Control Systems/Examples/Second Order Systems

Second Order Systems: Examples Example 1 A damped control system for aiming a hydrophonic array on a minesweeper vessel has the following open-loop transfer function from the driveshaft to the array. The gain parameter K can be varied. The moment of inertia, J, of the array and the force due to viscous drag of the water, Kd are known constants and given as: Tasks The system is arranged as a closed loop system with unity feedback. Find the value of K such that, when the input is a unit step, the closed loop response has at most a 50% overshoot (approximately). You may use standard response curves. Should K be greater or less than this value for less overshoot? Find the corresponding time-domain response of the system. The system is now given an input of constant angular velocity, V. For the limiting value of K found above, calculate the maximum value of V such that the array follows the input with at most 5° error. Task 1 First, let us draw the block diagram of the system. We know the open-loop transfer function, and that there is unit feedback. Therefore, we have: center|alt=R(s) instead of X(s) The closed-loop gain is given by: {| ||| |- | || |- | || |} We now need to express the closed-loop transfer function in the standard second order form. {| ||| |- | || |} We can now express the natural frequency ωn and damping ratio, ζ: We now look at the standard response curves for second order systems. center|500px We see that for 50% overshoot, we need ζ=0.2 or more. This is the maximum permissible value, thus K should be less than this value for less overshoot. We can now evaluate the natural frequency fully: Task 2 The output of the second order system is given by the following equation: {| ||| |- | || |- | || |} We can plot the output of this system: center|500px Task 3 The tracking error signal, E(s), is equal to the output's deviation from the input. Now, we can find the gain from the reference input, R(s) to the error tracking signal: The gain from the input to the error tracking signal of a unity feedback system like this is simply . {| ||| |- | || |} Now, R(s) is given by the Laplace transform of a ramp of slope V: We now use the final value theorem to find the value of E(s) in the steady state: {| ||| |- | || |- | || |} We require this to be less than