Classical System Identification

Abstract

The purpose of this chapter is to provide an introduction to system identification using prediction error methods (PEM). For a general treatment of system identification refer to any good textbook on the subject, like the two classic books [Lju87, SS89]. Here we will treat only the PEM approach. Furthermore we shall concentrate on results which are relevant for our purpose, namely estimation of model uncertainty. Specifically, the asymptotic distribution of the parameter estimates will be investigated under different assumptions. The representation and notation generally follows [Lju87, Chap. 9]. The general black-box model structure

$$ A(q)y(k) = \frac{{B(q)}}{{F(q)}}u(k) + \frac{{C(q)}}{{D(q)}}e(k) $$

((9.1))

will be assumed with

$$A(q) = 1 + {a_1}{q^{-1}} + {a_2}{q^{-2}} + \cdots + {a_{{n_a}}}{q^{-{n_a}}}$$

((9.2))

$$ B(q) = {b_1}{q^{-1}} + {b_2}{q^{-2}} + ... + {b_{{n_b}}}{q^{-{n_b}}} $$

((9.3))

$$F(q) = 1 + {f_1}{q^{-1}} + {f_2}{q^{-2}} + \cdots + {f_{{n_f}}}{q^{-{n_f}}}$$

((9.4))

$$ C(q) = 1 + {c_1}{q^{-1}} + {c_2}{q^{-2}} + \cdots + {c_n}_{_c}{q^{-{n_c}}} $$

((9.5))

$$ D(q) = 1 + {d_1}{q^{-1}} + {d_2}{q^{-2}} + \cdots + {d_{{n_d}}}{q^{-{n_d}}}$$

((9.6))

e(k) ∈ N(0, λ) denotes normal distributed zero mean noise with variance λ (white noise). q is the shift operator (qu(k) = u(k + 1)). By u(k) we mean the k’th element of the sequence {u(k)}. Let eg {u(k)} be a sequence obtained by sampling the continuous-time signal u_c(t) with sampling time T_s.