Table Of ContentLecture Notes ni
Control and
noitamrofnI Sciences
detidE yb amohT.M dna renyW.A
101
.P .E Crouch
A. .J nav der Schaft
Variational and Hamiltonian
Control Systems
galreV-regnirpS
nilreB Heidelberg weN kroY
nodnoL siraP oykoT
Series Editors
.M A. Thoma • Wyner
Advisory Board
L D, Davisson • A. G, .J MacFarlane • H. Kwakernaak
.J .L Massey • Ya Z, Tsypkin • A. .J Viterbi
Authors
.rD .P .E Crouch
Dept. of Electrical and Computer Engineering
Arizona State University
Tempe, AZ 85287
USA
Dr, A. .J van der Schaft
Dept, of Applied Mathematics
University of Twente
.P O. Box 217
?500 AE Enschede
The Netherlands
ISBN 3-540-18372-8 Heidelberg Berlin Springer-Verlag New kroY
ISBN 0-38?-18372-8 Springer-Verlag New Heidelberg Berlin York
Library of Congress Cataloging in Publication Data
Crouch, R .E
Variational and Hamiltonian control systems.
(Lecture noteisn control and information sciences; )101
Bibliography: p.
.1 Control theory.
2. Calculus of variations.
3. Hamiltonian systems.
.I Schaft, A. .J van der.
.II Title
III. Series.
QA402.3.C74 1987 629.8'312 8?-26421
ISBN 0-387-183"72-8 (U.S.)
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© Springer-Verlag Berlin, Heidelberg 198'7
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PREFACE
This monograph grew out of a combined effort to prow a conjecture) formulated by
the second author, concerning the characterization of Hamlltonlan control systems
in terms of their variational input-output behavlour. Thls conjecture was moil-
rated by the Hamiltonlan Realization Problem as well as by the Inverse Problem in
Classical Mechanics. In the course of proving a slightly modified version of this
conjecture we developed some concepts, whose interest seems not to be confined to
Hamiltonlan control systems. In particular the concepts of the prolonged system
and the Hamiltonlan extension, based on considering the variational and the
adjolnt varlablonal control systems to any nonlinear system, are, as we believe,
of independent interest to control theory.
The main concepts and results of this monograph are contained in chapters (I) to
(6). In chapter (0) we give a brief introduction to Hamiltonian control systems,
wlth particular emphasis on the relations between physical and control theoretic
notions. Indeed, the study of Hamlltonlan control systems Is one of the places
where (theoretical) physics and system and control theory meet. We conclude the
monograph wlth chapter (7) discussing some possible extensions to the theory
presented, as well as some open problems.
Tempe
Enschede, June 1987
CONTENTS
retpahC 0 INTRODUCTION
retpahC I THE HAMILTONIAN REALIZATION PROBLEM 12
retpahC 2 VARIATIONAL AND ADJOINT VARIATIONAL SYSTEMS 33
retpahC 3 MINIMALITY OF THE PROLONGATION AND
HAMILTONIAN EXTENSION 39
retpahC THE SELF-ADJOINTNESS CRITERION 47
retpahC 5 THE VARIATIONAL CRITERION 60
retpahC 6 GENERAL NONLINEAR SYSTEMS 85
retpahC 7 FINAL REMARKS DNA SOME OPEN PROBLEMS 96
References 116
0. INTRODUCTION
Of central importance in the modelling of physical systems are the classical
Euler-Lageange or Hamiltonian equations. These equations describe the dynamics of
a very large class of conservative physical systems, Including mechanical and
electromagnetic systems, and l el at the heart of the theoretical framework of most
physics. Although the conservation of energy is usually an idealiZation, in many
eases eht neglectlon of dissipation of energy (friction, damping) forms a natural
startlng point.
Let us consider, for example, a conservative mechanical system with n degrees of
freedom, locally represented by n (generalized) configuratlon variables ql,...,qn.
The Euler-Lagrange equations are the following well-known set oC second-order dlf-
ferentlal equations
(0.1) d (eL) _ B__~L = FI i = 1 ..... n
dt ~a l qa i
where L(q,q) - .... L(q|, qn,ql ..... qn) is the Lagrangimn of the system. In most
mechanical systems the Lagranglan is the dlfference of a kinetic energy T(q,q) and
a potential energy V(q)
(0.2) L(q,q) - T(q,q) - V(q)
where T(q,q) iS quadratic In the generalized v~locitles
(0.3) T(q,q) = ~ I qTM(q)q
for some posltlve-definite matrix M(q). In this case the Euler-Lagrange equations
specialize to
)4.0( ~ (3T) 3T av + ,iF n ..... i = I
d~ ~qi aqi ~ql
dna the terms ~ _ av represent the internal consorvatlve (i.e. derivable from a
potential) forces in the system. Flnally the vector F I = (F .... ,F n) denotes the
(generalized) external forces acting on the system while in configuration
(ql ..... nq )"
In the (mathematiCal) physics literature the external forces Usually are seen as
2
Riven functlons of time. Consequently, the external forces are often split into
two parts: one "maximal" component which is derivable from a potential function,
and so can be added to the internal forces, and remaining non-conservative forces.
