On July 16, 1993, the elevator in Yokohama's 70-storey Landmark Tower, operating at a peak speed of \(45 \mathrm{~km} / \mathrm{h}(28 \mathrm{mph})\), was inaugurated as the fastest super-fast elevator. To reach such a speed without leaving passengers' stomachs on the ground floor, and the lift accelerates for longer periods rather than precipitously. Going up, it reaches full speed only on the 27th floor; it begins to decelerate 15 floors later. The result is a peak acceleration similar to that of other skyscraper elevators - a bit less than a tenth of the force of gravity.

Admirable ingenuity has gone into making this safe and comfortable. Special ceramic brakes had to be developed; iron ones would melt. Computer-controlled systems damp out vibrations. The lift has been streamlined to reduce wind noise as it hurtles up and down. One proposed system for the elevator's vertical position is shown in the following diagram.

Determine the range of \(K\) for a stable system.

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Controller Elevator dynamics R(s) 1 Y(s) (K+1) Desired s(s+3s+3) Vertical vertical position position Elevator Vertical Position Control