Absolute stability of nonlinear control systems by Xiao-Xin Liao

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By Xiao-Xin Liao

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 advent to the Lurie challenge and the Lurie keep an eye on method, earlier than relocating directly to the easy algebraic enough stipulations for absolutely the balance of self sustaining and non-autonomous ODE platforms, in addition to numerous particular periods of Lurie-type structures. the focal point of the publication then shifts towards the hot effects and examine that experience seemed within the decade because the first version used to be released. This contains nonlinear keep watch over structures with a number of controls, period keep watch over platforms, time-delay and impartial Lurie regulate platforms, platforms defined by means of practical differential equations, absolutely the balance for neural networks, in addition to functions to chaos keep watch over and chaos synchronization.

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

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M; ci (xi ) xi ≥ 0, i = m + 1, . . , n; 3. There exist functions εi (xi ) > 0 (i = 1, 2, . . , m) such that ˜ b˜ i j )n×n + diag(ε1 (x1 ) · · · εm (xm )) 0 B( 0 0 n×n is negative semi-definite, where ⎧ ⎨ ci (xi )ai j fi j (x j )+c j (x j )a ji f ji (xi ) , x x = 0, i j ˜bi j (x) = i, j = 1, . . t. the partial variable y. Proof. t. the partial variable y. 33. 37. 18) satisfies 1. fi (xi ) xi > 0 for xi = 0 aii < 0, i = m + 1, . . , n, and fi (xi )xi ≥ 0, aii ≤ 0, i = 1, 2, . . , n; 2. There exist constants ci > 0 (i = 1, 2, .

Proof. t. y: n V (x) = ∑ i=1 0 xi Fii (xi ) dxi . 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.

34. 18) satisfies the following conditions: 1. 31; f (x ) 2. , i = j, i, j = 1, . . , n; 3. ⎤ ⎡ 1 −b21 · · · −bn1 ⎢ −b21 1 · · · −bn2 ⎥ A˜ 11 A˜ 12 ⎥ ⎢ , A˜ := ⎢ . .. ⎥ := A˜ A˜ . ⎣ . 21 22 . ⎦ −b1n −b2n · · · 1 where A˜ 11 , A˜ 12 , A˜ 21 , and A˜ 22 are m × m, m × p, p × m, and p × p matrices, ˜ ˜ −1 ˜ respectively, and A˜ 11 , A˜ 22 , I − A˜ −1 11 A12 A22 A21 are all M matrices. t. the partial variable y. Proof. For any ξ = (ξ1 , . . , ξm )T > 0, η = (η1 , . . t. c = (c1 , . . , cm )T and c˜ = (c˜1 , .

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