  In real world applications, uncertain parameters are the rule rather than the
exception. We present a reachability algorithm for linear systems with
uncertain parameters and inputs using set propagation of polynomial zonotopes.
In contrast to previous methods, our approach is able to tightly capture the
non-convexity of the reachable set. Building up on our main result, we show how
our reachability algorithm can be extended to handle linear time-varying
systems as well as linear systems with time-varying parameters. Moreover, our
approach opens up new possibilities for reachability analysis of linear
time-invariant systems, nonlinear systems, and hybrid systems. We compare our
approach to other state of the art methods, with superior tightness on two
benchmarks including a 9-dimensional vehicle platooning system. Moreover, as
part of the journal extension, we investigate through a polynomial zonotope
with special structure named multi-affine zonotopes and its optimization
problem. We provide the corresponding optimization algorithm and experiment
over the examples obatined from two benchmark systems, showing the efficiency
and scalability comparing to the state of the art method for handling such type
of set representation.
