By Eli Gershon

Multiplicative noise seems in platforms the place the method or dimension noise degrees depend upon the approach country vector. Such structures are correct, for instance, in radar measurements the place greater levels contain better noise point. This monograph embodies a entire survey of the correct literature with uncomplicated difficulties being formulated and solved by way of using quite a few recommendations together with video game idea, linear matrix inequalities and Lyapunov parameter-dependent capabilities. subject matters coated contain: convex H2 and H-infinity norms research of platforms with multiplicative noise; kingdom suggestions keep an eye on and country estimation of platforms with multiplicative noise; dynamic and static output suggestions of stochastic bilinear structures; monitoring controllers for stochastic bilinear platforms using preview details. a variety of examples which display the applicability of the speculation to functional regulate engineering difficulties are thought of; such examples are taken from the aerospace and tips regulate parts.

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**Additional info for H-infinity Control and Estimation of State-multiplicative Linear Systems**

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This term is a white noise sequence (in the discretetime case), or a Wiener process (in the continuous-time case). In the sequel we bring the description of a system with state-multiplicative noise and the signals involved in both the continuous- and the discrete-time setups. 11) where, for simplicity, we consider a single, zero-mean, Wiener process, denoted by β(t). We note that all the stochastic diﬀerential equations in this monograph are interpreted to be of the Ito type. e: x(t), w(t), u(t), y(t) and z(t) we introduce here the notations used for the continuous-time systems of Part 2.

The space of square-summable Rn − valued functions. on the probability space (Ω, F, P). (Fk )k∈N an increasing family of σ−algebras Fk ⊂ F . ˜l2 ([0, N ]; Rn ) the space of nonanticipative stochastic processes: {fk } = {fk }k∈[0,N ] in Rn with respect to (Fk )k∈[0,N ) satisfying N N ||fk ||˜2l = E{ 0 ||fk ||2 } = 0 E{||fk ||2 } < ∞, 2 fk ∈ ˜l2 ([0, N ); Rn ). ˜l2 ([0, ∞); Rn ) the above space for N → ∞ ˜ 2 ([0, T ); Rk ) the space of non anticipative stochastic processes: L f (·) = (f (t))t∈[0,T ] in Rk with respect to (Ft )t∈[0,T ) satisfying T T ||f (·)||2L˜ = E{ 0 ||f (t)||2 dt} = 0 E{||f (t)||2 }dt < ∞.

Note that the fact that Q(T ) = PT yields M (t) = 0. Using Ito lemma we have the following: d(xT Q(t)x) = xT ∂ T 1 ∂Q(t) xdt + (x Q(t)x)dxt + T r{2DxxT DT Q}dt. 11). We readily see that M (t) = xT0 Q(0)x0 − E{xT (T )PT x(T )} − γ 2 E{ −E{ T 0 T 0 ||w − γ −2 B1T Q(t)x||2 dt} (||z||2 − γ 2 ||w||2 )dt}, where we have used T r{QDxxT DT } = T r{xT DT QDx} and the fact that mixed terms involving x(t) and β(t) drop under expectation (see Appendix A). Adding and subtracting γ 2 xT0 R−1 x0 to the above M (t), we obtain that: M = −γ 2 E{ T 0 ||w−w∗ ||2 dt}−[E{ T 0 (||z||2 −γ 2 ||w||2 )dt}+E{xT (T )PT x(T )} +γ 2 xT0 R−1 x0 ] + xT0 [Q(0) − γ 2 R−1 ]x0 + ExT (T )[PT − Q(T )]x(T ) ∗ where w = γ see that: −2 B1T Qx.