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This paper provides the extension of the observability rank condition and theextension of the controllability rank condition to time-varying nonlinearsystems. Previous conditions to check the state observability andcontrollability, only account for nonlinear systems that do not explicitlydepend on time, or, for time-varying systems, they only account for the linearcase. In this paper, the general analytic conditions are provided. The papershows that both these two new conditions (the extended observability rankcondition and the extended controllability rank condition) reduce to the wellknown rank conditions for observability and controllability in the two simplercases of time-varying linear systems and time-invariant nonlinear systems. Theproposed 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 newconditions are used to study the observability and the controllabilityproperties of a lunar module. For this system, the dynamics exhibit an explicittime-dependence due to the variation of the weight and the variation of themoment 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 rankcondition introduced by this paper are required. The paper shows that, evenunder the constraint that the main rocket engine delivers constant power, thestate is weakly locally controllable. Additionally, it is weakly locallyobservable up to the yaw angle.

Rank Conditions for Observability and Controllability for Time-varying Nonlinear Systems