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

Sliding mode State Observer Mismatch equation Reduced order Luenberger observer

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

Observers for Time-varying Systems Block-Observable Form A i,i+1, y=y o

Time-varying Systems with disturbances The last equation with respect to y r depends on disturbance vector f(t), then v r,eq is equal to the disturbance. Simulation results: Disturba nces Estimates of Disturbances T

Observer Design

But matrix F k-1 is not constant

The Example The observer is governed by the equations Obswerver

Remark

Parameter estimation Lyapunov function Sliding mode estimator finite time convergence to

Sliiding mode estimator with finite time convergence of to zero Linear operator

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

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