Nonlinear Model Matching with an Application to Hamiltonian Systems

Copyright © IFAC l\'onlinear Control Systems Design, Capri , Italy 1989

NONLINEAR MODEL MATCHING WITH AN APPLICATION TO HAMILTONIAN SYSTEMS H.

J. c.

Huijberts

University of Twente, Department of Applied Mathel{wtics, p.a. Box 217 , 7500 AE Enschede, The N etherlands

Ab s t r act Some recent advances in the solution of the nonl inear Model Matching Problem are pursued further . First we study the Model Matching Problem making use of the notion of clamped dynamics or zero dynamics and a version of the modified Hirschorn algorithm. Then, making some extra assumptions, we generalize recent results on th~ nonlinear Model Matching Problem with prescribed tracking error for SISO-systems to MIMO-systems. Finally, our results are applied to the problem of matching Hamiltonian models by means of Hamiltonian plants. Special attention is paid to the structural properties of the system after we have solved the Model Matching Problem with prescribed tracking error. Keywords Nonlinear control systems, control system synthesis, Model Matching Problem, Hamiltonian systems.

loop

systems,

2 . THE MOOEL HATCHING PROBLEM

1 . INTRODUCTION

In this paper we consider a plant P described b y differential equations of the form

The Model Matching Problem consists of designing a compensator for a given system in such a way that the resulting input- output behavior matches that of a prespecified model. For linear systems the problem has been solved by several authors (Moore and Silverman, 1972; Morse, 1973; Emre and Hautus, 1980; Malabre, 1982) in different settings, while for nonlinear systems the problem has been treated by Isidori (1985a), Oi Benedetto and Isidori ( 1 986), Oi Benedetto (1988) .

(2.1) and a model H described by differentia l of the form:

z-

A(z) +

L u 1B i (z) ,

equat i ons

,(2.2)

YH - D(z)

1-1

where: x

Oi Benedetto (1988) investigates the problem of matching a linear model by using the notion of clamped or zero dynamics . This dynamics is calculated by means of a version of the modified Hirschorn algorithm (Isidori and Moog, 1988; Van der Schaft, 1988) . In section (2) of this paper we solve the problem posed by Oi Benedetto (1988) by means of another version of this algorithm in order to get conditions for solvability that are more in accordance wi th the sol vabil i ty condi tions in the linear case. Moreover,

closed

E

X, an open subset of ~n.

Z E Z, an open subset of ~r. f(x) 'gl(X)' . .. ,gm(x) ,A(z) ,Bl(z), ... ,B,,(z)

are smooth vectorfields. hex)

- (hl(x), .. . ,hp(x))T

is

a

smooth

is

a

smooth

function from ~n into ~p. D(z)

- ( Dl(z), ... ,Dp(Z))T

function from ~r into ~p.

connections between the two

solutions are given . i-1

i-I

Byrnes, Castro and Isidori (1988) give a solution to the Model Matching Problem with prescribed tracking error for SISO- systems. In Section ( 3) we generalize these results to MIMO- systems under some additional assumptions.

be abbreviated by g(x)u, B(z)v respectively.)

In Section (4) we apply the results of sections (2) and (3) to the matching of Hamiltonian models by means of Hamiltonian plants. Special attenti.on is paid to the structure of the limiting system (the system that remains if t ~ 00 after we have solved the Model Matching Problem with precribed tracking error). Some comments are made on the significance of the limiting system for stability of the o'lerall system. Moreover, it turns out that under ~ mild assumption the assumptions made in sections (2) and (3) and the conditions for solvability of MHP are always satisfied for simple Hamiltonian systems, a class of Hamiltonian systems that is of particular interest for ap pl ications.

A compensator Q used to control P is assumed to be

Remark In many cases we would like the model to be a linear system (Isidori, 1985a; Oi Benedetto , 1988).

a dynamic state- feedback compensator, with input s and v:- (v l ' ... ,V.,)T and output U:- (u l ' ... ,u..)T, of the following general form:

x

~ - a(~,x,v),

U -

b(~,x,v)

(2.3)

where: ~ E 3, an open subset of ~v a(~,x,v) a smooth vectorfield b(~,x,v) a smooth function from I/v-+n+m into 111".

