-
WA6 10:40
A N APPROACH TO NONLINEAR DISCRETE-TIME
H,-CONTROL*
H. Guillard*, S. Monaco** and D. Normand-Cyrot*
Laboratoire des Signaux & SystBmes
CNRS,SUPELEC, Plateau de Moulon
91190
Gif sur Yvette France
** Dipartimento di Informatica e Sistemistica
Universitk di Roma “La Sapienza”
via Eudossiana 18, 00184 Roma, Italy
Abstract
[12], [14]. In these recent works the existence of a controller solving the problem is shown t o be related to the
existence of a solution of a particular type of HamiltonJacobi equation, known as Isaacs equation. Moreover
a solution to such an equation exists under certain assumptions [14], [19].
Following recent works proposed in the continuous-time
context concerning the nonlinear equivalent of the I€,control problem and its connection with game theory
and Isaacs equation, the present paper sets and studies
the same problem in the discrete-time context. A digital static feedback law achieving closed-loop stability
and disturbance attenuation is firstly designed under
full information assumption and then by making use of
an observer.
1
The present paper deals with the nonlinear discretetime H,-problem.
We prove that the existence of a
controller providing a solution is related to the existence of a solution to a discrete-time version of Isaacs
equation. The solution proposed verifies some extra
assumptions which are strictly related to the discretetime context.
Introduction
H,-control theory gives ail answer to a major control
problem, which is to coiiceive controllers not designed
for a single plant under known inputs, but for a class
of plants under unknown inputs which can be disturbances, for instance.
The classic “full-information” case is firstly solved. On
this basis, sufficient conditions for the existence of a
solution to the H,coiitrol problem via measurement
feedback, using the same classic type of observer that
was used in the continuous-time case [13], are proposed.
H,-control, initiated by Zaines [22], only arised in the
beginning of the eighties.
The paper is organized as follows. The problem is formulated in a discrete-time context in section 2. Section
3 deals with the ”full-information” problem and we will
give a solution via measurement feedback in Section 4.
Initially designed in the frequency-domain, the €1,controller goal was to minimize the maximal norm, i.e.
the H,-norm of an input-output operator linking, for
example, an error to an unknown disturbance; the maximum being to be taken over the whole class of disturbances. The first solutions were thus elaborated in the
frequency-domain [7], [SI. Later works on a characterization of Ha-controllers in the time domain showed
that a certain Riccati equation was playing an essential role in the resolution of the linear problem [6], [9],
[Ill.
An extended study of this problem where complete
proofs are given can be found in [lo].
2
Coiisider a discrete-time system described by equations
of the form :
=
..‘k
=
yr. =
Xk+l
In parallel, in [l],[lG], [17], [21], setting the 11,-control
problem as an optimization one. a natural l i n k with differential linear quadratic game theory is proposed.
f(.k)
+ gl(2k)W + gz(2k)uk
h*(Zk)
hZ(2k)
+ kll(Xk)Wk + h ( X k ) U k
+ kZl(Zk)Uk
w“’
(2.1)
Tlie input variables are denoted by w E
(ex*
(control input). The
geneous input) and U E
outpiit variables are denoted by z E Wp’ (tracking
error) and y E RP3(measured variables). The mapPill@ f(z.1, !71(2), !72(2’), h l ( Z ) , h z ( Z ) , kll(Z), h2(+)
w””
In particular, in this context,new developinents in the
nonlinear continuous time case are ma.de possible [19],
‘Research supported in parts by grants froin MURST in Italy
and MEN in France
0191-2216/s3/$3.00 (Q 1993 IEEE
Problem forinulatioii
178
~~
c
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In our setting exogeneous sequence w stands for the
maximizing player, while control sequence U stands for
the miiiimizing player whose goal is to achieve (2.3).
This kind of situation is studied in [2], p. 254, where
the following solution is given :
and k21(x) are smooth mappings defined in a. neighbourhood of the origin in R".
