Game Development Reference

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i.e. verify:

Q
has no zeros in the right half-plane of the Laplace plane and only single zeros on

the imaginary axis (stable):

Re
(
q
11
)

≥

0,
Re
(
q
22
)

≥

0,

(9.4)

2

q
21
+

q
12

Re
(
q
11
)
Re
(
q
22
)

−

≥

0,

∀

ω

≥

0

2

We define the distribution matrix related to impedance matrix
Z
by the following

relation:

Id
]
−
1

S

=

[
Z

−

Id
]

·

[
Z

+

(9.5)

For a “2n poles'' LTI system, we thus have the following result:

Theorem 2:
if
S
(
s
) is the distribution matrix of the system considered, the latter is

passive if
S
(
s
) is stable and k
S
(
jw
)

∞

k

≤

1.

For a haptic system, the property of passivity helps to guarantee stability when the

latter is coupled to an operator and an environment, both considered to be passive.

Passivity is, of course, a very general characterisation, but one which is adequate in

practice. For a haptic system, the virtual world is not necessarily passive on account

of discretisation, but we can always assume the operator to be passive (Hogan, 1989).

Thus, we understand that an operator can increase the energy of the system being

considered by creating energy to carry out a task.

Nevertheless, if the latter is strictly passive, it will return to a state of equilibrium

when the sources of energy disappear. In addition, if the passivity criterion proves to be

adequate from a practical point of view, it also offers a simple theoretical framework

to implement. It is for this reason that the analysis and summary of the equalisers for

the haptic sense are based on passivity.

9.3.2 Stability

A passive system results from a parallel interconnection of passive systems and the

interconnection by feedback of two passive and/or strictly passive systems results in

a stable system in the
L
2
sense or asymptotically stable. We can refer to the works of

Micaelli (2002) for more specifications about the conditions to be satisfied by the inter-

connected systems. For a haptic system, we know that the operator does not directly

interfere with the environment, and as a result, a passivity type criterion can prove

to be somewhat conservative. For this, one would often prefer the “unconditional

stability in coupled mode'' criterion, which we can state as:

Definition 2:
A two-port linear system is called “unconditionally stable in coupled

mode'' if there are no passive 1-port terminations making the coupled system unstable

(Figure 9.5).

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