Tham khảo tài liệu 'adaptive control design and analysis part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 142 Chapter 3 Adaptive Parameter Estimation Perspective for Adaptive Control As in the following three chapters a linear error model to be derived for adaptive control of system with unknown parameters has the form e t p 0 t - r TC t p í - X í where p zm is an estimate of a constant parameter vector 0 which contains some unknown ideal controller parameters p t is an estimate of a constant parameter and c t c t t are some known signals. This error model is similar to that in except for the additional parameter p . Consider the quadratic cost function e2 W . where m t is a normalizing signal to be defined. As seen in Section a gradient algorithm updates 0 t and p t . sets ớ t and p t in the steepest descent directions of J f in terms of 0 and p. As specified in 151 325 the steepest descent direction of for 0 is the direction of It is clear that the sign of the unknown parameter p specifies the the direction of This motivates US to choose the adaptive update law for ớ í W where r rr 0 is a constant matrix and Ớ0 is an initial estimate of 0 . For p the steepest descent direction of J G p is _ẼZ _Í1 dp m2 Based on this result we choose the adaptive update law for p t as p i - ra2 i 7 PƠo Po t to where 7 0 is a constant scalar and po is an initial estimate of p . To specify the normalizing signal m t we need to ensure the boundedness of when ớ í and pit are bounded. This leads to the choice m i i l 7 t C i 2 t . Discussion 143 Analysis of this adaptive scheme is similar to that for by considering the positive definite function v 0 p h10rr-10 7-V where 0 i ớ í - 0 p t p t - p . The time derivative of v 8 p along the trajectories of and is V t 0. m2 t From this result we can conclude that the adaptive laws and ensure that G t G L pit G L e L2 n L ớ t G L2 n L and p t G L2 n L . These desired properties are standard in an adaptive system for either parameter estimation
