On the solution of a Hamilton-Jacobi type equation in nonlinear discrete-time H/sub ∞/-control PDF

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Proceedingsof the 33rd Conference on Decision and Control WP-16 3150 - Lake Buena Vista, FL December 1994 On the solution of a Hamilton-Jacobi type equation in nonlinear discret e-time Hw-Control H. Guillard, S. Monaco and D. Normand-Cyrot Abstract-The subject of this paper is to show that there tem...

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Proceedingsof the 33rd Conference on Decision and Control Lake Buena Vista, FL December 1994

-

WP-16 3150

On the solution of a Hamilton-Jacobi type equation in nonlinear discret e-time Hw-Control H. Guillard, S. Monaco and D. Normand-Cyrot Abstract-The subject of this paper is to show that there exists a smooth solution to the Hamilton-Jacobi type equation arising in nonlinear discrete-time H,-control provided that a solution to the Riccati equation associated to the linear approximated problem is available. An approximated solution of the Hamilton-Jacobi type equation can be then iteratively computed making use of algebraic expansions.

I. INTRODUCTION

A state-space approach of nonlinear H,-control

has recently taken a great interest in continuous-time ([l], [7], [lo]) as well as in discrete-time ([2], [3], [4], [5]).

In a continuous-time setting, when the system's equations are affine in the inputs, the existence of a controller solving the problem has been related to the solvability of a particular type of Hamilton-Jacobi equation ([l], [7], [lo]) whose solution leads to an explicit formulaof the controller. When the system's equations are nonlinear, the HamiltonJacobi and the controller equations are mixed up and have only solutions. In this set-up, based on a previ- implicit ~ U work S ([SI), a method has been derived t o find analytic solutions in terms of polynomial approximations ([7]). In a discrete-time setting, the problem has been treated independently in ([2], [3]) and ([4], [5]). In ([2], [3]), in a more general setting, the control law is characterized by two sets of equations not enabling a direct computation of the solution even if the system's equations are affine in the inputs. In [4], an extra and strong hypothesis of "quadraticity in the inputs" of the solution of the Hamilton-Jacobi type equation, allowed the present authors to give an explicit formula for the controller. Moreover, in [5], it is shown that, relaxing the quadraticity assumption, this controller still provides a solution to the problem if a strict inequality version of the Hamilton-Jacobi type equation is satisfied. The object of this paper is to characterize the solution of the Hamilton-Jacobi type equation and the related control. This is achieved in the "full information" case in terms of polynomial approximations of successive orders. In the first place, under the hypothesis that the system's equations are analytic, it is proved that the solutions of the implicit equations arising in the nonlinear context are also analytic by establishing a link between the solution of the HamiltonJacobi type equation and some standard hamiltonian sys-

tem. Then it is shown that the necessary and sufficient conditions known for solving the H,-control problem [6] associated to the linear approximation of the system are sufficient for computing iteratively approximated solutions. The method is developed on the basis of the approach proposed in ([2], [3]) for which no other way to compute the solution would be available. For convenience, the system's equations are supposed to be affine in the inputs but there is no difficulty to treat the general nonlinear case. The paper is organized as follows: section 2 sets out some algebraic preliminaries. Section 3 defines the "full information" discrete-time H,-control problem and recalls a solution. In section 4, according to ([7], [SI), one gives approximate solutions t o the Hamilton-Jacobi type equation and the related control. 11. ALGEBRAIC PRELlMlNARIES

Let us consider a nonlinear discrete-time dynamics on Rn with a control U E RP,

A ( . ) and B ( . ) are analytic functions of appropriate dimensions. Let moreover V ( . )be an analytic function from R" to R and U*(.) an analytic regular feedback on R".

e

ax IlfA.

0-7803-1968-0/94$4.00@1994IEEE

+

II+A.

terms of homogeneous polynomials in z. Their expressions are computed from the series expansions associated to the dynamics (1). For, denoting by ( . ) [ ' ] ( za) homogeneous i degree polynomial in 2 or a vector of such polynomials, one considers the expansions A ( z ) = C i l l A [ ' ] ( z ) ,B ( z ) =

Cil0B [ ' ] ( z )U, * ( Z )= Gill U * [ ' ] ( z )V, ( z )=

V['](z) and A * ( z ) = C i l l A * [ ' ] ( z with ) A * [ ' ] ( z )= i [ ' ] ( x )+ Ci-1 3 = O B b l ( 2 ) U * [ i - q z ); i 2 1. Denoting, as usual, the Lie derivative associated to a vector field ( A + = ( ( A + Bwa(z))i,l,.,,,nby LA+BU(')

H. Guillard and D. Normand-Cyrot are with the Laboratoire des Signaux et SystBmes, CNRS-ESE, Plateau de Moulon, 91190 Gif-surYvette, France. S. Monaco is with the Dipartimento di Mormatica e Sistemistica, Universitb di Roma "La Sapienza", Via Eudossiana 18, 00184 Rome, Italy. This work was partially supported by an italian MURST-40% funds.

