principal sheet)

0, ! < arg(s)< 3! , (sheet 2)

1 (= 3! 2), 3! < arg(s)< 5! , (sheet 3)

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1

xy

z

Fig. 1 Riemann surface for w= s1/3

!

"

#

$ %& #

$'

( first sheet

second sheet

second sheet

third sheet

"(w)

#(w)

Fig. 2 w-plane regions corresponding to the Riemann surface forw= s1/3

Thus, an equation of the type


This mapping is illustrated in Figure 1 and Figure 2 for the case of w = s1/3.

Figure 1 represents the Riemann surface that corresponds to the transformation

introduced above, and Figure 2 represents the regions of the complex plane w that

correspond to each sheet of the Riemann surface. These three sheets correspond to

k=

!

"

"

#

"

"

$

!1, !! < arg(s)< ! , (the principal sheet)

0, ! < arg(s)< 3! , (sheet 2)

1 (= 3! 2), 3! < arg(s)< 5! , (sheet 3)

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1

!1

!0.5

0

0.5

1

xy

z

Fig. 1 Riemann surface for w= s1/3

!

"

#

$ %& #

$'

( first sheet

second sheet

second sheet

third sheet

"(w)

#(w)

Fig. 2 w-plane regions corresponding to the Riemann surface forw= s1/3

Thus, an equation of the type


ans!n + an!1s

!n!1+ · · ·+ a0s!0 = 0, (20)

which in general is not a polynomial, will have an infinite number of roots, among

which only a finite number of them will be on the principal sheet of the Riemann

surface. It can be said that the roots which are in the secondary sheets are related

to time domain solutions (or responses) that are always monotonically decreasing

functions (they go to zero without oscillations when t""), and only the roots in the

principal sheet are responsible for a different dynamics: damped oscillation, oscilla-

tion of constant amplitude, or oscillation of increasing amplitude with monotonical

growth.

In general, it can be said that a fractional-order system, with an irrational-

order transfer functionG(s) = P(s)/Q(s), is bounded-input bounded-output (BIBO)stable if and only if the following condition is fulfilled (see [45] for more details):

#M, |G(s)|!M, $s #(s) " 0. (21)

The previous condition is satisfied if all the roots of Q(s) = 0 in the principal

Riemann sheet, not being roots of P(s) = 0, have negative real parts.For the case of commensurate-order systems, whose characteristic equation is a

polynomial of the complex variable $ = s! , the stability condition is expressed as

|arg($i)|> !%

2, (22)

where $i are the roots of the characteristic polynomial in $ . For the particular case

of ! = 1, the well known stability condition for linear time-invariant systems of

integer order is recovered:

|arg($i)|>%

2, $$i!Q($i) = 0. (23)

Nowadays we can find interesting studies on the stability of fractional-order

systems. There are even some attemps to develop polynomial techniques, either

Routh or Jury type, to analyze their stability. Of course, we can always use the

geometrical techniques of complex analysis based on Cauchy’s argument principle,

since they inform us about the number of singularities of the function within a

rectifiable curve by observing the evolution of the function argument through this

curve. For more details about the stability of fractional-order systems see [45], [79],

[57], [30], [36], [11].


In the case of discrete-time systems, the discrete-time transfer function will be of

the form

G(z) =bm

!

!!

z!1"""m + bm!1

!

!!

z!1"""m!1+ · · ·+ b0

!

!!

z!1"""0

an (! (z!1))#n + an!1 (! (z!1))#n!1 + · · ·+ a0 (! (z!1))#0, (15)

where!

!!

z!1""

is the Z transform of the operator $1h , or, in other words, thediscrete equivalent of the Laplace operator, s.

As can be seen in the previous equations, a fractional-order system has an

irrational-order transfer function in the Laplace domain or a discrete transfer func-

tion of unlimited order in the Z domain, since only in the case of #k " Z, there

will be a limited number of coefficients (!1)l!#kl

"

different from zero. Because of

this, it can be said that a fractional-order system has an unlimited memory or is

infinite-dimensional, and obviously the systems of integer order are just particular

cases.

In the case of a commensurate-order system, the continuous-time transfer func-

tion is given by

G(s) =

m

%k=0

bk(s# )k

n

%k=0

ak(s# )k. (16)

2.3.2 Dynamic behaviour and stability

In a general way, the study of the stability of fractional-order systems can be carried

out by studying the solutions of the differential equations that characterize them. An

alternative way is the study of the transfer function of the system (14). To carry out

this study it is necessary to remember that a function of the type

ans#n + an!1s

#n!1 + ....+ a0s#0 , (17)

with #i " R+, is a multi-valued function of the complex variable s whose domain

can be seen as a Riemann surface [24,82] of a number of sheets which is finite only

in the case of #i, #i " Q+, the principal sheet being defined by !& < arg(s) < & .In the case of #i " Q+, that is, # = 1/q, q being a positive integer, the q sheets ofthe Riemann surface are determined by

s= |s|ej' , (2k+ 1)& < ' < (2k+ 3)& , k =!1,0, · · · ,q! 2. (18)

Correspondingly, the case of k = !1 is the principal sheet. For the mappingw= s# , these sheets become the regions of the plane w defined by

w= |w|ej( , #(2k+ 1)& < ( < #(2k+ 3)& . (19)


Acciones Básicas de Control




3 Fractional-order Control

3.1 Generalized fractional-order control actions

Starting from the block diagram of Figure 3 ( [2], [10]), the effects of the generalized

basic control actions of type Ksµ for µ ! ["1,1] will be described in this section.The basic control actions traditionally considered will be particular cases of this

general case, in which µ = 0 for the proportional action, µ = "1 for the integralaction, and µ = 1 for the derivative action.

Ksµ G(s)! ! !!

""

R(s) E(s)Y (s)

Fig. 3 Block diagram of a closed-loop system with fractional-order control actions

As is known, the main effects of the integral action are those that make the system

slower, decrease its relative stability, and eliminate the steady-state error for inputs

for which the system has a finite error.

These effects can be observed in different domains. In the time domain, the

effects on the transient response consist of the decrease of the rise time and the

increase of the settling time and the overshoot. In the complex plane, the effects

of the integral action consist of a displacement of the root locus of the system

towards the right half-plane. Finally, in the frequency domain, these effects consist

of an increase of "20dB/dec in the slopes of the magnitude curves and a decreaseof !/2rad in the phase plots. In the case of a fractional-order integral, that is,µ ! ("1,0), the selection of the value of µ needs consideration of the effects

mentioned above. In the time domain, the effects of the control action can be studied

considering the effects of this action on a squared error signal. If the error signal has

the form

e(t) =N

"k=0

("1)ku0(t" kT ), k = 0,1,2, · · · ,N, (24)

where u0(t) is the unit step, its Laplace transform is

E(s) =N

"k=0

("1)ke"kTs

s. (25)

Thus, the control action, as shown in the block diagram of Figure 3, will be given

by


u(t) = L!1 {U(s)} = L

!1

!

K

N

!k=0

(!1)ke!kTs

s1!µ

"

= K

N

!k=0

(!1)k

"(1! µ)(t! kT )!µ

u0(t! kT ).

(26)

Figure 4 shows the function u(t) for the values µ = 0,!0.2,!0.5,!1; T = 30;N = 4. As can be observed, the effects of the control action on the error signal varybetween the effects of a proportional action (µ = 0, square signal) and an integral

action (µ =!1, straight lines curve). For intermediate values of µ , the control actionincreases for a constant error, which results in the elimination of the steady-state

error, and decreases when the error is zero, resulting in a more stable system.

0 2 4 6 8 10

0

1

2

3

4

5

µ

=!1

µ =!0.5

µ =!0.2

µ = 0

Controlaction

Time (sec)Fig. 4 Integral control action for a square error signal and µ = 0,!0.2,!0.5,!1

In the complex plane, the root locus of the system with the control action is

governed by

1+KsµG(s) = 0, (27)

or by the following equivalent conditions for the magnitude and phase:

|K|=1

|sµ | |G(s)|, (28)

arg [sµG(s)] = (2n+ 1)# , l = 0,±1,±2, · · · . (29)

Taking into account that

s= |s|ej$ =" sµ = |s|µ ejµ$ , (30)

the conditions of magnitude and phase can be expressed by

|K|=1

|s|µ |G(s)|, (31)


action corresponds to intermediate curves. It must be noted that the derivative action

is not zero for a constant error and the growth of the control signal is more damped

when a variation in the error signal occurs, which implies a better attenuation of

high-frequency noise signals.

0 2 4 6 8 10

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

1.2µ = 0

µ = 0.2

µ = 0.5

µ = 1

Time (sec)

Derivativecontrolaction

Fig. 5 Derivative control action for a trapezoidal error signal and µ = 0,0.2,0.5,1

In the frequency domain, the magnitude curve is given by (33) and the phase plot

by (34). As can be observed, by varying the value of µ between 0 and 1, it is possible

to introduce a constant increment in the slopes of the magnitude curve that varies

between 0dB/dec and 20dB/dec, and to introduce a constant delay in the phase plot

that varies between 0rad and !/2rad.

3.2 State of the art in FOC

Maybe the first sign of the potential of FOC, though without using the term

“fractional,” emerged with Bode ( [7], [8]). A key problem in the design of a

feedback amplifier was to devise a feedback loop so that the performance of the

closed loop was invariant to changes in the amplifier gain. Bode presented an

elegant solution to this robust design problem, which he called the ideal cutoff

characteristic, nowadays known as Bode’s ideal loop transfer function, whose

Nyquist plot is a straight line through the origin giving a phase margin invariant

to gain changes. Clearly, this ideal system is, from our point of view, a fractional-

order integrator with transfer function G(s) =!

"cg/s"#

, known as Bode’s idealtransfer function, where "cg is the gain crossover frequency and the constant phase

margin is $m = !!#!/2. This frequency characteristic is very interesting in termsof robustness of the system to parameter changes or uncertainties, and several design

methods have made use of it. In fact, the fractional-order integrator can be used as

an alternative reference system for control [80].


denoted as Bode’s ideal compared to the conventionalPID, and this fact allowed for

simpler tuning, better disturbance rejection, and better robustness to plant variations.

Though it can be considered as a seminal work, this controller has received few

attention. Thus, we will concentrate on the FoPID controller.

The integro-differential equation defining the control action of aFoPID controller

is given by

u(t) = Kpe(t)+KiD!!e(t)+KdD

µe(t). (40)

Applying Laplace transform to this equation with null initial conditions, the

transfer function of the controller can be expressed by

Cf(s) = Kp+Ki

s!+Kds

µ = k(s/"f)!+µ + s#fs

!/"f+ 1

s!. (41)

Figure 8 shows the frequency response of this controller for k= 1, "f = 1, #f = 1,and ! = µ = 0.5.

0

50

100

Mag

nit

ude

(dB

)

10!4

10!2

100

102

104

!45

0

45

90

Phas

e (d

eg)

Frequency (rad/sec)

Fig. 8 Frequency response of the FoPID controller with k = 1, "f = 1, #f = 1, and ! = µ = 0.5

As can be observed, this fractional-order controller allows us to select both the

slope of the magnitude curve and the phase contributions at both high and low

frequencies.

