By Jing Zhou, Changyun Wen
From the reviews:
"‘The e-book is beneficial to profit and comprehend the elemental backstepping schemes’. it may be used as an extra textbook on adaptive keep an eye on for complicated scholars. keep watch over researchers, specially these operating in adaptive nonlinear keep watch over, also will greatly reap the benefits of this book." (Jacek Kabzinski, Mathematical experiences, factor 2009 b)
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Extra info for Adaptive backstepping control of uncertain systems: Nonsmooth nonlinearities, interactions or time-variations
An (t)]T d(t) = [d1 (t), . . , dn (t)] ψ0 (y) = [ψ01 (y), . . 26) k = [k1 , k2 , . . 27) is chosen such that the matrix A0 is strictly stable. 19). We denote the ﬁrst m + 1 columns of Ω T by vm , . . , v1 , v0 and the remaining n columns by Ξ as follows Ω T = [vm , . . 29) and show that the equations for the ﬁrst n + 1 columns of Ω T are governed by v˙ j = A0 vj + en−j u, j = 0, . . 30) Due to the special structure of A0 , we have Aj0 en = en−j , j = 0, . . 41) Then we have x2 = ξ2 + Ω2T θ + 2 = ξ2 + [vm,2 , vm−1,2 , .
V1 , v0 and the remaining n columns by Ξ as follows Ω T = [vm , . . 29) and show that the equations for the ﬁrst n + 1 columns of Ω T are governed by v˙ j = A0 vj + en−j u, j = 0, . . 30) Due to the special structure of A0 , we have Aj0 en = en−j , j = 0, . . 41) Then we have x2 = ξ2 + Ω2T θ + 2 = ξ2 + [vm,2 , vm−1,2 , . . , v0,2 , Ξ2 ]θ + 2 = bm vm,2 + ξ2 + [0, vm−1,2 , . . 40). 44) T ω = [vm,2 , vm−1,2 , . . , v0,2 , Ξ2 + Ψa1 ] ω ¯ = [0, vm−1,2 , . . 1) is restricted to the ﬁrst ρ equations.
To show the global uniform stability, the boundedness of m = n− ρ dimension states ζ with zero dynamics should be guaranteed. 149) Ab = ⎢ .. ⎥ ⎢ ⎥ ⎢ . ⎦ ⎣ −b0 /bm 0 ... 0 ρ T = [(Ab ) e1 , . . , Ab e1 , Im ]. 150) With Assumption 2, we have that Ab is Hurwitz. 151) Now we deﬁne a Lyapunov function for the zero dynamics of the system as Vζ = ζ T P ζ. 152) are bounded, it can be shown that ζ is bounded. Thus all signals in the closed-loop are globally uniformly bounded. 144), it further follows that z(t) → 0 as t → ∞, which implies that limt→∞ [y(t) − yr (t)] = 0.