 
Summary: InputtoState Stability of PDcontrolled robotic
systems
David Angeli 1
Dip. Sistemi e Informatica
University of Florence
Abstract. A sufficient condition for Input to State Stability is presented
for systems admitting a Lyapunov function with negative semidefinite
derivative. An application to PDcontrolled robotic manipulators is also
described.
1 Introduction
This paper deals with external stability for nonlinear systems of the following form:
—
x = f(x; u)
y = h(x) (1)
with state x(t) evolving in R n , input u(t) 2 U ae R m and output y(t) 2 R p .
An important issue in systems theory is to provide suitable estimates of the state
variables in terms of past inputs. For stable linear systems, it is well known how
to separate the effect of inputs from that of initial conditions. In particular, the
state response can be split into two contributions, namely the free evolution (that
only depends upon initial conditions and tends exponentially to zero) and the forced