53

H. J. C. Huijberts

54

The system P controlled by Q is denoted by poQ, and

Then

its output by ypoQ

a(E,x,v) -

(2.4) Definition (Di Benedet , Q, 1988) Given a plant P - (f,g,h), a model H - (A,B,D) and a point (x o ' zo), the Hodel Hacching Problem (MMP) is solvable from (xo,zo) if there is an integer v, a compensator Q of the form (2.3) and an initial state Eo for Q such that for all c: /oQ(xo,Eo, c) - YM(zO' c) is independent of v If in addition the existence of an initial state of Q such that for all c:

E~

/oQ(xo,E~,C) - YM(ZO'C) - 0 is required, the problem will be referred to as the Scrong Hodel Hacching Problem (SMMP). 0 For the solution of (S)MMP we introduce an excended associated with the plant and the model (Di Benedetto, 1988): ;l - f'(x E) + i(xE)u + BE(xE)v (2.5) hE(XE) E with states x - (x,z), inputs u and v, and

a

fE(x) -

[~i;n,

BE(xE) _ [0

B(z)

]

i(x

,

E )

_

[g~X)],

(2. 9 ) Procedure E Define No as the set of points x - (x, z) such that hex) - D(z) - O. Assume that No is a smooth submanifold of X X Z. E E Step k Nk :- {x E Nk_1lsuch that f'(x ) E Step 0

E Tx E Nk - 1 + span(i(x )}}

If the plant and the model are linear systems, it is well known (Emre and Hautus, 1980) that ~lliP is solvable

if

and

only

if

ImBE

c

V* + ImgE

(2.6),

where V* is the maximal (f' ,gEl-invariant subspace E contained in Ker dh . (See Morse (1973), Malabre (1982) for equivalent but slightly different conditions.) Note that in the linear case MMP is solvable from any point if (2.6) holds. Furthermore, in the linear case solvability (lf MMP is equivalent to the solvability of SMMP. Di Benedetto and Isidori (1986) have made an attempt to generalize (2.6) to nonlinear systems. V* is generalized by means of fi*,

the maximal controlled E

invariant distribution contained in Ker dh . However, in this way condition (2.6) only yields sufficient conditions for solvability of MMP. Di Benedetto (1988) gives a solution to SMMP using the notion of clamped dynamics. This clamped dynamics is calculated by means of the modified Hirschorn algorithm (Isidori and Moog, 1988; Van der Schaft, 1988). A slightly modified version of the procedure of Di Benedetto (1988) for calculating the clamped dynamics is given by: (2.7) Procedure E Step 0 Define Ho as the set of points x - (x, z) such that hex) - D(z) - O. Assume that Ho Step k

is a smooth submanifold of X X Z. E E Hk :- {x E Hk_11f'(x ) + BE(XE)v E

X

X

o

Z.

We assume again that this procedure succeeds in producing smooth submanifolds at every step and that it converges in a finite number of steps, say

p . Denote

N*:- N

p

to

the

'

following modification of theorem

(2.10) Theorem SMMP is solvable from any x~ for all XoE

E

BE(x~)v

E N*

if and only if

N* we have: E

T

x

E

span(i(x~)} for all v.

N* +

o

o

In theorems (2.8) and (2.10) we have given necessary and sufficient conditions for solvability of SMMP. We will now give necessary and sufficient conditions for solvability of MMP under some extra conditions. We remark here that similar conditions have been given by Isidori (1985a) in a slightly different setting and a less explicit way. In order to state our assumptions we define characteristic numbers Pi (x), 0i (z) (i E !E), for the plant (2.1) and the model (2.2) respectively, in the following way : Pi(X) - min pj Em s.t. Lg L~hi(X) .. O} j kEIN 0i(Z) - min {3j Em s.t.

