Robust Stabilization in the Gap-topology by Luc C. G. J. M. Habets (eds.)

By Luc C. G. J. M. Habets (eds.)

The writer provides a topological method of the matter of robustness of dynamic suggestions keep watch over. First the gap-topology is brought as a distance degree among platforms. during this topology, balance of the closed loop method is a sturdy estate. moreover, it really is attainable to unravel the matter of optimally powerful regulate during this surroundings. The booklet might be divided into elements. the 1st chapters shape an creation to the topological process in the direction of strong stabilization. even though of theoretical nature, basically normal mathematical wisdom is needed from the reader. the second one half is dedicated to compensator layout. a number of algorithms for computing an optimally powerful controller within the gap-topology are offered and labored out. consequently we are hoping that the e-book won't in basic terms be of curiosity to theoreticians, yet that still practitioners will reap the benefits of it.

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Now Y is positive definite. ) < 0. So A o is also a stable matrix. 7). 7). a) ZoN 0 + YoDo = I ZoN0 = H1]2F(sI-Ao)-IK[(C-DF)(sI-Ac)-IB+D]H-1/2 = H112{F(sI_Ao)-IK(C_DF) (sI_Ac)-IB+F(sI_Ao)-IKD }H-112. YoDo = H1/2[I+F(sI-Ao)-1(B-KD)][I-F(sI-Ac)-1B]H-1/2 = H1/2 {I-F(sI-Ac)-IB+F(sI-Ao)-1(B-KD) -F(sI-Ao)-I(B -KD)F(sI-A ¢)-1B }H-I/2. 4B }H'412 = HI/2 {F(sI-Ao)-~ [(sI-Ao)-(sI-A~] (sI_A~-IB÷I+ -F(sI_Ac)-IB+F(sI_Ao)-IB }H=II2 = HI/2 {F(sI-Ac)-IB-F(sI-Ao)-IB+I_F(sI_Ac)-IB÷F(sI_Ao)-IB }H-I/2 = HI/2I H -1/2 = I.

Now suppose P0 is the nominal plant, and assume that C is a controller that stabilizes Po. £ = C. 5. becomes: if 8(P~,P 0) < w-l, then H(P~,,C) is stable, So all the plants in the ball around P0 with radius w-t are stabilized by C. Now w depends only upon Po and C, and P0 is fixed. So by changing C in S(P0) we can change the value of w-l. Now we are interested in a compensator CO e S(Po) for which w-t is as large as possible, because such a controller stabilizes all the plants in the largest ball around Po (the ball around P0 with radius w-l, where w-1 is as large as possible).

O+Do R = [I0 0]i . 41) becomes: w = H [(Yo-RNo),(Zo+PdSo)l )}. 42) we can now give a definition of the sharpest stability radius w~I. Let ~l~°n(s) denote the set of all stable mxn transfermatrices with elements in II(s): ~lm°n(s) :-- M(~-I ) ~ Utm*n(s). 3. we will show that this infimum is really achievable, and how a solution to Rg it, can be calculated. At present we simply assume that this is possible. 45) K(Po,e) := { P e ~a'm(s) I 8(P0'P) < E }. 5. 46) Cg := CYo-Rg~0)-I(Z0+Rg~0) stabilizes the whole set K(Po,w~t).

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