- Email: [email protected]
Modeling of Disturbed Flat System for Robust Control Design
IFAC
Copyright ~ IFAC Nonlinear Control Systems, St. Petersburg, Russia, 200 I
c:
0
[>
Publications www.elsevier.comllocatelifac
MODELING OF DISTURBED FLAT SYSTEM FOR ROBUST CONTROL DESIGN L. Lavigne, F. Cazaurang, B. Bergeon LAP-ARIA, EP 2026 CNRS Universite Bordeaux J, 35 J cours de la Liberation, 33405 Talence cedex, France Tel. 33-0556842415 - Fax: 33-56846644 e-mail:[email protected] Abstract: This paper aims at presenting a modelization of the uncertainty on the linear model obtained from the so-called linearizing feedback of dynamic flat systems. After such a linearizing feedback is computed to ensure good nominal tracking performance, a compact set of models can be computed by taking account of both state space disturbances and parametric errors. A robust linear teedback is then designed in order to guarantee a specified level of performance. This representation of the compact set of models makes use of the Linear Fractional Transformation. so that the global robustness of tracking performance is assessed by #analysis. Copyright © 2001IFAC
Key words: Nonlinear control, robust control, H-infinity control, J.l -analysis.
dynamic feedback. This dynamic feedback, (called endogenous feedback), is defined as a real-analytic function of state, input and a finite number of its derivatives. For a flat system there are m scalar functions Yi of state x, input u and a finite number of its derivatives such that the dynamic behaviour with input u and outputs y;. ... ,ym can be linearized from an input to state point of view. Outputs Y;. ....Ym' which might be regarded as fictitious outputs, are called linearizing or flat outputs. The major property of a flat system is that the state x and input u variables can be directly expressed, without integrating a differential equation, in terms of flat output Y and a finite number of its derivatives. This property is useful when dealing with trajectories (Fliess, 1999). From y trajectories, x and u trajectories are immediately deduced.
1. INTRODUCTION This paper aims at presenting a modelization of uncertainty resulting from the dynamic linearizing feedback of flat systems. As the measures on the so-called flat outputs, used in the linearizing feedback, are corrupted with noise and effects ofunmodelled dynamics or parametric errors, the state space trajectory of the actual plant is not identical to the reference state space trajectory. Then the exact linearization is not achieved. But, as long as the tracking error remains small enough, the actual quasi-Iinearised plant can be modelized as a disturbed linear plant. The behaviour of which is assumed to be described by one (at least) model belonging to a set defined by the Iinearized (from the input to state point of view) model and a model of uncertainty.
2.1 Definitions
Consider the following state space representation (I/):
This uncertainty makes it necessary to design a robust linear controller to achieve good tracking performances (prempain et al., 1998). Moreover a linear fractional transformation description of the so-obtained compact set of models is used to analyse, through ,u-analysis the performances of this controller on the actual plant.
x = f(x ,u) with x = (xI, ... ,xn)e Rn.u = (UI •...•Um) e R m (1) Where
f
and rank {
is regular function such as ./(0,0)
= 0
du (O,O)} = m .
This system is differentially flat if there are an output vector y named flat output composed by m fictitious outputs (Y1 , ...,yro) and three functions where y(a) is the derivative of order a of y such as (Fliess 1995):
After a brief presentation of the theory of flat systems in the section 2, the main contribution of this paper, namely the generation of the compact set of models, is detailed in section 3. The section 4 gives hint on the construction of the augmented plant used for the design of the H-infinity robust controller, and briefly present the ,u-analysis method. Finally the application to the control of the transient behaviour of the levels of a three-tank system illustrates the efficiency of the presented method.
x = A(y.y .... . /a) )
(2)
u = B(y,y .... . /a+I))
(3)
y = C(x.u.u•... •u(y) )
(4)
2.2 Path Planning
The nominal path planning approach determines input vector u(t) on a finite time (7) so that the state vector path x goes from the initial point x(O) to the final point x(7) . In the flat case, this is achieved without resolution of differential equations. It simply requires finding a
2. FLAT NONUNEAR SYSTEMS Flat systems correspond to a class of non-linear systems, which are equivalent to linear ones via a special type of
759
The nominal system and the endogenous feedback are represented by:
curve in the space (y, .... iQ) which verifies final and initial conditions of the state vector path x. It is preferable to take into account the structure and the physics of the system to obtain a curve adapted to the problem. The input u that brings the system of the state x(O} to the state x(T) is then obtained from the expression (2). It remains then to study singularities of functions A and B .
