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Unformatted text preview: ), where W ≻ 0 is the covariance matrix.
The matrix A ∈ Rn×n is called the dynamics matrix or the state transition matrix, and the matrix
B ∈ Rn×m is called the input matrix. You are given accurate measurements of the state and input signal, i.e., x(1), . . . , x(T ), u(1), . . . , u(T −
ˆ
ˆ
1), and W is known. Your job is to ﬁnd a state transition matrix A and input matrix B from these
data, that are plausible, and in addition are sparse, i.e., have many zero entries. (The sparser the
better.)
By doing this, you are eﬀectively estimating the structure of the dynamical system, i.e., you are
determining which components of x(t) and u(t) aﬀect which components of x(t + 1). In some
applications, this structure might be more interesting than the actual values of the (nonzero)
ˆ
ˆ
coeﬃcients in A and B .
46 By plausible, we mean that
T −1
t=1 ˆ
ˆ
W −1/2 x(t + 1) − Ax(t) − Bu(t) 2
2 ≤ n(T − 1) + 2 2n(T − 1). (You can just take this as our deﬁnition of plausible. But to explain this choice, we note that when
ˆ
ˆ
A =...
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 Fall '13
 F.Borrelli
 The Aeneid

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