Observer Design based on Observability Decomposition for Hybrid Systems with Linear Maps and Known Jump Times

Abstract

We propose an observer design method for hybrid systems with linear maps and known jump times based on decomposing the state into parts exhibiting different kinds of observability properties. Using a series of transformations depending on the time elapsed since the previous jump, the state may be decomposed into up to three parts, where the first one is instantaneously observable during flows from the flow output, the second one detectable at jumps from the jump output thanks to the combination of flows and jumps, and the remaining part naturally detectable at jumps still thanks to this combination of flows and jumps but implicitly from the flow output. An observer is then designed to estimate each part, relying on a flow-based Kalman-like observer exploiting the flow output for the first part, a jump-based observer exploiting the jump output for the second, and a jump-based observer exploiting a fictitious output for the third. Global exponential stability of the estimation error is proven using Lyapunov analysis.

Observer Design for Hybrid Systems with Linear Maps and Known Jump Times

“…This chapter deals with the third case, where the full state is not necessarily instantaneously observable during flows and not observable from the jump output only. It unifies and extends the work of [28,29]. More precisely, in Section 2, we start with an observability analysis allowing us to exhibit necessary conditions and sufficient conditions for observer design: first through hybrid Gramian conditions, and then via an observability decomposition.…”

“…• A flow-based observer with an innovation term during flows only, exploiting the observability of the full state during flows from 𝑦 𝑐 when 0 ∉ I [1, 6]; • A jump-based observer with an innovation term at jumps only, exploiting the detectability of the full state via the combination of flows and jumps from 𝑦 𝑑 available at the jumps only when I is bounded [6,17,12,24,26]; • An observer with innovation terms during both flows and jumps, exploiting the observability from both 𝑦 𝑐 and 𝑦 𝑑 and the combination of flows and jumps: this is done via a hybrid Kalman-like approach in [28], or via an observability decomposition in [29], or Lyapunov-based LMIs in [22,6].…”

“…Models of the form (1) include not only hybrid systems with linear maps described in the setting of [23], but also switched and/or impulsive systems with linear maps where the active mode is seen as an exogenous signal making (𝐹, 𝐽, 𝐻 𝑐 , 𝐻 𝑑 ) time-varying (see [1,17,25] among many other ones), and continuous-time systems with sporadic or multi-rate sampled outputs where the "jumps" correspond to sampling events, 𝐽 = Id, 𝑢 𝑑 = 0, 𝑦 𝑐 = 0, and 𝑦 𝑑 the outputs available at the sampling event [21,12,24]. See [23,28] for some examples of those classes of systems set in the framework of (1). The goal of this chapter is to present in a unified and more complete way recent advances concerning the design of an asymptotic observer for system (1), assuming that its jump times are known or detected.…”

“…• Gains that are computed offline (for example via matrix inequalities), according to all possible lengths of flow intervals in between jumps, i.e., they depend on each individual flow length, not the particular sequence of them. This is the path taken by [1,18,12,24,22,6,29]; • Or, gains that are computed online along the time domain of each solution of interest, i.e., they depend on the sequence of flow lengths in each particular solution. This is the path taken by all Kalman-like approaches in [17,26,28].…”

“…This is the path taken by [1,18,12,24,22,6,29]; • Or, gains that are computed online along the time domain of each solution of interest, i.e., they depend on the sequence of flow lengths in each particular solution. This is the path taken by all Kalman-like approaches in [17,26,28].…”