Discrete-Time High Order Neural Control: Trained with Kalman by Edgar N. Sanchez
By Edgar N. Sanchez
The target of this paintings is to provide fresh advances within the thought of neural regulate for discrete-time nonlinear platforms with a number of inputs and a number of outputs. the consequences that seem in each one bankruptcy contain rigorous mathematical analyses, in accordance with the Lyapunov process, that warrantly its houses; furthermore, for every bankruptcy, simulation effects are incorporated to ensure the profitable functionality of the corresponding proposed schemes. with a view to entire the therapy of those schemes, the ultimate bankruptcy provides experimental effects with regards to their program to a electrical 3 part induction motor, which exhibit the applicability of such designs. The proposed schemes might be hired for various functions past those awarded during this e-book. The e-book offers recommendations for the output trajectory monitoring challenge of unknown nonlinear platforms in response to 4 schemes. For the 1st one, a right away layout strategy is taken into account: the well-known backstepping strategy, less than the belief of whole sate dimension; the second considers an oblique process, solved with the block regulate and the sliding mode thoughts, lower than an identical assumption. For the 3rd scheme, the backstepping approach is reconsidering together with a neural observer, and at last the block keep an eye on and the sliding mode options are used back too, with a neural observer. all of the proposed schemes are constructed in discrete-time. For either pointed out keep watch over equipment in addition to for the neural observer, the online education of the respective neural networks is played through Kalman Filtering.
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Extra info for Discrete-Time High Order Neural Control: Trained with Kalman Filtering
5) becomes the standard extended Kalman ﬁlter [2, 8]. Usually Pi and Qi are initialized as diagonal matrices, with entries Pi (0) and Qi (0), respectively. It is important to remark that Hi (k), Ki (k), and Pi (k) for the EKF are bounded; for a detailed explanation of this fact see . 7) can be expressed as ei (k + 1) = wi (k)zi (x(k), u(k)) + zi . 4) is wi (k + 1) = wi (k) − ηi Ki (k)e(k). 10) Now, we establish the ﬁrst main result of this chapter in the following theorem. 1. 7) is semiglobally uniformly ultimately bounded (SGUUB); moreover, the RHONN weights remain bounded.
10) Now, we establish the ﬁrst main result of this chapter in the following theorem. 1. 7) is semiglobally uniformly ultimately bounded (SGUUB); moreover, the RHONN weights remain bounded. Proof. 1. 11), T ∆Vi (k) = [wi (k) − ηi Ki (k)ei (k)] [wi (k) − ηi Ki (k)ei (k)] + [wi (k)zi (x(k), u(k)) + − wi (k)wi (k) − e2i (k). 12) can be expressed as ∆Vi (k) = wiT (k)wi (k) − wiT (k)wi (k) + η 2 e2i (k)K T Ki (k) + 2 zi wi (k)zi (x(k), u(k)) + ziT (x(k), u(k))wiT (k)wi (k)zi (x(k), u(k)) + 2zi − 2ηi ei (k)wiT (k)Ki (k) − e2i (k), ∆Vi (k) ≤ |ei (k)|2 ηKi 2 − |ei (k)|2 − |2ηi ||ei (k)| wi (k) Ki (k) + | 2 zi | + |2 zi | wi (k) zi (x(k), u(k)) + wi (k) 2 zi (x(k), u(k)) 2 .
Vi (k) = wi (k)Pi (k)wi (k) + xi (k)Pi (k)xi (k), ∆Vi (k) = V (k + 1) − V (k), = wi (k + 1)Pi (k + 1)wi (k + 1) + xi (k + 1)Pi (k + 1)xi (k + 1) − wi (k)Pi (k)wi (k) − xi (k)Pi (k)xi (k). 14) can be expressed as ∆Vi (k) = wiT (k)Pi (k)wi (k) − wiT (k)[Bi (k)]wi (k) + η 2 xT (k)C T K T [Ai (k)]Ki (k)C x(k) + f T (k)Pi (k)f (k) − f T (k)[Bi (k)]f (k) + xT (k)C T giT [Ai (k)]gi C x(k) − wiT (k)Pi (k)wi (k) − xT i (k)Pi (k)xi (k), ∆Vi (k) ≤ x(k) 2 ηKi C 2 Ai (k) − x(k) 2 gi C 2 Ai (k) − x(k) 2 Pi (k) − wi (k) 2 Bi (k) + | zi |2 Ai (k) + 2 wi (k) + wi (k) 2 zi (x(k), u(k)) | zi (x(k), u(k)) 2 zi | Ai (k) Ai (k) , with Bi (k) = Di (k) − Qi , ∆Vi (k) ≤ − x(k) 2 Ei (k) − wi (k) 2 Fi (k) + | 2 zi | Ai (k) + 2Gi (k), with Ei (k) = Pi (k) − ηKi C 2 Ai (k) − gi C 2 Ai (k) , Fi (k) = Bi (k) − zi (x(k), u(k)) 2 Ai (k) , Gi (k) = wi∗ − wi max zi (x(k), u(k)) | zi | Ai (k) .