+

Denoting A * ( . ) A ( . ) B ( . ) U * ( +it) ,will be shown in this section that the composed analytic functions V ( I A * ) , av .B and BT .B admit series expansions in

=

" t=l

a

-(.)(A ax,

+ BU),

and by L2pBu its tensorial product, namely

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111. SOMERECALLS O N NONLINEAR H,-CONTROL I N

where 8 represents the tensorial product of matrices, one sets (see [9]): 1

eXP@LA+BU:= A A + B U := I + 121

DISCRETE TIME

A . Problem formulation Consider a nonlinear discrete-time system

7%! L 8A' t B U

where I represents the identity operator. According to [9], let us recall the following lemmas.

Lemma 1: V(I+A*) =AA.(V)

Yk

[ zt ]

=

(7)

I,

where B(z) [ Bl(Z) Bz(zc) D ( z ) = [ D11(z) D l z ( z ) Z k E R" is the state, the inputs are represented by wk E R"l(exogenous input) and 'uk E Rma (control input), and the outputs by E Rpl (tracking error) and ?Jk E RPa (measured variables).

1.

and

where line, is an operator on matrices with p columns. If L is a m x p matriz whose lines are denoted b y L i , i = 1 , . . . , m, then line,(L) is a line matriz of dimension mp defined b y line,(L) = [L1 . . .Lm].

The matrices A ( z ) , Bl(z),Bz(z), Cl(z), D11(z) et D I Z ( Z ) are analytic mappings of suitable dimensions defined in a neighbourhood of x = 0 in R". Standard assumptions are the existence of an equilibrium in 2 0 = 0, i.e. A ( 0 ) = 0 and, provided a suitable change of coordinates, Cl(0) = 0.

Lemma 2: The operator AA+SUis linear, namely

For convenience, one notes R(z)

aA+su

(E VI']) = E '22 i22

AA+BU

1,

[

(v['])

= DT(z)D(x) -

and S(z) = C T ( z ) D ( z )which are also an-

alytic.

Lemma 3:

The problem of disturbance attenuation via "full information" feedback can be expressed in the following terms: find a control u k = u ( Z k , W k ) achieving: Substituting the expansions of A* and V into (3), (4) and (5) and regrouping homogeneous terms one obtains

Proposition 1: The composed analytic function (3) admits the following ezpansion with respect to z V(z

+ A'(z)) =

1) Local asymptotic stability of the equilibrium zo = 0. 2) Disturbance attenuation in the sense of making the Lz gain of the closed loop system less than or equal to a prescribed real number 7 > 0, i.e. satisfying the inequality N

l7[31(5)

with ?bl(z) =

N ZzZk

322 k=O

E{=," Pi (Vb-'1(

5

y2

WrWk

VN

2O

k=O

ad-

for every sequence w = ( W O ,w1, ...) such that the resulting trajectory remains in a neighbourhood of xo = 0.

Proposition 3: The composed analytic function (5) admits the following ezpansion with respect to z

Partial solutions have been recently given independently in [2] and [4],[SI. The solution in [2], based on the implicit function theorem, seems to be a natural way to deal with the problem, although it provides only an existence result and no explicit formula for the control law.

Proposition 2: The composed analytic function mits the following ezpansion with respect to x

(4)

B. A solution with W b l ( z )=

Qi ( V b - i l ( z ) ) .

One considers a function H ( . , ., .) defined in a neighbourhood of (2, w , U ) = ( O , O , 0) in R" x Rml x Rm2by

with Y b l ( z )=

H(x,U)= V ( A ( z )+ B ( z ) U ) + z T z - y 2 w T w

0

T, V b - i J ( z )

The formal operators Pi, Qi and Ti, defined in appendix are of order i in the sense that they associate to any polynomial of degree j in z a polynomial of degree i j in z.

(8)

where V ( . ) is a C k (IC 2 2) function from R" to R and r

+

_

.

