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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 )
q 21 +
q 12
Re ( q 11 ) Re ( q 22 )
We define the distribution matrix related to impedance matrix Z by the following
Id ] 1
[ Z
Id ]
[ Z
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 )
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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