ABSTRACT
For control system design, it is useful to characterize multi-input, multi-output, time-invariant linear systems in terms of their transfer function matrices. The transfer function matrix of a real m-input, p-output continuous-time system is a (p×m) matrix-valued function G(s), where s is the Laplace transform variable; the corresponding object in discrete time is a (p×m) matrix-valued function G(z), where z is the z-transform variable. Things are particularly interesting when G(s) or G(z) is a proper rational matrix function of s or z, that is, when every entry in G(s) or G(z) is a ratio of two real polynomials in s or z whose denominator’s degree is at least as large as its numerator’s degree. In this case, the system has state space realizations of the form
x˙(t) = Ax(t)+Bu(t) y(t) = Cx(t)+Du(t)
or
x(k+ 1) = Ax(k)+Bu(k) y(k) = Cx(k)+Du(k),
where the state vector x takes values in Rn. Any such realization defines a decomposition G(s) = C(sIn −A)−1B+D or G(z) = C(zIn −A)−1B+D for the system’s transfer function matrix. A realization is minimal when the state vector dimension n is as small as it can be; the MacMillan degree δM(G(s)) or δM(G(z)) is the value of n in a minimal realization. A system’s MacMillan degree is a natural candidate for the “order of the system.”
