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Design and Simulation of Adaptive Observers for Linear and Time-Varying Systems

This study discusses state observers for linear systems, including conventional asymptotic observers and reduced-order Luenberger observers. It also covers sliding mode observers, adaptive Kalman filters, and estimation of disturbances in time-varying systems. The design and simulations of these observers are highlighted, showing their effectiveness in parameter estimation and convergence. The application of various observer equations is demonstrated, considering both known and unknown system parameters.

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Design and Simulation of Adaptive Observers for Linear and Time-Varying Systems

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  2. State Observers for Linear Systems Conventional Asymptotic Observers Observer equation Any desired spectrum of A+LC can be assigned Reduced order observer

  3. Sliding mode State Observer Mismatch equation Reduced order Luenberger observer

  4. Sliding mode State Observer Mismatch equation Reduced order Luenberger observer Variance Kalman filter without adaptation S.M. filter without adaptation Adaptive Kalman filter Noise intensity

  5. Observers for Time-varying Systems Block-Observable Form Ai,i+1, y=yo. . . . . . . .

  6. Time-varying Systems with disturbances The last equation with respect to yrdepends on disturbance vector f(t), thenvr,eqis equal to the disturbance. Simulation results: T Disturbances Estimates of Disturbances

  7. Observer Design

  8. But matrix Fk-1 is not constant

  9. The Example Obswerver The observer is governed by the equations

  10. Remark

  11. Parameter estimation Sliding mode estimator Lyapunov function finite time convergence to

  12. Sliiding mode estimator with finite timeconvergence of to zero Linear operator

  13. Example of operator Application: Linear system with unknown parameters X is known, A can be found, if component of X are linearly independent, as components of vector

  14. DIFFERENTIATORS The first-order system z x + f(t) u - Low pass filter The second-order system v u x + - + f(t) - s Second-order sliding mode uis continuous, low-pass filter is not needed.

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