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Let us note A [ l ] ( z )= A z , B[O](z) = B =

The equations arising in such a set up are

au ( 2 , U * ( 2 ) )= 0

Cpl(z) = CIS, D[OI(z) = D =

(9)

[ Dll

012

1, 1, 5f11(z)=

[ B1 B2

x T S = x T C D and R[O](z)= R = D T D -

and

The linear approximation of system (7) is H(Z,U'(Z)) = V ( 2 )

where U * ( z ) provides a saddlepoint solution to the function If(.,.) (see [2]). Equations (9) and (10) turn out to be

ax A * ( = ) . B ( z )+ 2S(2) + 2 U * T ( Z ) R ( 2 )= 0

(11)

It will be hereafter assumed that there exists a controller solving the H,-control problem for (16). According to [6], a necessary and sufficient condition is There exists a matrix X , 2 0 satisfyang the Riccati equation

(11) is the implicit equation for U*(z)and (12) is the discrete-time analogue of the Hamilton-Jacobi equation arising in a continuous-time setting, so one will refer to equation (12) as to a "Hamilton-Jacobi type" equation.

X, = A T X , A

+ C r C l - F T ( R+ B T X , B ) F

(17)

+

where F = - ( R + BTX,B)-'(BTX,A DTC1) is such that p ( A B F ) < 1 . Moreover there ezists a non sinB T X , B = P J T with gular matrix T satisfying R -I 0 J = [ 0 11.

+

The control solution can be given in terms of U*(z) by

+

If a solution for the linear approximation (16) exists, one will say that the linear approximated problem is solvable.

where

For the expansions (14) and (15) to be performed, it must be proved that V ( z )and U'(z) are analytic.

About the analyticity of the solutions It is sufficient to prove that V(z) is analytic since in this case U * ( z ) is the solution of the implicit analytic equation (11) and therefore is itself analytic.

Thus the remaining problem is to solve (1 I) and (12) simultaneously. In the present paper, we propose approximated solutions to these equations.

In order to prove the analyticity of V ( . ) ,one establishes a link between the solution of (12) and a certain hamiltonian is the local system. More precisely, one proves that (2, stable invariant manifold of the hamiltonian system defined on the basis of the hamiltonian function

According to [8]and, more recently [7], one finds in section 4 an approximation of the solutions U*(z)and V ( z ) of (9) and ( l o ) , or equivalently (11) and (12), as polynomials functions of z, that is

E)

IV. SOLUTIONS OF THE HAMILTON-JACOBI EQUATION

It is easy to prove that the standard assumptions imply that R (z ) is nonsingular. Then the control

AND THE ASSOCIATED FEEDBACK

All the matrices describing system (7) being analytic, one considers their expansions A ( z ) = & l A [ ' ] ( z ) , B ( z ) =

c;>1

B [ W ,Cl(Z) = C p ( 4 , D ( z ) = &o R[;](z). S(z) = CiZl S['](z) and R ( z ) =

D['](z),

satisfies

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z)

aH

This proves that (I, is an invariant manifold of the hamiltonian system which is, according to (19), described on this manifold by the locally asymptotically stable dynamics z k + 1 = A(2k) B(zk)U*(zk). Indeed, its linear approximation t k + l = ( A B F ) Z k is stable since the linear approximated problem is solvable.

-(PrZ,a*(z,P)) = 0 aU

+

and the related hamiltonian system is given by

=

zk+l

=

pk

(19)

($)T(Pk+l?zk>o*(l,.P*tl))

( ~ ) T ( p * + l ~ T * . ~ * ( z k l p k + l ) )

N~~ differentiating with respect to type equation (10)

(20)

g)

Rewriting (11) and (12) in terms of the operators introduced in section 2 and setting z tains

one obtains aH

az I=,U*(=)

+

av

au*

aH

+

AA @ L B ( V ) ( Z ) 2S(z)

aul=,U*(*)~axl= = GIz

+ A ( z ) =A A * ( z ) ,one ob-

+ 2U'T(z)R(z)= 0

(23)

and

that is, because of (9) --

(21)

Then, remembering that U * ( z ) verifies ( l l ) , it can be shown that equation (21) is nothing but

(22)

Suppose that, at the ICth step, the state ( x k , P k ) of the hamiltonian system belongs to the manifold (2, i.e.

g),

=

g)

One concludes that (2, is the local stable invariant manifold of the hamiltonian system, which is analytic. Thus (z, is also analytic.

the Hamilton-Jacobi Solutions of the Hamilton-Jacobi equation and the associated feedback

H(z,U*(z)) = V(z)

pk

+

AA( V )(z)- V ( z )+ C (z) +2S(z)U*(z) +U'T ( z ) R z)U* ( ( 2 )= 0 (24)

where C(z) = CT(z)C,(z). Expanding C(z) = C['](z) with C['](z) = Cf"lT(z)CI'-kl(z), substititing this expansion as well as (14) and (15) to the corresponding terms into (23) and (24) and regrouping terms of the same degree in z lead to k

