By Edward Ott, Tim Sauer, James A. Yorke

Brings jointly contemporary advances within the interpretive and functional purposes of chaos, which carry nice promise for huge applicability in the course of the actual sciences and engineering. DLC: Chaotic habit in platforms.

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8. 2 Applications 27 Fig. 9. Time evolution of ψ α (k) and its estimate (real in solid line and estimated in dashed line) Fig. 10. 3 Conclusions This chapter has presented the application of HONN to solve the tracking problem for a class of MIMO discrete-time nonlinear systems, using the backstepping technique. The training of the neural network is performed online using an extended Kalman ﬁlter. The boundness of the tracking error is established on the basis of the Lyapunov approach. The HONN training with the learning algorithm based in EKF presents good performance even in presence of larger bounded disturbances such as load torque variations and change on the plant parameters (resistance change).

Recently, other kind of observer has emerged: neural observers [3, 5, 6, 9, 10], for unknown plant dynamics. N. 1) can be rewritten as x(k) = x1 (k) . . xi (k) . . xn (k) , d(k) = d1 (k) . . di (k) . . dn (k) , i = 1, · · · , n, xi (k + 1) = Fi (x(k), u(k)) + di (k), y(k) = Cx(k). 2), we propose a Luenberger neural observer (RHONO) with the following structure: x(k) = x1 (k) . . xi (k) . . 8). 5) where xi is the ith plant state, zi is a bounded approximation error, which can be reduced by increasing the number of the adjustable weights [8].

2) where xi (i = 1, 2, . . , n) is the state of the ith neuron, Li is the respective number of higher-order connections, {I1 , I2 , . . , ILi } is a collection of nonordered subsets of {1, 2, . . , n + m}, n is the state dimension, m is the number of external inputs, wi (i = 1, 2, . . 8). 1) by the following discrete-time RHONN series–parallel representation [5]: xi (k + 1) = wi∗ zi (x(k), u(k)) + zi , i = 1, . . 3) where xi is the ith plant state, zi is a bounded approximation error, which can be reduced by increasing the number of the adjustable weights [5].