Ls

L~Di(Z)

..

O}

(2.11)

kEIN

Our first assumption is: (Al) Pi(X) and 0i(z) exist and are constants, Pi (x) - Pi < "", 0i(X) - 0i <

say

If (Al) holds we can define the decoupling matrix Ap(x) for (2.1) (Isidori, 1985b), in the following way:

E Tx E Hk - 1 + span(i(x )} for all v}

(Ap(x) ) ij - Lg L~ihi (x)

Assume that Hk is a smooth submanifold of X X Z.

that Nk is a smooth submanifold of

Assume

We come (2.8) :

hE(xE) _ hex) _ D(z)

with does

the job . Motivated by condi tion (2 . 6) for the linear case, we now consider the following procedure, which coincides with the zero dynamics algorithm (Isidori and Moog, 1988) applied to the extended system E (t,gE,h ) :

syscem,

i -

compensator of the form (2 . 3) A(O + B(Ov, b(E,x,v) - u*(E,x,v)

(2.12)

j

0

We assume that this procedure succeeds in producing smooth submanifolds at every step and that it converges in a finite number of steps, say Q .

Our second assumption is: (A2) Ap(x) has full rank for all x. (2.13) Lemma Assume

Denote: H*:- H .

that

(Al)

and

(A2)

hold.

Then

N*

is

a

Q

(2.8) Theorem SMMP is solvable from x~:- (xo,zo)

maximal maximal if and only if

E

u) contained in Ker dh . Moreover N* is given by:

x~ belongs to H*. Proof

See (Di Benedetto, 1988)

integral manifold of fi*, where fi* is the controlled invariant distribution (w. r. t.

N* -

o

{(x,z)IL~hi(X) - L~Di(Z); i

The sufficiency part of the proof is constructive: let u*(z,x,v) be a feedback that makes H* an invariant submanifold of the extended system (2.5).

Proof

Similar to 1985b)

E

E, k - O,l,,,,,pi}

(Van der Schaft,

1986;

(2.14) Isidori,

o

Nonlinear Model Matching with an Application

to

55

Hamiltonian Systems

(2.15) Theorem Assume that (Al) and (A2) hold. Then the following statements are equivalent: (i) MMP is solvable from ", ..y (ii) Vi El:'.: 0, ~ P!

x~ E X x Z

+ [Ap(I. + LAD.(z); k E '!!),'1)]i"U - L~i+1Di (z) -

0, <

for

Pi

i E I:'.

some

and

fix

x~ E X x Z.

- fdl. + LAD.(z) ,'1) (i - l, ... ,n-d)

- A(z) + B(z)v

rk

where

- L~,+lD,(za) + v~L~'D!(za) Hence y~

cannot be

made

of v,

independent

-

(r kO'

LAD. -

this implies that (2.1) anti (2.5) have the same structure at infinity (Nijmeij er and Schum .. cher,

. .. •

(D., .. .

E

'!!)

r ,p. ) ,L~'D.)

f,(~ ,'1) - L,'1,(~ ,'1)

which

implies that MMP is not solvable from x~. (ii) => (i) Assume that 0, ~ P, (i E E). As (AI) and (A2) hold

Then a compensator of the form: ~ - A(~) + B(~)v ",(x,/J) + -Yi(X,~)v

u, -

I [A;l(X)],j[L~jTlDj(/J)

",(x,~)

that makes 6" invariant (Di Benedetto and Isidori,

(Oij is the Kronecker delta)

Here

is

6"

the

E

distribution contained in Ker dh . By lemma (2.13) we know that this feedback also makes N" invariant. Hence the compensator mentioned below theorem (2.8) solves SMMP from any x~ E N-, (Hi) => (ii) Analogous to the proof of (i) => (ii) (Hi) _

(iv)

P i. • + ( L a,.(L,hi(x) - LAD,(/J»)O,j

.-0

-y,(x,~)

PRE-

(M)

involutive. m - p.

span(gl' . . . ,gm)

G:-

of the tracking error yE by a proper choice of the a'j if assumptions (AI), ... ,(A4) hold.

