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

By Jing Zhou, Changyun Wen

From the reviews:

"‘The e-book is useful to profit and comprehend the basic backstepping schemes’. it may be used as an extra textbook on adaptive regulate for complex scholars. keep watch over researchers, specially these operating in adaptive nonlinear regulate, also will generally take advantage of this book." (Jacek Kabzinski, Mathematical reports, factor 2009 b)

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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, . . , v − 1) ¯j , (j = 0, 1, . . 29) i=0 m j=0 we obtain ˙ ˙(t) = ζ˙ − ζˆ = A0 where A0 is a stable matrix. So we get limt→∞ (t) = 0 exponentially. 9). Let the state variable ζ(t) be partitioned as ζ = [ζ1T , . . , v ] , T T T [ξi,1 , . . , ξi,v ] , ξi,k ∈ Rr T T T [vj,1 , . . 34) We also define Θa = [−Av−1 , −Av−2 , . . , −A1 , −A0 ], Θb = [Bm , Bm−1 , . . , B1 , B0 ], T T T ξ(2) = [ξv−1,2 , .

Adaptive backstepping control for a class of linear MIMO systems was studied in [85, 86, 87]. In [85] there exists a restrictive assumption about the high frequency gain Bm that a matrix Sm must be known such that Bm Sm = (Bm Sm )T > 0. In [86] a similar restriction is relaxed using the factorization of high frequency gain. Recently a model-reference adaptive control was presented in [87] for MIMO linear systems without external disturbance using factorization of high frequency gain. In [88] a robust adaptive controller was designed for MIMO systems without disturbances by using switch functions.

Define χ = −w. Accordingly, w has no upper bound. 16) Thus, there must exist a monotone increasing variable {wi = w(ti )} with w0 = |w(t0 )| > 0, limi→∞ ti = tf , and limi→∞ wi = ∞. Following the procedure as in Case 1), we can also construct a subsequence that leads to a contradiction. Accordingly, we can claim that w has upper bound on [0, tf ). Since χ = −w, we know that χ has lower bound on [0, tf ). The above argument is true for all tf > 0. Therefore, χ must be bounded. t And also V (t) and 0 [g(τ )N (χ) + 1]χdτ ˙ are bounded on [0, tf ).

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