In a graphical way, the control possibilities using a FoPID controller are shown

in Figure 9, extending the four control points of the classical PID to the range of

control points of the quarter-plane defined by selecting the values of ! and µ .


denoted as Bode’s ideal compared to the conventionalPID, and this fact allowed for

simpler tuning, better disturbance rejection, and better robustness to plant variations.

Though it can be considered as a seminal work, this controller has received few

attention. Thus, we will concentrate on the FoPID controller.

The integro-differential equation defining the control action of aFoPID controller

is given by

u(t) = Kpe(t)+KiD!!e(t)+KdD

µe(t). (40)

Applying Laplace transform to this equation with null initial conditions, the

transfer function of the controller can be expressed by

Cf(s) = Kp+Ki

s!+Kds

µ = k(s/"f)!+µ + s#fs

!/"f+ 1

s!. (41)

Figure 8 shows the frequency response of this controller for k= 1, "f = 1, #f = 1,and ! = µ = 0.5.

0

50

100

Mag

nit

ude

(dB

)

10!4

10!2

100

102

104

!45

0

45

90

Phas

e (d

eg)

Frequency (rad/sec)

Fig. 8 Frequency response of the FoPID controller with k = 1, "f = 1, #f = 1, and ! = µ = 0.5

As can be observed, this fractional-order controller allows us to select both the

slope of the magnitude curve and the phase contributions at both high and low

frequencies.

In a graphical way, the control possibilities using a FoPID controller are shown

in Figure 9, extending the four control points of the classical PID to the range of

control points of the quarter-plane defined by selecting the values of ! and µ .

7 Fractional-order PI!Dµ Controllers 17

!

"

!

"µ µ

! !

PD PID

P PI

O

PD PID

P PI

O

µ=1

!=1

µ=1

!=1

(a) (b)

Fig. 9 FoPID vs classical PID: from points to plane: (a) integer-order controller and (b) fractional-

order controller

5 Tuning methods

5.1 Introduction

It is important to realize that there is a very wide range of control problems and,

consequently, also a need for a wide range of design techniques. There are already

many tuning methods available in the literature for fractional PI!Dµ controllers of

the form

C(s) = Kp+Ki

s!+Kds

µ . (42)

Since this kind of controller has five parameters to tune (Kp,Kd,Ki,! ,µ), up tofive design specifications for the controlled system can be met, that is, two more

than in the case of a conventional PID controller, where ! = 1 and µ = 1. It is

essential to study which specifications are more interesting as far as performance

and robustness are concerned, since it is the aim to obtain a controlled system robust

to uncertainties of the plant model, load disturbances, and high-frequency noise. All

these constraints will be taken into account in the tuning technique in order to take

advantage of the introduction of the fractional orders.

During the last decades, further research activities to define new effective tuning

methods for fractional-order controllers have been proposed as an extension of the

classical control theory, mainly for traditional PID controllers due to its widespread

industrial use. Some analytical methods, concerning phase and gain margins, flat

phase, or dominant poles, can be found in [79], [20], [43], [39], [38], as well as some

optimization-based methods in [12], [50] and tuning rules in [19], [5], [6], [75].

Recently, tuning methods for FoPID controllers based on AI tools such as Adaptive

Genetic Algorithms [16] and Particle Swarm Optimization [83], as well as Fuzzy

fractional-order PID controllers [3], have been proposed.

Here we will concentrate our attention in some methods that were the basis of

many others appeared in the literature (see [48], [77], [63] for a review).


Estrategias de Control Generalizadas



• Lineales:• PID• Espacio de estados

• No lineales:• Adaptativo: MRAC, Scheduling• Deslizante• Reset

• Robusto:• Loop Shaping - CRONE• QFT

• Óptimo:• LQR• Factorización Espectral - Wiener-Hopf

• Predictivo

Referencia: Monje, Chen, Vinagre, Feliu, Xue. Fractional Order Systems and Control, Springer 2010.


Efectos inducidos por la red


Retardos de transmisión Pérdidas de paquetes. Canales con ancho de banda limitado. Fallos de enlace/nodo. Cuantización. Errores de sincronización.

VENTAJAS: Modularidad, flexibilidad. Reducción del coste. Descentralización del control. Mantenimiento sencillo. ...

Degradación del rendimiento.

Reducción márgenes de estabilidad.

1. Introducción a los NCSs. Modelo de simulación 3. Estrategias de control aplicadas. Resultados 2. Estimación a través de redes con pérdidas con FKFs 4. Conclusiones y trabajo futuro

Sistema de control en red (NCSs del término en inglés): sistema en el que el lazo de control se cierra a través de una red de comunicación.



Modelo de simulación


Varias naturalezas del retardo:

Pérdidas de paquetes:


Modelo sólo con pérdidas de paquetes Modelo basado en el retardo



Estimación en redes con pérdidas de información (I)


Filtro deKalman (KF)

Filtro de KalmanFraccionario (FKF)

FKF Mejorado (ExFKF)

Extensión FKF (gFKF)

Extensión ExFKF (gExFKF)

Versionespara NCSs

Canales de comunicación ideales

Canales de comunicación con pérdidas


sistemas de

ordenfraccionario

forma dimensional

infinita sistema lineal fraccionario

Realizado en colaboración con Dr. Sierociuk (Universidad Técnica de Varsovia, Polonia). Signal Processing, 91:542--552, 2011.

(Dr. Sierociuk, 2006)



Filtro de Kalman Fraccionario (FKF)

Estimación en redes con pérdidas de información (II)



Forma generalizada sistema lineal discreto de orden

fraccionario con perturbaciones estocásticas

FKF Mejorado (ExFKF)

Sistema estocástico discreto de orden fraccionario en forma m-

finita (basado en la forma dimensional infinita)



Estimación en redes con pérdidas de información (III)



Extensión del Filtro de Kalman Fraccionario (gFKF)Extensión del Filtro ExFKF (gExFKF)

Simulación pérdidas de paquetes:

señal γ =

Simulaciones con la herramienta FSST (Fractional Order State-Space Systems) de Simulink.

0: pérdida 1: transmisión eficaz




Compensación del retardo - Planteamiento del problema

Condiciones redcambiantes (retardo)

COMPENSACIÓNFIJA (DISEÑO)

COMPENSACIÓNEN TIEMPO REAL

Ajustando ganancias controlador

Ajustando ganancias y orden

controlador

Retardo medio de la red

β (β>0) β (β>0), α (0<α<2)

1. Introducción a los NCSs. Modelo de simulación 3. Estrategias de control aplicadas. Resultados 2. Estimación a través de redes con pérdidas con FKFs 4. Conclusiones y trabajo futuro 1. Control Fraccionario 3. Sistemas Híbridos 2. Sistemas de Control en Red 4. Trabajo futuro



Características

Estrategias basadas en el análisis previo de la red. Útiles en aplicaciones en las que reemplazar un controlador por otro más eficiente es complicado. Técnicas generales: aplicables a cualquier controlador existente. Aplicaciones experimentales:

Control de velocidad de la plataforma “Smart Wheel” Compensación fija Control de velocidad de un vehículo Citroën C3 Programación ganancia Control de servomotor de velocidad Programación ganancia y orden

Análisis yModelado

Retardomedio

Distribucióndel retardo



Compensación fija del retardo (I)



Dinámica velocidad del eje de dirección:

Controladores:1. PI de orden fraccionario (FOPI):

2. PI método mejorado de Ziegler-Nichols (PIIZN):



Compensación fija del retardo (II)



3. Controlador predictivo generalizado fraccionario (FGPC):



Compensación fija del retardo (III)



Diseño:

τmedio red = 266 ms



Compensación fija del retardo – Resultados experimentales


Respuestas estables y similares con los tres controladores.

Oscilación en estado estacionario (perturbación en la carga debido a la asimetría del soporte de la SW).

Esta técnica de compensación puede ser eficiente cuando el retardo está caracterizado con precisión (valor medio, límites) y las condiciones de la red cambian dentro del rango caracterizado.



Controlador fraccionario con ganancia programable (FGSC) (I)


Controlador (fraccionario) nominal Programador de ganancia. Estimador del retardo. Sistema.



Controlador fraccionario con ganancia programable (FGSC) (II)


PASOS PARA EL DISEÑO:

1) Controlador (fraccionario) nominal.2) División rango de trabajo: τnet є(0,τnet,max].

3) Criterio estabilidad de Nyquist: βmax para τnet,i (aportación respecto a bibliografía consultada).

4) Definición función de coste J a minimizar.5) Cálculo de J en simulación para cada τnet,i

con β є(0,βmax].

6) Minimización de J para cada τnet,i:

βop = f(τnet,i)

Programador de ganancia



FGSC – Control de velocidad de un vehículo (I)


Dinámica longitudinal vehículo (baja velocidad):

Controlador local PIα:


1. Mp ≈ 0%2. tr ≈ 4s3. Robustez dinámicas no modeladas e imprecisión en medidas

Confort pasajeros (a ≤ 2m/s2)

Aproximación de 7º orden (método Oustaloup) Implementación digital filtro IIR (estructura paralela) en C++



FGSC – Control de velocidad de un vehículo (II)


Diseño del controlador FGSC (remoto):


βop = f(τnet)



FGSC – Control de velocidad de un vehículo. Resultados



Retardo fijo: τnet = 0.2s Mejora del 55.77% en error de velocidad

Retardo variable: τnet є [0.2,0.4]s Mejora del 83.85% en error de velocidad



Controlador fraccionario con ganancia y orden programables (FGOSC)



Controlador (fraccionario) nominal Programador de ganancia. Programador de orden. Estimador del retardo. Sistema.

Gain scheduler

Orderscheduler



FGOSC – Métodos de diseño (I)


Método I


Mínimo: αmin

Máximo: αmax

Máximo margen de ganancia: αGM

(máximo rango de exclusión en β).

α є[αmin,αmax]



FGOSC – Métodos de diseño (II)


Método I

Método II



FGOSC – Métodos de diseño (III)


Comparación métodos I y II para el caso de la SW


Método I Método II



FGOSC – Métodos de diseño (III)


Comparación métodos I y II para el caso de la SW


Otros métodos:1. Fijar α = αmin y minimizar J variando en β.

2. Minimizar J variando α є[αmin,αmax] y β є(0,βmax).

3. Minimizar J variando α є(0,2) y β є(0,βmax).

Diseño con el Método II



FGOSC – Control de la velocidad angular de un servomotor (I)



Aproximación de 4º orden (método modificado Oustaloup)

Control a través de LAN con la herramienta Instrument Control de MATLAB y tarjeta de adquisición de datos NI 6259.

Dinámica servo de velocidad:

Controlador nominal FOPI:



FGOSC – Control de la velocidad angular de un servomotor (II)



Diseño del controlador FGOSC (a partir de FGSC –método II–):

βop = f(τnet) αop = f(τnet)



FGOSC – Control de la velocidad angular de un servomotor (II)



Diseño del controlador FGOSC (a partir de FGSC –método II–):

βop = f(τnet) αop = f(τnet)τnet αop

(0,0.12) 0.8

[0.12,0.18) 0.9

[0.18,0.26) 1

[0.26,0.42) 1.1

[0.42,0.5) 1.2



FGOSC vs FOPI – Resultados experimentales


Las respuestas con el FGOSC son estables.

Mejora ~ 70%. La mejora en el

rendimiento es mayor cuanto mayor sea el retardo.


