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MIT6_079F09_hw07 - 6.079/6.975 Fall 2009-10 S Boyd P...

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�� 6.079/6.975, Fall 2009-10 S. Boyd & 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 0 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 ˆ and input matrix B ˆ 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) coefficients in A ˆ and B ˆ . By plausible, we mean that T 1 =1 t 2 W 1 / 2 x ( t + 1) ˆ Bu ( t ) Ax ( t ) ˆ 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 ˆ = A and B ˆ = B , the left-hand side is χ 2 , with n ( T 1) degrees of freedom, and so has mean n ( T 1) and standard deviation 2 n ( T 1).) (a) Describe a method for finding A
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  • Fall '06
  • boyde
  • Maximum likelihood, Type I and type II errors, state transition matrix, Covariance matrix, Convex Optimization, f3 partition Rn

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