New PDF release: Neural Network-Based State Estimation of Nonlinear Systems:

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By Heidar A. Talebi, Farzaneh Abdollahi, Rajni V. Patel, Khashayar Khorasani

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Extra resources for Neural Network-Based State Estimation of Nonlinear Systems: Application to Fault Detection and Isolation

Sample text

First, the identifier based on the parallel model was used. The simulation results are shown in Fig. 3. Figs. 3–b show the responses of θ1 and δ1 to a sin(t) input signal during learning. After the learning has been completed, the responses of θ1 , θ2 , θ˙1 , θ˙2 , δ1 , and δ˙1 are shown in Figs. 3–h, respectively. It can be observed that the neural network was able to learn the system dynamics. For all figures, the vertical axis units are rad for joint positions, rad/s for joint velocities, m for the flexible modes, and m/s for the velocities of the flexible modes.

In other words, the weights are updated based on the following dynamics 22 2 Neural Network-Based State Estimation Schemes ∂J W˙ˆ = −η1 ( ) − ρ1 y˜ Wˆ ∂ Wˆ ∂J V˙ˆ = −η2 ( ) − ρ2 y˜ Vˆ . 33) Let us define netvˆ = Vˆ x¯ˆ ˆ¯ netwˆ = Wˆ σ (Vˆ x). 35) can be computed as [42] ∂J ∂ J ∂ netwˆ = . ∂ netwˆ ∂ Wˆ ∂ Wˆ ∂J ∂ J ∂ netvˆ = . ∂ netvˆ ∂ Vˆ ∂ Vˆ On the other hand, we have ∂J ∂ J ∂ y˜ ∂ xˆ ∂ xˆ ∂ xˆ . = = −y˜T C. = −x˜T CT C. ∂ netwˆ ∂ y˜ ∂ xˆ ∂ netwˆ ∂ netwˆ ∂ netwˆ ∂J ∂ J ∂ y˜ ∂ xˆ ∂ xˆ ∂ xˆ . = = −y˜T C.

28) which guarantees the stability of the observer and the boundedness of the error estimation. 2. 28). , the estimation error, weights error, and the output error ˜ is are all bounded. In these equations, η1 , η2 > 0 are the learning rates, J = 12 (y˜T y) the objective function and ρ1 , ρ2 are positive numbers. 31) are the backpropagation terms and the second terms are the e-modification which incorporates proper damping in the equations. Then, the stability of the proposed observer is studied by Lyapunov’s direct method.

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