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Problem 7.61 PP
Consider the pendulum problem with control torque Tc and disturbance torque T&. 
&-+ 40 = Tc + Td.
(Here = 4.) Assume that there is a potentiometer at the pin that measures the output angle 0, 
but with a constant unknown bias b. Thus the measurement equation is y = 0 + b.
(a) Take the “augmented” state vector to be
where w is the input-equivalent bias. Write the system equations in state-space form. Give values 
for the matrices A, B, and C.
for the matrices A, B, and C.
(b) Using state-variable methods, show that the characteristic equation of the model is s(s2 4) 
= 0 .
(c) Show that w is observable if we assume that y = 0, and write the estimator equations for
Pick estimator gains [ I I 12 /^Tto place all the roots of the estimator error characteristic equation 
a t -10.
(d) Using full-state feedback of the estimated (controllable) state-variables, derive a control law to 
place the closed-loop poles at -2 ± j2.
(e) Draw a block diagram of the complete closed-loop system (estimator, plant, and controller) 
using integrator blocks.
(f) Introduce the estimated bias into the control so as to yield zero steady-state error to the output 
bias b. Demonstrate the performance of your design by plotting the response of the system to a 
step change in b; that is, b changes from 0 to some constant value.
S tep -by-s tep s o lu tio n
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