Moving-Horizon Predictive Input Design for Closed-Loop Identification

Machine Direction Adaptive Control on a Paper Machine

“…For a process described by an ARX model, the parameter vector θ is given by the expression θ = [ a 1 , ..., a n a , b 1 , ..., b n b ]. The trace of the Fisher information matrix R u = trace­( F ) is where n is the number output variables, m is the number input variables, n a is the order of A ( q , θ), n b is the order of B ( q , θ), n k is the input delay, and N is the number of IO samples …”

“…Since R u is convex, the maximum over a compact convex domain defined by the convex constraints in eq occurs at one of the vertices, i.e., when the constraints are active. The number of vertices grows exponentially with the dimension of the problem, so originally, a solution based on visiting each vertice individually was considered computationally infeasible . However, in this work, the number of input variables is m = 3, so with a small prediction horizon, the problem can be solved by determining the maximum at all of the vertices.…”

“…where n is the number output variables, m is the number input variables, n a is the order of A(q, θ), n b is the order of B(q, θ), n k is the input delay, and N is the number of IO samples. 53 The ID technique applied in this work can be summarized as follows: R y y t y…”

“…The number of vertices grows exponentially with the dimension of the problem, so originally, a solution based on visiting each vertice individually was considered computationally infeasible. 53 However, in this work, the number of input variables is m = 3, so with a small prediction horizon, the problem can be solved by determining the maximum at all of the vertices. The input sequence u q * is selected such that only combinations of inputs that obey the IO boundaries while taking the maximum allowable step size are considered.…”