We assume the existence of an equilibrium 20 = 0, i.e.
f(0) = 0, and providing a suitable change of coordinates, we assume that hI(0) = 0 and h,(O) = 0.
For a two-players, zero-sum, discrete-time dynamic
game of fixed duration k = 0 , . . . , N , the set of
strategies (w; (z),U ; (2)) provide a feedback saddle
point solution (i.e.l JN(u*,w) 5 J N ( u + , w * ) _<
J N ( u , w * ) Vu,Vw) if and only if there exists a function V k ( . ) . : R" + f# such that the following recursive
equation is satisfied :
Consider a controller described by equations of the
form :
where we assume t1ia.t B is defined in a neiglil~ourliood
p ( 0 , y) and p(0) are C kfunctions
of the origin in
(for some k 2 1).
w"',
The purpose of this controller is to provide :
e
Local asymptotic stability of the eclirilibrium
( Z , 6 ) = (0,O).
e
Disturbance attenuation in t,lie sense of sahfyiiig
the inequality :
N
Equation (2.4) is a discrete-time equivalent of Isaacs
equation.
Since we are interested in finding a time invariant control law, we consider only one function V .) : R" -+
Thus, Isaacs equation can be written in t e form :
N
V(r)
for every sequence w = ( W O , . . . , W N ) such t,liat t,he
resulting trajectory remains i n a, iieighbourhood of
1: = 0.
= V ( f ( r ) + 91(+U*(I)
+
w.
6
+g2(4u*(z))
- - y 2 w * ( z ) T w ' ( z(2.5)
)
Z=(I,U*(I))E(I,U*(I))
Moreover, we will not discuss Isaacs condition in the
sequel (i.e. interchangeability of operations max and
min), because we do not need to ensure that the couple (IO*(x), U* (x)) provides a saddle point solution.
The approach here chosen, has been niot,ivated by severa1 developments made i i i nonlinear coiit,iiiuoiis-tiiiie
t,lieand linear discrete-time contextas,relying
~- OII game
ory (see [l], [12], [21]).
In order to simplify the forthcoming developmeiits, we
assume that the mappings characterizing plant (2.1)
sa.tisfy the classic assumptions (see [14] for example) :
The idea is t o ass0ciat.e with the 11, discrete-t*iiiie
problem a two-players, zero-sum, difference game :
kll(Z) = 0
k L ( t ) k 1 2 ( z )=
I
k&(Z)hl(Z)
=0
(2.6)
We will first deal with the "full infortnation" prob~ N
exogeneous input
lem, where both state ( Z ~ ) ~and
( W ~ ) are
~ available
~ N for measurement.
of fixed duration k = 0 , . . . , Ai. with value fi~nctional
3
C4Fullinforimation" problem
As a result of the considered hypotheses, system (2.1)
is now described by the following equations :
where
U
(wk)k=o,
and w denote the sequences ( P Q ) ~ = o , , N and
, N . Moreover we note z(xk,uk) = hl(xk) +
+
= f(zk) G(zk)[t,"]
= h l ( z k ) -k k l 2 ( z k ) W
Yk = I;[
xk+i
zk
k12(Zk)Uk.
(3.1)
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where
Theorem 1 provides a solution to this particular case,
in a function V ( . ) which is supposed to verify Isaacs
equation (2.5),which will turn into a Hamilton-Jacobi
type equation by computation of (w*(z), u * ( z ) ) .
ax I
av(x)
GT (x)
X=,( I )
Sketch of proof:
The basic fact consists in finding a positive definite
function V ( . ) ,with V ( 0 ) = 0, such that :
Let us first give a discrete-time equivalent of a basic definition about detectability (see [3], [14] in the
continuous-time case), which will be used in the sequel.
V ( z k + l ) - V ( + e ) + % ~ t k - y 2 w ~ W5k O
Vk E N (3.7)
Summing (3.7) from k = 0 to IC = N , choosing 20 = 0
and recalling the positiveness of V (.), this all immediately leads to the disturbance attenuation (2.3).