If we have indeed initialized the compensator in the right way, the asymptotic behavior of plant + compensator will be given by:

is {

Recall from section (2) that MMP is solvable from any point if and only if 0, ~ P, (i E m). Since (AI) and (A2) hold, the set (~'j(x); i E '!!, j-O, ... ,p,), where ~'j(x):-L~h,(x), is a set of independent functions (Isidori, 1985b). Moreover, since (A3) holds, this set can be comple~ed with

functions fulfilling

'11(x), ... ,'1 n -d(X) the

(where

constraints:

d-

L. '1 ,(x) j

i - I , ... ,n-d)

(see

e.g.

Huijberts

L(P,+l»

°'-1(j

and

Van

coordinates: (I,j; i E '!!, j - 0, ... ,p,) Define

the matrix ~(z)

as

the

[~(z) J!j: - ~ L~'D, (z)

der

(3.1)

(m,m)-matrix with and

for

a

- fi(LADk(~); k E '!! ),'1)

/J' -

A(~)

YJ

Dj(~)

-

(i E '!!)

+ B(/J)v

(3.5)

(j E '!!)

The state equations of (3.5) just describe the clamped dynamics of the extended system (2.5) (which, since (A1) and (A2) hold are equal to the zero dynamics, in the sense of Van der Schaft (1988), see also (Isidori and Moog, 1988». The manifold on which the clamped dynamics evolve is

N;

N;: -

x Z, where {XI~ij(X) - 0; i E rn, j - 0, ... ,Pi}' the maximal controlled invariant submanifold contained in {xICi(x) - 0, i E '!!}.

it:-

E '!!,

Schaft (1987) . Now define: I,j:- ~ij - LiD, and choose as new (i E '!!, j - O, ... ,p)

elements

[A;l(X)]," ~(/J)

°

~

distribution

-

J

provides the required matching, with the i-th output of the extended system satisfying: ECp,+l) E(Pi) E Yi + a ip / ' + ... + aiay, (3.4)

Moreover we assume:

The

L~j+1hj(x)

Remark Note that we have implicitly assumed that the compensator is initialized at the initial condition of the plant (see definition (2.4) of MMP). 0

In this section we generalize the results of Byrnes, Castro and Isidori (1988) to (a class of) MIMO-systems. We consider a plant (2.1), a model (2.2) and we assume again that (AI) and (A2) hold.

(A3)

-

Hence, like in the SISO-case (Byrnes, Castro and Isidori, 1988), we can get any asymptotic behavior

Follows by comparison of procedures (2.7) and (2.9) and theorem (2.11). 0

3. MODEL HATCHING WITH SCRIBED TRACKING ERROR

-

j-1

(~,gE)-invariant

maximal

(3.3) (i E '!!)

where:

1985). Hence MMP is solvable from any x~ E X x Z (Di Benedetto and Isidori, 1986). (H) => (Hi) I f o!~p, (iEE), there is a feedback for (2.5)

1986).

J,.v

'1,

U So,)

(3.2) [AM(z)

z

Then: h~(k) (x~) (k

'!!)

L~,+lhi(I. + LAD.(z); k E '!!) ,'1) +

I,p, -

N"

Proof (i) => (ii) Assume that

E

l,p,_l- I !P,

x~ E N"

(iii) SMMP is solvable from any (iv) H" -

(i

matrix

j

H, let H," denote its i-th row. Then in coordinates (3.1) the extended system (2.5) becomes (Huijberts and Van der Schaft, 1987; Byrnes, Castro and Isidori, 1988):

4. APPLICATION SYSTEMS

TO

HAMILTONIAN

In this section we will apply the results of the foregoing sections to the matching of Hamiltonian models by means of Hamiltonian plants. First we will give a definition of an affine Hamiltonian control system and introduce some tools for studying Hamiltonian systems (for details we refer to Van der Schaft (1984), Abraham and Marsden (1978».

