Adaptive backstepping control of uncertain systems: by Jing Zhou, Changyun Wen

By Jing Zhou, Changyun Wen

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"‘The ebook is useful to benefit and comprehend the elemental backstepping schemes’. it may be used as an extra textbook on adaptive keep watch over for complex scholars. keep watch over researchers, specially these operating in adaptive nonlinear keep an eye on, also will generally make the most of this book." (Jacek Kabzinski, Mathematical studies, factor 2009 b)

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2) v−1 D(p) = p Ir + Av−1 p + . . + A1 p + A0 m N (p) = Bm p + . . + B1 p + B0 ⎡ ⎤ ⎤ ⎤ ⎡ ⎡ −Av−1 Ir 0 . . 0 ⎢ ⎥ 0 Bm ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ −Av−2 0 Ir . . 0 ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ . ⎥ . ⎥ .. .. . ⎥ ⎢ ⎥ Ag = ⎢ , Bp = ⎢ . ⎥ , Bg = ⎢ . . ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ B1 ⎥ ⎢ ⎥ ⎦ ⎦ ⎣ ⎣ ⎢ −A1 0 0 . . Ir ⎥ ⎣ ⎦ B0 Bp −A0 0 0 . . 0 Cg = [Ir 0 . . 4) where v is the observability index of G(p), rv = n, Ir is the r × r identity matrix, and Ai ∈ Rr×r , i = 0, . . , v − 1, and Bj ∈ Rr×r , j = 0, . . , m, m ≤ v − 1 are matrices.

An (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. Assumption 4. The system is minimum phase. Definition: System is said to be minimum phase if its zero dynamics, subject to appropriate initial conditions and a suitable control producing output identically zero, is stable.

The factorization Bm = S1 D1 U1 is not unique because the positive diagonal matrix D+ is arbitrary and so D1 is any diagonal matrix. 5), we choose the following matrix K = [k1 Ir , . . 16) where ki > 0, i = 1, . . 17) is stable. This is sufficient if sv +k1 sv−1 +. +kv−1 s+kv is a Hurwitz polynomial. 18) Preliminary Results 55 where ⊗ is the Kronecker product, and ei is the ith coordinate vector in Rv . We also denote vectors ξv (t), ξ(t), vj (t), as the outputs of the filters ξ˙v = A0 ξv + Ky ξ˙i = A0 ξi + Ev−i y i = 0, 1, .

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