(a) Consider the system i = Ax + bu where 1-1::) and b = (b1 b2)....
1 1 -2 Given the LTI system -Ax Bu where A3 3 2and B0 a) Check the controllability using i) the controllability matrix, and ii) the Hautus-Rosenbrock test. b) Identify the controllable and uncontrollable subspaces, and convert the system to a Kalman con- 0 trollable canonical form c) Suppose that we start from the initial state z(0) (1,1, 1)T. Is there a control u(t) that drives the state to (1(3,-1,1)7 at some time t? Is there a control u(t) that...
Consider a (continuous-time) linear system x=Ax + Bu. We introduce a time discretization tk-kAT, where ΔT = assume that the input u(t) is piecewise constant on the equidistant intervals tk, tk+1), , and N > 0, and N 1 a(t) = uk for t E [tk, tk+1). (a) Verify that the specific choice of input signals leads to a discretization of the continuous-time system x = Ax + Bu in terms of a discrete-time system with states x,-2(tr) and inputs...
1. Consider the following two ordered bases: Bu = {(5), (3) B2 = { 1-2) /1 1-2 ) B = 11-5)' ( 2 ) (a) Find the transition matrix from B1 to B2. (b) Use your transition matrix from part (a) to compute the Bi-coordinate vector for [(3 4)T]12
x(0) = 0 Consider the system defined by * = AX + Bu Where 1 A = (-6-3) BEG 1 and u=C)=6:10 [2.1(t) (5.1(t). Obtain the response x(t) analytically.
3. (6 marks) For b1,b2, 63, 64 € R, consider the linear system of equations + 21 + 22 -11 + 22 23 624 = bi 3.13 + 2014 b2 23 + 2:04 b3 603 + 20:24 = 54 *) 21 + 2:01 + 4.22 where 11, 12, 13, 14 ER. (a) Find a system of equations that bı, b2, 63, 64 must satisfy in order for (+) to be consistent. bi [ must satisfy so (+) is (b) Using...
1. Given the spring-mass-damper system in the figure below T3 T1 T2 b2 b1 k3 (a) Find the equations of motion for each of the masses 脳. Fi(s) (b) Assume F1 0 and find the transfer function (c) Assume Fs 0 and find the transfer function (d) Write the equations in matrix-vector form: M.ї + Bi + Kx-F where z is a 3 x 1 vector with the displacements r,2, r3 as components, M is the mass matrix, B is...
0 1 Let S span 1 1 1 0 }, a basis for S. Show that| (a) Let B1 { 1 0 1 1 0 is also a basis for S 0 B2 { 1 (b) Write each vector in B2 (c) Use the previous part to write each vector in B2 with respect to Bi (how many components should each vB, vector have?) (d) Use the previous part to find a change of basis matrix B2 to B1. What...
consider the system
X(t) = ax(1) + bu(t) with a = 0.001,b= 1,x(0) = 5. (a) Simulate this system using the Matlab command initial (b) Now use u(t) = -kx(t) where k is found as the optimal gain by minimizing the performance index J= ax (1) + ru (1) dt Use q=1, r=1 to simulate this system.
2. Consider the system 2n+4x2 + 8x3 + 12n = b2 (a) Reduce A b to Row Reduced Echelon Form Rx-c (b) Find the condition on bi, b2, bs for Ax b to have a solution (c) Find the nullspace of A as the span of special solutions (d) Find a particular solution when b- 3 6 9 and the general solution.
u iegression result presented in this 9.13. Consider the following model: Y; = B1 + B2 Di tui where D = 0 for the first 20 observations and D = 1 for the remainine 30 observations. You are also told that var(u) = 300. a. How would you interpret B and B2? b. What are the mean values of the two groups? c. How would you compute the variance of (Bi + B2)? Note: You are given that the cov...