A Direct Control Method For a Class of Nonlinear Systems Using Neural Networks

g…ihGpEsxpixqG„‚FTS g—m˜ridge …niversity ingineering hep—rtment „rumpington ƒtreet g—m˜ridge gfP I€ ingl—nd w—r™h IWWI e dire™t ™ontrol s™heme for — ™l—ss of ™ontinuous time nonline—r systems using neuE r—l networks is presentedF „he o˜je™tive of ™ontrol is to tr—™k — desired referen™e sign—lF „his o˜je™tive is —™hieved through inputGoutput line—riz—tion of the system with neur—l networksF „he network le—rning is ˜—sed on — st—˜ility type —lgorithm in whi™h le—rning —nd ™ontrol t—ke pl—™e simult—neouslyF „he method is —n—lysed in light of its ™lose rel—tion to —d—ptive ™ontrol methodsF „his —n—lysis provides —n interesting ˜ridge ˜etween well studiedD rigorous —d—ptive ™ontrol methods —nd the eld of neur—l network tr—iningF sn p—rti™ul—rD the import—n™e of the property of —nd its impli™—tions to le—rning with networks of lo™—lized re™eptive elds is dis™ussedF „he ™l—ss of systems we wish to ™ontrol is dened ˜y the dierenti—l equ—tions • a @ A C @ A C @ A a @ A a @ A @IA where X D X I —nd X I —re smooth fun™tions on the st—te sp—™e D —nd D I —re the system9s inputs —nd outputs respe™tivelyF „his type of system is en™ountered in m—ny —ppli™—tionsD eFgF rigid link ro˜ot m—nipul—tor ™ontrolF e v—riety of methods for ™ontrol of this type of system in the ™—se where the fun™tions D D I —re knownD h—ve ˜een developed ˜y nonline—r ™ontrol theoristsF e good survey of these methods is —v—il—˜le in ‘R“F ed—ptive ™ontrol methods for line—riz—tion —nd ™ontrol of this system h—ve —lso ˜een proposed ‘IPD IQ“F „hese methods —re ˜—sed on the —ssumption th—t —ll of the system9s nonline—r fun™tions ™—n ˜e expressed —s line—r ™om˜in—tions of known fun™tionsF sn wh—t follows — dire™t ™ontrol method for this systemD using neur—l networks @xxA is presentedF „he method is ™lose in n—ture to the —d—ptive ™ontrol —ppro—™hF st diers in th—t it does not —ssume —ny knowledge of pl—nt nonline—ritiesF por the simpli™ity of present—tion we will rst dis™uss the single input single output @ƒsƒyA ™—seD iFeF when a IF sn se™tion S the method will ˜e extended to wswy systemsF „he system equ—tions in the ƒsƒy ™—se —re simplied to • a @ A C @ A a @ A @PA „he ™ontrol method is ˜—sed on inputGoutput line—riz—tion of this system with st—ti™ st—te feed˜—™kD where the feed˜—™k is gener—ted ˜y two neur—l networksF „he netE works9 role is to …