Hence the variance of the estimation error converges

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Unformatted text preview: Friedland in 1989 (among others) these eigenvalues are the transmission zeros of the plant with respect to the noise input to the process. Hence the variance of the estimation error converges to zero if the plant is “minimum phase” with respect to the noise input. For purposes of robustness the noise distribution matrix F should include a term proportional to the control distribution matrix B , i.e., F = F + q 2BB′ In this case, the zero-variance observer gain would satisfy H = B2 − KB1 = 0 (26) as q → ∞. If (26) is satisfied the observer poles are located at the transmission zeros of the plant. Thus, in order to use the gain givenby (26), it is necessary for the plant to be minimumphase with respect to the input. In 1982 Rynaski has defined observers meeting this requirement as robust observers which, as discussed below, have remarkable robustness characteristics. When a reduced-order observer is used, it is readily established that the closed-loop dynamics are given by ⎡ x ⎤ ⎡ A − BG ⎢e ⎥ = ⎢ 0 ⎣ 2⎦ ⎣ BG2 ⎤ A 22 − KA12 ⎥ ⎦ ⎡x⎤ ⎢e ⎥ ⎣ 2⎦ (27) and hence that the eigenvalues of the closed-loop system are given by s I − A + BG s I − A 22 + KA12 = 0 Thus the separation principle also holds when a reduced-order observer is used. ©Encyclopedia of Life Support Systems (EOLSS) (28) CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. VIII - Reduced-Order State Observers - Bernard Friedland Robustness can be assessed by carrying out an analysis for a reduced-order observer similar to the analysis for a full-order observer. It is found that the characteristic polynomial for the closed-loop control system, when a reduced-order observer is used and the actual control distribution matrix B = B + δB differs from the nominal (design) value B , is given by sI − A = sI − F + ΔG2 ΔG −BG2 sI − A c (29) where Δ = Kδ B1 − B2 U SA NE M SC PL O E– C EO H AP LS TE S R S (30) It is seen that the characteristic polynomial of the closed-loop system reduces to that of (28) when Δ=0 (31) It is noted that (31) can hold in a single-input system in which the loop gain is the only variable parameter. In this case δ B1 = ρ B1 , δ B2 = ρ B2 (32) and thus ( ) Δ = ρ K B 1 − B 2 = −ρ H ( ) Hence, if the observer is designed to satisfy (26) H = B2 − KB 1 = 0 the separation principle holds for arbitrary changes in the loop gain, thus justifying Ryn...
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This document was uploaded on 03/09/2014 for the course EL 5021 at Institute of Technology.

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