Download Solutions for Midterm Exam - Special Topics in Mechanical Engineering | ME 4803 and more Exams Mechanical Engineering in PDF only on Docsity! -lrpa13 lly ua~? aq IOU II~M JaMsm la3paumu a~dwrs asam3 lsour q -laMsutt ~noi JOJ uosn~ aqq o!aldxa Jomoa JaMsott ~noX qq!~ JOM ~noX moqs plnoqs noA 53uqm uo paprhold aq llr~ auros 'ladad lauoyppe paau norZ JI -ura~qoid ayl T~OM 01 lr .%UIMOIIOJ . ayl uo pm luauralqs ura~qold ayl ylaauaq paprhoid ladtld ayl asn .pasn aq bur qooq ON .lolaln3la= a pm sap~s yloq uo uanp salou JO laays a am Lam noA xmxa sw ala~duro:, 01 salnurur 0s pm smoy ZJO umqxaur a ahay noA :suopnqsq 1. The system diagrammed below can be analyzed by Newton's laws to produce two differential equations: m i ~ l =-b(Y,-Y2)-bl +f m2~2 = b(Y1 - Y2 ) Produce a 31d order matrix state space equation i = Ax + Buy y = Cx that describes the system using the following definitions for states X I , x2 and x3: x1 'Y1 x2 = Y1 x3 = Y2 The output y is the difference in velocity of the two masses. The masses are constrained to mover horizontally. The yl and y2 variables in the equations are the horizontal displacement of the masses ml and m2, respectively, b is the damping coefficient and k is the spring constant. The applied force is f is the system input u . 4. For the system in problem 2 only the single state x, is available for feedback control, so it is proposed to use an observer. Assume that a program was made to find the state feedback gains K for a general controllable problem with arbitrary A and B matrices. The program call statement is of the form: K = poleplace(A, B, pc) where pc is a vector of the desired pole positions of the controlled system. (a) How would you use this program to place the poles of the observer (instead of the state feedback controlled system as was required in problem 2) at positions stored in a vector called po? Stated differently, if Lt = poleplace(M1, M2, M3) What would you input for the three M matrices (make clear what the numerical values of MI and M2 are. M3 is dealt with in part d) and what would you do with Lt? Give the observer parameters in terms of the program output and write the equation that would describe the observer. (b) How would you use the observer in the overall control system to achieve the ends of the original controller design? (c) How many observer poles would you have in a fill order state observer? What is the minimum number you would have in a reduced order state observer? (d) What would be reasonable values to place in the vector po to get good estimator rcsplse? - -- - \ - AQJnJ tL observer eW&fu\ e 3, A 4 c - L C % + Z I A,: A - L C 5. For the system in problem 3, (a) find the transfer function assuming 242 is the input and that the output is x,. (b) Find the poles and zeros of the above transfer function and simplifL the transfer function to the extent possible. \