Alternatlvsly, in stochastic mathematical physics the external forces are modelled
as stoohastlo variables [Bi]. In systems and control theory the approach, however,
is quite different. Usually (some 0£) the external forces will be interpreted as
control Or input variables; i.e., "arbltrary" functions of time. Instead of consl-
derlng the Influmnce of the environment on the system as given, one is primarily
interested in the way the system will react to difiCerent external forces. Of
course this is intimately related to the fact that in control theory one wishes to
prescribe the beha vlor of the system, instead of only describing It [BI].
In general not all degrees of freedom are dlrectly accessible to control action,
resul~ing in Euler-Lagrange equations of the form
d )__L_B( LB '( iU i = 1,...,m
(0.5)
i = m+1,...,n
3q i
where now u - (uI,...,U m) are the controls or Inputs (i.e., "arbitrary" functions
of time). eW call (0.5) a Lagranglan control system.
As is well-known, the Hamlltonlan equations of motion are obtalned from the Euler-
Lagrangs equations (0. )I by defining the generalized momenta
(0.6) Pl i a6
In most oases the transformation from (41 ..... 4 n) to (Pl ..... pn ) is a (local)
diffeomorphlsm, allowing us to transform the Lagrangian L(q,q)Into the Hamil-
n
tonlan H(q,p) = ~ plql -L(q,q) (the Legendre transformation), and the (second-
i=I
order) Euler-La~range equations into the set of flrst-order differential equations
~i = &M- lP~
(0.7) i = I ..... n
+
~ql Fi
which are called the Hamiltonlan equations of motion. In case the Lagranglan is
gi%~en as in (0.2)-(0.3) the Hamlltonlan becomes
(0.8) H(q,p) = ~ 1 pTM-1 (q)p + V(q)
and so denotes the (Internal) energy. Finally the Lagranglan control system (0.5)
results in the Hamlltonian control system
aH i = I ...,n
=
(0.9)
~H +ful I = 1 ..... m
Pl =-~ql Lo i=m*l .....
For (0.5) as well as (0.9) we have assumed for simplicity that the inputs i, i u =
1,...,m, are directly coupled to the first m degrees of freedom. Of course hhls
particular form Is not invarlant under a nonslngular change o£ configuration coor-
dinates
(o.1o) % = ~i(~, ..... ~n ) i : I ..... n
with the Jacoblan De(q) everywhere non-singular. In fact, as can b~ easily
checked, under such a coordinate transformation the Lagrangian control system
transforms into
- ~ = uj ~ i = 1,...,n
(o. tl) EL-~-iJ I a~ j~1 a~ i
with L(q,q) = L(q,q), while the Hamiltonlan control system (0.9) becomes
(0.12) I = 1 .... ,n
aH m j~a
where ~(~,~)
= H(q,p).
Let us from now on concentrate on Hamlltonlan control systems. Notice that (0.12)
suggests we enlarge the class of Hamiltonlan control systems to systems of the
form
~H o
(0.13) I : 1,...,n
0 ~H m ~Hj
÷ Uj
with p) the HO(q, internal Hamiltenian, and H~(q), j ~ 1,...,m, arbitrary (smooth)
functions. In particular, this form iS clearly invarlant under a change of confi-
guration coordinates.
We shall even go a little bi~ further. Part of the power of Hamiltonlan formalism
is to regard the generalized momenta Pl on the same footing as the generallzed
configuration coordinates qi" Consequently, one does not only allow for transfor-
8L
motions of the configuration coordinates ql (with Pl ~ --T resulting in a brans-
forma~lon of the pl), but one considers all coordinate ~ql trans~v ~^rma tl s on {q,p)
(q,p), which leave the Hamiltonlan form of tha equations Invarlant, i.e. the
canonical transformations. Under such a general canonical transformation, the
functions HI,.o.,H m become functlons of .~nd q p, and therefore we define a general
(afflne) Hamiltonlan control system as
3H O m ~Hj
: - ~ Uj 3p i
(0.14) i : 1,...,n
0 ~H n jH~
+ Uj "~
where the functions H 0,H I,...,H m are all arbitrary functions of q and p. (See
Example 3 for a physical interpretatlon.)