56

H.]. C. Huijberts

(4.1) Definition Let S be a symplectic manifold with symplectic form w. Then a system

(4.5) Lemma '1k,l -

0,1, ... :

(4.2) are

Hamiltonian

vector fields

with Hamiltonian functions H,C 1 , •.• ,Cm' defined by setting w(X s '-) - -dH, w(Xc ,-) - -dCj (j Em), is j called an affine Hamiltonian control system. 0 Remark In the sequel Hamiltonian system will always mean an affine Hamiltonian control system. 0 By Darboux's theorem there exist local coordinates (ql'··· 'qn'Pl'··· ,Pn ) for a symplectic manifold (S,w), called canonical coordinates, such that

I

w -

1\

to

the

aH

In

dqi'

such

familiar

coordinates

expression

x - XH(x)

qi -

aH

api (q,p),

Pi - - aqi (q,p),

(i E '2), and similarly for XCj ' Note that this implies that a symplectic manifold is necessarily even dimensional. One particular subclass of Hamiltonian systems often encountered in practice, e.g. mechanical systems, is given below:

(4.3) Definition Let S be a symplectic manifold of the form T·Q with Q the configuration manifold with coordinates (ql' ... ,qn)' Let H(q,p) - K(q,p) + V(q), where

±I

n

K(q,p) -

gij(q)PiPj

and

the

matrix

(gij(q»)

i. j - l

is

positive

Cl' ... ,Cm:

definite

for

all

q,

and

let

Q ~~. Then the Hamiltonian system (2.4)

is called simple. The term K(q,p) kinetic energy and the term V(q) potential energy of the system.

is called the is called the 0

Given two symplectic

real valued functions F, G on a manifold (S,w), we define their Poisson-bracket by: (F,G): - Lx/ - w(Xr,X G ). In

I (~ .~ - ~.~) aqi api .

i-l api aqi

ad~G - G, a~G - (F,a~-lF) (k - 1,2, ... ). The Poisson bracket satisfies the Jacobi-identity: (F,(G,H)) + (H,(F,G)) + (G,(H,F)) - 0 for any F,G,H: S ~~. By the definition of the Poisson-bracket, for a Hamiltonian system (4.2) the characteristic numbers Pi (if we assume then to be constants) can be Pi - min (3j E 'E: (Cj,ad~iCi) .. 0) kEJN are finite all characteristic numbers decoupling matrix has elements defined by:

aij (x) -

(C j ,

submanifold

.

If

the

ad~iCi j(x) . T

c S

submanifold of S if

is

called

(-l)PiH(Ci,ad~iCj)

I f (Cj,ad~iCi) ,. 0, then Pj

(ii)

:s

o

P,

(4. 7) Theorem Consider a Hamiltonian system (4.2) and assume that its decoupling matrix has full rank. Assume that we have arranged the outputs in such a way that

.s ... .s

Pm'

Define numbers k l • . . . ,kH1w 1 •... ,,,,,Nld l • . . . IdN in the following way: (*) kl < ... < kN are the values taken by the characteristic numbers (*) wi is the number of characteristic numbers taking value k i i

(*) d i

-

I

wj

.

j-l Then: if k i is even, then wi is even (i E ~). Proof From corollary (4.6) it follows that A(x) has the following form: (*) A(x) has N diagonal block Di(x), where Di(x) is a (wi,wi) -matrix . (*) The elements of A(x) above the diagonal blocks are zero. (*) The elements of A(x) below the diagonal blocks may be non-zero. Hence, since A(x) has full rank, the diagonal block DI (x) must have full rank. Now assume that k i is even. Then by corollary (4.6) we have that Di(x) is skew-symmetric. Hence: Di(x) is askew-symmetric matrix of full rank, which implies that Di (x) is even-dimensional (Abraham and Marsden, 1978). Hence wi is even. 0

a

Furthermore we define inductively:

A

(4.6) Corollary (i) (Cj,ad~'Ci) -

Remark 1. If A(x) does not have full rank, k i is even does not have to imply that wi is even. Consider e.g.

canonical coordinates: (F,G) -

o

Using lemma (4.5) we can prove quite easily:

PI

dPi

i-l reduces

. .'

