Computation of Lft Uncertainty Bounds with Repeated Parametric Uncertainties

REPEATED PARAMETRIC UNCERTAINTIES K. B. Lim , D. P. Giesyy NASA Langley Research Center, MS 161, Hampton, Virginia, 23681-0001 Abstract A new methodology in which linear fractional transformation uncertainty bounds are directly constructed for use in robust control design and analysis is proposed. Existence conditions for model validating solutions with or without repeated scalar uncertainty are given. The approach is based on minimax formulation to deal with multiple non-repeated structured uncertainty components subject to xed levels of repeated scalar uncertainties. Input directional dependence and variations with di erent experiments are addressed by maximizing uncertainty levels over multiple experimental data sets. Preliminary results show that reasonable uncertainty bounds on structured non-repeated uncertainties can be identi ed directly from measurement data by assuming reasonable levels of repeated scalar uncertainties. 1 Problem De nition Figure 1 shows measured input, u 2 Cnu and output, y 2 Cny , across a true plant. These inputs and outputs are assumed to be discrete Fourier transforms obtained from discrete time records. The corresponding output, ~ y, from an upper linear fractional transformation model, Fu(P; ), depends on a given augmented plant, P , and an assumed structured uncertainty of the form = blk{diag( 1; . . . ; ); i 2 Cmi ni ; i 2 T (1) where T := (1; . . . ; ). Consider a set of measurements from ne independent experiments whose inputs and outputs are denoted by (u(j); y(j)) for the jth experiment. De ne the set Tne := fu(j); y(j) : j = 1; . . . ; neg (2) Henceforth, consider input and output pairs that satisfy (u; y) 2 Tne . At each frequency, the signals and depend on input vector u and in particular its direction so that a ratio of their norms will also be dependent on the particular input direction. To account for this dependence, consider satisfying model validating conditions [1] [5] for all available experiments: = (3) = P11 + P12u (4) ey = y P21 P22u = 0; 8(u; y) 2 Tne (5) The basic problem we address is to determine smallest bounds among all uncertainties which are model validating with respect to available input/output measurements. From equation (3), the uncertainty bound can be written as a ratio of 2-norms k k2 k k2 sup 0 k 0k2 k 0k2 = sup 0 k 0k2 k 0k2 := ( ) (6) Research Engineer, Guidance & Control Branch, Flight Dynamics & Control Division, [email protected] yMathematician, Guidance & Control Branch, Flight Dynamics & Control Division, [email protected] -