(a) τnet є [0.05,0.12]s

(b) τnet є [0.12,0.18]s

(c) τnet є [0.18,0.25]s

Forma de variación del retardo



FGOSC vs FGSC – Resultados experimentales (I)



Respuestas similares. Sensible mejora del rendimiento con el FGOSC: 2.43 – 5.76%.

(a) τnet є [0.05,0.12]s

(b) τnet є [0.12,0.18]s

(c) τnet є [0.18,0.25]s

Forma de variación del retardo



FGOSC vs FGSC – Resultados experimentales (II)



Cambio en el valor medio de τnet a los 5s

τnet є (0,0.05]s

(a) τadd = 0.05 s

(b) τadd = 0.12 s







τnet є (0,0.05]s

(a) τadd = 0.05 s

(b) τadd = 0.12 s

τnet αop (0,0.12) 0.8

[0.12,0.18) 0.9

[0.18,0.26) 1

[0.26,0.42) 1.1

[0.42,0.5) 1.2






Respuestas similares.Mejoraría aún más si en la implementación de orden variable se

evitasen “saltos” de orden??.


τnet є (0,0.05]s

(a) τadd = 0.05 s

(b) τadd = 0.12 s



Introducción (I)




Sistema Híbrido: Interacción de dinámica continua (ej.: ecuación diferencial) y eventos discretos (ej.: autómata)

◦ Ejemplos◦ Sistemas continuos con funcionamiento por fases: Bouncing ball, robots

caminantes .....◦ Sistemas continuos controlados con lógica discreta:termostatos, plantas químicas

con válvulas, bombas, etc.Chemical plants with valves, pumps...◦ Procesos coordinados: sistemas de transporte ...

32 IEEE CONTROL SYSTEMS MAGAZINE » APRIL 2009

Some models of hybrid systems explicitly partition the state of a system into a continuous state j and a discrete state q, the

latter describing the mode of the system. For example, the val-ues of q may represent modes such as “working” and “idle.” In a temperature control system, q may stand for “on” or “off,” while j may represent the temperature. By its nature, the discrete state can change only during a jump, while the continuous state of-ten changes only during fl ows but sometimes may jump as well. These systems are called differential automata [82], hybrid au-tomata [S2], [51], or simply hybrid systems [S1], [9]. All of these systems can be cast as a hybrid system of the form (1), (2).

The data of a hybrid automaton are usually given by a » set of modes Q, which in most situations can be identi-fied with a subset of the integersa » domain map Domain: Q SS Rn, which gives, for each q [ Q , the set Domain(q) in which the continuous state j evolvesa » flow map f : Q 3 Rn S Rn , which describes, through a differential equation, the continuous evolution of the continuous state variable ja » set of edges Edges ( Q 3 Q , which identifies the pairs (q, q r ) such that a transition from the mode q to the mode q r is possiblea » guard map Guard : Edges SS Rn, which identifies, for each edge (q, q r ) [ Edges, the set Guard1q, q r 2 to which the continuous state j must belong so that a transition from q to q r can occura » reset map Reset : Edges 3 Rn S Rn , which de-scribes, for each edge (q, q r ) [ Edges, the value to which the continuous state j [ Rn is set during a transi-tion from mode q to mode q r. When the continuous state variable j remains constant at a jump from q to q r, the map Reset 1q, q r, # 2 can be taken to be the identity.

Figure S1 depicts part of a state diagram for a hybrid automaton. The continuous dynamics of two modes are shown, together

with the guard conditions and reset rules that govern transitions between these modes.

We now show how a hybrid automaton can be modeled as a hybrid system in the form (1), (2). First, we reformulate a hybrid automaton as a hybrid system with explicitly shown modes. For each q [ Q , we take

Cq5 Domain (q ) , Dq5 d(q, qr)[Edges

Guard (q, q r ) ,

Fq (j ) 5 f(q, j ) , for all j [ Cq,

Gq (j ) 5 d5qr:j[Guard(q, qr)6(Reset(q, q r, j ) , q r ) , for all j [ Dq.

When j is an element of two different guard sets Guard (q, q r ) and Guard (q, qs ) , Gq (j ) is a set consisting of at least two points. Hence, Gq can be set valued. In fact, Gq is not neces-sarily a function even when every Reset(q, q r, #) is the identity map. With Cq , Fq , Dq , and Gq defi ned above, we consider the hybrid system with state (j, q ) [ Rn 3 R and representation

j#5 Fq (j ) , q [ Q, j [ Cq,

(j1, q1 ) [ Gq (j ) , q [ Q, j [ Dq.

Example S1: Reformulation of a Hybrid Automaton Consider the hybrid automaton shown in figures S2 and S3, with the set of modes Q5 51, 26 ; the domain map given by

Domain (1 ) 5 R#0 3 R, Domain (2 ) 5 506 3 R;

the fl ow map, for all j [ R2, given by

f(1, j ) 5 (1, 1 ) , f(2, j ) 5 (0,21 ) ;

the set of edges given by Edges5 5 (1, 1 ) , (1, 2 ) , (2, 1 ) 6 ; the guard map given by

Guard (1, 1 ) 5 R$0 3 R#0, Guard (1, 2 ) 5 R2

$0, Guard (2, 1 ) 5 506 3 R#0;

and the reset map, for all j[R2, given by

Reset(1, 1, j ) 5 (25, 0 ) , Reset(1, 2, j ) 5j, Reset(2, 1, j ) 5 2j.

The sets Guard (1, 1 ) and Guard (1, 2 ) overlap, indicating that, in mode 1, a re-set of the state j to (25, 0 ) or a switch of the mode to 2 is possible from points

FIGURE S1 Two modes, q and q r, of a hybrid automaton. In mode q, the state j evolves according to the differential equation j

#5 f(q, j ) in the set Domain1q 2. A transition

from mode q to mode q r can occur when, in mode q, j is in the set Guard1q, q r 2 . During the transition, j changes to a value j1 in Reset1q, q r, j 2 . Transitions from mode q to other modes, not shown in the figure, are governed by similar rules.

! !Guard(q, q")

! !Guard(q", q)

!+ !Reset(q, q", !)

!+ !Reset(q", q, !)

#

#

! !Domain(q")

q"! = f (q", ! )

! !Domain(q)

q! = f (q, ! ). .

Hybrid Automata

Authorized licensed use limited to: Universidad de Extremadura. Downloaded on June 19, 2009 at 07:41 from IEEE Xplore. Restrictions apply.


Introducción (II)





Coordinación de vehículos (I)




!"#!"$%&'()*# #+*%,

-. /0" '#+11 !#+%('2 +3 20" *(2+$+4+(1 ($&2 %&#"'2&+$ 5"'2+#

6!!. 7&20 20" '+44+$ ($&2 %&#"'2&+$ 5"'2+# 6!!". 4(12

8" !+1&2&5"9 &,",9 20" 4*$(*) 5"0&')" &1 &$ 20" #&:02;0*$%

1":4"$2 +3 20" !"#!"$%&'()*# #+*%,

<. /0" '#+11 !#+%('2 +3 20" *(2+$+4+(1 ($&2 %&#"'2&+$ 5"'2+#

6!!. 7&20 20" 4*$(*) ($&2 %&#"'2&+$ 5"'2+# 6!". 4(12

8" $":*2&5"9 &,",9 20" 4*$(*) 5"0&')" &$ 20" #&:02;0*$%

1":4"$2 +3 20" !"#!"$%&'()*# #+*% &1 *!!#+*'0&$: 20"

'#+11#+*%,

/0" '+$2#+))"# 7&)) *)1+ 8" *'2&5*2"% 70"$ 20" '*#1 *#" &$

20" #*%&+1 +3 <= 4"2"#1 +3 &$2"#1"'2&+$ !+&$2,

>&:, ?, @"'2+# #"!#"1"$2*2&+$ &$ 20" )"32;0*$% '++#%&$*2" 1A12"4,

!" #$%&'$(()' *)+,-%. /0" '+$2#+))"# &1 %"1&:$"% 8*1"%

+$ 20" 0A8#&% 12#*2":A, /0" %A$*4&'1 +3 0A8#&% 1A12"41 &$

20&1 '*1" '*$ 8" 1(44*#&B"% 7&20 * 0A8#&% *(2+4*2+$ 6*1

&))(12#*2"% &$ 1"'2&+$ CC.9 70&'0 '+$2*&$1 20" 3+))+7&$: "$2&2&"1

D

! " ! !##$ #!$ #""$! % ! "&$ '$ (# # !$$! )*+, ! !##"$!& # !

$ $ ,! - ","%%#& ,! . ","&%#"$! / ! ' 0# ()*"(#$ 0# *+,"(#$ 0#

!

$-$ 70"#" & *$% ' *#"

20" '++#%&$*2"19 ( &1 20" 5"0&')" +#&"$2*2&+$9 0# &1 20"

5")+'&2A +3 :(&%*$'" !+&$2 *$% 1 &1 20" %&12*$'" 8"27""$

3#+$2 *$% #"*# *E"1,

! 2"### ! !% # !$ $ ,! - "," % %# & ,! . "," & %#"$! 2"#!# ! !% # !$ $ "," % %# ' ,! ' "," &%# . / '&%!' 01"$

! 2"#"# ! !% # !$ $ ","%%# ' ,! ' ","&%# . &%!-

/ . &%"- /"$

! 3 ! !"##$ #!#$ "#!$ ###$ "##$ #"#$ "#"$ ###$ "#!$ #"#" $! 4"##$ #!# ! !% # !

$ $ "," % %# ' ,! ' "," &%# . / ' &%

!' 01"$

! 4"##$ #"# ! !% # !$ $ "," % %# ' ,! ' "," &

%# . &%!- / . &%

"- /"$

! 4"#!$ ### ! 4"#"$ ### ! !% # !$ $ ,! - ","%%#& ,! ."," & %#"$

! 4"#!$ #"# ! !% # !$ $ &%!- / . &%

"- /"$

! 5"##$ #!$ %# ! 5"#!$ ##$ %# ! 5"##$ #"$ %# !5"#"$ ##$ %# ! 5"#!$ #"$ %# ! !%"$

70"#"9 % &1 20" '+$2&$(+(1 12*2" *$% " &1 20" %&1'#"2"

12*2" &$ 20" 0A8#&% *(2+4*2+$, /0" "11"$'" +3 20" %"'&1&+$

4*F&$: 1A12"4 '*$ 8" "E!)*&$"% 7&20 >&:, G, /0&1 4+%()" &1

*$ *(2+4*2+$ 70"#" "*'0 12*2" &1 * %"1&#"% '*# 8"0*5&+#, /0"

2#*$1&2&+$1 8"27""$ 12*2"1 %"!"$% +$ !#+'"11"% &$3+#4*2&+$ +3

20" !+1&2&+$ *$% 1!""% +3 20" '*#1 &$5+)5"% &$ 20" '#+11&$:,

H+1&2&+$ *$% 1!""% +3 20" *(2+4*2&' '*# *#" (1"% 2+ '+4!(2"

20" 2&4" $""%"% 2+ '#+11 20" &$2"#1"'2&+$ 70"#" ,! !&!

"

'"9

," !&!