Proof By induction, using the Jacobi-identity.

a

symplectic

w:-

wiT is non-degenerate .on T. Now consider the special case that a submanifold T is given by T - {xIF 1 (x) - ••• - F 2k (x) - O}, where F 1 (x), ... ,F2k (x) are independent real valued functions on (S,w). Then T is a sympleccic submanifold of S if and only if: 'Ix E T,i E 2k 3j E 2k: (Fi,Fj)(x) ,. 0 (4.4) For the proof of this we refer to Van der Schaft (1987) .

Hamiltonian

system

on

T·~2

with

H(q,p)-

P2ql + p~eql+q2, C1 (q,p) - ql' C2 (q,p) - q2' 2. Theorem (4.6) and the above remark answer a question posed by Grizzle & Nijmeijer (1987) 0

We will return to the Model Matching Problem. Assume that the plant and the model are Hamiltonian systems evolving on symplectic manifolds (Sp,wp), (SM'~) respectively and described in local coordinates by (2.1) and (2.2), where: f(x):- XHp(x), gj(x):- XCj(x) , hj(x):- Cj(x) , A(z) : -

X~(z),

B j (z):- XOj (z).

Moreover,

we assume

that (Al), (A2), (A3) hold. Note that for a Hamiltonian system (A4) is satisfied by definition. Moreover, (A3) is not very restrictive: if all characteristic numbers of the plant are larger than zero, it is already satisfied (Huijberts and Van der Schaft, 1987). Now

let

S:- Sp x SM'

w:-

1f;Wp -

1r~,

where

?fp.

1fH

denote projections on Sp, SM respectively. Then (S ,w) is a symplectic manifold (Abraham and Marsden, 1978). We denote the Poisson brackets on (Sp,wp), (SM'~)' (S,w) by ( , )p, ( ')M' (,) respectively. Assume that MMP is solvable from any point , 1. e. G , ~ Pi (i E 'E)' Then it is obvious that after application of the compensator (3.3) in general the

57

Nonlinear Model Matching with an Application to Hamiltonian Systems system will not be Hamil.tonian anymore. However. the limiting system (3.5) will have a Hamiltonian structure. as we will show in the sequel. From section (3) we know that the dynamics of the limiting system (3.5) evolve on N; - {xl acfa p Ci(x); i

N":- N;

'E. k - 0 ....

E

x SM' where

,Pi}'

(4.8) Theorem

N;

is a symplectic submanifold of (Sp.wp). Proof Assume that Pi :s P2 :s ... :s Pm' Using the notation of theorem (4.7) we see: dim(N;) -

L (pj+l)

dim(Sp) -

Lwi(ki+l).

dim(Sp) -

j-l

1-1

Pi -

As

Dr(x)

such

that

kr ·

has

full

Pj - Pi

rank

N;

there

Then. using lemma (4.5) ~nd characteristic numbers we have:

must

be

a

(Cj.ad~~Ci)(X)" O.

and

the

definition

of

(-l)Pi(ad~~Cj,Ci)(X) .. O.

-

that (acfaci.ad!Cj)(x) .. 0. p

'E. k E

E

{O •... ,Pi}>

{O •...• Pj})

Thus by (4.4):

p

N;

symplectic submanifold of (Sp.wp ).

such is a 0

"~p~p -,,~

wL : -

projection on submanifold of

N" •

on

where

"N p

N;. is

Then (S.w) with

on

the

symplectic

is a symplectic wiN" - ~ (Abraham and

submanifold

(N".w L )

and Marsden. Mathieu-transformation. (Abraham 1978). Denoting the kinetic energy matrix of the plant

{:

X HM (Jl) -

where 6:-

VjXOj(Jl)

(4.l0b)

'E)

(4.l0c)

(ad~Dk(Jl);

(j k E

E

'E.

j

of

_g~i(q).

