Chapter 7 Design of Robust Compensators based on u Theory There are two main limit ations in the use of Hoo theory for compensator design. First, only full complex perturb ations A(s)E Cnxm can be treated in a non-conservative way in an Hoo robust st ability test. Second, robust performance can only be handled in a conservative way even for full compler perturbations since st ability and performance can not be separated in the Hoo structure. The conservatism depends on the uncert ainty structure and on the condition number k of the sy stem. In this chapter, it will be demonstrated, that these limit ations can be avoided by using the structured singular value u First, the analysis problem will be considered, i.e. how given a compensator K(s) robust stability and robust performance is verified using 4. Then, the synthesis problem will be discussed, i.e. how to find a compensator, which is optimal with respect to u 7.1μ Analysis 7.1.1 Robust Stability In the sequel, control problems that can be represented in the block diagram structure shown in Figure 7.I will be considered. This structure will be refered to as the NAK structure The similarity between the NAK and the 2 x 2 block structure is obvious Here, however, A(s will not be restricted to be a full complex block. Instead, it is assumed that A(s has a cert ain block diagonal structure. Indeed, assume that A(s belongs to the following bounded B△={△(8)∈△|l(△(ju)<1} here△ is defined as: {diag(61L1,…,矶n,n,l,m,+x,…,bm。L, ∈R,∈C,△∈C Thus, both real and complex perturb ations which influen ces the nominal system via the NAk structure are considered. Very general robust st ability problems can be formulated via this structure, e.g. parametric uncert ainty, see Example 7. 1 on the following page. Obviously, the
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78 of △(s) N(8) K(8) Figure 7. 1: NAK formulation of the robust stability problem block diagonal structure of A(s) allow for a much more detailed uncert ainty description, than if A(s) simply consist s of one full complex block. Note, that a single full complex block of course is just a special case of the set A Example 7.1 Diagonal perturbation formulation D) This example is a slightly modified version of an example given in/Hol94 Ass ume that the system G(s) is given b g G(s)= (7.3) 68+1 where the dC gain a and the time constant B only are known with 10 9 uncertainty a=[27.0,33.0 B=[0.9,1.1 Erp ressing a and B by their nominal ual ues along with two perturbations Aa and As for which a, B< 1 can be obtained as (7.5) (7.6) △a∈[-1,+1 (77) Let ba denote the set[-1,+l. Then, the transfer function G(s) can be written as G(s 30(1+0.1△a) (78) (1+0.1△8)8+ with ,△a∈B△ (7.9) In block diagram form, G(s can be rep resented as shown in Figure 7.2 on
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Pher ammo il 0.1 F2gueg 7.2: Example 7.1: Block diagram rep res entation of G(s) o determine the Ns K formulation, thes block in Figure 7.2 is d, and the transfer functions from the three inputs ia(s), iB(s), and u(s) to the three Oa(s),og(s), and y(s) are determined. Standard block diagmm manipulation in matrix form gives 3(s)N (7.10) y(s) (8) The uncertainty blocks are given a (s) sB」Los(s) Nou, let w(s), 8, N(s), and s(s be given b w(s)N (s) (712) N (7.13) N(8)N oer er so er (7.14) The perturbed system can now be described as (8) y(s) u(8) (8)Ns(8)z(s) 717) and can immediately be put into the Ns K structure
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