Notlc9 that a general Hamiltonlan system (0.14) can also be regarded as a set of
tlme-varylng Hamlltonlan dlf ferantlal equations governed by the tlme-varylng
Hamiltonlan
m
40.15) Ho{q,p)- .~ uj(t)Hj(q,p)
j-1
This can be interpreted in the foilowlng abstract way (see also [BuI,Bu2, VI])° The
possibility of oontrolllng the system with HamIltonlan (internal energy) HO(q, p)
rests on the ability to exchange energy wlth the environment along some external
channels. This can be regarded as the physical basis of conbol. The exchangeable
energy along the j-th channel is of the form Hj(q,p), and i u denotes the strength
of thlS energy exchange. Indeed, It is ~sily deduced that
oHd - ~ uj(t)
(0.16)
dt j~1
In physics the Hamiltonians Hj(q,p), j are called interaction or
coupling Hamlltonlans.
Up to now we have not yet defined outputs y of a Hamlltonlan control system. Of
course nothing forbids us to consider as outputs arbitrary functions of the state
x : (q,p). However, there Is a natural set of outputs associated to every Hamllto-
nian control system (0.14), namely the interaction Hamlltonians themselves:
(0.17) yj : Hj(q,p) J = I ..... m
There are many good reasons for doing this. First of all with this choice of
outputs we obtain From (0.16) the energy balance
dHo ~ "
(0.18) ~---d = Ujyj.
j=1
Hence the decrease or increase of the internal energy of the system Is a function
of the inputs and (the tlme-derlvatlvos of) the outputs only_. (Compare with the
work of Wlllems on dlsslpatlveness [W3].) For example in the simple case (0.9)
where yj = qj, J = 1,...,m, we have
dH0 ~ "
(0.19) ~'--d " ujqj
j=1
m
and ~ ujqj equals the instantaneous external work performed on the system.
j=1
Secondly, with thls particular choice of outputs we obtain the following symmetry
or reclprcclty between inputs and outputs. The external "forces" ul,...,u m in-
fluence the system via the external channels corresponding to the outputs
H ,I . ..,Hm, which are the "dlsplac~ments" caused by these excitations along the
same llne of action. For example in case Hj(q) = qj, j = 1,...,m, the input ju
~quals the external force corresponding to the J-th degree of freedom qj. Hence If
qj Is a Cartesian coordlnate, then uj will be a translational force, while if qj
is, say, an angular coordinate, then u. will be the corresponding external torque
3
(see also Example 2). Notice also that in the original Euler-Lagrange or Hamilto-
nlan equations I) (0. and (0.7) the vector F - (F I .... ,F) represents the external
forces as measured in the configuration (ql,...,qn). Hence in order to define the
external forces we need to know the configuration coordinates {ql,...,qn ) which
are (if we interpret i F as lnputs) just the natural outputs of the system. Let us
also remark that in order to define a general coupling Hamiltonian we need this
6
to boa function of the observations made on the system, i.e. a function of the
natural OUtpUts. A third argument for choosing yi as in (O.]6) has a more system-
theoretlo flavouP. Consider for the Hamiltonlan system 40.13) a state feedback law
ui = ai(q, p) + v~, with v~ the new inputs. When is the system after feedback again
Hamlltonlan? This is the case ([VI]) if and only if there exists a function S such
that
40.20) ~j(q,p) : ~-a~S j(H1(q,p) ..... Hm(q,p)) j = I ..... m
o output
l.e., if and only if the feedback is feedback with respect to the natural
outputs and furthermore has the special form as in (0.20).
From a mathematical point of view this dlscusslon can be summarized by noting that
the space of Inpubs and natural outputs can bs given the structure of a cotangent
bundle T'Y, where Y is the output manifold wlth local coordinates (yl,...,ym), end
where the coordinates of the fibers of this b~idle are the inputs (or external
forces) U .i Concludlng, we define a general (afflne) Hamiltonlan input-output
system as a Hamiltonlan control system (0.14) together with the natural outputs
(0.17).
Finally let us notice the close similarity wlth the description of electrical
networks wlth external ports. In thls case each external port carries two "dual"
Variables, current and voltage, which are also needed for stating an energy
(in-)equallty. In our formallzatlon of Hamlltonian systems the external channels
involve the dual variables % ("force") and yj = Hj ("displacement"). For more
details, including a partial treatment of the theory of interconnectlons of Hamil-
ionian input-output systems we refer to IV1].
Some examples
.I Consider the following linear mass-sprlng system (without frlotlon)
I k 2 k
'2m 1
ql q2 = y
The Hamlltonlan 0 H here is the sum of the kinetic and potentlal energies of
both masses I m and m .2 If u is the external force on mass 2 m then the natural
output y is the displacement q2 of this second mass. The same holds for the
first mass m .I
