By Xiao-Xin Liao
Following the hot 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 off with an advent to the Lurie challenge and the Lurie keep an eye on procedure, earlier than relocating directly to the straightforward algebraic adequate stipulations for absolutely the balance of self sufficient and non-autonomous ODE platforms, in addition to a number of exact sessions of Lurie-type platforms. the focal point of the booklet then shifts towards the hot effects and learn that experience seemed within the decade because the first version was once released. This contains nonlinear keep watch over platforms with a number of controls, period keep an eye on platforms, time-delay and impartial Lurie keep watch over platforms, platforms defined via sensible differential equations, absolutely the balance for neural networks, in addition to purposes to chaos keep watch over and chaos synchronization.
This e-book is aimed toward undergraduates and teachers within the components of utilized arithmetic, nonlinear regulate structures and chaos regulate and synchronisation, yet can also be precious as a reference paintings for researchers and engineers. The ebook is self-contained, even though 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.
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Additional info for Absolute stability of nonlinear control systems
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, . . , m), c j ≥ 0 ( j = m + 1, . . , n) such that ⎧ n ⎪ ⎪ −c |a | + ci |ai j | < 0, j = 1, . . , m, ⎪ j j j ∑ ⎨ ⎪ ⎪ ⎪ ⎩ −c j |a j j | + i=1,i= j n ∑ ci |ai j | ≤ 0, j = m + 1, . . t. the partial variable y. Proof. We construct the Lyapunov function n V (x) = ∑ ci |xi |.
Obviously, conditions (1) and (2) imply that m xi V (x) ≥ ∑ i=1 0 ϕi (xi ) dxi := ϕ (y) → +∞ as y → +∞. t. 18), the Dini-derivative of V (x) is given by D+V (x) ⎧ n m ⎪ ⎪ ϕ (x ) i i ⎪∑ ∑ fi j (x j ) at the continuous points of ϕi (xi ), ⎪ ⎪ i=1 j=1 ⎪ ⎪ i = 1, . . 18) = n m n m ⎪ ⎪ ⎪ max ϕ (x +0) f (x ), ϕ (x −0) fi j (x j ) i i i j j i i ⎪ ∑ ∑ ∑ ∑ ⎪ ⎪ i=1 j=1 i=1 j=1 ⎪ ⎩ at the discontinuous points of ϕi (xi ), i = 1, . . , n. 18) ≤ −ψ (y). t. the partial variable y. 30. t. all variables. 31 given below, for m = n, the statement follows from the global stability of all variables.
M; fii (xi ) xi ≤ 0, i = m + 1, . . , n; 2. ±∞ fii (xi ) dxi 0 = −∞, i = 1, 2, . . , m; 3. There are constants ci > 0 (i = 1, 2, . . , m), c j ≥ 0 ( j = m + 1, . . , n), ε > 0 such that ε Im×m 0 A(ai j )n×n + is negative semi − deﬁnite, 0 0 n×n where ⎧ ⎨ 1 ci fi j (x j ) c j f ji (xi ) + , xi x j = 0, ai j (x) = i, j = 1, 2, . . t. the partial variable y. 26 2 Principal Theorems on Global Stability Proof. We construct the Lyapunov function n V (x) = − ∑ xi i=1 0 ci fii (xi ) dxi . t. y. This is because m xi V (x) ≥ − ∑ i=1 0 ci fii (xi ) dxi := ϕ (y) → +∞ as y → +∞.