C ( Q ~ - - Z ( V [ ~ - ~ + ~ ] ( Z ) ) + W *+[2~s][ k~](( Z z=))oR(25) [~--'~(Z)) 1=1

in (20). Then it can be computed that z k

Then, assuming

one can apply the implicit function theorem to deduce that equation (20) has an unique solution p k + l ( . ) satisfying p k + l ( z k = 0) = 0. But setting z = zk in (22), it shows precisely that pk

=

ax l : * ( X k )

is a solution of (20) with

. Since the solution is unique, one has Xk

Pk+l

=

-ax 1 aV

=

av -ax 1

One immediately observes that at the first step (for k = 1) (25) and (26) restore the linear control providing a saddlepoint solution

A*(=,)

T A(3k)+B(zk)t(*(ZkrPk+~)

Theorem If t h e l i n e a r approzimated p r o b l e m is solvable, t h e n equations (25) a n d (26) are solvable tlk 2 1 . Sketch of proof

T

Thus, according to (18), one obtains U * ( z k ) ,showing that Pk+l

Then, one can state the following result.

U.*(zk,Pk+l)

_T - av ax

l=k+l

=

U * [ ' ] ( z=) F z

and the linear Riccati equation

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A Moreover, for any k > 2, standard manipulations of operators Q; and R; in (25) lead to ~ V [ ~ l (2zU)*+I k I T ( z ) ( R + B T X , B ) = 0

From these formulae one can deduce an approximation of the control solution. In order to do so, one needs an assumption on the linear approximation which is quite usual in the linear context.

A2 ~ ( D i z=) m2 where M['](x)depends on U*[l](z),...,U*["l](z) and Then one can state the following result. vf21($1,...,V['+l] ( 2 ) only. Corollary U n d e r A 2 , if the linear approximated problem Since R + BTX,B is invertible, one has i s solvable, then the control i n (13) providing a solution for the nonlinear problem can be computed at any desired I ~ * [ k l (= ~ )--(R+ B T x , B ) - ~ M [ ~ ~ ~ ( ~ ) ( 2 7 ) degree. 2 Analogous calculations show that the coefficient of U*[']in (26) is equal to 2(BTX,(A

To calculate U(z,w ) , one considers, instead of (13)

+ B F ) + RF + S)z

which is zero because of the expression of U*['](z) = Fx. It results that V['++'](x) verifies Vik+']( ( A + B F ) z ) - V [ k + l ] ( z+) N l k + l ] ( z )= 0

where

Proof

+

Now, considering G(z, w) = U+'(x) iiz(z)w, to solve (30) is the same as solving

(28)

N['t'](z) depends on V[2](x),...,V['](x) and

~ * [ + ' ] (.z..,~*['-'](z) ), only.

According to this, one can compute recursively V['+'l(x) from (28) and then U'['](Z) V L 2 1 from (27) using the following sequence

+

Since R 2 2 = DT2D+'2 BrX,Bz is invertible, it is very easy to see that, after expanding the different entities and regrouping the terms of the same order, one has

-

which is computed at the previous steps.

k-I

R~-'l(z)aI'l(z))

The problem is now to discuss the solvability of ( 2 8 ) . Reminding that gree i in x can is a line vector Vk+l.z@'+' and becomes

any homogeneous polynomial P ( z ) of debe written as P(x) = Pi.z@' where P; of dimension n i l one sets V[kt'](z) = N['+'](x)= N'+~.z@'+~.Equation (28)

or equivalently, with

with

I@'++' = I n k + l , n k + l

v k + l ( ( A 4- B F ) @ k + '-

= -Nk+,

where

Tz2and Ti2' are operators defined in the appendix.

Then one can recursively compute U['] starting with

+

One can show that assumption p ( A B F ) < 1 implies p ( ( A BF)@'+') < 1 so that ( A BF)@'+' - I@'++' is invertible . Then one gets immediately

+

+

which provides the linear control solution.

A

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Remark: The solution V ( . )of the Hamilton-Jacobi equation is required to be positive definite (see [2]). A sufficient condition is to suppose X, invertible. Indeed V ( . )would be positive definite at least in a neighbourhood of z = 0 in R" .

line,,

c

3 -2

( B , Ta2V ( = ) ~ ~ .~l(z))= =+A*(=)

320

T?(v~--I](=))

l = - ~

where

V. CONCLUSIONS The contributions of the present paper are, first, to prove that the solutions of the equations associated to the discrete-time H,-control problem are analytic, by explaining their...