1. Introduction

The dynamics of a system can be formulated using either the Newtonian, the Lagrangian or the Hamiltonian approaches. In the Newtonian and the Lagrangian formulations it is assumed that if all the coordinates and velocities are simultaneously specified, the state of the system can be completely determined and its consequent motion calculated. This gives rise to solving a system of second order ordinary differential equations. On the other hand, the Hamiltonian formulation, we seek to describe the motion in terms of first order equations of motion parametized by generalized coordinates and momenta. The transition from the Newtonian and Lagrangian formulations to the Hamiltonian formulation corresponds to changing the variables in the system from to where are the generalized coordinates and are the generalized momenta [1]. However the Hamiltonian methods are not superior to the other methods for direct solutions of mechanical problems. Rather we gain another more powerful method of working with physical systems. The usefulness of the Hamiltonian viewpoint lies in providing a framework for theoretical extension in many areas of physics such as statistical mechanics and quantum mechanics.

A Hamiltonian system is characterized by an existence of a symplectic structure on a smooth even-dimensional manifold [2]. The symplectic approach allows one to extend the local description of a dynamical system to a global description. [3] has shown how network modelling of lumped-parameter physical systems naturally leads to a geometrically defined class of systems, called port-controlled Hamiltonian systems with dissipation. The structural properties of these systems were discussed, in particular the existence of Casimir functions and their implications for stability. [4] has shown the geometric property and structure of the Hamilton--Jacobi equation arising from nonlinear control theory are investigated using symplectic geometry. [5] made an analysis on Hamiltonian systems and his results revealed a systematic geometric frame for generalized controlled Hamiltonian systems. The pseudo-Poisson manifold and the ω-manifold are proposed as the state space of the generalized controlled Hamiltonian systems were established. However the above results did not consider the external forces as basic variables.

In this paper it is therefore intended to show that external forces should. It will be shown that it is necessary to maintain forces as basic variables and those have to be included in the definition of mechanical systems.

2. Hamilton’s Equation

Let be the Lagrangian function for a given system. Then the Hamiltonian function for the system is defined by ([1], [6]).

(1)

The differential of is given by

(2)

Also

(3)

Using from the generalized momenta, equation (3) reduces to

(4)

Comparing (4) and (2) we get

(5)

(6)

Equations (5) consist of a set of first order ordinary differential equations called the Hamilton’s equations of motion and are called canonical coordinates.

2.1. Hamiltonian Control Systems

According to [2] we let

● be a manifold of the state space in a manifold with symplectic form ,

● be a manifold of the space of external variables with symplectic form (‘e’ for external). In local coordinates, where are the external forces (inputs) and are the observations (outputs),

● A fiber bunder be over ,

● A smooth function (smooth meaning ) such that ( is a tangent bundle over .

Then

(i) with and symplectic manifolds is called a full Hamiltonian system if is a Lagrangian submanifold of where

is derived from the local coordinates.

(ii) is called degenerate Hamiltonian system if there exists a full Hamiltonian system such that is a submanifold of .

The definition of Hamiltonian control system depends on the submanifold and not on and separately. It can be observed that the set of external variables can be split into inputs and outputs . The inputs are the external forces (controls) and the outputs are the observations. If the external forces are constant, then the dynamics of the system are described by a Hamiltonian vector field on .

2.2. Affine Hamiltonian Control Systems

Let be a symplectic manifold. Let be an observation manifold. Define the symplectic form on by ( is the cotangent bundle of , and are pullbacks of and by and respectively [2]. Then according to [7] an affine Hamiltonian system is given by a submanifold such that

(i) can be parametized by the coordinates of and the coordinates of the fibres of ,

(ii) is a Lagrangian submanifold of ,

(iii) The value of the -coordinates of a point on is a function only of the -coordinates of this point.

This system is thus given by ([8])

(7)

In vector form we denote the system (7) by

(8)

Because of (ii) above, has a generating function. Because of (i) and (ii), this generating function has then form with canonical coordinates for and natural coordinates for . Therefore coordinates of are given by

(9)

and the coordinates by

This is equivalent to linearizing with respect to . We note that without condition (iii), the external forces enter the system in a nonlinear way and the generating function of is which locally gives

(10)

This is just the usual general input-output Hamiltonian system. We note that if there are no dynamics i.e. no state space , then is just a Lagrangian manifold of and this describes statistic mechanical systems. If there are no inputs and outputs i.e. no , then is a Lagrangian submanifold of . This describes a locally Hamiltonian vector field.