Since Gp(q) is posit ,i ve definite for all q we have:

( ii)

Hence

0

i

~

Pi

(i E

'E)'

'E). and by theorem (2.15): MMP

is solvable.

0

5. CONCLUSIONS

j·l

YJ - Dj(m)

choice

of

(I•. lOa)

L

this

yieldS:

g~i(q) > 0

(S.w) are given by:

- XBp ('1. 6 )

Gp(q):-(g~j(q»).

by

coordinates

Analogous to (i) we can prove that 0i - 1 (i E

Marsden. 1978). The dynamics of the limiting system (3.5)

0)

the

N"

N;.

(4.11) Theorem Assume that the plant (2.1) and the model (2.2) are both simple Hamiltonian systems satisfying (AS). Then: 0) (Al) ..... (A4) are automatically satisfied (H) MMP is always solvable Proof

and

N;.

canonical coordinates for In these coordinates we will have: £«(.'1) - XHp «(''1). on

m

and hence Pi - 1. Thus. (Al) is satisfied. Moreover. the decoupling matrix of the plant consists of the first m rows and colunns of Gp(q) . Thus. by the fact that Gp(q) is po :;itive definite for all q. the decoupling matrix ha:; full rank for all (q .p). Hence (A2) holds. Furth ,~rmore (A3) holds because Pi - 1 > 0 (Huijberts and Van der Schaft. 1987). As remarked before. (A4) is always satisfied for Hamiltonian systems.

Hence we can choose (local) canonical coordlnates Sp such that (ql.··· .qn.Pl.··· .Pn ) for (r :s n) are (Local) '1:- (qr+l.··· .qn.Pr+l.··· .Pn )

Define the symplectic forms wNp:- wpl N;

dim span{dD 1 •. • dD m } -

Consider the plant. As (AS) holds. we can find (local) canonical coordinates such that Ci (q) - qi (i E 'E) (e.g. by using a so-called

(Cj.ad~~Ci)(X) - -(adHpCj.ad~~- lCi)(X) -

Hence for all x and acfapc i (i E 1 there is an ad H / j (j E 'E. 1

-

Hence

by theorem (4.7): the dimension of is even. Now choose a kr (r E!!.) and let i E m be such that

j E!!.

We will syecialize to the case that the plant as well as the model are simple Hamiltonian systems satisfying the additional assumptions: (AS) dim span{dC 1 •. • dCm }

N

-

at most critically stable. but never asymptotically stable (Huijberts and Van der Schaft. 1987). Hence bounded input bounded state stability of (4.l0a) will in general not be achieved. This implies that for Hamiltonian systems we will have to look for another notion of stability than is advocated by Byrnes . Castro and Isidori (1988). It is hoped that the Hamiltonian structure of the system will be of help for such a further analysis. This remains for further research.

- O •...• Pk).

Hence the limiting system consists of an affine Hamiltonian control system mimicking the behavior

N;

of the model and a general Hamiltonian system on (the maximal controlled invariant submanifold contained in (xICi(x) - o. i E m) which is unobservable and driven by the outputs of the affine Hamiltonian system and its derivatives. As already pointed out by Byrnes. Castro and Isidori (1988). the dynamics of (4.10) are of great importance for the stability properties of the closed loop system. Especially (4. lOa) • which describes the dynamics of the inverse system of the plant (Byrnes. Castro and Isidori. 1988) is of importance. As (4.l0a) describes the dynamics of a general Hamiltonian system (Van der Schaft. 1984). where we can interpret 6 as inputs. the stability discussion becomes quite delicate. One first observation reveals that (4.l0a) for 6 - 0 will be

