Model (In-)Validation from a H∞ and μ perspective

We give a short overview on methods of Model (In-)Validation, that fit to the robust control framework. The idea is that the mismatch between a measured datum and an expected datum is explained by a disturbance signal w and an error model ∆, representing unmodelled dynamics. The key question is if there exists a pair (w,∆), sufficiently small, that can produce the measured datum. In particular, we view the different approaches by Smith et.al. and Poolla et.al., their numerical solution and given examples. 1 Problem Setup The general setup for robust control is depicted in figure 1(a): a generalized plant P with the inputs control signal u and disturbance w and an error ∆, representing unmodelled dynamics. The plant P is given (modelling, identification) and we have a measured datum (umeas, ymeas). The question is: Does the datum fit to the model? The main idea is that the (possible) mismatch betweeen a measured datum and the expected (output-)datum is explained by a disturbance signal w and an error model ∆. The relation between inputs and output is given by ULFT: ( z y ) = ( P11 P12 P13 P21 P22 P23 ) ·  v w u  =: P ·  v w u  (1) ⇒ y = ( P21∆(I − P11∆)[P12, P13] + [P22, P23] ) · ( w u ) =: FU (P,∆) · ( w u ) (2) Figure 1(b) shows a simplification of the general case: a weighted additive error. As a special case, it ”disables” the feedback (P11 = 0) and fixes the location of the disturbance (P12 = 0). These two simplifications make the optimization problem (presented in the next section) convex. We state the Problem, treated in the following sections: Problem-Definition: Suppose the setup in figure 1(b). The (scaled) model P (plant and weights) is given, also given a measured datum (umeas, ymeas). Do there exist ||w|| ≤ 1 and ||∆|| ≤ 1 so that eqn.(2) holds? Scaling is possible by exploiting FU (γP,∆) = γFU (P, γ∆). ∗Division of Automatic Control, Dept of Electrical Engineering, Linköping University, S-581 83 Linköping, Sweden, email: [email protected], http://www.control.isy.liu.se/̃ wolle/