Dissipative Systems

Dissipative systems are of particular importance in the field of engineering. The dissipation hypo-

thesis, which separates such systems from the general dynamical systems, results in fundamental

constraint on the dynamic behaviour of such systems. A common example of this type of systems

is electrical networks in which part of the supplied energy is dissipated in the resistors as heat.

The theory of dissipative systems is important in control engineering due to its implications on

the stability of control systems. One of the results of the stability theory states that a passive and

stable feedback system consists of a passive dynamical systems in its feed forward and feedback

loops. Furthermore, the Lyapunov function of such a system can be the sum of the stored energies in

those loops. The existence of the stored energy function is simple to establish due to its equivalence

to the passivity assumption. However, there is a range of possible functions for a system with a

predefined input - output behaviour [Willems, 1972].

Given a dynamical system Σ defined with the state space model

˙

x = f (x) + G(x)u

y = h(x) + J (x)u (5.33)

along with the supply rate w : U × Y → R1_{, where U and Y are the input and output spaces}

respectively. For any u ∈ U , y ∈ Y and (t0, t1) ∈ R+ where t1 > t0, the function w(t) =

w(u(t), y(t)) is locally integrable. i.e.

ˆ t1 t0

|w(t)|dt < ∞ (5.34)

Definition 1. A dynamical system Σ with supply rate w is defined to be dissipative if there exists

a non-negative function S : X → R+ _{called storage function, such that for any u ∈ U , y ∈ Y and}

(t0, t1) ∈ R+ where t1 > t0, S(x0) + ˆ t1 t0 w(t)dt ≥ S(x1) (5.35) 1

The following notation is used in this section:R = the real numbers; Rn = n − dimensional Euclidean space;

5.4.Dissipative Control 109

where x0 is some initial state of the system at time t0 and x1 is the system state at time t1.

The definition above follows from the fact that the dynamical system is assumed to be dissip-

ative and that a storage function exists. Another important quantity termed the available storage

plays an important role in determining whether a system is dissipative or not [Willems, 2007].

The available storage is defined as the maximum amount of energy that can be extracted from a

dynamical system at any time. This concept is a generalization of the available energy concept

defined in [Willems, 1971] [Baker, 1969] [Estrada, 1971].

Definition 2. The available storage, Sa, of system Σ with supply rate w is the function Sa: X → R

defined by Sa(x0) = − sup T >0 ˆ T 0 w(t)dt (5.36)

where x(0) = x0 and the supremum is taken over all the admissible inputs u ∈ U.

For the system to be dissipative, the available storage Sa has to be finite for all x ∈ X and

0 ≤ Sa ≤ S. The proof of this theorem is derived in [Willems, 1972]. This definition and theorem

provide a method that can be used to verify whether a system is dissipative or not without requiring

knowledge of its storage functions. In other words, it is an input / output test. The inequality

0 ≤ Sa≤ S only states that the available storage can be the actual storage function of the system.

In the case of a dynamical system where its available storage is its actual storage, an interesting

property is observed. That is all the system internal storage is available to the outside world

through its external terminals.

The available storage concept examines what happens when the system starts from a particular

state. This leads to the logical examination of the system when it ends up in a particular state

introducing the concept of required storage. [Willems, 1972] [Willems, 2007] . Assuming that for the

system Σ defined in Eq. 5.33 there exists an equilibrium state x? _{∈ X such that S(x}?_{) = min} x∈XS(x) .

Furthermore, assume the following reachability assumption. For all x ∈ X, there exists a t ≥ 0 and u ∈ U such that x(t) = x?. i.e. each state can be reach from and steered towards the equilibrium state.

Definition 3. The required storage Sr for a dissipative system with supply rate w is the function

Sr(x) = inf T ≤0

ˆ 0 T

w(t)dt (5.37)

with the infimum taken over all the admissible input u ∈ U.

For dissipative system the storage function must satisfy the a priori inequality Sa≤ S ≤ Sr, i.e.

a dissipative system can only supply a portion of its stored energy and store a fraction of what it

has been supplied with. The available and required storage always satisfy the dissipation inequality

as shown in [Willems, 1972]. It must be noted that not every function bounded by this inequality

is a possible storage function.

For a dissipative dynamical system Σ with supply rate w and storage function S, if S is a

differentiable function of x, then

˙

S ≤ w(u, y) (5.38)

along any trajectory of the system. To turn the above into an equality, introduce a real valued

function d : X × U → R called the dissipative function (dissipation rate) defined as

d = ˙S − w (5.39)

The function d satisfies

S(x0) +

ˆ t1 t0

(w + d)dt = S(x1) (5.40)

It is clear that given d ≥ 0 being non-negative implies the dissipativeness of the system. The

dissipation rate d can uniquely determine storage S, provided the appropriate reachability and

controllability conditions are met.

Stability of Dissipative Systems

Willems [Willems, 1972] defines the stability of dissipative systems in the context of Lyapunov

stability theory. The Lyapunov functions are generalised energy functions, that correspond to the

storage function S. The storage function S of a stable dissipative dynamical system Σ must be

continuous and attains a strong local minima at the equilibrium point x∗. The function S is a

5.4.Dissipative Control 111

If the available storage Sa attains a strong local minima at x∗, then so does the storage function

S. This condition can be verified without explicit knowledge of S. Also the fact that x∗ is an

equilibrium point in itself follows from the fact that the system Σ is continuous in t for all t ≥ t0

and that the supply rate w(t) ≤ 0 for all x ∈ X.

The above definition of the stability may be refined in several ways. Some of which are explained

here:

• The local minima of the storage function defines the stable equilibria and vice versa.

• Local maxima of the storage function will define an unstable equilibrium, if all trajectories

in its neighbourhood involve some dissipation.

• Under strong dissipative assumptions, it may be concluded that all trajectories approach the

point of minimum storage. The system is strong dissipative in the sense that no trajectory

is free of dissipation as defined in [David Hill, 1976].

• If the system supply rate w(u, y) = 0 for all y ∈ Y then, local minima and maxima of the storage function define stable equilibria.