In this paper we have given necessary suffic ient condi tions for sol vabili ty of nonlinear SMMP that are in accordance with conditions for solvability of the problem in

and the the the

linear

and

case,

Moreover ,

we

gave

necessary

sufficient conditions for solvability of the nonlinear MMP under some extra conditions. These results were achieved by using the notion of clamped or zero dynamics. It is expected that this notion will be of great use for deriving necessary and sufficient conditions for the general solution of the nonlinear MMP. Also further study of the modified Hirschorn algorithm is required in order to wipe away the assumptions made in procedures (2.7) and (2.9). This remains for further research. In section (3) recent results on the nonlinear MMP with prescribed tracking error for SISO-systems have been generalized to MIMO-systems. making some extra assumptions. In section (4) the results of sections (2) and (3) have been applied to Hamiltonian systems. It turned out that the topic of internal stability of the closed loop after we solved the nonlinear MMP with prescribed tracking error becomes quite delicate in this case. Further research on this topic is required. Moreover it is shown that for simple Hamiltonian systems. a class of systems that is of particular interest for

58

H.]. C. Huijbens

practical purposes, under an extra assumption the assumptions of sections (2) and (3) are always satisfied and that the nonlinear MMP is always solvable. Here it should "2 noted that from a practical point of view this extra assumption is quite a weak assumption.

ACKNOWLEDGMENTS I would like to thank A.J. van der Schaft for some very useful discussions.

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Emre, E. and M.L.J. Hautus, (1980). A polynomial characterization of (A,B)-invariant and reachability subspaces. SIAH J . Control & Opt . , ll, 420-436. Grizzle, J.W. and H. Nijmeijer, (198 7 ) . Personal communication.

Huijberts, H.J.C. and A.J. van der Schaft, (1987). Input-output decoupling with stability for Hamiltonian systems. Memo nr. 648, Department of Applied Mathematics, University of Twente , Enschede. Submitted for publication. Isidori, A., (1985a) . The matching of a prescribed linear input-output behavior in a nonlinear system. IEEE Trans. Aut. Contr., 30, 258-265. Isidori, A., (1985b). Nonlinear Control Systems: An Introduction, Lecture Notes in Control and Information Sciences 72 . Springer, Berlin. Isidori, A. and C.H. Moog , (1988). On the nonlinear equivalent of the notion of transmission zeros. In C. Byrnes and A. Kurzhanski (Eds . ) , Modelling and Adaptive Control, Lecture Notes in Control and Information Sciences 105. Springer, Berlin. Malabre, M., (1982) . Structure a l'infini des triplets invariants. In A. Bensoussan and J.L. Lions (Eds.), Anal y sis and Optimizati on of Systems , Lecture Notes in Control and Information Sciences 44. Springer, Berlin. Morse, A.S., (1973) . Structure and design of linear model following systems . IEEE Trans. Aut. Contr., 18, 346-354 . Moore , B.C. and L.M. Silv erman, (1972). Model matching by state feedback and dl namic compensation . IEEE Trans. Aut. Contr., 11 , 491-497. Nijmeijer , H. and J.M. Schumacher, (1985). Ze r os at Infinity for Affine Nonlinear Control Sy,tems. IEEE Trans. Aut. Contr., 30, 566 - 573. Van der Schaft, A.J. , (1984) . System theoretic descriptions of phy sical s y stems, 011 Tract J. CWI, Amsterdam. Van der Schaft, A. J., (1986). On feedback control of Hamiltonian systems. In C. By rnes and A. Lindquist (Eds.), Theory and Applications of Nonlinear Control Systems. Elsevier, Amsterdam . (1986) . Van der Schaft , A.J. , ( 1987 ) . Equations of motion for Hamiltonian systems with constraints . J. Phys. A., 20, 3271-3277.

Van der Schaft, A.J., (1988). On clamped dynamics of nonlinear systems. In C.I . Byrnes, C.F . Martin and R. E. Saeks (Eds.), Analysis and Control of Nonlinear Systems. Elsevier, Amsterdam.