On state estimation and control in discrete-time systems of linear type (Q1115400)

The note presents a unified approach to state estimation and control for time-delayed systems \[ \left[ \begin{matrix} x_{k+1}\\ y_{k+1}\end{matrix} \right]=\sum^{a(k)}_{j=0}\left[ \begin{matrix} A_ 1(k,j)\\ A_ 2(k,j)\end{matrix} \right]x_{k-j}+\sum^{b(k)}_{j=0}\left[ \begin{matrix} B_ 1(k,j)\\ B_ 2(k,j)\end{matrix} \right]y_{k-j}+ \] \[ +\sum^{c(k)}_{j=0}\left[ \begin{matrix} C_ 1(k,j)\\ C_ 2(k,j)\end{matrix} \right]w_{k-j}+\sum^{d(k)}_{j=-1}\left[ \begin{matrix} D_ 1(k,j)\\ D_ 2(k,j)\end{matrix} \right]v_{k-j},\quad k=0,1,..., \] with non-white noises whose correlation matrices satisfy the following conditions \(E(w_ kw_ 1')=0\) for \(k-1>q_ k\), \(E(v_ kv_ 1')=0\) for \(k-1>r_ k\), \(E(w_ kv_ 1')=0\) for \(k-1>s^ 1_ k\), \(E(v_ kw_ 1')=0\) for \(1-k>s^ 2_ k\), and \(q_ k,r_ k,s_ k^ 1,s_ k^ 2\), are nonnegative numbers possibly dependent on k, \(1\leq k\), \(k,1=0,1,....\) The state estimation problem is solved by the orthogonal projection method and state augmentation. We prove the separation theorem solving the control problem with an arbitrary cost. Our solutions are given in recurrence form which is very useful in practice.