( )=(f(Xm,Um(Xm'~m'Vm))) m ( ) ( ~m)=f.emxm'':>m'V Xm J:
~m'
xm'~m,vm
Ym = hm{xm,Um(xm'~m' Vm ))= h~(xm'~m' vm) The global dynamic system Ie.mcan be extended to the infinite dimensional state space and the global dynamic system definition F •. m is :
2.3 Endogenous dynamic feedback After the flat output determination (Yh ... , ynJ, an endogenous state feedback and a diffeomorphism that transfonns system into an integrator chain, formed from (y), ... , ynJ , are used.
Fe.m(xm'~m,um,um,iim'···)= fe.m(xm'~m,um}-;j.-x ... 00
zeR
q
veR
m
m(x,;, v))) ( ;~) = fe(x.;. v) =(f(x,U )(m(x.;,v)
(6) Then, the extended system is linearizable by static feedback.
a(x.z)+b(x.z)v
With
k1,. .. ,km
(y), ... , ynJ·
commandability
indice
y
Fe(x,~,u,u,ii"")=fe(x, ~,u)~+ i:u{Jt+l)
to
... v(a+l) ... ) 'm
(14)
y- Ym = h~(x . ;. ii)-h~(xm.;m. ii)
_ --a;ah~ ( ~ _~_ ah~ ( ,; _~" x, ,v po' + a~ x , ,v po. ...
Y - Ym -
... +a~ Y-
Ym =
(xm ,';m' ii)5ii +0(1&1
2
+I~n
(15)
Dhm(xm'~m' ii{ ~]+ 0(1&1 +I~n 2
With the following designation: (8)
And for k= a+ 1 the gap is given by: Y
Denote:
rn'
and
m'; m
This endogenous feedback leads to the following relations between the flat outputs Ym.i and the inputs vrn,i: \a,+l) = v . 'Vi E {I ... m} otherwise y (a+l ) = V Ym,l m,l " m m
V = (v
Xm '~m
For k = 0, the first order lagrangian development of Y on the vicinity of nominal path x gives:
The endogenous feedback determined on the previous nominal flat system {El} is described by:
-U m {xrn ' "='m' J: V )-u- -( ... u(r) ... ) m - m - \U, , m'
d{Jt) (13)
du
&=x-xm &; =~ -~m (jV = v-vm' v-vm' ... )
The endogenous feedback is computed from the measured flat outputs, without considering the effects of disturbances, the actual trajectory of the flat outputs do not coincide with the specified ones , so that the actual plant, including the linearizing feedback, is not perfectly the desired linear one. The definition of a compact set of model aims at containing certainly a good description of the disturbed linearized plant . The approach is based on differential geometric theory of jets and prolongation of infinite order developed in particular by Vinogradov (1989). This approach is also used by Fliess et al. (1999) to generalise the differentially flat non linear systems to orbitally flat systems.
{~m -Xm(xm'~m,vm)
iJ=O
Denote the distances between the nominal disturbed x,~ state space co-ordinates.
