Consider a scalar system (i.e. x(k) E R) described by x(k+ 1) = 2x(k) + u(k)+w(k),...

 

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Consider a scalar system (i.e. x(k) E R) described by x(k+ 1) = 2x(k) + u(k)+w(k), y(k) = x(k) + v (k) where x(0) is a Gaussian random variable with its mean and variance given by E[x (0)] = 0, E[x² (0)] = 1. w(k) and v(k) are Gaussian white random processes with v(k)~N(0,1) and w(k)~N (0,1). x(0), w (k) and v(k) are all independent to each other. a) Consider the output feedback control u(k)= -Ky(k). Find the state feedback gain K such that lim E[(x(k) - mx(x))²](i.e. the steady-state variance of x(k)) is minimized. k→∞o b) Now consider that we design the Kalman filter for x(k). Find the equation for dynamics of the estimation error covariance Z(k) = E [(x(k) − x(k|k))²]. Namely, find the difference equation that relates Z(k+ 1) to Z(k). Using this dynamics equation, find the first three values of Z(k), i.e. Z(0), Z (1), and Z(2). Also, find the steady-state value of Z(k), i.e. Z, = lim Z(k). k-00 Consider a scalar system (i.e. x(k) E R) described by x(k+ 1) = 2x(k) + u(k)+w(k), y(k) = x(k) + v (k) where x(0) is a Gaussian random variable with its mean and variance given by E[x (0)] = 0, E[x² (0)] = 1. w(k) and v(k) are Gaussian white random processes with v(k)~N(0,1) and w(k)~N (0,1). x(0), w (k) and v(k) are all independent to each other. a) Consider the output feedback control u(k)= -Ky(k). Find the state feedback gain K such that lim E[(x(k) - mx(x))²](i.e. the steady-state variance of x(k)) is minimized. k→∞o b) Now consider that we design the Kalman filter for x(k). Find the equation for dynamics of the estimation error covariance Z(k) = E [(x(k) − x(k|k))²]. Namely, find the difference equation that relates Z(k+ 1) to Z(k). Using this dynamics equation, find the first three values of Z(k), i.e. Z(0), Z (1), and Z(2). Also, find the steady-state value of Z(k), i.e. Z, = lim Z(k). k-00

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Solution a To minimize the steady State variance of xK we can Use the LQR Linear Q... View the full answer