#

'#%"$+2" 20" *##&5*) 2&4" 2+ &$2"#1"'2&+$ +$ *$% &%

!9

&%"%"$+2" 20" %&12*$'" 2+ &$2"#1"'2&+$ +$ *$% 0!$ 0" %"$+2"

20" 5")+'&2A +3 20" *(2+4*2&' '*# *$% 4*$(*) '*#9 #"1!"'2&5")A,

>&:, G, IA8#&% *(2+4*2+$D '#+11&$: 1*3")A

J1 &2 &1 4"$2&+$"% &$ IA8#&% *(2+4*2+$9 20"#" *#" 20#""

%&1'#"2" 12*2"1 " ! !##$ #!$ #"", ## &1 20" &$&2&*) 12*2" +3 20"0A8#&% *(2+4*2+$ 70&'0 &2 &1 4"*$2 20" '*# &1 &$ &21 $+#4*)

1!""% *$% 3+))+71 20" #"3"#"$'" 1!""%, #! &1 20" 12*2" &$ 70&'0

20" '*# 12*#21 2+ #"%('" &21 1!""% *$% #" &1 20" 12*2" 70"#"

* '+))&1&+$ #&1F "E&121 *$% 20" '*# 7&)) 8" #"%('" &21 1!""%

12#+$:)A *$% "5"$2(*))A &2 12+!, K+4*&$ +3 "*'0 12*2" &1 10+7$

&$ >&:1, L *$% M, K+4*&$ +3 ## '*$ 8" "*1&)A %"2"#4&$"% 8A

>&:, L 8(2 %+4*&$ +3 #! *$% #" 7&)) 8" %"2"#4&$"% 8A >&:1, L

*$% M9 1&4()2*$"+(1)A,

>&$*))A9 %&1'#"2" #"3"#"$'" &$!(2 &$ "*'0 12*2" 7&)) 8"

*'0&"5"% *1 D

0(",& 2# !

!"

#

6%&'$ &3 " ! ##1730(",#$ &3 " ! #!17/0(",#$ &3 " ! #"

6?.

70"#" 6%&' &1 20" 4*E&4(4 5*)(" 3+# 20" 5")+'&2A *$% 0# 7&))

8" *'0&"5"% 8*1"% +$ )+$:&2(%&$*) %A$*4&'1 &,", NO, 6<.,

/" 0,12(3&,$% ')+2(&+. P&4()*2&+$ 0*1 8""$ '*##&"% +(2 8A

PCQRSCTU (1&$: 12*2"V+7, Q*E&4(4 5")+'&2A 3+# *(2+4*2&'

'*#1 *#" 1"2 *2 01 F4W0 , /0" 1&4()*2&+$ #"1()2 &1 10+7$ &$>&:, X, /0" 2+! !)+2 10+71 20" 5*#&*2&+$ +3 20" 5")+'&2A +3 "*'0

'*# %(#&$: 20" '#+11&$:, /0" 8+22+4 !)+2 !#"1"$21 20" %&12*$'"

8"27""$ "*'0 '*# 2+ &$2"#1"'2&+$9 70"#" 20" B"#+ 5*)(" +3 20&1

0 1 2 3 4 50

1

2

3

4

5

6

7

tA

t M

q0

q1 or q

2

q1 or q

2

!"#$ %$ &'()"* '+ !! )*, !!"" !#"

0 5 10 15 20 25 30 350

5

10

15

20

xA

i

xMi

q1

q2

!"#$ -$ &'()"* '+ !" )*, !#

,"./)*01 (1)*. /21 0)3 ". #'"*# /' 03'.. /21 "*/13.10/"'*$ 4* /2".

0).1 /21 ()5"(6( 718'0"/9 )*, ,"./)*01 :1/;11* )6/'()/"0

0)3 )*, ()*6)8 0)3 /' "*/13.10/"'* )31 .1/ )/ !"# <(=2! $% (&)*, !'% <(=2! (# (&> 31.?10/"7189$ @. "/ 0)* :1 .11* +3'( !"#$A> /2131 ". 3".< '+ 0'88"."'* )*, /21 ,10"."'* :8'0< ". ,10",1,

/' 31,601 /21 .?11, '+ )6/'()/"0 0)3 )/ " ! )' .> )*, /21 0)3.)31 03'..1, .)+189$

!" !#$% &'() &'( #*+(,#+-. .#/$ #01 (0% ,#0*#2 .#/

4* /2". 0).1> /2131 )31 /;' )6/'()/"0 0)3. )*, '*1 ()*6)8 0)3

;2"02 2)71 /' 03'.. .)+189$ @6/'()/"0 0)3. )31 ?13+'3("*# )*

),)?/"71 036".1 0'*/3'8 B@CCD$ E2131+'31> ,1."#*"*# ?3'01,631

2). /;' ?)3/.> 03'.."*# )*, /3)0<"*#$

34 !(0+/(22%/ 1%$-50) C3'.."*# 0'*/3'8813 ". /21 .)(1 6.1,

"* ?317"'6. .10/"'* )*, ". )??8"1, +'3 :'/2 )6/'()/"0 0)3.$ 4*

'3,13 /' ,1."#* ) 0'*/3'8813 /' F5 ) .)+1 .?)01 :1/;11* /;'

)6/'()/"0 0)3.> ) 29:3", ./3)/1#9 ". )??8"1,$ G9:3", @6/'()/'*

'+ /2". 0'*/3'8813 ". .2';* "* !"#$ H$

0 5 10 15 20 25 300

2

4

6

8

Velo

city(m

/s)

0 5 10 15 20 250

50

100

time(s)

Dis

tance t

o

inte

rsection (

m)

Automatic Car

Manual Car

!"#$ A$ I"(68)/"'* 31.68/J 0).1 '*1

!"#$ H$ G9:3", @6/'()/'*J E3)0<"*#

E21 .)(1 ). ?317"'6. .10/"'* 29:3", )6/'()/'* ;"88 :1

+'3(68)/1, :9 BKD> ;2131

! # * "$!! $"! $##!! % * !&!!

! '!!! (& $ !$!

! )*+" * "$!# % "& $ !$ + ,

"!!

"!& &!!

"

& ,!"!!

"!- )&#!

! , * . -!!/01!(&! -!!

123!(&! -!!

"

#4!

! .!$!& * "% $ !$ + ,"!!

"!& &!!

"

& ,!"!!

"!- )&#!

! .!$"& * "% $ !$ + &!!

"

/ ,"!!

"!#!

! .!$#& * "% $ !$ + &!!

"

0 ,!"!!

"!- )&#!

! 1 * "!$!! $"&! !$"! $!&! !$!! $#&! !$#! $!&! !$"! $#&! !$#! $"&# !! 2!$!! $"& * 2!$#! $"& * "% $ !$ + &!!

"

/ ,"!!

"!#!

! 2!$!! $#& * 2!$"! $#& * "% $ !$ + &!!

"

0 ,!"!!

"!- )&#!

! 2!$"! $!& * 2!$#! $!& * "% $ !$ + ,

"!!

"!& &!!

"

&

,!"!!

"!- )&#!

! 3!$!! $"! %& * 3!$"! $!! %& * 3!$!! $#! %& *3!$#! $!! %& * 3!$"! $#! %& * "%#4

0 2 4 6 8 100

1

2

3

4

5

6

7

8

vA2

xA12 q

0

q2

q1

!"#$ %&$ '()*"+ (, -*./ 01*1- ,(2 .*0- 3 4

'()*"+ (, -*./ 01*1- "0 0/(5+ "+ !"#$ %&$ 6+ 1/"0 .*0- 7*0-8

(+ 1/- 12*,9. :*5; 1/- 8"01*+.- 7-15--+ 15( *<1()*1". .*2 "0

9=-8$ 6+ 1/"0 :*5 1/- 0*,- 8"01*+.- "0 8-9+-8> (+- .*2 :-+#1/ ,(2

-?-2@ !" A)B/ (, 0C--8$ D/"0 ?*:<- "0 .(+0"8-2-8 *0 )"+")<)8"01*+.- *+8 1/- )*=")<) 8"01*+.- "0 1/- )"+")<) 8"01*+.-

C:<0 :-+#1/ (, 1/- .*2$ E-+#1/ (, 1/- .*2 "0 .(+0"8-2-8 # )"+ 1/- 0")<::*1"(+ F:-+#1/ (, G"12(-+ GH 5/"./ "0 $! %&$ ))I$D/-2- "0 +( ./*+#- (, ?-:(."1@ ", 1/- 8"01*+.- "0 7-15--+ 1/-

)"+")<) *+8 )*=")<) ?*:<-; "$-$ 1/- 01*1- "! "0 *.1"?-$ 6, 1/-

8"01*+.- "0 :-00 1/*+ )"+")<) ?*:<- 1/- .(+12(::-2 5":: 2-8<.-

1/- 0C--8; "$-$ 1/- 01*1- "" "0 *.1"?-$ 6, 1/- 8"01*+.- 5*0 )(2-

1/*+ )*=")<) ?*:<- 1/- .(+12(::-2 5":: "+.2-*0- 1/- 0C--8 "$-$

1/- 01*1- "# "0 *.1"?-$ D/-2-,(2-; 1/- 2-,-2-+.- ?-:(."1@ 5":: 7-

*./"-?-8 *0>

#!"!'$( !) *

!"

#

#!"!'$)! ", % * "!

"&+#!"!'$)! ", % * ""

!&!#!"!'$)! ", % * "#

FJI

D/- 0*)- *0 C2-?"(<0 0-.1"(+ #"!.*+ -*0":@ (71*"+-8 <0"+#

FHI$

!" #$%&'()$*+ ,-.&')./ D/- 0")<:*1"(+ 2-0<:1 "0 0/(5+ "+

!"#$ %%$ 6+ 1/"0 0")<:*1"(+; "+1-20-.1"(+ "0 .(+0"8-2-8 *0 *

0-#)-+1 (, J )-1-20 :-+#1/$ K..-:-2*1"(+ "0 :")"1-8 7-15--+

!& *+8 (& 7*0-8 (+ 2-*: 8*1* ,2() GKL$ D/- .*2 )(8-:

"0 G"12(-+ GH$ !<21/-2)(2-; "1 "0 *00<)-8 1/*1 1/-2- *2-

15( "+1-20-.1"(+0; "$-$ *,1-2 .2(00"+# 1/- 9201 "+1-20-.1"(+ 1/-

.*20 0/(<:8 .2(00 *+(1/-2 "+1-20-.1"(+ *7(<1 !"" )-1-20 :*1-2$K<1()*1". .*2 M($ % .2(00 *1 $ * !!&$ 0 5/-+ 1/- )*+<*: .*2"0 *7(<1 %4 )-1-20 1( .2(00 1/- "+1-20-.1"(+ +<)7-2 % *+8

)*+<*: .*2 5":: .2(00; *1 $ * !&&, 0 5/-+ 1/- *<1()*1"..*2 M($ 4 "0 *7(<1 N )-1-20 1( 1/"0 "+1-20-.1"(+$ D( 1-01 1/-

C-2,(2)*+.- (, 1/"0 .(+12(::-2; 0C--8 (, )*+<*: .*2 "0 2-8<.-8

*1 $ * !, 0 *+8 "1 "0 "+.2-*0-8 *1 $ * &" 0 "+ .2(00"+#"+1-20-.1"(+ +<)7-2 4 *+8 *<1()*1". .*20 *8*C1-8 1/-)0-:?-0

1( 1/"0 ?*2"*1"(+ 7@ 2-8<."+# 1/-"2 0C--8$ K+8; K<1()*1". .*2

M($ % .2(00 *1 $ * &$ 0 5/-+ 1/- *<1()*1". .*2 M($ 4 "0 *7(<1%H )-1-20 1( .2(00 1/- "+1-20-.1"(+ +<)7-2 4 *+8 *<1()*1". .*2

