Indirect adaptive control of dynamic processes that are nonminimal phase for noise (Q1108986)

The paper deals with the problem of adaptive control for a dynamic process described by the difference equation \[ (1)\quad x(n)=A(q)x(n)+qB(q)u(n)+C(q)\xi(n). \] Here \(n=0,1,...\), \(u(n)\in {\mathbb{R}}^ 1\) is a control, \(\xi(n)\in {\mathbb{R}}^ 1\) is a random action, \(x(n)\in {\mathbb{R}}^ 1,\) q is a delay operator \((qz(n)=z(n-1))\) and \(A(q),\) \(B(q),\) \(C(q)\) are polynomials with \(C(0)=1\), \(A(0)=0\). Let at first the coefficients \(A(\cdot)\), \(B(\cdot)\), \(C(\cdot)\) be known. For a wide class of optimal control problems for the plant (1) the optimal controller is described by the equation \[ (2)\quad V(q)u(n)=U(q)r(n)+W(q)x(n). \] Here U, V, W are polynomials, \([U(\cdot),V(\cdot),W(\cdot)]={\mathcal F}[A(\cdot),B(\cdot),C(\cdot)]\), r(n) is a given stationary random process. Let \(x(n)=x\) 0(n), \(u(n)=u\) 0(n) be a solution of (1), (2). Now let the coefficients of A, B, C be unknown. Therefore, it is impossible to use the controller (2). The problem is to find an adaptive controller (instead of (2)), such that \(\lim_{n\to \infty}| x(n)-x^ 0(n)| =^{P=1}0\) (here the symbol \(=^{P=1}\) means equality with probability 1). This problem is solved under some assumptions. Moreover, an identification algorithm is constructed such that \[ A_ n(q)=^{P=1}A(q),\quad B_ n(q)\to^{P=1}B(q),\quad C_ n(q)\to^{P=1}C(q), \] where \(A_ n(q)\), \(B_ n(q)\), \(C_ n(q)\) are estimates of the unknown A(q), B(q), C(q). The adaptive controller is described by \[ V_ n(q)\tilde u(n)=U_ n(q)r(n)+W_ n(q)x(n),\quad \tilde u(n)=G_ n[u(n)], \] where \([U_ n(\cdot),V_ n(\cdot),W_ n(\cdot)]={\mathcal F}[A_ n(\cdot),B_ n(\cdot),C_ n(\cdot)]\) and \(G_ n(u)\) is a certain function. Under some assumptions it is proved that for some \(\alpha >0:\) \(\lim_{n\to \infty}n^{\alpha}| x(n)-x^ 0(n)=^{P=1}=0.\)