By Jing Zhou, Changyun Wen
From the reviews:
"‘The publication is beneficial to benefit and comprehend the basic backstepping schemes’. it may be used as an extra textbook on adaptive keep an eye on for complex scholars. regulate researchers, particularly these operating in adaptive nonlinear keep watch over, also will commonly take advantage of this book." (Jacek Kabzinski, Mathematical studies, factor 2009 b)
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Additional resources for Adaptive backstepping control of uncertain systems: Nonsmooth nonlinearities, interactions or time-variations
2cρ , } > 0. Due to the utilization of projection operations for θˆ and qˆ, the boundedness of θ˜ and q˜ can be guaranteed. Together t the boundedness d(t), q and θ˙ ,the boundedness of Mρ and 0 f2 (θ˜T Γ −1 θ˜ + t q˜2 )e−f (t−τ ) dτ + 0 Mρ e−f (t−τ )dτ can be guaranteed. 4), f0 is selected as the upper bound of Vρ (0)+ 0 f2 (θ˜T Γ −1 θ+ t Mρ e−f (t−τ ) dτ, g(t) = bm (t). 1, we can conclude that Vρ (t) 0 and χ(t), hence zi , (i = 1, . . , ρ), θ, q and a are bounded. Since z1 and yr are bounded, y is also 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. 1) satisfying Assumptions 1-4 such that the closed-loop system is stable and the system output can track a given reference signal yr (t) as close as possible. 2 Preliminary Result In order to cope with the unknown sign of high-frequency gain, the Nussbaum gain technique is employed in this chapter.
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. 5) Bp y = Cg x where x ∈ Rn is the system state, A ∈ Rn×n is the matrix Ag with the ﬁrst r columns equal to zero, Ap ∈ Rn×r are the ﬁrst r columns of Ag and BP Problem Formulation 53 ¯ ∈ R(m+1)r×r , d(t) = [0 Bp ]T d(t) = [dT1 , .