3. COMPACT SET OF MODELS
m
=hm(x,Um(x,;, v)) = h~(x,;. iI)
dx
~m :Um(xm'~m,vm)
(12)
The global disturbed dynamic system le can be extended to the infinite dimensional state space Sm then:
(7)
involve
(11)
Consider now a disturbed plant. As a first consequence, the extended disturbed field F. is different from the nominal one F..", . Secondly the actual state space trajectory is in a vicinity of the nominal one. The global disturbed model is described by :
The diffeomorphism E on extended state space ~ is given by:
z
°m
dum
J.l=o
(5)
~ =(~) =(f(x.a(x.z) + /3(x.z)v)) ; e Rn+q
d
(P+I)
"'+~>m ~
Let us give the endogenous state feedback as:
i:=a(x.z)+b(X.Z)V {u =a(x.z) + /3(x.z)v
(10)
(a+l) _ y(a+l) = L(a+l)h' '"
Fe
1ft
(x " v)- L(a+l)h' ( " v) (17) F~ m Xm'~,"' m ,"',
y'''') - y!'.') • Lt')·; « •• <•. ,). DL~·')'; « ••<•. ,{~
(9)
,
... -LF•• h~(x .. ,~ ... v)+0~arI2
760
+1&;1 2 )
l
(18)
y(a+I) - y:+I) = c5(L~+I)h~ Xx,.,~,", v)+ ...
1
(nLt"'; +.(nL~"'';' l)·•. (•.,{~
--
(19)
-
.+0(&12 +1&;12) Then, the flat output Ym is governed by the new input Vm and y~+l) = vm . The compact set of models is given by (Cazaurang F., 1997): y (a+l) =
Vm
Figure 1 : robust tracking design
+0~(L(a+l)h",')():-) F. Xm''>m'V
i DLt'i.;,.
+
4.ill-anaiysis: The structured singular value jll>. (Doyle ,1982) is a positive real-valued function of a matrix M and a specified perturbation block structure A with the following definition:
o( DL~+'i.;,. ) )x•. ,•.•{; }. (20)
+0(1&1 2 + io;n
f\ = (c5rl l'i ,... ,c5 rJ rk ,c5c /
With the following notations:
LFti1m is the Lie derivative of hmon the Fe field.
15,.,
,
o~;+l)~Xxm'~"" v)=D;+l)~(xm'~"" v)- D;+l)Ji",(xm'~m' v)(21) e t." t
....
'X, c5 c
"
E
C, ~c
·,c5ck ICk '~CI ' '' ''~Ck )(22)
E C10dim(cl) ,}
and
II~II ~
.. ~ ~r
The performance robustness is equivalent to the structured stability robustness specified by the following diagram and relation (23).
Dl:.+l)Ji",(xm'~m' v)=D;+l)Ji",(x",,~, v)_D;+l)Ji",(X,~, v) e,'"
E
c1 , ..
"._
o(D#+l)Ji",h", ~m' v) = D#+l)h",(x m,;"" v)-D#+I)h",(xm,~m' ii) .. et" t
,(a+llJ:( ): -) ,(a+lh:( ): -) .Ia+llJ:( ):-) D1.1:: '-mXm'~m'V =LF., "nrx",,"'m,V -LE:, ,.",x,~,v <
The input v includes the reference input Vm and the linear regulator output liv. The effect due to the distance between the nominal extended field F•.m and disturbance extended field F. (second term on the right hand side) is reduced by the output regulator action. It contributes also to reducing the effect of the gap between the nominal path and the disturbed path (third term on the right hand side).
If II~II.. < 1the robust stability is assured if: .u~ (F, (p(s ~ K))::; 1
(23)
Where FlP,K) represents the lower L.F.T. interconnection and jll>. represent structured singular value.
4. LINEAR FRACTIONAL REPRESENTATION AND Jl-ANALYSIS
5. APPLICATION ON HYDRAULIC SYSTEM
The two degree-of-freedom design methodology is based on a decoupling scheme, according to the Youla parametrization (Youla ,1985). A two step methodology has been developed initially for the synthesis of robust tracking control in the case of linear multi variable systems (Prempain et aI., 1998). Here, a similar diagram (fig. 1) is proposed for flat systems in which x is the state space vector, u the input vector and y is the socalled flat output (Cazaurang et ai, 2000).
