  Ensuring the correctness of critical real-time systems, involving concurrent
behaviors and timing requirements, is crucial. Timed automata extend
finite-state automata with clocks, compared in guards and invariants with
integer constants. Parametric timed automata (PTAs) extend timed automata with
timing parameters. Parameter synthesis aims at computing dense sets of
valuations for the timing parameters, guaranteeing a good behavior. However, in
most cases, the emptiness problem for reachability (i.e., whether the emptiness
of the parameter valuations set for which some location is reachable) is
undecidable for PTAs and, as a consequence, synthesis procedures do not
terminate in general, even for bounded parameters. In this paper, we introduce
a parametric extrapolation, that allows us to derive an underapproximation in
the form of linear constraints containing not only all the integer points
ensuring reachability, but also all the (non-necessarily integer) convex
combinations of these integer points, for general PTAs with a bounded parameter
domain. We also propose two further algorithms synthesizing parameter
valuations guaranteeing unavoidability, and preservation of the untimed
behavior w.r.t. a reference parameter valuation, respectively. Our algorithms
terminate and can output constraints arbitrarily close to the complete result.
We demonstrate their applicability and efficiency using the tool Rom\'eo on two
classical benchmarks.