Definition : Suppose f(0) = 0, h(0) = 0. The pair
{h, f } is said to be detectable if there exists a neighbourhood U of 0 such that if 10 E U, any trajectory
of zk+l = f ( z k ) is such that h ( z k ) is defined for all
IC E N and verifies :
To this end consider :
H(zkrWk,Uk)
T
= %k
+ v(zk+l)
%k - y 2 W F W k
1
According to (2.6 and (3.2), and applying to V the
Taylor expansion ormula, we obtain :
H(Zk,Wk,Uk)
Theorem 1 Suppose ;
(i) (hl(x),f(z)) locally deteciable around L' = 0.
(ii) There exists a sm.ooth positive definite fmction.
V ( . ) , with V ( 0 ) = 0, defin.ed i n a iieigh.bourh.ood of
x = o of R", satisfying:
*
V (f ( x )
hT(lk)hl(lk)
+ V(f("k))
Now it is easy to see that a couple
I;::*.:[
(3.6) is such that :
+ G(x)[,"]) is q,uadmtic in [E]
BH(z,,
(3.2)
*
=
Wk,t i k )
a[::]
Hamilton-Jacobi type equation :
I
(ck
given by
=o
tu;)
(The reversibility required by the computation of such
a couple is guaranteed by assumption (3.4)).
Then, applyin to H the Taylor expansion formula and
remembering f3.4), we can immediately check that :
H(XkrWk*4)
=
-1ITll(lk)(Wk
+ IITzI(~~)(w~
+
There exists a non singular inatriz. of sinooth
functions, defined in a neighbourhood of x = 0,
(3.8)
Choosing now U as in (3.5) leads to :
H(Q,
2Uk,ii(2'kr W k ) )
- H ( X k , W * ( l k ) , U*(Xk)) 5 0
(3.9)
Using the fact that V verifies Hamilton-Jacobi equa) ) V ( x k ) ,which
tion (3.3), i.e. H ( x k , w * ( x k ) , ~ * ( x ~ =
is Isaacs equation (2.5), it clearly implies (3.7).
G(x)= TT(z)JT(x)
:I.
- W*("k))l12
~ * ( z k ) ) 7 ' 2 2 ( ~ k ) ( u k - uC(~k))1I2
*
with. J =',[
H(.k,W*(Zk),U*(Zk))
(3.4)
To prove stability, note that inequation (3.7) with
wk = 0 Vk E NIand definite positiveness of V (.) clearly
prove that V (.) is a Lyapunov function for the system.
Then, if we initialize system (3.1) in xo = 0, it is possible t o find a controller resolving the H,-coiitrol "fullin.formation" problem for this system..
This controller is given by :
Finally, asyinytotic stability follows from local detectability o r ( h l ( z ) ,f(z)). The reasoning involving La
Salle's invariance principle is rather classic (see [12]).
C ( z k , wk) = u*(zk)- (T22(zk))-'T21(zk)(Wk- w * ( Z h ) )
A.
(3.5)
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4
Remark 1 : Choosing V ( t )= t T X z in (3.3)and (3.4)
in a linear setting, leads to the Riccati equation and the
,Im,,,,,,factorization of condition (a) in Theorem 3.1 in
[ll]. So, in a nonlinear discrete-time context, a factorization of the form (3.4)is needed, contrarily t o the
nonlinear continuous-time case. In fact, less restrictive
assumptions than (2.6)would require also a factorization in the coiitinuoustilne problem, but not involving
the function V(.). (See [15]).