M($ 4 5":: .2(00; *1 $ * &#&, 0 5/-+ 1/- )*+<*: .*2 "0 *7(<1 %H

0 5 10 15 20 25 300

50

100

Dis

tanc

e t

o

inte

rsec

tion

(m)

0 10 20 30 40 500

5

10

Vel

ocity

(m/s

)

0 10 20 30 40 50-4

-2

0

2

4

Acc

eler

atio

n (m

/s2 )

0 10 20 30 40 5010

15

20

25

time (s)

Dis

tanc

e be

twee

n tw

o

auto

mat

ic c

ar (

m)

Automatic Car # 1

Automatic Car # 2

Manual Car

Intersection segment

!"#$ %%$ O")<:*1"(+ 2-0<:1> .*0- 15(

)-1-20 1( 1/"0 "+1-20-.1"(+$ K0 "1 .*+ 7- 0--+ 7(1/ *<1()*1". .*2

.2(00-8 "+1-20-.1"(+0 0*,-:@$ 6+ 1/- 0")<:*1"(+ 1/- *..-:-2*1"(+

:")"10 *2- *:0( ?-2"9-8$ P(2-(?-2; 1/- 8"01*+.- 7-15--+ 15(

*<1()*1". .*20 "0 9=-8 7-15--+ "10 )*=")<) *+8 )"+")<)

?*:<- "$-$ !& ' !$&,- ' !.! *,1-2 .2(00"+#$ D/-2-,(2-; 1/- .*20.2(00 0*,-:@ *+8 0-.(+8 *<1()*1". .*2 "0 A-C1 "10 8"01*+.- "+

1/- *::(5-8 ?*:<-0$

6Q$ GRMGESO6RM

K+ "+1-::"#-+1 .2(002(*8T12*?-20"+# 0@01-) *")-8 *1 ")C2(?T

"+# 12*,9. U(5 "0 C2-0-+1-8 "+ 1/"0 C*C-2$ D/- .(+12(::-2 "0

8-0"#+-8 7@ /@72"8 *<1()*1(+$D/- C<2C(0- (, 1/"0 C*C-2 /*0

7--+ 15(,(:8> %I O*,-:@ .2(00"+# ,2() "+1-20-.1"(+ *+8 4I

*8*C1"?- .2<"0- .(+12(:$ D/<0; (<2 0@01-) "0 .*C*7:- (, +(1 (+:@

0*,- .2(00"+# 1/- "+1-20-.1"(+ 7<1 *:0( A--C 1/- 0*,- 8"01*+.-

7-15--+ .*20$ O")<:*1"(+ 2-0<:1 0/(50 1/- -,9."-+.@ (, 1/-

.(+12(::-2$ L-*: -=C-2")-+10 *2- "+ C2(#2-00$


Coordinación de vehículos (II)




• Distribuido: cada vehículo ejecuta de manera independiente el controlador y sólo conoce la información de los vehículos que están en su radio de comunicación.

• Híbrido: la estrategia de navegación se implementa mediante una máquina de estados y en cada estado existe un controlador en bucle cerrado para dirigir el vehículo. Estados: Libre (azul), Detenido (morado), Buscando (naranja), Acoplado (marrón)

Estados

• Libre. La orientación hacia el punto destino está dentro del rango de orientaciones en que puede moverse el vehículo: v=vn, w=PID.

• Detenido. El vehículo no tiene lugar hacia el que moverse y se detiene por completo: v=0,w=0.• Buscando. El vehículo busca la orientación de salida, que es el resultado de las intersecciones que se

estén produciendo con los discos reservados de otros vehículos: v=0, w=PID.• Acoplado. El vehículo acoplado gira alrededor del disco reservado de uno de los vehículos adyacentes.

El giro siempre se realiza dejando a la izquierda el vehículo adyacente: v=vn, w=PID.


Coordinación de vehículos (III)




Estados

• Libre. La orientación hacia el punto destino está dentro del rango de orientaciones en que puede moverse el vehículo: v=vn, w=PID.

• Detenido. El vehículo no tiene lugar hacia el que moverse y se detiene por completo: v=0,w=0.• Buscando. El vehículo busca la orientación de salida, que es el resultado de las intersecciones que se

estén produciendo con los discos reservados de otros vehículos: v=0, w=PID.• Acoplado. El vehículo acoplado gira alrededor del disco reservado de uno de los vehículos adyacentes.

El giro siempre se realiza dejando a la izquierda el vehículo adyacente: v=vn, w=PID.

v: velocidad de desplazamiento deseada; vn: velocidad de desplazamiento nominalw: velocidad de giro deseada

PID: regulador PID empleando como señal de error la diferencia entre la orientación que tiene el vehículo y la orientación deseada. En el estado Libre la orientación deseada es la orientación hacia el punto destino. En el estado Buscando, la orientación deseada es el límite derecho de las intersecciones de los discos reservados. En el estado Acoplado la orientación deseada es la tangente al disco del vehículo sobre el que se gira.


Coordinación de vehículos (IV)





!"#$ %&'()*#( +", -.-"' /)*&0 1- /()2-" ') 0-3"- '(- #+!' /,/&-4

!' !2 #-"-5+&&, /)".-"!-"' ') *2- '(- !"2'+"' +' 6(!/( )"- 7))' /)"8

'+/'2 '(- #5)*"0 9:!"!'!+& /)"'+/':;$ <7 !' !2 0-/!0-0 ') 2'+5' 6!'(

!"!'!+& /)"'+/' )7 '(- 5!#(' 7))'4 '(-" '(- /,/&- 6!&& /)"'!"*- *"'!&

'(- 5!#(' 7))' /)"'+/'2 '(- #5)*"0 +#+!"$ =(- &-7' 7))'4 )7 /)*52-4

#)-2 '(5)*#( ->+/'&, '(- 2+?- 2-5!-2 )7 -.-"'2 +2 '(- 5!#('4 1*'

0!2@&+/-0 !" '!?- 1, (+&7 + /,/&-$ =(-5- 2-.-" ?+A)5 -.-"'2 9<"!8

'!+& /)"'+/'4 B@@)2!'- ')- )774 C--& 5!2-4 B@@)2!'- !"!'!+& /)"'+/'4

=)- )774 D--' +0A+/-"'4 =!1!+ .-5'!/+&; '(+' 2*10!.!0- '(- #+!' /,/&-

!"') 2-.-" @-5!)024 7)*5 )7 6(!/( )//*5 !" '(- 2'+"/- @(+2-4 6(-"

'(- 7))' !2 )" '(- #5)*"04 +"0 '(5-- !" '(- 26!"# @(+2-4 6(-" '(-

7))' !2 ?).!"# 7)56+50 '(5)*#( '(- +!5$ =(- 2'+"/- @(+2- !2 +&2)

2*10!.!0-0 !"')E

F$ G)+0!"# 5-2@)"2-4

H$ I!082'+"/-4

J$ =-5?!"+& 2'+"/-4

K$ L5-826!"#$

=(- 26!"# @(+2- &+2'2 75)? ')- )77 ') '(- "->' !"!'!+& /)"'+/'$

<' !2 2*10!.!0-0 !"')E

F$ <"!'!+& 26!"#4

H$ I!0826!"#4

J$ =-5?!"+& 26!"#$

=(- 0*5+'!)" )7 + /)?@&-'- #+!' /,/&- !2 M")6" +2 '(- /,8

/&- '!?-4 6(!/( !2 0!.!0-0 !"') 2'+"/- '!?- +"0 26!"# '!?-$ <"

'(- 2'+"/- @(+2- '(- /)"'5)&&-5 6!&& &)/M '(- ?)')5 ")' ') ?).-

+"0 &-' '(- @+'!-"' /)"'5)& (!2 M"--2$ %"0 !" '(- 26!"# @(+2-

'(- /)"'5)&&-5 6!&& +0)@' (!?2-&7 ') 7)&&)6 '(- 5-7-5-"/- 6(!/( !2

+/(!-.-0 *2!"# @&+"+5 2-"2)52$ D!#$ F 2()62 6(-" '(- /)"'5)&&-5

2()*&0 &)/M '(- ?)')5 92'+"/-; +"0 6(-" '(- /)"'5)&&-5 (+2 ') 7)&8

&)6 '(- 5-7-5-"/- 926!"#;$ <" '(- 0)*1&- 2'+"/-4 1)'( M"--2 +5-

&)/M-0$

!"#$%& '$ =!?!"# )7 2!"#&- +"0 0)*1&- 2*@@)5'

G!M-6!2-4 D!#$ H 2()62 + 2!?@&!3-0 0!+#5+? )7 (*?+" 6+&M8

!"# #+!'4 6!'( '(- '-5?2 '(+' 6!&& 1- *2-0$

!"#$%& ($ N!)?-/(+"!/2 )7 6+&M!"#

D!#$ J 2()62 '(- 2+#!''+& @&+"- +"#&-2 +' '(- (!@ +"0 M"--

A)!"' 7)5 '(- 5!#(' &-# 2()6!"# A)!"' +"#&- 7)5 (!@ +"0 M"-- O->8

!)"P->'-"2!)" ?)'!)"2 0*5!"# &-.-&8#5)*"0 6+&M!"#$ =(-2- 0+'+

+5- +/(!-.-0 7)5 '(- 5!#(' &-# !" + #+'- /,/&- QRS$

!"#$%& )$ T+#!''+& @&+"- A)!"' +"#&-2 90-#5--2; 0*5!"# + 2!"#&- #+!'

/,/&- )7 5!#(' (!@ 9+->!)" @)2!'!.-; +"0 M"-- 9+->!)" @)2!'!.-;$ <U V

!"!'!+& /)"'+/'W B= V )@@)2!'- ')- )77W CX V (--& 5!2-W B< V )@@)2!'-

!"!'!+& /)"'+/'W =B V ')- )77W D% V 7--' +0A+/-"'W =Y V '!1!+ .-5'!/+&$

*+,%"- -+./0"12 3! /22"24&- #/"4

=) /)"'5)& +" )5'()2!2 ') !"'5)0*/- +" +@@&!/+1&- 5-7-5-"/-

!2 "-/-22+5,$ =(-5-7)5-4 7)*5 @&+"+5 2-"2)52 6!&& 1- *2-0 !" '(-

)5'()2!2 ') 3"0 @-5!)02 +"0 '(-" + &!"-+5 5-7-5-"/- 6!&& 1- !"'5)8

0*/-0 7)5 -+/( @-5!)0$ L&+"+5 2-"2)52 +5- BZPBDD 2-"2)52 6(-5-

F ?-+"2 '(+' '(- &-# !2 !" ')*/( 6!'( '(- #5)*"0 +"0 [ 5-7-52 ')

'(- /)"'5+5, 2!'*+'!)"$ \2!"# '(- @&+"+5 2-"2)52 +"0 1+2-0 )" '(-

(*?+" 6+&M!"# 1!)?-/(+"!/2 92-- D!#$ H; 6- 6!&& 2-- -!#(' 0!78

7-5-"' @+5'2 !" -+/( #+'- /,/&- )7 6+&M!"#$ =(- +/'!.+'!)" )7 '(-

H U)@,5!#(' /! H[FF 1, %TI]

Órtesis activa (I)




!"#$%& '! "#$#%#&'# (&## )&*+# $,% -.# +#$- +#*

/0- 10# -, -.# $)'- -.)- 2)-3#&-4 .)5# ',&-%,+ ,$ -.# .32 604'+#47

8.3'. )+4, -)(# 2)%- 3& 9#:3,& ,$ -.# (&##7 -.# &#'#44)%; -,%<0#

34 63&,%7 )/,0- =!>? @6A(* B)22%,:36)-#+; >C!C @6 $,% ) 2#%4,&

,$ ?= (*D! E.#%#/;7 -.# )'-0)-,% 604- 2%,531# -.34 -,%<0# 5)+0#!

