Modeling of Endpoint Feedback Learning Implemented Through Point-to-Point Learning Control

Generalized Iterative Learning Control Using Successive Projection: Algorithm, Convergence, and Experimental Verification

“…Algorithm 1. Given system dynamics (1), input constraint set Ω, output constraint set Φ, extended reference r e , any initial values u 0 ∈ Ω andr 0 ∈ l ℓ 2 [0, N ], the input sequence {u k } k 0 defined by the updating law (30) followed by the projections (32) iteratively solves the generalized ILC problem (19). Note that G s is a linear operator defined by…”

“…Implementation of Algorithm 1 consists of two steps: the ILC update (30) and projection steps (31), (32). The update (30) can be directly implemented as a feedforward solution using u k and e s k to constructũ k+1 .…”

“…Proposition 1. The ILC update (30) in Algorithm 1 can be implemented in a feedforward plus feedback solution…”

“…In addition, the input u k is obtained from the projection operator P Ω at each trial so u k ∈ Ω. Then, substitutex k = (ỹ e k ,ỹ k ,ũ k ) and x k = (r e ,r k , u k ) with the update solution (30) into the monotonic convergence condition (26) to give J k+1 J k , and substitute x * = (r e , y * , u * ) ∈ S 1 ∩ S 2 and x k = (r e ,r k , u k ) into the monotonic convergence condition (29) to giveJ k+1 J k , which completes the proof.…”

“…However in many application domains, the output may only be available at certain time instants, or only certain time instants may be relevant to capture the task requirement, e.g. high-acceleration positioning tables [25], robotic manipulators [26], two-mass systems [27], electro-mechanical systems [28], [29] and human motor system [30]. In recent years, the 'classical' ILC problem has been extended to meet the above needs leading to the so-called intermediate point (or point-to-point) ILC.…”