A Lyapunov approach to detectability of nonlinear systems

Given a system with outputs, it is usually desirable to design an algorithm (called an observer), estimating the current state of the system based only on the past history of inputs and outputs. Detectability is the property making the design of an observer possible.;For linear systems detectability is dual to controllability and thus is easily reduced to a simple algebraic condition.;There are two natural ways of generalizing detectability to non-linear systems. One is based on a well-known notion of input-to-state stability (ISS), which can be dualized (for systems without controls) to output-to-state stability (OSS) by replacing inputs by outputs in the state estimate provided by ISS. Combining ISS and OSS results in the proposed notion of input- output-to-state stability (IOSS).;Another (Lyapunov-theoretic) approach, preferred in controller design, consists in finding a proper and positive definite function, decaying along trajectories of a system when the magnitudes of inputs and outputs are small in comparison with the magnitude of the state.;This presentation proves the equivalence of the two mentioned approaches. It is easy to establish an IOSS property for a system that admits an IOSS-Lyapunov function. The proof of the converse implication is far more complex. The most difficult step is finding a continuous Lyapunov function for a "stable modulo outputs" system without controls. No obvious guesses (such as integral along the trajectory or first hitting time) provide a continuous function. The goal is achieved by introducing artificial controls and changing the dynamics of the system near the part of the state space where output is sufficiently smaller than the state in magnitude. This results in an optimal control problem with a continuous value function.;As the IOSS property can be seen as a generalization of global asymptotic stability of ODEs, input to state stability of nonlinear control systems, and output to state stability of ODEs with outputs, the converse Lyapunov theorem presented here generalizes the Lyapunov characterizations for all these particular cases.;As a corollary to the presented results, construction of norm-estimators is provided, and integral characterizations of detectability (in terms of finite energy estimates) are obtained.