MIT6_079F09_hw07

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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 left-hand side is 2...
View Full Document

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online