By Haddad W., Chellaboina V.
ISBN-10: 0691133298
ISBN-13: 9780691133294
Nonlinear Dynamical structures and keep watch over provides and develops an in depth remedy of balance research and keep watch over layout of nonlinear dynamical platforms, with an emphasis on Lyapunov-based tools. Dynamical procedure concept lies on the middle of mathematical sciences and engineering. the appliance of dynamical structures has crossed interdisciplinary limitations from chemistry to biochemistry to chemical kinetics, from drugs to biology to inhabitants genetics, from economics to sociology to psychology, and from physics to mechanics to engineering. The more and more complicated nature of engineering structures requiring suggestions keep an eye on to acquire a wanted procedure habit additionally provides upward thrust to dynamical platforms. Wassim Haddad and VijaySekhar Chellaboina supply an exhaustive therapy of nonlinear structures conception and keep watch over utilizing the top criteria of exposition and rigor. This graduate-level textbook is going well past usual remedies through constructing Lyapunov balance idea, partial balance, boundedness, input-to-state balance, input-output balance, finite-time balance, semistability, balance of units and periodic orbits, and balance theorems through vector Lyapunov services. an entire and thorough remedy of dissipativity thought, absolute balance thought, balance of suggestions structures, optimum keep an eye on, disturbance rejection keep watch over, and powerful regulate for nonlinear dynamical structures can also be given. This booklet is an quintessential source for utilized mathematicians, dynamical structures theorists, keep an eye on theorists, and engineers.
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Let {xn }∞ n=0 ⊂ R . Then {xn }n=0 is convergent if and ∞ only if {xn }n=0 is a Cauchy sequence. Proof. Assume {xn }∞ n=0 is convergent, and hence, xn → x as n → ∞ for some x ∈ Rn . Given ε > 0, let N ∈ Z+ be such that xn − x < ε/2 whenever n > N . Now, if m > N , then xm − x < ε/2. 42) which shows that {xn }∞ n=0 is a Cauchy sequence. Conversely, assume that ∞ {xn }n=0 is a Cauchy sequence. 3), S has an accumulation point p ∈ Rn . Now, since {xn }∞ n=0 is Cauchy, given ε > 0, there exists N such that xn − xm < ε/2 whenever n, m > N .
As a consequence, the resulting solution to the modified optimal control problem guarantees robust stability and performance for a class of nonlinear uncertain systems. In Chapter 12, we extend the framework developed in Chapter 11 to nonlinear systems with nonlinear time-invariant real parameter uncertainty. Robust stability of the closed-loop nonlinear system is guaranteed by means of a parameter-dependent Lyapunov function composed of a fixed (parameterindependent) and variable (parameter-dependent) part.
8. The next results show that the finite intersection and infinite union of open sets is open, whereas the finite union and infinite intersection of closed sets is closed. 6. The following statements hold: i) Let S1 , . . , Sn be open subsets of Rn . Then S = S1 ∩ · · · ∩ Sn is an NonlinearBook10pt November 20, 2007 30 CHAPTER 2 open set. ii) Let {Si : i ∈ I} be an indexed family of open sets contained in Rn with the index set I being finite or infinite. Then S = i∈I Si is an open set. Proof. i) The result is obvious if S = Ø.
Nonlinear dynamical systems and control: A Lyapunov-based approach by Haddad W., Chellaboina V.
by Charles
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