Suppose a system P as in Fig 1, which is described by some unknown dynamics and a set of noisy data (ui, i) where ui,i are the input and the output of the system P at k = i, respectively. Assume that the collected data come from a model described by the following equation k+1 = aoyk + + apyk-p+bouk + +bpuk-p+Wk+1 (1) where ao, ..., ap, bo,...bp are the unknown parameters of the system P and wk is i.i.d~ N(0, 1). Given the set of data (uk, k) for k = 0, 1, ...n and assuming we start the system at rest, rewrite the previous form as k+1 = 60+ wk+1, k = 0, 1, ..., n - 1 = = where of [yk,... Yk-p, uk, ..., Uk-p] is called the feature vector and T R2p+2 is the vector of the real parameters of the system, which are unknown. u System P Figure 1: Input-Output System a) (7 points) By formulating the least squares problem n n-1 y = arg min (k+1 - 670) 0 k=0 n-1 compute the unique estimator On of 0, given that (7)- exists. k=0 (2) ,bp] [ao,..., ap, bo,...,l (3) b) (8 points) The inputs of the systems are variables that we can control. Assume now that uk = Kyk. What issue arises in this case? Conclude that even for this simple identification problem, the nature of the inputs of the system is of extreme importance. This problem is called persistent excitation in the input signal u.