Download Absolute stability of nonlinear control systems by Xiao-Xin Liao PDF

By Xiao-Xin Liao

ISBN-10: 1402084811

ISBN-13: 9781402084812

Following the hot advancements within the box of absolute balance, Professor Xiaoxin Liao, at the side of Professor Pei Yu, has created a moment variation of his seminal paintings at the topic. Liao starts off with an creation to the Lurie challenge and the Lurie regulate procedure, earlier than relocating directly to the easy algebraic adequate stipulations for absolutely the balance of independent and non-autonomous ODE platforms, in addition to numerous specific periods of Lurie-type platforms. the focal point of the ebook then shifts towards the recent effects and study that experience seemed within the decade because the first variation was once released. This contains nonlinear keep watch over structures with a number of controls, period keep watch over structures, time-delay and impartial Lurie keep an eye on platforms, platforms defined through sensible differential equations, absolutely the balance for neural networks, in addition to functions to chaos keep an eye on and chaos synchronization.

This publication is geared toward undergraduates and teachers within the components of utilized arithmetic, nonlinear regulate structures and chaos regulate and synchronisation, yet can also be invaluable as a reference paintings for researchers and engineers. The ebook is self-contained, notwithstanding a easy wisdom of calculus, linear method and matrix thought, and traditional differential equations is needed to realize an entire knowing of the workings and methodologies mentioned inside of.

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Extra info for Absolute stability of nonlinear control systems

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7. , U(x) is positive semi-definite on the hyperplane σ = 0); (2) cT H H T (AT c + 2 PTb) ≤ 0 (where H T B H = I). 7 is quite lengthy (see [180]), yet the condition (2) is still not easy to verify. If one can determine the lowest bound of U(x) on σ = cT x = 0, then one can use infx∈{x|σ =0} U(x) ≥ 0 to replace the condition (1). 7, Zhu [184] proved the following theorem. 8. [184] Suppose A is a Hurwitz matrix. t. 13) where d = P b + 12 AT c. Proof. Since B is a symmetric positive definite, so is B−1 , and there exists matrix H such that B = H T H, B−1 = H −1 (H −1 )T .

The assumptions can be reduced to y z T Thus we have y z A11 − ε Im×m A12 A21 A22 T A11 A12 A21 A22 y z y z ≥ 0. m ≥ ∑ ε x2i . i=1 This implies our claim. 29. If there exist functions ϕi (xi ) on (−∞, +∞) (i = 1, 2, . . , at the discontinuous points, the left- and right-hand limits of ϕ (xi ) exist) such that 1. ϕi (xi ) xi > 0 for xi = 0, i = 1, 2, . . , m; ϕ (xi ) xi ≥ 0, i = m + 1, . . , n; 2. 0±∞ ϕi (xi ) dxi = +∞, i = 1, 2, . . , m; 3. t. the partial variable y. 7 Systems with Separable Variables 25 Proof.

17) is unique. Let y = (x1 , x2 , . . , xm )T and z = (xm+1 , xm+2 , . . , xn )T . 17) can be written as y˙ = z˙ = n n j=1 n j=1 ∑ f1 j (x j ), . . , ∑ fm j (x j ) n T , ∑ f(m+1) j (x j ), . . , ∑ fn j (x j ) j=1 T . t. partial variables [68]. A11 A12 is symmetric, where A11 , A12 , A21 , Let us assume that A(ai j )n×n = A21 A22 and A22 are m × m, m × p, p × m, and p × p matrices, respectively, and m + p = n. 27. t. y if there are constants εi > 0 (i = 1, 2, . . , m) such that m m xT A x ≥ ∑ εi x2i xT A x ≤ − ∑ εi x2i .

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