Mathematical Methods in Robust Control of Linear Stochastic by Vasile Dragan, Toader Morozan, Adrian-Mihail Stoica PDF

Mathematical Methods in Robust Control of Linear Stochastic by Vasile Dragan, Toader Morozan, Adrian-Mihail Stoica PDF

By Vasile Dragan, Toader Morozan, Adrian-Mihail Stoica

Linear stochastic structures are effectively used to supply mathematical versions for genuine procedures in fields resembling aerospace engineering, communications, production, finance and economic climate. This monograph offers an invaluable technique for the keep watch over of such stochastic structures with a spotlight on powerful stabilization within the suggest sq., linear quadratic keep an eye on, the disturbance attenuation challenge, and powerful stabilization with appreciate to dynamic and parametric uncertainty. platforms with either multiplicative white noise and Markovian leaping are covered.

Key Features:

-Covers the mandatory pre-requisites from chance conception, stochastic techniques, stochastic integrals and stochastic differential equations

-Includes unique therapy of the basic homes of stochastic platforms subjected either to multiplicative white noise and to leap Markovian perturbations

-Systematic presentation leads the reader in a typical solution to the unique results

-New theoretical effects followed by means of exact numerical examples

-Proposes new numerical algorithms to resolve coupled matrix algebraic Riccati equations.

The detailed monograph is geared to researchers and graduate scholars in complex keep an eye on engineering, utilized arithmetic, mathematical platforms concept and finance. it's also available to undergraduate scholars with a primary wisdom within the thought of stochastic systems.

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Extra info for Mathematical Methods in Robust Control of Linear Stochastic Systems

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22). Proof Let U(t, to) : S^ -^ S^ be defined by (U(t, to)iH))(i) = £[0*(r, to)H(rj(t))(t, to)\r](to) = H H e Si, ieVj> to. Taking H e 5^, we define v{t, jc, /) = JC*//(/)JC, X eWJ eV,t> 0. 22), we obtain 7'' jc*(W(r,ro)(//))(/)jc-x*//(/)jc = x * / and to the equation (U(sJo)(C%s)H))ii)dsx. 12) it follows that U(t,s) = T*(t,s), t > s, and the proof is complete. 9). Remark 4. 9), then the following hold: (i) T{t, to) > 0, T*(t, to) > 0 for all t > to. r, ^o G I; (ii) ift -> Ak(t) are bounded functions, then there exist 8 > 0,y > 0 such that T(t, to)J'^ > 5^-^^^-'oV^ T*{t, to)J'^ > Se-^^'-'^^J"^ for all t > to, ^ to e T.

The following are equivalent: (i) The system (AQ, . . , A^; Q) is stable. (ii) There exist the constants fii > I, a > 0 such that \\T'(t,to)\\ to, t, to e X. (iii) There exists a constant 8 > 0 such that [ \T\s,t)\\ds<8 for all t eX. (iv) There exists 8 > 0 such that f T\s,t)J'^ds <8J'^ for all t e X. 29) dt has a bounded and uniform positive solution on X. (vi) For each H: X ^^ S^ continuous, bounded, and uniform positive function, the affine differential equation on S^, — Kit) + C(t)K(t) + Hit) = 0, dt has a bounded and uniform positive solution defined on X.

50 2 Exponential Stability and Lyapunov-Type Linear Equations (iii) =^ (i) Let / / : J -> 5f be a continuous and bounded function. It follows that the real constants 8i, 82 exist such that 81J^ < H(s) < 82 J^ for all s el. Since T(t,s) is a positive operator defined on 5f, we deduce 8\T{t,s)J^ < T(t, s)H(s) < 82Tit, s)J'^ for all r > 5 > ro, to e I. Hence 81 I T{t,s)jUs< I T(t,s)H(s)ds JtQ <82 j JtQ T{t,s)jUs JtQ for all t > to, to el. Thus, if (iii) holds we deduce that the real constants 81,82 exist such that 81J"^ < / T(t,s)H(s)ds <527^ /• JtQ for all t > to, to e J , which shows that t -^ f!

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