Widely tunable RF-signal generation with mutually coupled DFB-lasers

Kwakemaak, M.H.; Krishnan, J.; Abeles, J.H.; Largent, Craig C.; Rangwala, Z.

doi:10.1109/cleo.2002.1034146

Cited by 1 publication

(2 citation statements)

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“…where L q = (V12+T.yj1CrT)Vm-' and ql~r,lsO is solution of equation (4.0). Remark : Theorem 3 is more precise that the analogous t h e m of Kwakcr~ak [3], [ll] in case 1 and U MW in case 2.…”

Section: Sinouur Rxccall Equationmentioning

confidence: 98%

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Reduced order H/sub 2/ and H/sub ∞/ observers-singular measurement noise case

Chevrel

Bourlès

Proceedings of 32nd IEEE Conference on Decision and Control

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Abilrnct : In this paper. both Hz and 9f-singular estimation problems are solved in a new way. The results are obtained by taking the Limit from the regular case. It is shown that at the neighbohood of the singular case. the Riccati equdion can be shared into two parts. The fust one is a reduced Riccati equation and the second one consists of quantities tending to 0. The asymptotic gain which then obtained tends to iufiinity with a special shucture. Some obsener mode: become infiiite with a finite imaginary part. Finally, the resulting observer becomes singular in singular pertubation sense. It can be calculated by solving a reduced standard problem. INTRODUCnONThe notion of "best estimation" of the state of a linear time-invariant (L.) system is usually referred to the Kalman filtering theory [l]. The Kalman filter, indeed, minimizes the state estimation error's covariance. This approach is quadratic and can also be formulated as an Hz optimization problem. Another approach consists of minimizing the 4-norm of the state estimation error in the worst case of a normalized disturbance (i.e a unity &-norm disturbance). This second approach is equivalent to solve an optimization problem in the ff-Hardy space [Z]. In these two approaches, the resulting "central" filter (dual of the "central controller") is of same order than the system. However, when the measurement noise is singular (i.e. when some components of this noise are zero), this obsuver becomes a reduced-order one.The H2estimation problem has been widely treated in this singular case. Kwakernaak et al. [3] transform by differentiation the singular problem into a nonsingular one. Halevi [4] makes use of optimal projection. The solotion then consists of one Riccati equation and two coupled Lyapunov equations. An other way is explored by Shaked et al. [5]. who solve the problem in the fwquency domain, by using a spedrpl factorization approach.More recently, the singular H-"standard problem" [2] has been solved by Stoorvogel [a) (through geometric approach) and by Copeland et al. [7] (by using the generalized eigenstruchve of two Hamiltonian system matrix pencils).All these papers do hot completely analyze the behaviour of the Riccati equation at the neighborhood of the singularity, as opposed to which we do here. The work proceeds as follows. %ion 2 gives preliminary defmitions. Section 3 to 5 highlight what happens when the obsewation noise becomes singular. The Riccati equation, the feedback gain and the observer poles are inspected when close to singularity. It is then shown h t the observer becomes of a reduced d e r at the limit and can be calculated by solving a reduced standard problem. The theorem are proved in appendix. The notation is standard see e A. Singular system Let us consider the following LTI system :where x(t)ERn, y(t@ Rp. U(t)ERm. p(t)ER1, and A, B, C, r. H are constant matrices of appropriate dimensions. We also define : {(tk[vT(t) wT(t)IT. Let us now dislinguish twocBbcF depending on the nature of signal p and of the optimization metho...

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Section: Sinouur Rxccall Equationmentioning

confidence: 98%

“…Kwakernaak et al [3] transform by differentiation the singular problem into a nonsingular one. Halevi [4] makes use of optimal projection.…”

Section: Introducnonmentioning

confidence: 99%