2.3. Linear Hamiltonian Control Systems

The Linear system in state form given by

(11)

is a linear Hamiltonian system if and are symplectic linear spaces. It is a result in symplectic geometry that there exists a nondegenerate skew-symmetric bilinear form and on the state space and on the set of external variables respectively ([2]). In the same language o symplectic geometry if and then there exists bases of and such that in these bases

[9] has established and proved by the following theorem the conditions required for the system above to be a full linear Hamiltonian system.

Theorem 1:

Let be a linear system given by (11) above. If is injective and are linear symplectic spaces, then is a full Hamiltonian system if and satisfy the following:

(11)

If the feedback given by , is applied to , then necessarily has to satisfy

(12)

where

The condition that is injective is similar to the condition that is an embedding for the case of nonlinear systems.

3. Controllability and Observability for Hamiltonian Systems

We shall consider the ideas of controllability and observability only for affine nonlinear Hamiltonian systems. Consider the system given by

(13)

Defined on a symplectic manifold where is a locally Hamiltonian vector field i.e. the Lie derivative . And is the Hamiltonian vector fields such that ([10]).

Definition 1

Consider the affine system (13) above. We define and . This system is locally weakly observable if satisfies Here is the linear subspace of spanned by with [11].

For Hamiltonian systems, since and there exists such that then the defined above satisfy with the affine subspace of vector space of functions on . ([6])

We define Controllability and Observability as follows:

Controllability: Controllability is the set of points reachable from in time by applying input functions with initial condition contains an open subset of for every and for every ([9]).

Observability:

For every and every there exists a neighbourhood of such that for and in and , there exists input functions such that, if we denote the solutions of on corresponds to initial conditions and by and respectively, then the output functions and are different while the trajectories and remain in ([9]).

4. Feedback for Hamiltonian Systems

Define a function such that , where is a closed one-form on the output , and consider the output feedback given by

(14)

Let be the graph of and be a Lagrangian submanifold i.e. . Accordingly we say that the output feedback given by equation (14) is Hamiltonian ([5]).

Proposition 1

Let be an affine Hamiltonian system given by

with and so and i.e. ([5]). Let be a feedback for this system. After feedback, this system will again be an affine Hamiltonian system.

iff is a Hamiltonian feedback i.e. there exists a function such that and satisfy

Hence Hamiltonian feedback adds a potential function which is only a function of the output.

Let us now consider a solution of Disturbance decoupling by observation feedback (DDOF). The formulation of DDOF is as follows: Let be a Hamiltonian system on a symplectic space where . It is assumed that there are disturbances in this system and it is intended to control the state space. The system can be described by

with the disturbances and the variables which are to be regulated. We shall call with Hamiltonian and a Hamiltonian system with disturbances. Then the DDOF problem is to find a compensator

Such that the closed-loop system

decouples the disturbances from .

We shall require to posses a symplectic form and

Proposition 2

Let be a Hamiltonian system with disturbances. Then

(i) DDOF is solvable iff there exists an -invariant subspace contained in and which is coisotropic ([10]).

(ii) DDOF is solvable if the pullback is coisotropic ([7]).

Let be a Hamiltonian system with disturbances. Let be -invariant and Lagrangian, so DDOF is solvable by static output feedback . Then also the Hamiltonian output feedback solves DDOF. ([7]).

5. Discussions and Conclusions

In the past, most treatments in classical mechanics have dealt only with analytical mechanics. This part of mechanics confines itself to the study of mechanical systems without external influences. When forces are present they are assumed as coming from a potential field. In this context one observes the motions, makes classifications etc. i.e. one does only descriptive work but cannot influence the behaviour of the system. This restriction entails a heavy loss of generality because external forces do come up at various places for instance experimental devices and technical applications and mostly cannot be derived from a potential function. Control theory on the other hand does prescriptive work. One attempts to express all models in input/output form so that the variables which may be manipulated and observed are clearly distinguished. Also one tries to find methods for regulating the response of systems by altering the equations of motion. Problems of this type are very natural i.e. in engineering, operational research, and in economics where the problems are laid out in terms of the decision variable (inputs) which have to be chosen on the basis of certain observations (outputs) and the mathematical model involves the interrelations between these variables. The Newtonian and Lagrangian approaches are as good as any other but it is more advantageous to develop the Hamiltonian point of view because it reveals certain structural features e.g. we can anlyze the non-linear systems globally; conservation laws follow easily. In this paper, a framework in which we can incorporate external forces in the systems prescription with emphasis on Hamiltonian systems with external forces and on the consequences of external forces has been elaborated.