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This article presents a novel geometric framework for the design of extended state observers (ESOs) using the immersion and invariance (I&I) method. The ESO design problem of a class of uncertain lower‐triangular nonlinear systems is considered for joint state and total disturbance observation. This problem is formulated as designing a dynamical system, as the observer, along with an appropriately defined manifold in the system‐observer extended state‐space. The ESO convergence translates into the attractivity of this manifold; that is, the convergence of the system‐observer trajectories to a small boundary layer around the manifold. The design of both reduced‐order and full‐order ESOs is studied using the I&I formulation. Moreover, an optimization method based on linear matrix inequalities is proposed to establish the convergence of ESOs. It is shown that the I&I‐based method leads to a unifying framework for the design and analysis of ESOs with linear, nonlinear, and time‐varying gains. Detailed simulations are provided to show the efficacy of the proposed ESOs.

We introduce (γ,δ)-similarity, a notion of system comparison that measures to what extent two dynamical systems behave similarly in an input-output sense. This behavioral similarity is characterized by measuring the sensitivity of the difference between the two output trajectories in terms of the external inputs to the two potentially nondeterministic systems. As such, (γ,δ)-similarity is a notion that characterizes approximation of input-output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the L 2 signal norm, (γ,δ)-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of (γ,δ)-similarity as a linear matrix inequality feasibility problem and derive its interpretations in terms of transfer matrices. Our study on the compositional properties of (γ,δ)similarity shows that the notion is preserved through series and feedback interconnections. This highlights its applicability in compositional reasoning, namely abstraction and modular synthesis of large-scale interconnected dynamical systems. We further illustrate our results in an electrical network example.

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