  This paper provides the extension of the observability rank condition and the
extension of the controllability rank condition to time-varying nonlinear
systems. Previous conditions to check the state observability and
controllability, only account for nonlinear systems that do not explicitly
depend on time, or, for time-varying systems, they only account for the linear
case. In this paper, the general analytic conditions are provided. The paper
shows that both these two new conditions (the extended observability rank
condition and the extended controllability rank condition) reduce to the well
known rank conditions for observability and controllability in the two simpler
cases of time-varying linear systems and time-invariant nonlinear systems. The
proposed new conditions work automatically and can deal with any system,
independently of its complexity (state dimension, type of nonlinearity, etc).
Simple examples illustrate both these conditions. In addition, the two new
conditions are used to study the observability and the controllability
properties of a lunar module. For this system, the dynamics exhibit an explicit
time-dependence due to the variation of the weight and the variation of the
moment of inertia. These variations are a consequence of the fuel consumption.
To study the observability and the controllability properties of this system,
the extended observability rank condition and the extended controllability rank
condition introduced by this paper are required. The paper shows that, even
under the constraint that the main rocket engine delivers constant power, the
state is weakly locally controllable. Additionally, it is weakly locally
observable up to the yaw angle.
