By Jing Zhou, Changyun Wen
From the reviews:
"‘The booklet is useful to benefit and comprehend the basic backstepping schemes’. it may be used as an extra textbook on adaptive regulate for complicated scholars. keep an eye on researchers, particularly these operating in adaptive nonlinear regulate, also will generally make the most of this book." (Jacek Kabzinski, Mathematical experiences, factor 2009 b)
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Additional info for Adaptive backstepping control of uncertain systems: Nonsmooth nonlinearities, interactions or time-variations
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, . . 20) v˙ j = A0 vj + Ev−j u j = 0, 1, . . 21) T T T where ξv = [ξv,1 , . . , ξv,v ] . 23) j=0 ¯ i = diag[Bi , . . , Bi ] ∈ Rn×n . where A¯i = diag[Ai , . . 3. 25) exponentially. Proof. Based on the special structures of A¯i and A0 , we have A¯i A0 = A0 A¯i , (i = 0, 1, .
70) where Γ is a positive deﬁnite matrix and γ is a positive constant. 76) |z1 | < δ 4 ¯ is bounded. 2 Now, we evaluate the dynamics of the second state z2 . 79) where Π = [ξ T , V ec(Ξ)T ]T . 85) Remark 3. Note that M2 contains s4 P H 4 and this term may not be bounded. As seen from our analysis, 8l12 s4 P H 4 disappears in M2 due to the use of V at step 2. If we use V at each step, this term will disappear in Mρ of the last step. i (i = 3, . . ) is a smooth projection operation to ensure the estimates belong to compact sets for all time.
We need to design ﬁlters to estimate x and generate some signals for controller design. 100) vi = Ai0 λ, i = 0, 1, . . 101) where k = [k1 , . . , kn ]T such that all eigenvalues of A0 = A − keT1 are at some desired stable locations. 102) 24 Adaptive Backstepping Control Note that xˆ is unavailable due to the unknown parameters θ and b, so the estimate x ˆ cannot be used in the later controller design. Instead, it will be used for stability analysis. 107) This Lyapunov function guarantees that → 0, which implies xˆ(t) → x(t).