The experimental study is based on a pilot three-tank system are depicted on Fig. 2. FlowQ,
TIDkT,
TIDkT,
TIDkT,
The global design problem uses Linear Fractional Transformation representation. This representation allows an augmented model (P') including a compact set of models, and performance specifications (some frequency limitations are associated with fictitious uncertainties blocks ~ro and weighting function W) . A linear feedback controller K is designed through the Hinfinity mixed sensitivity method (Balas J.G., 1993). Suppose that hJ>h3>h2. then a model is given by:
761
Sh l = -a zl S n J[2g(h l
hJ] + QI Sh 3= azl Sn J[2g(h l -h3 )] -a z3Sn J[2g(h 3 -h2 )] Sh 2 = a z3S n J[2g(h 3 - h2 )] - a z2 S n J(2gh 2) + Q2
Mu-graph (fig. 3) represents for each frequency the ~ value. All values are lower than I, so robust stability and robust performances specifications are validated.
-
(24)
6. CONCLUSIONS
Where az}, az2 and az3 parameters are various numerical flow coefficients. Flatness output vector is composed by h} and h3 tanks level. Applying the so-called linearising feedback leads to the input to state linear system:
.
I
I
'
Tools for the modelization of uncertainty on input to state linearized flat systems have been presented. Aiming at characterising the discrepancy between the nominal input to state linearized model and the nonlinear (quasi-linearized) plant, this uncertainty can be represented through standard linear fractional transformation on the linearized input to state model. This representation can be used to check the performance robustness of an H-infinity linear controller, by using standard J.l -analysis.
2
With y\ = h3' Y2 = h3' y\ = hI and I
QI = aZIS n (2g(hl - h3))2
Q2
+ SVI
(26)
= (a z,SS2g, (h 3 -h2 )~ -2a z3S n (2g(h 3 -h 2 ))~
AKNOWLEDGEMENTS
a z3 S n (2g)l:
-az,Sn(2g)~[(h3 -h2 Xh, -hJ~ -(h,-hJ~]
The authors wish to thank Professor Jean Levine for his help to establish the main result in the infinite dimension frame.
(27)
REFERENCES Balas G.J., J.C. Doyle, K. Glover, A. Packard, and R. Smith, (1995). J.l -analysis and synthesis Toolbox (for use with MATLAB), The Math Works Inc. Bergeon B., F. Cazaurang, and S. Ygorra (2001). Methodologie de commande robuste lineaire , APIIJESA, 35, WI, pp. 85-106 In a first time, calculus of disturbance is determined for the following az2 parameter variation.
a z 2 =a z 2nom
+Oz2.1.p
with
II.1. p llS;1
Cazaurang F., (1997). Commande robuste des systemes plats, Application a la commande d 'une machine synchrone, PhD Thesis, Universite Bordeaux I.
(28)
Cazaurang F., B. Bergeon, and S. Ygorra (2000). Robust Control of flat nonlinear system, In Proceedings of Workshop lagrangian and hamiltonian methods for nonlinear control IF AC Princeton U.S.A, ppI67-169.
Corresponding disturbance term is given by.
Doyle J.C. (1982). Analysis of feedback systems with tructured uncertainties, In Proceedings of .Inst. Elec .Eng. 129 (part D) , pp.242-250.
(29)
Fliess M., 1. Levine, Ph. Martin and P. Rouchon (1995). Flatness and defect of non-linear systems: introduction theory and examples. Int. Journal of Control, 61,1327-1361.
Linear fractional transfonnation model of the controlled plant exhibits the following uncertainties structuration:
Fliess M., J. Levine, Ph. Martin and P. Rouchon (1999), A Lle-BackIund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control, 44 W5, 922-937.
Where .1. p is the real-block representing the effect of the perturbation pert, .1... .1. 2, .1. 3 are the fictitious complex uncertainty related to the H-infinity synthesis.
Prempain E. and B. Bergeon (1998). Multivariable twodegree of freedom control methodology, Automatica, 34,1601-1606. Vinogradov A. M. (1989), Symetries of partial Differential Equations, Kluwer, Dordrecht. Youla D.e., H. A. Jabr and J. J. Bongiorno (1985), A feedback theory of two degree of freedom optimal Wiener-Hopf design, IEEE Transaction on Automatic Control 1985, 30, N°5 , 662-665.
