By Ali A. Jalali, Craig S. Sims, Parviz Famouri

ISBN-10: 354034358X

ISBN-13: 9783540343585

This monograph offers an in depth and unified therapy of the speculation of decreased order platforms. coated issues comprise decreased order modeling, diminished order estimation, decreased order regulate, and the layout of diminished order compensators for stochastic structures. specified emphasis is put on optimization utilizing a quadratic functionality criterion. either non-stop and discrete time linear dynamical platforms are thought of, and nation house approach representation is used through the booklet. It presents a coherent view of the new thought of diminished order idea and its purposes together with quite a lot of program difficulties, suggestions and unresolved issues.

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There are several ways of avoiding this difficulty by different problem formulations. We choose to take this as an opportunity to show, by example, how to formulate and solve problems in a discrete setting. The discrete setting has the advantage that the performance measure does not become unbounded when discrete white noise enters the control. 7 A Discrete Control Problem 53 where x j is the state at time t j and u j is the corresponding control. 2) The process noise wj and measurement noise v j are independent zero mean discrete white noise terms with covariance expressions ˆ δ E {wjwkT } = Q d jk E {v jvkT } = Rˆ δ jk .

The free filter parameter and controller parameters are contained in the matrices G 2 respectively. 25) ∂L =0 ∂G 2 ˆ θTCTΩT +Ω Rˆ ΩT ⎤ = ⎡(L −Ω C)θ Q ˆ θTCTΩT −Ω Rˆ ΩT −Γ P ΓT ⎤ . 26) Very conveniently, the Lagrange multipliers have been removed from consideration. 29) Rˆ Ω T . 30) ⎦ ⎣ ⎦ ⎣ where the expressions for the filter coefficient matrices are −1 F = Γ1 + (Ξ1 − Γ1Pr ΓT2 )(Γ2 PrΓ T2 + Ξ2 ) Γ 2 −1 K = Ω1 + (Ξ1 − Γ1Pr ΓT2 )(Γ2 Pr ΓT2 + Ξ2 ) Ω2 . 32) In this section we have demonstrated how to solve discrete stochastic optimization problems.

Thus far we have looked at stochastic control without the aid of a dynamic compensator. 19) tells us that we cannot use a controller of the form u = K c m . The practical reason is that even if one had actuators that could respond to a wide band signal, it does not seem appropriate to inject such noise into the system, and the high power level would be unsatisfactory. The mathematical reason is that the quadratic control penalty in the performance measure would be unbounded. 20) where zˆ is driven by the sensor information zˆ = Fzˆ + Km + LBu .

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