Feedback Control Systems
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1 ME Homework #0 Feedback Control Systems Last Updated November 06 Text problem 67 (Revised Chapter 6 Homework Problems- attached)
2 65 Chapter 6 Homework Problems 65 Transient Response of a Second Order Model (Section 6) For problems 6-5: a Compute the natural frequency n and damping ratio b Compute the eigenvalues and c State whether the system is underdamped or overdamped d Compute the time constant for each mode e If there is one mode and it is oscillatory compute the damped frequency of oscillation d and the damped natural period Td f Confirm by simulation that the time constant and frequency of oscillation accurately predict the response Use initial conditions of (0)= for your simulation In your simulation code set the time range from t=0 to five times the longest time constant ( 0) 0 75
3 65 Concepts of Feedback Control (Section 6) 66 Consider a DC electric motor connected to an rotational inertial load I shown at the right A voltage V(t) is applied to the electric motor and the armature possesses a rotational inertia of Ia a Derive a model for the electric V(t) motor and rotational inertia load I I Assume that rotational damping with coefficient ct is used to model frictional torque applied to the armature In your model use the symbols Kt Kb R and L to represent the torque coefficient back-emf coefficient armature resistance and armature inductance respectively and (t) to represent the angular displacement of the armature b It is desired to develop a feedback control system for the motor and inertial load A sensor is used to measure the angular velocity of the armature The measurement of the angular velocity is used by a controller and amplifier that implements integral control to supply the voltage V(t) to the motor of V ( t) Ki ( t) dt where Ki is the integral gain and (t) is the intended speed of the motor Neglecting the armature inductance L show that the model for the motor inertial load and controller with amplifier reduces to KiKt KiKt KtKb ( t) c I I T a R R R c Using the model for part b above show that the damping ratio and natural frequency can be computed from the following formulas ct KtKb R KiKt I I K K R RI n I a d Design a controller ie choose an integral gain Ki that will achieve a damping ratio of =04 As will be the case in part e the desired speed will be (t)=0 for t<0 and then switch to a constant value (t)==50 deg/sec for t0 Assuming that R=5 Kt=075 Nm/A Kb=075 Vs Ia=005 kgm I=6 kg/m ct=05nms compute the anticipated natural frequency n percent overshoot Mp settling time ts and rise time tr that will characterie the response of the motor as it transitions from (0)=0 to (t)= at steady state e With the controller designed in part d above simulate the system over the time range 0t0 sec using a value of =50 deg/sec as the desired speed You may assume that the initial conditions for the simulation are ( 0) 0 i t P 66 Ia a 76
4 ( 0) 0 rad/sec and rad/sec Plot the motor speed versus time t Using this plot verify that the percent overshoot Mp rise time tr and settling time ts computed in part d reasonably characterie the response of the motor 67 Consider a DC electric motor coupled to a rack of mass m by a pinion gear of radius r A voltage V(t) is applied to the electric motor and the armature possesses a rotational inertia of Ia The purpose of the system is to control the position of the rack P 67 V(t) c Ia I r m a Derive a model for the electric motor and the rack In your model use the symbols Kt Kb R and L to represent the torque coefficient back-emf coefficient armature resistance and armature inductance respectively and (t) to represent the angular displacement of the armature and x(t) to measure the displacement of the rack b It is desired to develop a feedback control system to control the position x(t) of the rack A sensor is used to measure the instantaneous position x(t) of the rack The measurement of the position of the rackis used by a controller and amplifier that implements proportional control to supply the voltage V(t) to the motor of V ( t) k px ( t) x where kp is the proportional gain and X(t) is the intended position of the rack Neglecting the armature inductance L the armature rotational inertia Ia and the pinion inertia I show that the model for the motor rack and controller with amplifier reduces to k pkt k pkt KtKb X ( t) x c x m x rr rr r R c Using the model for part b above show that the damping ratio and natural frequency can be computed from the following formulas c k K T KtKb r R p t n m K k rr rrm d Design a controller ie choose a proportional gain kp that will achieve a damping ratio of =04 As will be the case in part e the desired speed will be X(t)=0 for t<0 and then switch to a constant value X(t)=X for t0 Assuming that R=5 Kt=05 Nm/A Kb=05 Vs c= Ns/m m=0 kg compute the anticipated natural frequency n percent overshoot Mp settling time ts and rise time tr that will characterie the response of the t p 77
5 motor as it transitions from x(0)=0 to x(t)=x at steady state e With the controller designed in part d above simulate the system over the time range 0t0 sec using a value of X=50 cm as the desired speed Use intial conditions of m and m/sec Plot the rack displacement x(t) versus time t Using this plot verify that the percent overshoot Mp rise time tr and settling time ts computed in part d reasonably characterie the response of the motor x( 0) 0 x( 0) 0 65 Solution of Linear Homogeneous Model for Systems of Arbitary Order (Section 64) For the following problems 68-7: a Compute the eigenvalues and eigenvectors b How many exponential and oscillatory modes are there (ie what are Nr and Nc)? c Assess the stability of the system d Compute the time constants for stable exponential and oscillatory modes For all oscillatory modes (stable or unstable) compute the frequency of oscillation and the period of oscillation e Assemble the matrix [(t)] using the eigenvalues For the complex eigenvalues make sure and use the Euler formula to convert the complex exponential to sines and cosines in [(t)] as described in Section 6 f Compute the matrix [E] - g Compute the solution {q(t)}=[e][(t)][e] - {q(0)} 68 Initial conditions q(0)=- q(0)= q 56q 69 Initial conditions q(0)=0 q(0)= 0895q 687q 60 Initial conditions q(0)=0 q(0)=- 6 Initial conditions q(0)=0 q(0)= 085q 0q 0059q q 474q 095q 0q 07q 78
6 08q 458q 6 Initial conditions q(0)= q(0)= 85q 0q 098q 048q 566q 4q 6 Initial conditions q(0)=- q(0)= and q(0)=0 78q 59q 7q 89q 077q 0q 64 Initial conditions q(0)=- q(0)= and q(0)=5 5q 0075q 05q 558q 08q 79q 65 Initial conditions q(0)=0 q(0)= and q(0)=0 5q 644q 686q 706q 967q 97q 66 After completing parts a and b of Problem 54 9q 577q 54q q 6q 08q 75q 07q 08q a Compute the eigenvalues and eigenvectors b How many exponential and oscillatory modes are there (ie what is Nr and Nc)? c Assess the stability of the system d Compute the time constants for stable exponential and oscillatory modes For all oscillatory modes (stable or unstable) compute the frequency of oscillation and the period of oscillation 67 After completing part a of Problem 5 and using parameters specified in part b a Compute the eigenvalues and eigenvectors b How many exponential and oscillatory modes are there (ie what is Nr and 79
7 Nc)? c Assess the stability of the system d Compute the time constants for stable exponential and oscillatory modes For all oscillatory modes (stable or unstable) compute the frequency of oscillation and the period of oscillation 80
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