In Chapter 2, we developed adaptive state tracking control schemes for linear time-invariant plants with actuator failures. For a controlled plant % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaca % GaaiikaiaadshacaGGPaGaeyypa0JaamyqaiaadIhacaGGOaGaamiD % aiaacMcacqGHRaWkcaWGcbGaamyDaiaacIcacaWG0bGaaiykaaaa!435C!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\dot x(t) = Ax(t) + Bu(t)$$ whose input u(t) may have failed components, to achieve desired plant-model state dynamics matching in the presence of actuator failures, it is necessary that there exist constant vectors k* s1i ∈ R m and nonzero constant scalars k* s2i ∈ R, i = 1, ... , m, such that A + b i k* s1i T = A M , b i k* s2i = b M , where b i , i = 1, ... , m, is the ith column of B, and A M and b M are a pair of reference model matrices independent of A and B.