Problem # 1 Consider a closed-loop, rotary, speed-control system with a proportional controller K p, as shown below. The inertia of the rotor is J. The damping coefficient B in mechanical systems is usually difficult to estimate accurately. Because the computed gain value depends on the estimate of B, the actual system performance can be quite different than the predicted value if our estimate of B is very inaccurate. T d (s) Ω ( s) Ω ( s ) d + - S K p + + 1 Σ Js + B 1
Suppose the nominal value of B = 1. Discuss the sensitivity of the steady-state response to uncertainties in B, for a step command input. Specifically, compare the sensitivity for K p =1 and for K p = 9. Discuss the sensitivity of the closed-loop transfer function to uncertainties in B (whose nominal value is B = 1). Assume J = 1. Here, the sensitivity is a function of the Laplace variable s. Use the magnitude frequency response plot to compare the sensitivities for K p = 1 and K p = 9. Consider now the position control system shown below with K p = 0.05, J = 20, and B = 1. T d (s) R(s) + - S K p + + Σ 1 ( + B) s Js C(s) 2
The inertia J of the object to be positioned changes during the control process. The control of the angular position of a satellite (to aim a telescope, for example) must account for the change in the satellite s inertia due to the fuel consumption of the control jets. The inertia of a tape reel or a paper roll changes as it unwinds, and the controller must be designed to handle this variation. Investigate the sensitivity of this design if J can vary by ± 10%. 3
Problem # 2 Satellites often require attitude control for proper orientation of antennas and sensors with respect to the earth. This is a three-axis, attitude-control system. To gain insight into the three-axis problem we often consider one axis at a time. For the one-axis problem, the plant transfer function is G(s) = 1/s 2. This results from the equation of motion: J is the moment of inertia of the satellite about its mass center, T is the control torque applied by the thrusters, and θ is the angle of the satellite axis with respect to an inertial reference frame, which must have no angular acceleration. A lead controller that gives the closed-loop system a damping ratio ζ 0.7 and an undamped natural frequency ω n 6 rad/sec is given by: ( ) J θ= T ( s+ 3) ( + ) Gc s = 121.7 s 18.23 4
Assuming the controller is to be implemented digitally, approximate the time lag from the D/A converter to be: 2/T s+ 2/T Determine the closed-loop system root locations for sample rates ω s = 5 Hz, 10 Hz, and 20 Hz, where the sample period T = 1/ω s seconds. Plot the unit step responses for each sample rate and compare. State your observations regarding closed-loop stability. The closed-loop block diagram is shown below: R(s) 2/T ( s+ 3) 121.7 s 18.23 ( + ) S 2 + s + 2/T s - 1 C(s) 5
The analog implementation of the closed-loop system has a rise time of about 0.3 sec (t r 1.8/ω n ). How fast do you think one should sample in order to have a reasonably smooth response? 6
Problem # 3 Suppose a radar search antenna at an airport rotates at 6 rev/min and data points corresponding to the position of flight 1986 are plotted on the controller s screen once per antenna revolution. Flight 1986 is traveling directly toward the airport at 540 mi/hr. A feedback control system is established through the controller who gives course corrections to the pilot. He wishes to do so each 9 miles of travel of the aircraft, and his instructions consist of course headings in integral degree values. What is the sampling rate, in seconds, of the range signal plotted on the radar screen? What is the sampling rate, in seconds, of the controller s instructions? Identify the following signals as discrete or continuous in amplitude and time: The aircraft s range from the airport The range data as plotted on the radar screen 7
The controller s instructions to the pilot The pilot s actions on the aircraft control surfaces Show that it is possible for the pilot of flight 1986 to fly a zigzag course which would show up as a straight line on the controller s screen. What is the lowest frequency of a sinusoidal zigzag course which will be hidden from the controller s radar? 8
Problem # 4 Consider the two mass one spring system, shown below. An external force F is applied to mass M 1. Derive the equations of motion for this system. Determine the transfer functions: X1( s) X2( s) and Fs Fs ( ) ( ) X 1 X 2 F K M 1 M 2 Frictionless Surface 9
Identify the poles and zeros of the two transfer functions. Physically what do these poles and zeros represent? Why are the poles of the two transfer functions the same? Why are the zeros of the two transfer functions different? Write the equations of motion in state-space form with the state variables chosen to be the positions and velocities of the two masses, and the output chosen to be the position of mass M 1. The input is the applied force F. Draw by hand a Matlab/Simulink block diagram for this system that could be used to predict the dynamic behavior of this system. Do not run the simulation. 10
Problem # 5 Shown is a computer-controlled motion control system. M = 0.001295 lbf-s 2 /in B = 0.259 lbf-s/in K mf = 0.5 lbf/a K pa = 2.0 A/V Sensor Gain = 1.0 V/in Computational Delay = 0.008 s pure/ideal mass and damper 11
A MatLab/Simulink block diagram of this digital control system is shown below. Explain all the elements in this block diagram. Simulate this system in MatLab/Simulink for various proportional control gains (e.g., 10, 20, 30) and computational delays (e.g., 0,.001,.002). State your observations. Step Sum 1 Kp Proportional Controller Comp Delay Quantizer 1 10 V - 8 bits Resolution 0.039063 Zero-Order Hold 1 T = 0.005 sec Kpa*Kmf Gain 1 Sum 2 1/M Gain 2 B acc vel pos 1/s 1/s Integrator 1 Integrator 2 dpos Gain 3 dpos Ks step input Clock t time Quantizer 2 1 V - 12 bits Resolution 0.000244 Zero-Order Hold 2 T = 0.005 sec Sensor pos position 12