No abstract

Some recently proposotl gradiont motliods for minimax or near minimax approximation are applied to producing optimal second-order and third-order models of a highorder system. The Fletcher-Powell method, a more recont method by k'letchcr and a mcthod hy Jacobson and Oksman are employcd with least pth appro xi ma ti or^, using largc valocs of p , as proposed by Ranrller and Charalambous and critically compared with t,hc graaor search tcchniquc of minimas approximation by Bandler ct al. The solutions ohtniztccl nre shown to satisfy t,hc necessary conditions for a minimax optimum. IntroductionThe purpose of determining low-order models for high-order systems is t o simplify preliminary design and optimization of such systems. System modelling using least-squares approximation has been investigated recently by Bn,ndler et ul. ( 1072 a). They compared, in particular, the relative cffeciencies of three gradient minimiza,tion methods as applied t o least squares problems. The system modelled was a seventh-order system which represents tlic control system for the pitch rate of a supersonic transport aircraft.In the present work, the maximum value of the error between the step responses of the above seventh-order system and the model is effectively minimized. This may be accomplished either by directly minimizing the maximum error (minimax), or by least ptli approximatiou techniques which, by selecting a large enough value for p, gives, for practical purposes, a minimax solution. The direct minimax method, called the grazor search teclmique, has been recently proposed by Bandler, Srinirasan and Charalambous (1972 c). The strategy is based on steepest descent directions found by linear programniing. The least ptli approximation approach was based on a paper by Handler and Charalambous (1971) where very large values of p-up to 113'~-11ave been successfnlly used. For the presmt problem the value of p was ehoscn as 1000, on the bask of acceptable alinost minimax results and reasonable eomputcr central processing time. A comparison is made between three