F'',%13&* -, -.# 6#13')+ 42#'3G')-3,&4 $,% 3&',62+#-#

423&)+ ',%1 3&H0%#1 2)-3#&-47 -.# 2%,2,4#1 1#53'# 604- 2%,531#

)4434-)&'# 3& -.# 483&* 2.)4# )&1 )+4, 604- /# +,'(#1 3& 4-)&'#

2.)4# 3& 4)$# ',&13-3,&4! E.34 IJKFLM )+4, .)4 -, )5,31 2+)&-)%

9#:3,& 3& 483&* 2.)4# 10# -, -.# *)3- )44,'3)-#1 83-. -.34 (3&1

,$ 3&H0%3#4! N#&'#7 -.# 1#43*& ,$ -.# ,%-.,434 34 13531#1 3&-, -8,

4-)*#4O ,& -.# ,&# .)&17 -.# 1#43*& ,$ -.# )&(+# 6,10+# )&17

,& -.# ,-.#%7 -.# ',%%#42,&13&* -, -.# (&## 6,10+#! @#:-7 /,-.

1#43*& 83++ /# 1#4'%3/#1!

()*+, -./0+,

F&(+# 6,10+# B4## L3*! PD 34 /)4#1 ,& ) ',66#%'3)+ 2)4Q

435# ,%-.,434 FLM7 8.3'. 8)4 6,13G#1 -, )1)2- )& #&',1#% ,&

-.# )%-3'0+)-3,& -, 6#)40%# -.# )&*0+)% 5)%3)-3,&! E.34 #&',1#%

34 )& F5)*, FRSF TT== EFE 6,1#+7 83-. ) %#4,+0-3,& ,$ C=UP

JV"7 8.,4# %,-)-3,& ):34 34 H,3&#1 -, -.# ):34 ,$ -.# )%-3'0+)-3,&7

)&1 -.# 6,0&- 34 G:#1 -, -.# 4-3%%027 4, -.# %,-)-3,& %#',%1#1 34

$,,- ,5#% 4.)&(! E.34 2)%- ,$ -.# ,%-.,434 34 ',&4-3-0-#1 /; -8,

)+063&306 /)%47 ,&# ,& #)'. 431# ,$ -.# +#*7 6)(3&* -.# $0&'-3,&

,$ 4022,%-7 8.3'. )%# )1H04-#1 -, -.# +#* /; 5#+'%, -)2#4 F K+#&Q

W)'( 1#53'# ,& #)'. 431# ,$ -.# )&(+# )5,31 1,%439#:3,& 3& 483&*

2.)4#! F 6#-)++3' 4-3%%02 83-. )& 3&4,+# ',&4-3-0-#4 $,,- 4022,%-

8.3'. 34 3&4#%-#1 ,& -.# 4.,#4! E.34 ,%-.,434 ')& %#*0+)-# -.# )&-3

1,%439#:,% -,%<0# /; -.# )1H04-6#&- ,$ -.# 42%3&* 8.3'. .)4 )

4'%#8 -.)- ',&-%,+4 3-4 +,&*3-013&)+ 136#&43,& E.34 6,10+# )+4,

.)4 ) /+,'( 6#'.)&346 -, %#4-%3&'- 9#:3,& 6,5#6#&- -, ) 6):Q

3606 ,$ X= 1#*! V+)&)% 4#&4,%4 )%# 3&4-)++#1 ,5#% -.# 3&4,+# 3&

,%1#% -, ',&-%,+ -.# 6,-,% 0&3- 3& -.# (&## 6,10+#! E.,4# 2+)&-)%

4#&4,%4 )%# Y,-3,&Z)/ 483-'.#4 YF>[T! L0%-.#% $0&'-3,&)+3-3#4

)%# 1#4'%3/#1 3& -.# $,++,83&* 4#'-3,&4!

!"#$%& 1! F&(+# 6,10+# 83-. ) 1#-)3+ ,$ -.# #&',1#%

2),, 3./0+,

E.# 6#&-3,&#1 )+063&06 /)%4 ',&-3&0# -, -.# (&## )%-3'0Q

+)-3,&7 8.#%# -.#; H,3& 83-. -.# (&## )%-3'0+)-3,&! F&,-.#% 2)3%

,$ /)%4 ',5#%4 -.# -.3*.7 *353&* ) &#8 4022,%-3&* 2,3&-! F 5#+'%,

-)2# G:#4 -.# ,%-.,434 -, -.# -.3*.! E.# (&## 6,10+# 34 ',62,4#1

/; ) ',66#%'3)+ )%-3'0+)-3,& ,& -.# #:-#%&)+ 431# )&1 ) ',66#%Q

'3)+ /+,'(3&* 4;4-#6 3& -.# 3&-#%&)+ 431#7 )4 ')& /# ,/4#%5#1 3&

L3*! \! E.# )%-3'0+)-3,& 34 6,13G#1 -, )1H04- -.# )'-0)-3,& 4;4Q

-#6 8.3'. ',&434-4 ,$ ) SJ 6,-,% 83-. ) *#)%.#)1 -, ,/-)3& -.#

2%,2#% -,%<0# 3& #)'. 6,6#&- ,$ -.# *)3- ';'+#! E.# +,'(3&* 4;4Q

-#6 2%,531#4 -.# &#'#44)%; +,'( ,& -.# (&## 10%3&* -.# 4-)&'#

2.)4#! ]& -.34 ,%-.,4347 8# 04# -.# @R^"M E"M@]J _7 8.3'.

2%,531#4 -.# )1#<0)-# +,'( ,& 4-)&'# 2.)4# 3& 4)$# ',&13-3,&4! M&

-.# #:-#%&)+ 431#7 ) ',66#%'3)+ )%-3'0+)-3,& 34 04#17 8.3'. 8)4

)+4, 6,13G#1 -, 3&'+01# ) SJ 6,-,%O -.# RJ C[ 9)- $%,6 Y):,&

Y,-,%! E.34 6,-,% 34 )44,'3)-#1 83-. ) 2+)&#-#%; *#)%.#)1 )+4,

$%,6 Y):,& Y,-,%7 -.# `V CX J7 )&1 -.# ',%%#42,&13&* #&',1#%

-, %#',%1 -.# )&*0+)% 5)%3)-3,& /#-8##& -.# -.3*. )&1 -.# 4.)&(!

E.34 *%,02 .)4 ) 46)++ 8#3*.7 )/,0- [?= *7 )&1 2%,531#4 ) -,%<0#

,$ >=!?? @6 3& ',&-3&0,04 )- T![C F 83-. ) 5#+,'3-; ,$ TT!P[ %26

)- >? a! ]& '#%-)3& '3%'064-)&'#47 -.# -,%<0# ')& /# 3&'%#)4#1 -,

%#)'. -.# &#'#44)%; >C!C @6 6#&-3,&#1 3& -.# 42#'3G')-3,&4!

4567%588&% 9&:"#6

F2)%- $%,6 -.# 6#'.)&3')+ 1#43*&7 ',&-%,+ ,$ -.# ,%-.,434 34

3& 3&-#%#4- ,$ -.34 2)2#%! E8, $%)'-3,&)+ ,%1#% ',&-%,++#% B]! BVSD)&1 V]" D )%# ',&431#%#1 -, ',&-%,+ -.# (&## )&*+#! b,-. ',&Q-%,++#% )%# ',62)%#1 $,% -.# 4)6# 42#'3G')-3,&! E.# ',62+#-#

',&-%,++#1 4;4-#6 /+,'( 34 4.,8& 3& L3*! ?! F4 ')& /# ,/4#%5#17

-.# ,0-20- ,$ -.# 2+)&)% 4#&4,%4 -,*#-.#% 83-. +#$- )&1 %3*.- (&##

C J,2;%3*.- '! X=>> /; FIYR

!"#$ %&'()*#( +", -.-"' /)*&0 1- /()2-" ') 0-3"- '(- #+!' /,/&-4

!' !2 #-"-5+&&, /)".-"!-"' ') *2- '(- !"2'+"' +' 6(!/( )"- 7))' /)"8

'+/'2 '(- #5)*"0 9:!"!'!+& /)"'+/':;$ <7 !' !2 0-/!0-0 ') 2'+5' 6!'(

!"!'!+& /)"'+/' )7 '(- 5!#(' 7))'4 '(-" '(- /,/&- 6!&& /)"'!"*- *"'!&

'(- 5!#(' 7))' /)"'+/'2 '(- #5)*"0 +#+!"$ =(- &-7' 7))'4 )7 /)*52-4

#)-2 '(5)*#( ->+/'&, '(- 2+?- 2-5!-2 )7 -.-"'2 +2 '(- 5!#('4 1*'

0!2@&+/-0 !" '!?- 1, (+&7 + /,/&-$ =(-5- 2-.-" ?+A)5 -.-"'2 9<"!8

'!+& /)"'+/'4 B@@)2!'- ')- )774 C--& 5!2-4 B@@)2!'- !"!'!+& /)"'+/'4

=)- )774 D--' +0A+/-"'4 =!1!+ .-5'!/+&; '(+' 2*10!.!0- '(- #+!' /,/&-

!"') 2-.-" @-5!)024 7)*5 )7 6(!/( )//*5 !" '(- 2'+"/- @(+2-4 6(-"

'(- 7))' !2 )" '(- #5)*"04 +"0 '(5-- !" '(- 26!"# @(+2-4 6(-" '(-

7))' !2 ?).!"# 7)56+50 '(5)*#( '(- +!5$ =(- 2'+"/- @(+2- !2 +&2)

2*10!.!0-0 !"')E

F$ G)+0!"# 5-2@)"2-4

H$ I!082'+"/-4

J$ =-5?!"+& 2'+"/-4

K$ L5-826!"#$

=(- 26!"# @(+2- &+2'2 75)? ')- )77 ') '(- "->' !"!'!+& /)"'+/'$

<' !2 2*10!.!0-0 !"')E

F$ <"!'!+& 26!"#4

H$ I!0826!"#4

J$ =-5?!"+& 26!"#$

=(- 0*5+'!)" )7 + /)?@&-'- #+!' /,/&- !2 M")6" +2 '(- /,8

/&- '!?-4 6(!/( !2 0!.!0-0 !"') 2'+"/- '!?- +"0 26!"# '!?-$ <"

'(- 2'+"/- @(+2- '(- /)"'5)&&-5 6!&& &)/M '(- ?)')5 ")' ') ?).-

+"0 &-' '(- @+'!-"' /)"'5)& (!2 M"--2$ %"0 !" '(- 26!"# @(+2-

'(- /)"'5)&&-5 6!&& +0)@' (!?2-&7 ') 7)&&)6 '(- 5-7-5-"/- 6(!/( !2

+/(!-.-0 *2!"# @&+"+5 2-"2)52$ D!#$ F 2()62 6(-" '(- /)"'5)&&-5

2()*&0 &)/M '(- ?)')5 92'+"/-; +"0 6(-" '(- /)"'5)&&-5 (+2 ') 7)&8

&)6 '(- 5-7-5-"/- 926!"#;$ <" '(- 0)*1&- 2'+"/-4 1)'( M"--2 +5-

&)/M-0$

!"#$%& '$ =!?!"# )7 2!"#&- +"0 0)*1&- 2*@@)5'

G!M-6!2-4 D!#$ H 2()62 + 2!?@&!3-0 0!+#5+? )7 (*?+" 6+&M8

!"# #+!'4 6!'( '(- '-5?2 '(+' 6!&& 1- *2-0$

!"#$%& ($ N!)?-/(+"!/2 )7 6+&M!"#

D!#$ J 2()62 '(- 2+#!''+& @&+"- +"#&-2 +' '(- (!@ +"0 M"--

A)!"' 7)5 '(- 5!#(' &-# 2()6!"# A)!"' +"#&- 7)5 (!@ +"0 M"-- O->8

!)"P->'-"2!)" ?)'!)"2 0*5!"# &-.-&8#5)*"0 6+&M!"#$ =(-2- 0+'+

+5- +/(!-.-0 7)5 '(- 5!#(' &-# !" + #+'- /,/&- QRS$

!"#$%& )$ T+#!''+& @&+"- A)!"' +"#&-2 90-#5--2; 0*5!"# + 2!"#&- #+!'

