New algorithm for minimal solution of linear polynomial equations

plays central role in the linear control theory. It occurs in optimal control synthesis, in dead-beat as well as in quadratic criteria [1, 2, 3]. In this equation, a(X), b(X), c(X) are given polynomials, x(X), y(X) are unknown ones. We suppose a(X), b(X) are not both identically zero. Main tool for studying this Euclidean equation is the Euclidean algorithm. It yields general solution and can serve as a computing algorithm for it. In control theory applications, some particular solutions are of interest: those with degree of x or y (or both) minimal. Algorithms are known for selecting the minimal solution from the general one. In this paper, a new algorithm is described which computes the minimal solution directly. This way is more efficient than the general solution computation. The method is also based on the Euclidean algorithm. The indeterminate X can represent either the derivative operator s in the continuous control theory or the delay operator £ in the discrete case. We shall also write a instead of a(X) for brevity. By da we mean the degree of polynomial a(X), da = — GO means zero polynomial a(X) = 0. The leading coefficient of a(X) is denoted by ada. By gcd (a, b) we mean the greatest common divisor of polynomials a, b.