1_:..:'----,,-:;.. -""'•.--::"/~_:_..::. - ,7,-...J,t Fig.3 : Mu...,graph
762
Copyright ~ IFAC Nonlinear Control Systems, St. Petersburg, Russia, 200 I
c:
0
[>
Publications www.elsevier.comllocatelifac
MODELING OF DISTURBED FLAT SYSTEM FOR ROBUST CONTROL DESIGN L. Lavigne, F. Cazaurang, B. Bergeon LAP-ARIA, EP 2026 CNRS Universite Bordeaux J, 35 J cours de la Liberation, 33405 Talence cedex, France Tel. 33-0556842415 - Fax: 33-56846644 e-mail:[email protected] Abstract: This paper aims at presenting a modelization of the uncertainty on the linear model obtained from the so-called linearizing feedback of dynamic flat systems. After such a linearizing feedback is computed to ensure good nominal tracking performance, a compact set of models can be computed by taking account of both state space disturbances and parametric errors. A robust linear teedback is then designed in order to guarantee a specified level of performance. This representation of the compact set of models makes use of the Linear Fractional Transformation. so that the global robustness of tracking performance is assessed by #analysis. Copyright © 2001IFAC
Key words: Nonlinear control, robust control, H-infinity control, J.l -analysis.
dynamic feedback. This dynamic feedback, (called endogenous feedback), is defined as a real-analytic function of state, input and a finite number of its derivatives. For a flat system there are m scalar functions Yi of state x, input u and a finite number of its derivatives such that the dynamic behaviour with input u and outputs y;. ... ,ym can be linearized from an input to state point of view. Outputs Y;. ....Ym' which might be regarded as fictitious outputs, are called linearizing or flat outputs. The major property of a flat system is that the state x and input u variables can be directly expressed, without integrating a differential equation, in terms of flat output Y and a finite number of its derivatives. This property is useful when dealing with trajectories (Fliess, 1999). From y trajectories, x and u trajectories are immediately deduced.
1. INTRODUCTION This paper aims at presenting a modelization of uncertainty resulting from the dynamic linearizing feedback of flat systems. As the measures on the so-called flat outputs, used in the linearizing feedback, are corrupted with noise and effects ofunmodelled dynamics or parametric errors, the state space trajectory of the actual plant is not identical to the reference state space trajectory. Then the exact linearization is not achieved. But, as long as the tracking error remains small enough, the actual quasi-Iinearised plant can be modelized as a disturbed linear plant. The behaviour of which is assumed to be described by one (at least) model belonging to a set defined by the Iinearized (from the input to state point of view) model and a model of uncertainty.
2.1 Definitions
Consider the following state space representation (I/):
This uncertainty makes it necessary to design a robust linear controller to achieve good tracking performances (prempain et al., 1998). Moreover a linear fractional transformation description of the so-obtained compact set of models is used to analyse, through ,u-analysis the performances of this controller on the actual plant.
x = f(x ,u) with x = (xI, ... ,xn)e Rn.u = (UI •...•Um) e R m (1) Where
f
and rank {
is regular function such as ./(0,0)
= 0
du (O,O)} = m .
This system is differentially flat if there are an output vector y named flat output composed by m fictitious outputs (Y1 , ...,yro) and three functions where y(a) is the derivative of order a of y such as (Fliess 1995):
After a brief presentation of the theory of flat systems in the section 2, the main contribution of this paper, namely the generation of the compact set of models, is detailed in section 3. The section 4 gives hint on the construction of the augmented plant used for the design of the H-infinity robust controller, and briefly present the ,u-analysis method. Finally the application to the control of the transient behaviour of the levels of a three-tank system illustrates the efficiency of the presented method.
x = A(y.y .... . /a) )
(2)
u = B(y,y .... . /a+I))
(3)
y = C(x.u.u•... •u(y) )
(4)
2.2 Path Planning
The nominal path planning approach determines input vector u(t) on a finite time (7) so that the state vector path x goes from the initial point x(O) to the final point x(7) . In the flat case, this is achieved without resolution of differential equations. It simply requires finding a
2. FLAT NONUNEAR SYSTEMS Flat systems correspond to a class of non-linear systems, which are equivalent to linear ones via a special type of
759
The nominal system and the endogenous feedback are represented by:
curve in the space (y, .... iQ) which verifies final and initial conditions of the state vector path x. It is preferable to take into account the structure and the physics of the system to obtain a curve adapted to the problem. The input u that brings the system of the state x(O} to the state x(T) is then obtained from the expression (2). It remains then to study singularities of functions A and B .
