Tham khảo tài liệu 'adaptive control design and analysis part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 222 Chapter 5 Continuous-Time Model Reference Adaptive Control System with Disturbances In the presence of an input disturbance du i and an output disturbance dj t the plant model is expanded as p s y - dy t kpZ s u d t . An important issue in adaptive control has been the robustness of the signal boundedness and tracking properties with respect to these disturbances. As illustrated in 23 147 151 282 340 without modification of an adaptive law a model reference adaptive controller applied to the plant with general disturbances d t and dy t may not ensure signal boundedness. When the controller that is Ií t ớf t wi t ớ t w2 i Ớ2ũ í ỉ t ỡ3 t r t is applied to the plant the tracking error equation becomes e t y i - m i X ÃTk e0 t where 0 t and w t are defined in - e TỀ w t 5-i60 and e0 t is an exponentially decaying term due to initial conditions. From we express the estimation error e i e t p t t in as e t ựT t C í p t i d t e0 t where í ớT t C t - y ớTw í C t jr w i as in and and p f p t p with p kp. L2 Robustness of MRAC Consider the plant with d t e L nL2 and dy t e L r L2. From the definition of d t in and L2 stability theory see Lemma it follows that d t L nL2. Since e0 i G L OL2 we can ignore it in the presence of d t . With the adaptive law - the time derivative of the positive definite function v 0 p ip 6 Tr_ 10 7 1p2 along is v 2e2ft I 2e W e2Ợ I m2 t m2 t m2 t m2 t Robustness of MRAC 223 From this expression it follows that f dT 2v e í dT 2V O . 0 s C1 5 163 J J Illi i I J J tiff I I i for some constant Cl 0 as d t G L2. From we conclude that G L2 and G L that is ớ t p t G L . It can then be shown that G L and ỡ t j t G L n L2. Following the stability proof of Theorem it can be shown that all closed-loop signals are bounded and e í G L2. If in addition linii oo dyft 0 then lim oo d t 0 so that lim oo e t 0. Note that for du t G L2 .
