Question

4. Consider the position control of a rigid body, Figure, where u(t) is the control force. An analog PID control law is described as de u)kpe)k kije0)dt; e({)= x4C)x) dt 0 kp. ky only a position sensor is available. are controller gains. Also, xq (t) is the desired position of the mass. It is assumed that kJ where and Derive a difference equation for the implementation of this PID control law on a digital computer. Use backward difference and trapezoidal rules for derivative and integral, respectively a. Clearly draw the block diagram of your digital control system and determine the closed-loop transfer b. function in z-domain. If the desired position xd(t) is a unit step function, analytically determine and plot the output x(kT) c. where T is the sampling period. Verify your solution by directly solving the difference equation in Matlab Take T 0.01s m = lkg., k, =10N/meter, k, =1N-s/meter and kj 0.1N (meter - s) d. Verify the steady state value in part c via the final value theorem. A->x(t) ult

Answer 1

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