By Xiao-Xin Liao

Following the new advancements within the box of absolute balance, Professor Xiaoxin Liao, together with Professor Pei Yu, has created a moment variation of his seminal paintings at the topic. Liao starts with an advent to the Lurie challenge and the Lurie keep watch over process, earlier than relocating directly to the easy algebraic enough stipulations for absolutely the balance of self reliant and non-autonomous ODE platforms, in addition to numerous certain sessions of Lurie-type structures. the focal point of the ebook then shifts towards the recent effects and examine that experience seemed within the decade because the first version used to be released. This comprises nonlinear keep an eye on platforms with a number of controls, period keep watch over platforms, time-delay and impartial Lurie keep an eye on structures, structures defined via practical differential equations, absolutely the balance for neural networks, in addition to purposes to chaos keep watch over and chaos synchronization.

This publication is aimed toward undergraduates and teachers within the parts of utilized arithmetic, nonlinear keep an eye on platforms and chaos regulate and synchronisation, yet can also be valuable as a reference paintings for researchers and engineers. The publication is self-contained, although a easy wisdom of calculus, linear approach and matrix concept, and traditional differential equations is needed to achieve a whole knowing of the workings and methodologies mentioned inside of.

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**Extra resources for Absolute stability of nonlinear control systems**

**Sample text**

31, and is thus omitted. 36. 18) satisfy the following conditions: 1. fi (xi ) xi < 0 for xi = 0 aii > 0, i = m + 1, m + 2, . . , n, and fi (xi )xi ≤ 0, aii ≥ 0, i = 1, 2, . . , n; 2. There exist n functions ci (xi ) (i = 1, 2, . . , n), which are continuous or have only ﬁnite discontinuous points of the ﬁrst or third kind, and satisfy ci (xi ) xi > 0 for xi = 0 and 0±∞ ci (xi ) dxi = +∞, i = 1, 2, . . , m; ci (xi ) xi ≥ 0, i = m + 1, . . , n; 3. There exist functions εi (xi ) > 0 (i = 1, 2, .

5 Popov Frequency Criterion In this section, we brieﬂy introduce the Popov frequency method and the simpliﬁed Popov criterion. 1 The Classical Popov Criterion In the late 1950s, Popov developed the popov frequency criterion [120–123]. First, we give three Lemmas. 13. Suppose the function f (t) is deﬁned for t ≥ 0, and is continuous and piecewise continuously differentiable. Further assume that f (t) and f (t) are bounded. The function G(x) is deﬁned for all x, and is continuous, positive deﬁnite for all x.

34 2 Principal Theorems on Global Stability Then m xi V (x) ≥ ∑ i=1 0 Fii (xi ) dxi := ϕ (y). Clearly we have ϕ (y) → +∞ as y → +∞. t. y. For any x = ξ ∈ Rn , without of loss of generality we can assume that k n ∑ ∏ ξi = 0, i=1 ξi2 = 0, m ≤ k ≤ n. i=k+1 Then, it follows that G(t, ξ ) = ≤ k k i=1 j=1 ∑ Fii(ξi ) ∑ Fi j (t, ξ j ) k m ∑ aii (t, ξ )Fii2(ξi ) + ∑ ε Fii2(ξi ) i=1 i=1 k ∑ + ≤ m ai j (t, ξ )Fii (ξi )Fj j (x j ) − ∑ ε Fii2 (ξi ) i, j=1,i= j m − ε Fii2 (ξi ) < i=1 ∑ i=1 0. t. the partial variable y.