Extreme properties of an adaptive control system with identification (applying the identifiability equations). I: A one-input, one-output plant (Q1120527)

The plant and the controller are described by the equations \[ A^{(1)}(q)y_ t+q^ kA^{(2)}(q)u_ t+\sum^{k- 1}_{j=3}A^{(j)}(q)\psi_{j\quad t}=A^{(k)}(q)\epsilon_ t, \] \[ B_ t^{(1)}(q)y_ t+B_ t^{(2)}(q)u_ t+\sum^{k-1}_{j=3}B_ t^{(j)}(q)\psi_{jt}=0. \] Here \(t=0,1,...\); \(y_ t,u_ t,\psi_{jt}\in {\mathbb{R}}\), \(\psi_{jt}\) are nonlinear functions, q is the shift operator \((qz_ t=z_{t-1})\), \(A^{(j)}(q)\) are the polynomials \(B_ t^{(j)}(q)=b_{jt}^{(0)}+b_{jt}^{(1)}\cdot q+b_{jt}^{(2)}\cdot q^ 2+..\). are two polynomials with time-varying coefficients, \(\epsilon\) is nonobservable white noise. The coefficients \(b_{jt}^{(h)}\) are defined according to a certain rule which is connected with estimates of the system's unknown parameters. The plant may be nonidentifiable. It is supposed that these estimates are convergent in some probabilitic sense. The asymptotical properties of the adaptive system are examined. Some sufficient conditions of plant identifiability are given.