  Backward reachability analysis is essential to synthesizing controllers that
ensure the correctness of closed-loop systems. This paper is concerned with
developing scalable algorithms that under-approximate the backward reachable
sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm
sequentially linearizes the dynamics, and uses constrained zonotopes for set
representation and computation. The main technical ingredient of our algorithm
is an efficient way to under-approximate the Minkowski difference between a
constrained zonotopic minuend and a zonotopic subtrahend, which consists of all
possible values of the uncertainties and the linearization error. This
Minkowski difference needs to be represented as a constrained zonotope to
enable subsequent computation, but, as we show, it is impossible to find a
polynomial-sized representation for it in polynomial time. Our algorithm finds
a polynomial-sized under-approximation in polynomial time. We further analyze
the conservatism of this under-approximation technique, and show that it is
exact under some conditions. Based on the developed Minkowski difference
technique, we detail two backward reachable set computation algorithms to
control the linearization error and incorporate nonconvex state constraints.
Several examples illustrate the effectiveness of our algorithms.
