State Estimation of Continuous-Time Dynamical Systems With Uncertain Inputs With Bounded Variation: Entropy, Bit Rates, and Relation With Switched Systems

Abstract

We extend the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">estimation entropy</i> of autonomous dynamical systems proposed by Liberzon and Mitra [1] to nonlinear dynamical systems with uncertain inputs with bounded variation. We call this new notion the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -estimation entropy of the system and show that it lower bounds the bit rate needed for state estimation. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -estimation entropy represents the exponential rate of the increase of the minimal number of functions that are adequate for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -approximating any trajectory of the system. We show that alternative entropy definitions using spanning or separating trajectories bound ours from both sides. On the other hand, we show that other commonly used definitions of entropy, for example the ones in [1], diverge to infinity. Thus, they are potentially not suitable for systems with uncertain inputs. We derive an upper bound on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -estimation entropy and estimation bit rates, and evaluate it for two examples. We present a state estimation algorithm that constructs a function that approximates a given trajectory up to an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> error, given time-sampled and quantized measurements of state and input. We investigate the relation between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -estimation entropy and a previous notion for switched nonlinear systems and derive a new upper bound for the latter, showing the generality of our results on systems with uncertain inputs.