( )=(f(Xm,Um(Xm'~m'Vm))) m ( ) ( ~m)=f.emxm'':>m'V Xm J:
~m'
xm'~m,vm
Ym = hm{xm,Um(xm'~m' Vm ))= h~(xm'~m' vm) The global dynamic system Ie.mcan be extended to the infinite dimensional state space and the global dynamic system definition F •. m is :
2.3 Endogenous dynamic feedback After the flat output determination (Yh ... , ynJ, an endogenous state feedback and a diffeomorphism that transfonns system into an integrator chain, formed from (y), ... , ynJ , are used.
Fe.m(xm'~m,um,um,iim'···)= fe.m(xm'~m,um}-;j.-x ... 00
zeR
q
veR
m
m(x,;, v))) ( ;~) = fe(x.;. v) =(f(x,U )(m(x.;,v)
(6) Then, the extended system is linearizable by static feedback.
a(x.z)+b(x.z)v
With
k1,. .. ,km
(y), ... , ynJ·
commandability
indice
y
Fe(x,~,u,u,ii"")=fe(x, ~,u)~+ i:u{Jt+l)
to
... v(a+l) ... ) 'm
(14)
y- Ym = h~(x . ;. ii)-h~(xm.;m. ii)
_ --a;ah~ ( ~ _~_ ah~ ( ,; _~" x, ,v po' + a~ x , ,v po. ...
Y - Ym -
... +a~ Y-
Ym =
(xm ,';m' ii)5ii +0(1&1
2
+I~n
(15)
Dhm(xm'~m' ii{ ~]+ 0(1&1 +I~n 2
With the following designation: (8)
And for k= a+ 1 the gap is given by: Y
Denote:
rn'
and
m'; m
This endogenous feedback leads to the following relations between the flat outputs Ym.i and the inputs vrn,i: \a,+l) = v . 'Vi E {I ... m} otherwise y (a+l ) = V Ym,l m,l " m m
V = (v
Xm '~m
For k = 0, the first order lagrangian development of Y on the vicinity of nominal path x gives:
The endogenous feedback determined on the previous nominal flat system {El} is described by:
-U m {xrn ' "='m' J: V )-u- -( ... u(r) ... ) m - m - \U, , m'
d{Jt) (13)
du
&=x-xm &; =~ -~m (jV = v-vm' v-vm' ... )
The endogenous feedback is computed from the measured flat outputs, without considering the effects of disturbances, the actual trajectory of the flat outputs do not coincide with the specified ones , so that the actual plant, including the linearizing feedback, is not perfectly the desired linear one. The definition of a compact set of model aims at containing certainly a good description of the disturbed linearized plant . The approach is based on differential geometric theory of jets and prolongation of infinite order developed in particular by Vinogradov (1989). This approach is also used by Fliess et al. (1999) to generalise the differentially flat non linear systems to orbitally flat systems.
{~m -Xm(xm'~m,vm)
iJ=O
Denote the distances between the nominal disturbed x,~ state space co-ordinates.