Assuming ( z ~ ) L . ~and
N the exogeneous input ( w L . ) ~ ~ N
are no longer available for measurement, we consider
the following system :
= f(ZL.1 + G ( X k ) C y
= hl(zk) + k l Z ( 2 k ) U k
Yk = M z k ) + k2l(zk)Wk
2k+1
zk
Remark 2 : What essentially differs from the linear
discrete-time case is conditio11 (3.2),which disappears
in the linear problem by choosing a quadratic function
V(z] = zTXz which gives the ex ected results. It
wou d be interesting to examine wfiether or not this
y a d r a t i c condition is necessary for the resolution of
t e H,nonlinear discrete-time problem. Note that, if
V ( . ) is a function verifying Isaacs equation (2.5), we
obtain :
V ( m
+ c(z)[$:;;l,
=
+
V(z)
Disturbance attenuation via
measurement feedback
In order t o use our last result, we introduce an observer
(see, for instance, [13])which is an exact copy of the
dynamics with a term proportionnal to the error introduced by such a choice. This observer is therefore
described by equations of the form :
ek+l
rZW*=(z)W*(4
-u*=(r)u*(z)- h T ( x ) h l ( z )
+ G(x)c:{zi])
This equation shows that V (f (2)
quadratic in
.c = 0 in W".
[:.*):/I
for every
t
(4.1)
U(&)
is
=
+
=
fwk)+ g l ( e k ) w ( e k )+ g 2 ( e k ) 4 e k )
wek)[Yk
- h2(8k) - k 2 l ( ~ k ) w ( e k ) l
(4.2)
.(ek,w(ek))
With regards to w ( & ) , which still has to be chosen,
it seems reasonable, as noticed in [13],to take the
worst perturbation, namely w ( & ) = w'(0k) according
to (3.8).
in a neigIil>ourhood of
Remark 3 : If we do not make any of the assumptions
(2.6), considering :
We are now able to state our main result, for which we
will give a sketch of the proof using observer (4.2).
Theorem 2 Suppose :
( i ) ( h l ( t ) ,f(z)) locally detectable.
(ii) There exists a smooth positive definite function
V ( . ) , with V ( 0 ) = 0 , defined in a neighbourhood of
w",
= 0 in
satisfying the same assumptions as in
Theorem 1.
(iii) There exists a n x p2 matrix of smooth functions
M ( 4 ) defined in a neighbourhood of 0 = 0 which locally
renders th.e equilibrium 0 = 0 of ihe following sysiem
asym.pto2ically stable :
E
The same arguments of proof lead to a similar result.
Remark 4 : It is possible, as it has been done in the
continuous time in [14],to present these results in a
discrete-time dissipative setting, since (3.7)is a dissipative inequality with supply rate y'wrwk - z T z k and
storage function V ( . ) (see [20]for concepts of disipativity). Then, the Ha-control problem call be solved
by trying to render the closed-loop system dissipative,
with storage function V (.). In order to do so,instead of
verifying Isaacs equation, V (,) would suppose to satisfy
an inequality of the type :
=
!(ek) + g l ( e k ) w * ( e k )
--Awk)[h,(ek)
+ k21(ek)W*(~k)1(4.3)
(iw) There exists a smooth positive semi-definite function W(.,
.) defined in a neighbourhood of (z,e) = (0,O)
in R" x R", satisfying :
*w(o,e)> o ve # o
(4.4)
*!v(fobr
(z)+ gobs( + ) w ) 2s quadratic in 44.5)
*w(z,e)= CT(z,e)a(z,e)+ f o b a ( X , e )
H(Qr w * ( z k ) , u * ( z . k )I
) V(Q)
In our setting, this inequality would become a
Hamilton-Jacobi type inequality :
-l.*T(t,B)Rob,(z,B)r*(t,
0)
0
*Rott,(z,e)
(44
(4.7)
wh.ere
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which is nothing but 4.6, we can easily compute, taking
wk = 0 for all L in the formulae :
- U ( z k , e k )= - I I ~ ~ ( z ~-)I II IG~( ~ ~ , W * ( W I I ~
4- ( r ( 2 . k )- T'("k,ek))TRob,(tkrek)(T(lli)
- r * ( z k ~ e k ) ) (4.10)
Remembering (4.7), this clearly shows that U is a Lya-
punov function for the system. For asymptotic stability, we proceed as in the proof of Theorem 1, using
local detectability of ( h l ( z ) ,f(z)),(4.3), a well-known
property of casca.de systems (see [IS]) and La Salle's
invariance principle.
