The design of stable adaptive model reference control systems using sensitivity coefficients†

Abstract

A procedure is described for the design of stable adaptive model reference control systems. This procedure combines the Lya.punov method with the use of sensitivity coefficients. A basic algorithm is. first obtained and the resulting system is shown to be asymptotically stable in the sense of 'eventual stability' as used by La Salle and Rath (196:3) . Modifications of this algorithm are then proposed which arc easier to implement. One of these uses sensitivity coefficients and is studied in some detail. Notation ex, f3 = positive constants a.e. = almost everywhere AND = coincidence operator c = p-dimensioned plant and unadjustable controller parameter vector e, e = scalar and vector error signals k c ' k p = controller gain and process gain k i = value of ith adjustable parameter -\ = maximum rate of adjustment of a parameter A = a positive definite matrix M i = maximum rate of adjustment of the ith parameter ,....( . ) = Lebesgue measure 11' 11 = norm in n-dimensional Euclidean space ill = subset of R" on which the dynamics of the system arc defined D u = product space D 1 x TI 4>(t) = a bounded measurable function on TI r, r = scalar and vector inpu t signals R" = r-dimensioned Euclidean space TI = the interval (0, (0) T/, T_I = subsets of Tl ( .)T = transpose of a vector or matrix X = uncontrollable factors on v(e)