3. COMPACT SET OF MODELS
m
=hm(x,Um(x,;, v)) = h~(x,;. iI)
dx
~m :Um(xm'~m,vm)
(12)
The global disturbed dynamic system le can be extended to the infinite dimensional state space Sm then:
(7)
involve
(11)
Consider now a disturbed plant. As a first consequence, the extended disturbed field F. is different from the nominal one F..", . Secondly the actual state space trajectory is in a vicinity of the nominal one. The global disturbed model is described by :
The diffeomorphism E on extended state space ~ is given by:
z
°m
dum
J.l=o
(5)
~ =(~) =(f(x.a(x.z) + /3(x.z)v)) ; e Rn+q
d
(P+I)
"'+~>m ~
Let us give the endogenous state feedback as:
i:=a(x.z)+b(X.Z)V {u =a(x.z) + /3(x.z)v
(10)
(a+l) _ y(a+l) = L(a+l)h' '"
Fe
1ft
(x " v)- L(a+l)h' ( " v) (17) F~ m Xm'~,"' m ,"',
y'''') - y!'.') • Lt')·; « •• <•. ,). DL~·')'; « ••<•. ,{~
(9)
,
... -LF•• h~(x .. ,~ ... v)+0~arI2
760
+1&;1 2 )
l
(18)
y(a+I) - y:+I) = c5(L~+I)h~ Xx,.,~,", v)+ ...
1
(nLt"'; +.(nL~"'';' l)·•. (•.,{~
--
(19)
-
.+0(&12 +1&;12) Then, the flat output Ym is governed by the new input Vm and y~+l) = vm . The compact set of models is given by (Cazaurang F., 1997): y (a+l) =
Vm
Figure 1 : robust tracking design
+0~(L(a+l)h",')():-) F. Xm''>m'V
i DLt'i.;,.
+
4.ill-anaiysis: The structured singular value jll>. (Doyle ,1982) is a positive real-valued function of a matrix M and a specified perturbation block structure A with the following definition:
o( DL~+'i.;,. ) )x•. ,•.•{; }. (20)
+0(1&1 2 + io;n
f\ = (c5rl l'i ,... ,c5 rJ rk ,c5c /
With the following notations:
LFti1m is the Lie derivative of hmon the Fe field.
15,.,
,
o~;+l)~Xxm'~"" v)=D;+l)~(xm'~"" v)- D;+l)Ji",(xm'~m' v)(21) e t." t
....
'X, c5 c
"
E
C, ~c
·,c5ck ICk '~CI ' '' ''~Ck )(22)
E C10dim(cl) ,}
and
II~II ~
.. ~ ~r
The performance robustness is equivalent to the structured stability robustness specified by the following diagram and relation (23).
Dl:.+l)Ji",(xm'~m' v)=D;+l)Ji",(x",,~, v)_D;+l)Ji",(X,~, v) e,'"
E
c1 , ..
"._
o(D#+l)Ji",h", ~m' v) = D#+l)h",(x m,;"" v)-D#+I)h",(xm,~m' ii) .. et" t
,(a+llJ:( ): -) ,(a+lh:( ): -) .Ia+llJ:( ):-) D1.1:: '-mXm'~m'V =LF., "nrx",,"'m,V -LE:, ,.",x,~,v <
The input v includes the reference input Vm and the linear regulator output liv. The effect due to the distance between the nominal extended field F•.m and disturbance extended field F. (second term on the right hand side) is reduced by the output regulator action. It contributes also to reducing the effect of the gap between the nominal path and the disturbed path (third term on the right hand side).
If II~II.. < 1the robust stability is assured if: .u~ (F, (p(s ~ K))::; 1
(23)
Where FlP,K) represents the lower L.F.T. interconnection and jll>. represent structured singular value.
4. LINEAR FRACTIONAL REPRESENTATION AND Jl-ANALYSIS
5. APPLICATION ON HYDRAULIC SYSTEM
The two degree-of-freedom design methodology is based on a decoupling scheme, according to the Youla parametrization (Youla ,1985). A two step methodology has been developed initially for the synthesis of robust tracking control in the case of linear multi variable systems (Prempain et aI., 1998). Here, a similar diagram (fig. 1) is proposed for flat systems in which x is the state space vector, u the input vector and y is the socalled flat output (Cazaurang et ai, 2000).