/,/&- )7 5!#(' (!@ 9+->!)" @)2!'!.-; +"0 M"-- 9+->!)" @)2!'!.-;$ <U V

!"!'!+& /)"'+/'W B= V )@@)2!'- ')- )77W CX V (--& 5!2-W B< V )@@)2!'-

!"!'!+& /)"'+/'W =B V ')- )77W D% V 7--' +0A+/-"'W =Y V '!1!+ .-5'!/+&$

*+,%"- -+./0"12 3! /22"24&- #/"4

=) /)"'5)& +" )5'()2!2 ') !"'5)0*/- +" +@@&!/+1&- 5-7-5-"/-

!2 "-/-22+5,$ =(-5-7)5-4 7)*5 @&+"+5 2-"2)52 6!&& 1- *2-0 !" '(-

)5'()2!2 ') 3"0 @-5!)02 +"0 '(-" + &!"-+5 5-7-5-"/- 6!&& 1- !"'5)8

0*/-0 7)5 -+/( @-5!)0$ L&+"+5 2-"2)52 +5- BZPBDD 2-"2)52 6(-5-

F ?-+"2 '(+' '(- &-# !2 !" ')*/( 6!'( '(- #5)*"0 +"0 [ 5-7-52 ')

'(- /)"'5+5, 2!'*+'!)"$ \2!"# '(- @&+"+5 2-"2)52 +"0 1+2-0 )" '(-

(*?+" 6+&M!"# 1!)?-/(+"!/2 92-- D!#$ H; 6- 6!&& 2-- -!#(' 0!78

7-5-"' @+5'2 !" -+/( #+'- /,/&- )7 6+&M!"#$ =(- +/'!.+'!)" )7 '(-

H U)@,5!#(' /! H[FF 1, %TI]


Órtesis activa (II)




!"#$%& '! "#$%& #' ()% *+,-./ 01#0#2%$! /3 ()% &%'( 24$% 42 1%05

1%2%3(%$ 6 $%(64& #' ()% 67(86(4#3 43 ()% 93%%! /3 ()% 14:)( 24$% 42 2)#;3

6 $%(64& #' ()% &#7943: 2<2(%=!

63:&%2 > ! !" 63$ ! !#? 61% 7#324$%1%$ 62 6 '%%$@679 (# ()% 7#35(1#&&%1! A)% 64= 42 (# 43(1#$87% 6 01#0%1 4308( 24:36& 43 #1$%1 (#

(1679 ()% 1%'%1%37% 43 %67) 0%14#$! A)% $%7424#3 @62%$ 7#3(1#&&%1

;4&& 1%7#:34B% ;)47) 061( #' ()% =#C%=%3( 42 67(4C% 63$ 2%3$ ()%

01#0%1 1%'%1%37% (# ()% 7#3(1#&&%1! D%24$%2E 43(1#$8743: ()% 1%'5

%1%37% C6&8% '#1 ()% $%7424#3 @62%$ 7#3(1#&&%1 ;4&& $%74$% 4' ()%

7#3(1#&&%1 2)#8&$ @% 67(4C% #1 3#(! -2 4( 763 @% 2%%3 '1#= .4:! FE

$%7424#3 @62%$ 7#3(1#&&%1 ;4&& 2%3$ 6 7#==63$ (# &#79 ()% =#(#1

;)%3 4( 42 43 2(637% 0)62% 63$ 6&2# 67(4C6(% ()% 7#3(1#&&%1 63$ =#5

(#1 (# '#&&#; ()% 76&78&6(%$ 1%'%1%37% C6&8% ;)%3 ()% =#C%=%3(

42 43 2;43: 0)62%! G3 3#1=6& 7#3$4(4#32E 43 2(637% 0)62% ()% 065

(4%3( 43H%7(2 =#1% %3%1:< 43 ()% 93%%2 '#1 1%242(43: ;%4:)5@%6143:

2#E 43 ()6( ;6<E ()% =#(#1 3%%$2 =#1% %3%1:< (# 7#3(1#& ()% #15

()#242! A)%1%'#1%E 43 ()42 762% 43 #1$%1 (# #C%17#=% ()42 01#@&%=E

()% =#(#1 ;4&& @% @&#79%$ 2# 62 (# 9%%0 ()% 93%% 63:&% 7#32(63(!

I%37%E ()% #1()#242 $<36=472 43 2(637% 0)62% 763 @% 1%01%2%3(%$

62J

K! ! ! $"! !!! %#!! % !&'()*" >L?

-3$ 43 ()% 2;43: 0)62% ()% $<36=472 #' ()% 2<2(%= 763 @% '#15

=8&6(%$ 62J

K! ! ! $"! !!! %#!K! % ! + "! !!! %#$,-

!! %!! %#-

">M?

;)%1% ! %#- 42 ()% 1%'%1%37% C6&8% 43 %67) 0%14#$ 63$ - 1%01%2%3(2%67) 0%14#$! $""# 63$ + ""# 61% '837(4#32 ;)47) $%0%3$ #3 ()% #15()#242! G3 #1$%1 (# $%24:3 6 7#3(1#&&%1E 6 N+ =#(#1 =#$%& 42 82%$

(#:%()%1 ;4() 6 &#6$ ;)47) 42 ()% =%7)63476& 061( #' #1()#242 >2%%

.4:! OP?! -2 =%3(4#3%$ @%'#1%E 43 ()42 =#$%& 4( 42 7#324$%1%$ ()6(

()% )40 763 @% =#C%$ @< ()% 06(4%3( 63$ ()% =#(#1 ;4&& 7#3(1#&

()% =#C%=%3( #' ()% 93%%!

!"#$%& (! D&#79 $46:16= #' ()% 7#3(1#&&%$ 2<2(%=

!"#$%& )! N%7424#3 D62%$ +#3(1#&&%1

*(6(% 2067% $<36=472 #' ()% =#(#1 ;4() #1()#242 62 6 &#6$

Q +#0<14:)( 7" LPOO @< -*"R

!"#$%& '! "#$%& #' ()% *+,-./ 01#0#2%$! /3 ()% &%'( 24$% 42 1%05

1%2%3(%$ 6 $%(64& #' ()% 67(86(4#3 43 ()% 93%%! /3 ()% 14:)( 24$% 42 2)#;3

6 $%(64& #' ()% &#7943: 2<2(%=!

63:&%2 > ! !" 63$ ! !#? 61% 7#324$%1%$ 62 6 '%%$@679 (# ()% 7#35(1#&&%1! A)% 64= 42 (# 43(1#$87% 6 01#0%1 4308( 24:36& 43 #1$%1 (#

(1679 ()% 1%'%1%37% 43 %67) 0%14#$! A)% $%7424#3 @62%$ 7#3(1#&&%1

;4&& 1%7#:34B% ;)47) 061( #' ()% =#C%=%3( 42 67(4C% 63$ 2%3$ ()%

01#0%1 1%'%1%37% (# ()% 7#3(1#&&%1! D%24$%2E 43(1#$8743: ()% 1%'5

%1%37% C6&8% '#1 ()% $%7424#3 @62%$ 7#3(1#&&%1 ;4&& $%74$% 4' ()%

7#3(1#&&%1 2)#8&$ @% 67(4C% #1 3#(! -2 4( 763 @% 2%%3 '1#= .4:! FE

$%7424#3 @62%$ 7#3(1#&&%1 ;4&& 2%3$ 6 7#==63$ (# &#79 ()% =#(#1

;)%3 4( 42 43 2(637% 0)62% 63$ 6&2# 67(4C6(% ()% 7#3(1#&&%1 63$ =#5

(#1 (# '#&&#; ()% 76&78&6(%$ 1%'%1%37% C6&8% ;)%3 ()% =#C%=%3(

42 43 2;43: 0)62%! G3 3#1=6& 7#3$4(4#32E 43 2(637% 0)62% ()% 065

(4%3( 43H%7(2 =#1% %3%1:< 43 ()% 93%%2 '#1 1%242(43: ;%4:)5@%6143:

2#E 43 ()6( ;6<E ()% =#(#1 3%%$2 =#1% %3%1:< (# 7#3(1#& ()% #15

()#242! A)%1%'#1%E 43 ()42 762% 43 #1$%1 (# #C%17#=% ()42 01#@&%=E

()% =#(#1 ;4&& @% @&#79%$ 2# 62 (# 9%%0 ()% 93%% 63:&% 7#32(63(!

I%37%E ()% #1()#242 $<36=472 43 2(637% 0)62% 763 @% 1%01%2%3(%$

62J

K! ! ! $"! !!! %#!! % !&'()*" >L?

-3$ 43 ()% 2;43: 0)62% ()% $<36=472 #' ()% 2<2(%= 763 @% '#15

=8&6(%$ 62J

K! ! ! $"! !!! %#!K! % ! + "! !!! %#$,-

!! %!! %#-

">M?

;)%1% ! %#- 42 ()% 1%'%1%37% C6&8% 43 %67) 0%14#$ 63$ - 1%01%2%3(2%67) 0%14#$! $""# 63$ + ""# 61% '837(4#32 ;)47) $%0%3$ #3 ()% #15()#242! G3 #1$%1 (# $%24:3 6 7#3(1#&&%1E 6 N+ =#(#1 =#$%& 42 82%$

(#:%()%1 ;4() 6 &#6$ ;)47) 42 ()% =%7)63476& 061( #' #1()#242 >2%%

.4:! OP?! -2 =%3(4#3%$ @%'#1%E 43 ()42 =#$%& 4( 42 7#324$%1%$ ()6(

()% )40 763 @% =#C%$ @< ()% 06(4%3( 63$ ()% =#(#1 ;4&& 7#3(1#&

()% =#C%=%3( #' ()% 93%%!

