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Unformatted text preview: 6.079/6.975, Fall 200910 S. Boyd &amp; P. Parrilo Homework 7 additional problems 1. Identifying a sparse linear dynamical system. A linear dynamical system has the form x ( t + 1) = Ax ( t ) + Bu ( t ) + w ( t ) , t = 1 ,...,T 1 , where x ( t ) R n is the state, u ( t ) R m is the input signal, and w ( t ) R n is the process noise, at time t . We assume the process noises are IID N (0 ,W ), where W is the covariance matrix. The matrix A R n n is called the dynamics matrix or the state transition matrix, and the matrix B R n 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 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) coecients in A and B . By plausible, we mean that T 1 =1 t 2 W 1 / 2 x ( t + 1) Bu ( t ) Ax ( t ) 2 n ( T 1) 2 2 n ( T 1) , where a b means the interval [ a b,a + b ]. (You can just take this as our definition of plausible. But to explain this choice, we note that when A = A and B = B , the lefthand side is 2...
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 Fall '06
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