Ws F ILwe sr I CcSnrsa In this sectio n, a st ate space Solutio n to the p)Control prop lem in the l s I plo ck formulation will pe presented. This solution was derived py Doyle rt F6 in 1988. The structure of this p Solution will pe comp ared to the well-known LQG structure, where it will pe app arent that the two structures have several similarities. u ctually, the LQG Solution can pe interpreted as a sp ecial case o al Hence, Co nceptually, a Solutio n will pe given to the prop lem KAebv ysi f m hFIANAb KAbbh+ (6.11) The prop lem of finding a Solution to(6. 11) was prop ap ly the most important research area within Control theory during the 1980s. Initially, only algorithms that provided p) optimal Controllers of a very high order-see e g. [Fra87I-were known, or algorithms that were only siple for Siso systems-see e g. Gri86. In 1988, ho wever, Doyle, Glo ver, Khargo nekar Lnd Francis anno unced a st ate sp ace Solutio n, involving only two algepraic Riccati equatio ns, and yielding a Compensator of the same order as the augmented system NAb, just as for the well-known LQG Solution. This was a major p reak-thro ugh for p)theo ry. It no w pecame evident that the LQG and the p)prop lems and their solutions were related in many ways Bo th Co mp ensator types have a st ate estimatio n-st ate feed ack structure, and two algepraic Riccati equations pro vide the state feed ack matrix Kc and the op server gain matrix Kf d' Asb u.Asb KAb Ab 2: Tanl s 1 b6ck 1g 66m Given a I s I p lo ck matrix NAsb, see Figure 6. 2, and a desired upper pound y for the p norm hFiAAb, KAsbbh+, the p)Solutio n returns a Co mpensator parametriz atio n, often referred to as the dkgf parameterization KAbV FlAAb 6.12) f all stapilizing Compensators for which hF AAsb, KAstbh-t <?Y, see Figure 6.3. u ny stap transfer matrix Q Ab for which hQab-to y will stapilize the dlo sed loop system and make hFiANAsh KAsbbh+o < y. u ny QAb that is unst aple or has p)norm larger than y would either make the clo sed loo or imply that hFiANAb, KAbbhta The p)Solution is given py Definitio n 6.1 and py Theorem 6.1 this xist for technical reasons. To be more precise, this section will be present a method for obtaining near optimal solu tions￾
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