Absolute Stability of Nonlinear Control Systems by Xiao-Xin Liao

By Xiao-Xin Liao

This quantity provides an outline 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, equivalent to Lyapunov services, K-class capabilities, Dini-derivatives, M-matrices and the imperative theorems on international balance. bankruptcy 2 provides absolutely the balance thought of self sustaining keep an eye on structures and the well known Lurie challenge. bankruptcy three provides a few easy algebraic invaluable and enough stipulations for absolutely the balance of numerous specified keep an eye on structures. bankruptcy four discusses nonautonomous and discrete regulate platforms. bankruptcy five bargains with absolutely the balance of keep an eye on structures with m nonlinear regulate phrases. bankruptcy 6 devotes itself to absolutely the balance of regulate platforms defined by way of useful differential equations.
The publication concludes with an invaluable bibliography.
For utilized mathematicians, and engineers whose paintings consists of keep watch over systems.

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Proo/. The conditions imply that there exists t with O < t « 1 such that 45 2. 3. 7) k+, (k ~ , - cTR-1d) 2 - cTR-1c(RR-1d + (JcTb) > O. 1l = xTRx + 2R X/(I1) + rr(l1) + a/(I1) ( 11 t:. S(X,I1) + a/(I1) ( 11 - - L " +1 , /(11») ~ ,/(11) ) , k where d = - [Pb r =- + ~ (ac + (JATc)] , ({JcTb _ _ a k+, ) . Obviously, R is positive definite. Thus, the conditions (2. 3. 7) and (2. 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.

Liapunuu function be of tiu! 4) and O ~ f«(1)(1 < MI. 5) is negative definite. 1) ta be absolutely stable in tiu! Hurwitz angle [O,k) are 49 2. 4. Direct Control Systema {! -cTB-ld~O, _1_(.! _ cTB-1d) I cTB-1c Il In case of It = + 00, i. e. , JTB-1d - 2{kTb ~ _ ~ O• ! = O, we can prove the following corollary along the line of proving Theorem 2. 4. 2 and Corollary 2. 4. 3. Corollary 2. 4. 4. Let the liapunou famction be of the form (2. 4. 4) (for any fE F). 4. 5) is negaJive defmite. 4. }) to be absolutely stable are { CTB-ld~ O, cT ~-lC (cTB-1d)1 - JT B-1d - 2{kTb ~ o.

39 2. 2. Nec_ry and Sufficient Conditiona for AbeoIute Stability Clearly, Vis radially unbounded positive definite for Q = {x:11 = = Xl + 2xz + X3 O}. 7) is absolutely stable. Example 2. 2. 12. 8) p/(I1) , J:"" /(I1)dl1 = + 00. 1) O is unstable, the Liapunov matrix equation - 1 O ATP+PA=-G has no positive definite matrix solution for any positive definite matrix G . So the traditional Lurie method cannot be applied. Instead, we can use Theorem 2. 2. 7. (i) Let /(11) = 11. 8) changes into dXt Te = dxz Te 11, Xz - =- dI1 dt = Xl ).

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