Finally, in order to prove disturbance attenuation, note
that if we do not take Wk = 0 for all k , (4.10) becomes:
r*(z,O) =
w*(z),1L(t), T l l ( t ) and
orem 1.
T32(z)are
defined as in The-
U(fk+l,ek+l)
- U(zkrek)
=
-llh1(zk)112
+ ?"IIWkl12
-Ila(@kt w*(ek))1l2
Then, if we initialize the closed-loop system in
(O,O), the controller resolving the H,control problem f o r system ( 4 . 1 ) is given b y :
+(P(xk, W k )
( ~ 0 ~ 6 0 =
)
6k+l
=
f ( e k )+ g1(sk)w*(ok)+ g z ( e k ) u . ( o k )
u(6,)
=
ii(Bk,W*(Bk))
+ w e k ) [ y k - / t 2 ( e k )- ~ ~ ~ ( e , ) w * ( e , ) 1
Sketch of proof
5
:
1'(2kr W k )
= Tll(Zk)(Wn: - W * ( Z k ) )
= ~ ( t k , ~ k+ )T 2 1 ( z k ) 1 ; ; 1 ( I k ) r ( 2 ' k , w k )
Then, applying to Hobs the Taylor expansion formula,
we obtain :
+ ('("k~
Wk)
- r*("k
t
=
Hnbs("k> e k , T * ( S k I e k
e k ) ) T R o b s ( x k t ek ) ( T ( z k , W k )
- '*("k
I
ek))
Coiiclusioiis
R e fe re iices
1)
[l] T. Basar, A dynamic games approach to controller
design: disturbance rejection in discrete time, Proceedings of the 28th IEEE Conference on Decision
and Control, 407-414, Tampa, 1989.
- r* (rlre k ) )
Define a function U ( ., .) from [w'z x [w" in [w, as :
Wk)
Before sending the final version of this paper, the authors became awa.re of a work from C. Byrnes and W.
Lin (see [4]) where nonlinear H,-control problem in
discrete-time via state and full information feedback is
solved without quadraticity requirement.
According to (4.5), HobS is quadratic in r ( x k , wk). It is
easy to verify that r * ( z , o )given by (4.8) is such that:
H o b s ( Z k 7 e k ! T ( z k >B k ) )
e k ) ) T R o b s ( z k , ek)(T("kr
In this preliminary paper, we have presented sufficient conditions for the existence of a static control
law which solves the H,-control problem for a nonlinear discrete-time affine system satisfying some classic assumptions. Some of the results concerning nonlinear 11,-control set in the continuous-time case are
proposed for discrete-time systems and difference game
theory. The analogies but also the difficulties with respect to the continuous-time situation or the linear case
are briefly discussed. Moreover, pursuing the study,
one can also easily set the nonlinear discrete-time H,control problem in a dissipative context.
It is immediate to verify that a.fter applying observer
(4.2)to system (4.1), the dynamics is described by :
v(Zk,ek,tok)
- T*(zkr
Summing this last inequality from L = 0 to L =
N , using (4.7), the positiveness of U ( . , .) and taking
(IO,
00) = (0,O) immediately leads to the desired disA.
t,urba.nce a,t,t,enuation(2.3).
(4.9)
+
U ( Z , 6 ) = V ( z ) W(Z, 8)
p2] T . Basar, G.J. Olsder, Dynamic noncooperative game theory, Academic Press (London, New
York), 1982.
Hypotheses (ii) and (iii)(4.4) of Theorem 2 show that
U ( Z ,6) > 0. Recalling (3.8), (4.9) and according t.0 the
fact that W verifies :
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Feedback equivalence, and the global stabilization
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