Conjugated and symmetric polynomial equations. II. Discrete-time systems

(i) a(c-)^(c) + «(c)x(c-) = HO + Kc-) where a(C), b(Q are given real polynomials of indeterminate C or £""-, x(C) is an unknown polynomial. The equation plays the same role in the discrete control theory [1] as the corresponding equation [2] does in the continuous one, see Part I. A related equation (2) a(0x(c-) + Kr 1 )X0 = <0 + ^(c-) with two unknown polynomials is also studied. Unlike the continuous case, the equation above cannot be made equivalent to a polynomial equation (3) «(X)l;(X) + f}(X)r,(X) = y(X) whose theory is well developed [3]. General solution of (3) contains polynomials of arbitrary high degree. As will be seen, general solution of (l) or (2) is always of finite dimension. Mathematical tools for (l), (2) must be built separately from those for (3) but they are very similar to them. This is done in the sequel.