Tham khảo tài liệu 'adaptive control design and analysis part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 102 Chapter 3 Adaptive Parameter Estimation To develop an adaptive algorithm based on the available signals y t and 0 i to estimate the unknown parameter vector 0 in on-line a parametric error model is needed. Letting ớ t be the estimate of 0 to be obtained from an adaptive update law we define the estimation error e t ớT t t - y t t to. Using we have e t ớT í ự t ỡ t ỡ t ớ t to which is a linear parametric error model linear in the parameter error ỡ t . For the parametric model or we define the estimation error W - ỉ W - for e i ỉ 3-21 _ OW - y t - Al -Pi y t for where 0p t and ỡ í are the estimates of s and s respectively. The linear error model is common in adaptive control and system identification for which the key task is to develop an adaptive law to update the parameter estimate ớ t based on the knowledge of the measured signals y t and to ensure some desired properties for ớ t . Next we present two of the most commonly used adaptive parameter update laws a normalized gradient algorithm and a normalized least-squares algorithm. Normalized Gradient Algorithm A normalized gradient algorithm for updating the parameter estimate ớ í is to choose the derivative of ớ t in a steepest descent direction to successively generate ớ t to minimize a normalized quadratic cost function where e j y as in and m t is a so-called normalizing signal which does not explicitly depend on t . A preferred choice of m t is the one which ensures the boundedness of see Lemma below. The steepest descent direction of J ff is 325 . For the above chosen J 0 which suggests the adaptive update law for 0 t ẽ t 3-23 m2 t Normalized Gradient Algorithm 103 where r rT 0 is a gain matrix ỚQ is an initial estimate of Ớ and mịt ựl t0T í 0 t with K 0 being a design parameter. Lemma The adaptive algorithm guarantees i 0 i 0 t are bounded and ii and ff t belong to L2. Proof i Introducing the positive definite function v