!"#$%& (! D&#79 $46:16= #' ()% 7#3(1#&&%$ 2<2(%=

!"#$%& )! N%7424#3 D62%$ +#3(1#&&%1

*(6(% 2067% $<36=472 #' ()% =#(#1 ;4() #1()#242 62 6 &#6$

Q +#0<14:)( 7" LPOO @< -*"R

!"#$%& '(! "#$%&'($&) *+,%-.(. /-0#)

$&' 1# +#2+#.#',#0 &. 3-))-4.5

6!! "!"#$%&

"!

!

"""""#

!'()(

!*+)(

7 7 7*,-(

!#+("+$-(

!*-(

+-(

*-(

7 8 7 7 7

7 +-.

*-.

!#+"+.$-.

!*-.

7 7 7 8 7

$

%%%%%&!

#!'8")( 7 7 7 7

(/! 0! ! . ! ! 1"! 2#

4%#+#9 ! !'%! 6!(! !(! 6! .! ! .

(&+# ,%# .,&,#. 4%($% 6!( &'0 ! .

&+# ,%# /-,-+ &'0 -+,%-.(. &':)# &'0 % (. ,%# &+/&,;+# $;++#',!

)( &'0 '( +#2+#.#', ,%# #)#$,+($ ('0;$,&'$# &'0 +#.(.,&'$# &'0

*, &'0 *+ &+# ,-+<;# &'0 1&$= #/3 $-'.,&',.! -( &'0 -. &+# ,%#

/-/#',. -3 ('#+,(& -3 ,%# +-,-+ &'0 ,%# -+,%-.(.! +( &'0 +. &+# ,%#

0&/2(': +&,(-. -3 ,%# /#$%&'($&) .>.,#/ 3-+ /-,-+ &'0 -+,%-.(.

)('=9 +#.2#$,(?#)>! * &'0 + &+# ,%# .2+(': $-'.,&', &'0 0&/2(':

3-+ $-''#$,(-' -3 +-,-+@-+,%-.(. )('=9 +#.2#$,(?#)>!

A%# 2&+&/#,#+. -3 ,%# +#&) /-,-+ &'0 ,%# -+,%-.(. &+# :(?#'

(' A&1)# B!

C.(': A&1)# B9 ,%# /-0#) $&' 1# #&.()> +#0;$#0 &.5

3#4$ !! .$%&

!B#DE

4#7#784"8$# FGH

)*+,& -! "-,-+ 2&+&/#,#+.

"-,-+ *+,%-.(.

)( 7!BII /J -. 8!BD#KG L:/I

'( 7!G8B ! +. 7!78DG M/.@+&0

-( 8!BD#KD L:/I * 877 M@/

+( 8!DG#KB M/.@+&0 + 7!7778 M/.@+&0

*+ IN!O#KD P.@+&0

*, 7!7ID8 M/@Q

"" ./01 2345637786 936 :;8<7 3=84 733=

R' -+0#+ ,- $-',+-) ,%# -+,%-.(.9 & 3+&$,(-'&) -+0#+ $-',+-))#+

(!#! R" FSTH 4()) 1# 0#.(:'#0! A%(. $-',+-))#+ 4()) 1# 0#.(:'#01&.#0 -' & $)-.#0 )--2 +#3#+#'$# /-0#) #U2+#..(': ,%# 0>'&/($&)

&'0 +-1;., 2#+3-+/&'$#.!

Q' (0#&) -2#' )--2 ,+&'.3#+ 3;'$,(-' (' ,%# 3-+/ -3 & 3+&$K

,(-'&) (',#:+&,-+ (. $-'.(0#+#0 %#+#9 4(,% ,%# 3-+/5

356 7 #4$ !#56 74""8

!8

$56 7 4""8! 7$ " $ 8# FDH

A%#+#3-+#9 ,%# +#)&,(-' 1#,4##' ,%# +#3#+#'$# /-0#) -3 ,%#

.>.,#/ &'0 $-',+-))#0 -2#' )--2 .>.,#/ $&' 1#9

356 7 #4$ !8 #4$3#4$

C.(': .>.,#/ 0>'&/($. 3#4$ &'0 (0#&) +#3#+#'$# /-0#)356 7 #4$9 ,%#' ,%# $-',+-))#+8#4$ $&' 1# #&.()> &$%(#?#0 &.9

8 #4$ !#$4"8$9$56 7 4"

! FVH

4%($% $&' 1# +#4+(,,#' &.5

8 #4$ !8

4"#*:"*;4$ FNH

4%#+#*; !$

9$56 7 &'0*:!8

9$56 7 &+# $)&..($&) ST $-#3W$(#', &'0

8 #4$ +#2+#.#', & ST $-',+-))#+ $-/2-.#0 4(,% & 3+&$,(-'&) -+K0#+ (',#:+&,-+ FR" FSTHH! A%(. $-',+-))#+ ('$+#&.#. ,%# ,>2# -3 ,%#.>.,#/ 4(,% ,%# 3+&$,(-'&) (',#:+&,-+ 4%($% #)(/('&,#. ,%# .,#&0>

.,&,# #++-+ 3-+ ,%# +&/2 +#3#+#'$# F.## X(:.! G &'0 DH!

V Y-2>+(:%, $" I788 1> QZ"[

10-1

100

101

102

103

-50

0

50

100

Mag

nitu

de (

dB)

Bode plot

10-1

100

101

102

103

-141

-140.5

-140

-139.5

-139

Frequency (rad/s)

Pha

se (

deg)

!c=100

PM=40

!"#$%& ''! "#$% &'#( #) *#+(,#''%$ -.-(%/ 01(2 3! 4567 *#+(,#''%,

10-1

100

101

102

103

-50

0

50

100

Mag

nitu

de (

dB)

Bode plot

10-1

100

101

102

103

-180

-160

-140

-120

-100

Frequency (rad/s)

Pha

se (

deg)

!c=100

PM=40

!"#$%& '(! "#$% &'#( #) *#+(,#''%$ -.-(%/ 01(2 53" *#+(,#''%,

)*+),$-"*+-

3+ (21- &8&%,9 8 /%*28+1*8' /#$%' 8+$ $%-1:+ #) ;<=>?@

#,(2%-1- 1- -(A$1%$! > 2.B,1$ /#$%' 1- 1+(,#$A*%$! 3+ #,$%, (# *#+C

(,#' (2% #,(2#-1- D+%% 8+:'% 8 '1+%8, ,%)%,%+*% 1- 1+(,#$A*%$ A-1+:

)#A, &'8+8, -%+-#, 1+ (2% )##(! > ),8*(1#+8' #,$%, *#+(,#''%, A-1+:

(2% *#/B1+8(1#+ #) *'8--1* 56 8+$ 8 ),8*(1#+8' 3+(%:,8(#, 1- &,#C

&#-%$! E21- *#+(,#''%, 1- +#( -%+-1(1F% (# (2% +#1-% 8+$ %'1/1+8(%-

(2% -(%8$. -(8(%- %,,#, 021*2 8 *'8--1*8' 56 $#+G(! 3+ 8$$1(1#+9

8 53" *#+(,#''%, 1- &,#&#-%$ 8+$ (2% *#+(,#''%, &8,8/%(%,- 8,%(A+%$ A-1+: &,#&%, -&%*1H*8(1#+-! E2% -1/A'8(1#+ ,%-A'(- -2#0

(2% %)H*1%+*. #) (2% *#+(,#''%,-!

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Rig

ht

Kn

ee a

ng

le (

rad

)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

Gate cycle

Left

Kn

ee a

ng

le (

rad

) Controlled angle

Reference angle

!"#$%& '.! ;1/A'8(1#+ ,%-A'( #) *#+(,#''%$ -.-(%/ 01(2 3! 4567!

/)0+*1,&2#3&+4

E21- 0#,D 1- -A&&#,(%$ B. (2% ;&8+1-2 I1+1-(,. #) ;*1%+*%

8+$ 3++#F8(1#+ A+$%, (2% &,#J%*( 653KLLMCNOPOQC<LOCLK 8+$

653KLLMCNOPNQC<LO! E2% -A&&#,( 1- :,8(%)A''. 8*D+#0'%$:%$!

%&!&%&+)&-

RNS >!I! 6#''8,9 8+$ T! T%,,! >*(1F% #,(2#-%- )#, (2% U#0%,C

U1/B-V <28''%+:%- 8+$ ;(8(% #) (2% >,(9 5,#*%%$1+:- #) (2%

3WWW NL(2 3+(%,+8(1#+8' <#+)%,%+*% #+ X%28B1'1(8(1#+ X#C

B#(1*-9 MYQCMZZ9 KLLZ!

RKS >! <A''%''9 [!<! I#,%+#9 W! X#*#+9 >! ?#,+%,C<#,$%,#9 8+$

[!U! 5#+-! "1#'#:1*8''. B8-%$ $%-1:+ #) 8+ 8*(A8(#, -.-(%/

)#, 8 D+%%C8+D'%C)##( #,(2#-1-9 I%*28+1-/ 8+$ I8*21+%

E2%#,.9 !!9 QLYCQZK9 KLLM!

ROS E! \8D1/#F1*29 W!6! U%/81,%9 8+$ [! =#)/8+! W+:1+%%,1+:

$%-1:+ ,%F1%0 #) -(8+*%C*#+(,#' D+%%C8+D'%C)##( #,(2#-%-9

[#A,+8' #) X%28B1'1(8(1#+ X%-%8,*2 ] 6%F%'#&/%+(9 !"9

K^ZCKYZ9 KLLM!

RPS ;!I! <81+9 =! W! _#,$#+9 8+$ 6! ?%,,1-9 U#*#/#(#, 8$8&C

(8(1#+ (# 8 &#0%,%$ 8+D'%C)##( #,(2#-1- $%&%+$- #+ *#+(,#'

/%(2#$9 [#A,+8' #) `%A,#W+:1+%%,1+: 8+$ X%2B1'1(8(1#+9 PC

PQ9 KLLZ!

R^S a! I8(J8b*1c*9 >! @'%+d%D9 E! "8J$9 "1#/%*28+1*8' *28,8*C

(%,1e8(1#+ 8+$ *'1+1*8' 1/&'1*8(1#+- #) 8,(1H*18''. 1+$A*%$

(#%C08'D1+:V 61))%,%+*%- B%(0%%+ &A,% -#'%A-9 &A,% :8-C

(,#*+%/1A- 8+$ *#/B1+8(1#+ #) -#'%A- 8+$ :8-(,#*+%/1A-

*#+(,8*(A,%-9 [#A,+8' #) "1#/%*28+1*-9 OM9 K^^CKYY9 KLLY!

Q <#&.,1:2( *! KLNN B. >;IW


Trabajo futuro



Generalización de estrategias de control óptimo y robusto mediante el uso de cálculo fraccionario. Aplicaciones en biomecánica y robótica móvil. (GECORFOC):

LQG/LTR. Factorización espectral - Wiener-Hopf - sistemas de orden commensurable. SMC. Model-free, incluyendo controladores GPI, i-PID y observadores GPIO. FGPC - sistemas híbridos y en red. Inclusiones diferenciales fraccionarias: dinámica, estabilidad, .etc.. Aplicaciones experimentales en robótica móvil y biomecánica


intranet.ceautomatica.esintranet.ceautomatica.es/sites/default/files/upload/13/files/04... ·...

Documents