4. A certain system is described by the model Set the value of the damping constant c so that both of the following specifications i and ii) are satisfied. Give priority to the overshoot specification. If both cannot be satisfied state the reason. Time is measured in seconds. Use the Figure 8.3.5 graphs provided. 1 Maximum percent overshoot should be as small as possible and no greater than 20% ii 100% rise time should be as small as possible and no greater than 3 s. (a) Calculate wn (b) Because of the overshoot specification, what is the range of ? (c) Because of the rise time specification, what are the restrictions on (d) What is the damping constant c? ?

We determine the natural frequency ωn of the system. The natural frequency is related to the damping constant c and the undamped natural frequency wn0 by

ωn = ωn0 × √(1 - c^2)

From the graph provided, we can see that the maximum percentage overrun is 20%. This means that the response of the system to a step input oscillates around steady state with an amplitude of 20% of steady state. You can find ωn using the percent overshoot formula for a second-order system.

% overshoot = e^(-c × ωn × π ÷ √(1-c^2))

Solving this equation for ωn gives us:

ωn = -ln(20%) / (c × π / √(1-c^2))

(a) The natural frequency of the system is ωn = -ln(20%) ÷ (c × π / √(1-c^2)).

(b) Because of the overshoot specification, the range of c is c >= -ln(20%) ÷ (ωn × π ÷ √(1-c^2)).

(c) Because of the rise time specification, the restrictions on c are c <= 3 × √(1 - c^2) ÷ π and ωn >= 3 ÷ π.

(d) The damping constant c is the intersection of the range determined by the overshoot specification and the constraint determined by the rise time specification. Both specifications are not met because the range specified by the overshoot specification is larger than the limit specified by the rise time specification.

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