Introduction
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 design observers for different types of systems, including linear, nonlinear, time-delay, and systems with unknown inputs.
Key Insights
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Piecewise Linear Systems:
- Luenberger-type observers can be designed for bi-modal piecewise linear systems, ensuring global asymptotic stability of the observation error dynamics in continuous and discrete time.
- Switched observers are effective for non-smooth and discontinuous mechanical systems, achieving global asymptotic stability in estimation error dynamics.
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Nonlinear Systems:
- Observers for nonlinear systems can be designed using the off-line solution of a Riccati equation, particularly for systems with globally Lipschitz nonlinearities.
- Observer error linearization is a viable approach for nonlinear systems, providing necessary and sufficient conditions for the existence of a linearization transformation.
- For special classes of nonlinear systems with triangular structures, observers can be designed under certain regularity assumptions, applicable to both single-output and multi-output cases.
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Systems with Unknown Inputs:
- Observers can be designed by decomposing the state into known and unknown components, using projection operators and sliding modes to estimate the unknown state components.
- High-gain control strategies can be employed to design observers for systems with unknown inputs, ensuring the auxiliary system tracks the output of the original system.
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Time-Delay Systems:
- For linear systems with time delays, observer design involves finding a coordinate change that injects all time-delay terms through the system's output, guaranteed by a rank condition on the observability matrix.
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Discrete-Time Descriptor Systems:
- The observer design for discrete-time descriptor systems can be formulated using linear matrix inequalities, providing necessary and sufficient conditions for the existence and convergence of the observer.
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General Linear Systems:
- Function observers, which estimate a linear combination of the state vector, can be designed with lower order than state observers, simplifying the design problem to solving a set of linear equations.
Conclusion
Observer design encompasses a variety of methodologies tailored to different system characteristics, including piecewise linear, nonlinear, time-delay, and systems with unknown inputs. Key strategies involve Luenberger-type observers, Riccati equation solutions, observer error linearization, projection operators, sliding modes, high-gain control, and linear matrix inequalities. These approaches ensure accurate state estimation, crucial for effective control and monitoring of complex systems.