A distributed compensator for a nonlinear control problem

Many systems exhibiting self-excited oscillations of determinate amplitude may be modelled by a system of differential equations which includes the scalar nonlinear equation (cf. discussion in [1]) y + h(???, y, y)y + k(???, y) = u(t) + g*x(t) (I) where h, k are smooth functions involving a bifurcation parameter ???. We will assume that, uniformly in ???, yk(???, y) > 0, ?k/?y(???, 0) > 0 and that ?0 ? k(???, y)dy = ?. Further, we assume that the function h may be expressed in the form h(???, y, y) = -h1(???, y)y + h2(???, y)(y)3... with h1 (???, y) > 0 uniformly throughout the y interval of interest. Also, we assume that there is a value of ???, call it ???0, such that (???-???0)h2 (???, y) > 0, in the same region. Under these assumotions one can show that for ??? > ???0 there is a unique periodic solution yp(t, ???) near y = y = 0 with weriod T(???), a smooth function of ??? with-T(???0) equal to the common period of all solutions of the linear oscillator equation y + ?k/?y(???0, 0)y, having average amplitude A(???) =1/T(???) ?0 T(???)((yp(t, ???))2+(yp(t, ???))2)dt The question which we address initially is: suppose u(t) = 0, g = 0 in (I) and we have a single measurement (t) = ay(t) + by(t) available from the system (I). Assuming fixed for the present, ?p(t, ???) = ayp(t, ???) + byp(t, ???) will be the data obtained from the periodic solution yp(t, ???). How may we obtain an estimate of the system state vector (yp(t, ???), yp(t, ???)) from present and recorded bast values of ?p(t, ???)? With T = T(???) assumed fixed, let us consider the system with delay w(t) = v(t) v(t) = v(t-T). (II) The oeriodic function ?p (t, ???) is a particular output corresponding to a particular solution of this system. For w(t) = yp (t, ???), w(t) = v(t) = yp(t, ???) will satisfy (II) if yp (t, ???) is periodic with period T. Then ?(t) = ?p (t, ) = aw(t) + bv(t). (III) The question now becomes one of constructing an observer (cf. [2]) for the system (II) based on the observation (III),- so that the estimated state will tend, asymototically, to the state (yp(t, ???), yp(t, ???)) in an appropriate sense. Our paper will concern the construction of the estimator and the nature of the convergence of the estimate. We will also discuss feedback synthesis of the control u(t) from this estimate. We will also deal with the case wherein a controlled elastic system x = Ax + cy(t) + dy(t) + fu(t) (IV) is coupled to the nonlinear oscillator (I).