By Jing Zhou, Changyun Wen

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"‘The ebook is beneficial to benefit and comprehend the basic backstepping schemes’. it may be used as an extra textbook on adaptive regulate for complex scholars. keep an eye on researchers, particularly these operating in adaptive nonlinear keep watch over, also will commonly reap the benefits of this book." (Jacek Kabzinski, Mathematical stories, factor 2009 b)

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Extra resources for Adaptive backstepping control of uncertain systems: Nonsmooth nonlinearities, interactions or time-variations

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1), the following assumptions are imposed. Assumption 1. The uncertain parameter vector θ is inside a compact set Ωθ , where θ = [bm (t), . . , b0 (t), θa1 (t), . . , θan (t)]T . In addition, there exists an unknown bounded positive constant q so that θ˙ ≤ q. Also q is inside a compact intervals Ωq = [I − , I + ] and bm (t) = 0, ∀t. Assumption 2. The relative degree ρ is fixed and known. Assumption 3. The reference signal yr and its (ρ − 1)th order derivatives are assumed to be known and bounded.

Where Θ = [Θa , Θb ] = ⎢ .. ⎥, Bm = ⎢ .. ⎥. 4. Note that the traditional filters in [1] cannot deal with the unknown disturbance generated from an unknown exosystem. 43) are introduced to achieve disturbance rejection. 5. 44) where F = diag{F1 , . . , Fr }, G = diag{G1 , . . , Gr }. Proof. 43), it can be shown that ⎡ ⎤ ⎡ ⎤ F1 e1 G1 2,1 ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ e˙ = ⎢ .. ⎥ − ⎢ .. ⎥ ⎣ ⎦ ⎣ ⎦ Fr er Gr 2,r With the auxiliary error e, we can express q2 as ⎡ ⎤ ψ1T η1 ⎥ ⎢ ⎢ . ⎥ q2 = ⎢ ..

R Fr λr + Gr ω T 58 Multivariable Adaptive Control ⎤ ⎡ ⎤ T F1 λv,1 + F1 G1 vm,1 λ˙ v,1 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ λ˙ v = ⎢ .. ⎥ = ⎢ ⎥ . 43) m,1 T T T where ω = [ξ(2) + (E1 y)T , [−(K1 vm,1 )T , vm−1,2 , . . , v0,2 ]]T . We define the auxiliary error ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ T λ1 Θ1T G1 y1 G1 Bm,1 vm,1 λv,1 Bm,1 e1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ e = ⎢ .. ⎥ = η − δ + ⎢ .. ⎥ − ⎢ .. ⎥ + ⎢ ⎥+⎢ ⎥ . ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ T er λr ΘrT Gr yr Gr Bm,r vm,1 λv,r Bm,r ⎡ ⎤ ⎡ ⎤ Θ1 Bm,1 ⎥ ⎥ ⎢ ⎢ ⎢ .

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