AE470F23Homework6

iii ) Now consider the same UO model but with velocity output y = x 2 and the V 1 and V 2 covariance matrices V 1 = 0 . 1 0 . 1 0 . 1 0 . 1 , V 2 = 0 . 01 . Consider the same observer described in part ( ii ) with the same initial conditions for both the system and observer but with only w 1 = 0. Let w 2 be modeled as white noise with variance V 2 . Plot the trajectories of x 1 and ˆ x 1 on one figure and x 2 and ˆ x 2 on another figure. Hint: In MATLAB, σ × randn(1,k) generates a normally distributed vector of length k with variance σ 2 . iv ) Repeat part ( iii ) with the same covariance matrices and initial conditions. However, consider 2 separate cases for w 2 . One case where w 2 has a variance >> V 2 and one where w 2 has a variance << V 2 . Based on these results and those from part ( iii ), discuss how the variance of the sensor noise w 2 affects the estimates ˆ x . Hint: Recall what V 2 represents. Problem 3 [30 points]. Consider the UO again with position output, but with an input, disturbance, and sensor noise ˙ x = 0 1 − 1 0 x + 0 1 u + 1 2 × 1 w 1 , y = 1 0 x + w 2 . i ) Manually compute the controllability matrix C ( A, B ). Is this system controllable? ii ) Given initial conditions x (0) = [2 1] T , design R 1 and R 2 weighting matrices such that the output y converges within 20 seconds and the input u does not exceed ± 1. Plot the input u and output y . What are the eigenvalues of your closed loop A + BK ? Hint: Assume no disturbance or measurement noise. iii ) Given the observer ˙ ˆ x = A ˆ x + Bu + F ( y − ˆ y ) , ˆ y = C ˆ x, design V 1 and V 2 weighting matrices such that the estimate ˆ y converges to y within 2.5 seconds given initial conditions x (0) = [2 1] T and ˆ x (0) = [0 0] T . For simplicity, assume all elements of V 1 are the same. Assume no input, disturbance, or sensor noise when testing your simulation. On a single figure, plot y and ˆ y . What are the eigenvalues of your closed observer loop A − FC ? iv ) Provide the numerical values for the matrices ˜ A , ˜ B , ˜ C , E , and G for the augmented system ˙ x ˙ ˆ x = ˜ A x ˆ x + ˜ Bu + E w 1 w 2 , (3) y ˆ y = ˜ C x ˆ x + G w 1 w 2 . (4) v ) Using your gain matrix K from part ( ii ) where u = K ˆ x , reduce (3) to closed-loop form ˙ x ˙ ˆ x = ˜ A CL x ˆ x + E w 1 w 2 , (5) y ˆ y = ˜ C x ˆ x + G w 1 w 2 . (6) What are the eigenvalues of ˜ A CL ? Are they the same as the eigenvalues of A + BK and A − FC ? vi ) Simulate your closed loop system (5) and (6) given the same initial conditions in part ( iii ). Include disturbance and sensor noise w 1 and w 2 where w 1 and w 2 can be modeled as white noise both with variance σ 2 = 0 . 001. On one figure plot y and ˆ y . On a separate figure, plot the input u . How do the combined results compare to the individual design considerations? 3