Observer design

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Observer Design: A Comprehensive Overview

Introduction to Observer Design

Observer design is a critical aspect of control theory, focusing on estimating the internal states of a system from its outputs. This is essential for systems where not all states are directly measurable. Various methodologies have been developed to address observer design for different types of systems, including linear, nonlinear, time-delay, and systems with unknown inputs.

Observer Design for Non-Smooth and Discontinuous Systems

Piece-Wise Linear Observers

For non-smooth and discontinuous systems, piece-wise linear observers have been effectively implemented. These observers are particularly useful for systems with one-sided restoring characteristics, such as a harmonically excited flexible steel beam with a one-sided support. They ensure global asymptotic stability of the estimation error dynamics, both in theory and practice.

Dynamic Rotor Systems

In dynamic rotor systems, which serve as benchmarks for motion systems with friction and flexibility, observers have been developed to guarantee global asymptotic stability. These systems benefit from switched observers that achieve state reconstruction despite the non-smooth and discontinuous nature of the system.

Nonlinear Observer Design

Observer Error Linearization

The observer error linearization approach is a robust method for designing observers for nonlinear systems. This method provides necessary and sufficient conditions for the existence of a linearization transformation, which simplifies the observer design process. This approach is applicable to both systems with and without inputs, offering a comprehensive solution for nonlinear observer design.

Riccati Equation-Based Observers

For a class of nonlinear systems characterized by globally Lipschitz nonlinearities, observers can be designed using an off-line solution of a Riccati equation. This method is versatile, providing globally valid results for certain systems and approximate observers for more general nonlinear systems. An example application includes a single-link flexible joint robot, where the observer estimates the robot link variables based on joint measurements.

Linear Time-Delay Systems

Generalized Coordinate Change

In linear systems with time delays, observer design can be achieved by finding a generalized coordinate change. This transformation ensures that all time-delay terms are injected by the system's output, which is guaranteed by a rank condition on the observability matrix. This method is particularly novel in multiple-output cases, where dimensional expansion in the coordinate change may be necessary.

Hybrid Systems Approach

Aperiodic Sampled-Data Systems

For linear systems with aperiodic sampled-data measurements, a hybrid observer design is proposed. This observer uses two output injection terms: one at the sampling instants and another providing intersample injection. The error dynamics are analyzed using hybrid system tools, and Lyapunov theory is employed to ensure convergence. This approach is optimized using LMI-based design for observer gains.

Piecewise Linear Systems

Luenberger-Type Observers

For bi-modal piecewise linear systems, both continuous and discrete time, Luenberger-type observers are effective. These observers ensure that the observation error dynamics are globally asymptotically stable when the system dynamics are continuous over the switching plane. For discontinuous dynamics, conditions are derived to ensure that the relative estimation error remains asymptotically small.

Data-Driven Observer Design

Neural Canonical Observer Structures

An indirect data-driven approach using neural canonical observer structures is proposed for nonlinear discrete-time systems. This method employs an artificial neural network auto-encoder trained with recorded state, input, and output data. The neural approximations are used to construct an observer with linear error dynamics, demonstrated on both academic and industrial examples.

Systems with Unknown Inputs

Projection-Based and Sliding Mode Observers

For systems with unknown inputs, two observer design approaches are proposed. The first approach decomposes the state into known and unknown components, using a dynamical system to estimate the unknown component. The second approach combines sliding modes and Lyapunov's second method to create a nonlinear observer that forces the observation error into a sliding mode along a manifold in the observation error space.

Decentralized Design for Distributed Systems

Completely Decentralized Observers

A decentralized design for distributed state observers in continuous-time linear time-invariant plants is presented. Each agent in the system compensates for the lack of detectability through local communication with neighboring agents. This design allows for plug-and-play operation, enabling new agents to join or leave the network without disrupting the overall system.

Conclusion

Observer design is a multifaceted field with various approaches tailored to different system characteristics. From non-smooth and discontinuous systems to nonlinear, time-delay, and systems with unknown inputs, each methodology offers unique advantages. The continuous development in this field ensures robust and efficient state estimation, crucial for advanced control systems.