The experimental study is based on a pilot three-tank system are depicted on Fig. 2. FlowQ,
TIDkT,
TIDkT,
TIDkT,
The global design problem uses Linear Fractional Transformation representation. This representation allows an augmented model (P') including a compact set of models, and performance specifications (some frequency limitations are associated with fictitious uncertainties blocks ~ro and weighting function W) . A linear feedback controller K is designed through the Hinfinity mixed sensitivity method (Balas J.G., 1993). Suppose that hJ>h3>h2. then a model is given by:
761
Sh l = -a zl S n J[2g(h l
hJ] + QI Sh 3= azl Sn J[2g(h l -h3 )] -a z3Sn J[2g(h 3 -h2 )] Sh 2 = a z3S n J[2g(h 3 - h2 )] - a z2 S n J(2gh 2) + Q2
Mu-graph (fig. 3) represents for each frequency the ~ value. All values are lower than I, so robust stability and robust performances specifications are validated.
-
(24)
6. CONCLUSIONS
Where az}, az2 and az3 parameters are various numerical flow coefficients. Flatness output vector is composed by h} and h3 tanks level. Applying the so-called linearising feedback leads to the input to state linear system:
.
I
I
'
Tools for the modelization of uncertainty on input to state linearized flat systems have been presented. Aiming at characterising the discrepancy between the nominal input to state linearized model and the nonlinear (quasi-linearized) plant, this uncertainty can be represented through standard linear fractional transformation on the linearized input to state model. This representation can be used to check the performance robustness of an H-infinity linear controller, by using standard J.l -analysis.
2
With y\ = h3' Y2 = h3' y\ = hI and I
QI = aZIS n (2g(hl - h3))2
Q2
+ SVI
(26)
= (a z,SS2g, (h 3 -h2 )~ -2a z3S n (2g(h 3 -h 2 ))~
AKNOWLEDGEMENTS
a z3 S n (2g)l:
-az,Sn(2g)~[(h3 -h2 Xh, -hJ~ -(h,-hJ~]
The authors wish to thank Professor Jean Levine for his help to establish the main result in the infinite dimension frame.
(27)
REFERENCES Balas G.J., J.C. Doyle, K. Glover, A. Packard, and R. Smith, (1995). J.l -analysis and synthesis Toolbox (for use with MATLAB), The Math Works Inc. Bergeon B., F. Cazaurang, and S. Ygorra (2001). Methodologie de commande robuste lineaire , APIIJESA, 35, WI, pp. 85-106 In a first time, calculus of disturbance is determined for the following az2 parameter variation.
a z 2 =a z 2nom
+Oz2.1.p
with
II.1. p llS;1
Cazaurang F., (1997). Commande robuste des systemes plats, Application a la commande d 'une machine synchrone, PhD Thesis, Universite Bordeaux I.
(28)
Cazaurang F., B. Bergeon, and S. Ygorra (2000). Robust Control of flat nonlinear system, In Proceedings of Workshop lagrangian and hamiltonian methods for nonlinear control IF AC Princeton U.S.A, ppI67-169.
Corresponding disturbance term is given by.
Doyle J.C. (1982). Analysis of feedback systems with tructured uncertainties, In Proceedings of .Inst. Elec .Eng. 129 (part D) , pp.242-250.
(29)
Fliess M., 1. Levine, Ph. Martin and P. Rouchon (1995). Flatness and defect of non-linear systems: introduction theory and examples. Int. Journal of Control, 61,1327-1361.
Linear fractional transfonnation model of the controlled plant exhibits the following uncertainties structuration:
Fliess M., J. Levine, Ph. Martin and P. Rouchon (1999), A Lle-BackIund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control, 44 W5, 922-937.
Where .1. p is the real-block representing the effect of the perturbation pert, .1... .1. 2, .1. 3 are the fictitious complex uncertainty related to the H-infinity synthesis.
Prempain E. and B. Bergeon (1998). Multivariable twodegree of freedom control methodology, Automatica, 34,1601-1606. Vinogradov A. M. (1989), Symetries of partial Differential Equations, Kluwer, Dordrecht. Youla D.e., H. A. Jabr and J. J. Bongiorno (1985), A feedback theory of two degree of freedom optimal Wiener-Hopf design, IEEE Transaction on Automatic Control 1985, 30, N°5 , 662-665.
1_:..:'----,,-:;.. -""'•.--::"/~_:_..::. - ,7,-...J,t Fig.3 : Mu...,graph
762