Download Absolute Stability of Nonlinear Control Systems by Xiao-Xin Liao PDF

By Xiao-Xin Liao

This quantity offers an summary of a few fresh advancements on absolutely the balance of nonlinear keep an eye on systems.
bankruptcy 1 introduces the most instruments and the primary effects utilized in this e-book, corresponding to Lyapunov capabilities, K-class services, Dini-derivatives, M-matrices and the crucial theorems on worldwide balance. bankruptcy 2 offers absolutely the balance thought of independent keep watch over structures and the well known Lurie challenge. bankruptcy three provides a few easy algebraic precious and enough stipulations for absolutely the balance of numerous detailed keep watch over structures. bankruptcy four discusses nonautonomous and discrete keep watch over platforms. bankruptcy five bargains with absolutely the balance of regulate platforms with m nonlinear keep an eye on phrases. bankruptcy 6 devotes itself to absolutely the balance of keep watch over platforms defined via sensible differential equations.
The publication concludes with an invaluable bibliography.
For utilized mathematicians, and engineers whose paintings includes regulate systems.

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Extra info for Absolute Stability of Nonlinear Control Systems

Example text

3. 2. k]. • Example 2. 3. 2 indicates that the original S-method losed efficiency. In the following, we still adopt this example ta iIlustrate that the modified S-method easily judge the absolute stability. Taking we derive dVI CZ. 3. j [:: ]- 1/2 1 -1/2 -1/2 = O. /(x,) 1 -2r(xz) u,/(r,). 46 2. S) 1 x ,/(x,), ~- -2 is negative definite for (J = {XI(1 = X, = O}. 3) is absolutely stable. Evidently, the modified S-method is more general than the standard one. 2. 4. 1) (1 = cT x, where A E R"x", A is stablel b,c E R" and / E F [/ EFI]' Assume that there exist an n X n symmetric positiue definite Theorem 2.

The unit matricea on both sides of the equality can be different order. Indeed. det(l + EF)= det (1+FEF - det ( 1 F = det(1 - O) = det (1O 1 E) (1 E) 1 + FE). O 1 ( det 1 F O) 1 + FE Now. we prove cTA-1b ~ O. Since A and A + dx T are stable (E > O). for" det(l" - (A = + dx » = O. we have T det(1" - A) • det[I - E(l" - A)-lbcT] = det(l" - A) • det[I - reT(l" - A)-lb] If cTA-1b < O. then there must exist Eo > Osuch that 1 - EocT ( - A)-lb = 1 + EocTA-1b = =i= o. (2. 1. 3) O. 3). As a result. cT A-1b ~ O.

R stands for a given n X n symmetric positive definite matrix and {J ~ O is a constant. 1. 1) is negative definite. the zero solution of (2. 2. 1) is absolutely sta bie. (tI) and estimate its sign by means of the Selvester condition. Thus. (2. 3. (tI) if and only if R det - ( - ( ~{JATc + Pb) ~{JATC+ Pbf >0. 2) can never be satisfied. It can be shown that R det - ( ~{JATC+ Pbf - ( ~{JATc + Pb) ~O. Hence. (tI). This was pointed out by Xie 41 2. 3. The S-Method anei Modified S-Method Huimin [2]. To overcome this difficulty a new method called S-process was discovered.

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