EE249 Homework 4 TA Alessandro Pinto apinto eecs berkeley edu November 2 2003 Goal of HW4 is to give you an example of hybrid system modeling General Comments there is not right wrong answer to a question I would rather consider your homework as solutions to design problems I could argue that your solution is inefficient and you could argue the same about mine Grade is established mostly on the rationale you follow to answer questions Hence it is in your interest to justify all your statements You can use any kind of sources as long as you reference them all Homework are created by the TA from scratch It is then reasonable that he may forgot assumptions or make some that are not necessary You can add or remove assumptions at you convenience as long as you justify your decisions Problem description We want to build a simplified model of an elevator As you may have noticed there is a specific pattern that the elevator follows to go from one floor to another After you step in the elevator and the doors close it starts increasing it s speed until a cruise speed is reached When the distance to the destination is smaller that a certain value the speed decreases until a landing speed is reached and finally the elevator stops at the desired floor We can define few parameters To reach the cruise speed the elevator moves with a constant acceleration equal to a 0 5g where g 9 81m s2 The cruise speed is vc 4km h The breaking distance the distance needed by the control algorithm to stop the elevator is xb 1m Finally the landing speed is vl 0 1m s The behavior of the elevator and control algorithm can be informally described as follows The elevator waits a new command of the form go to position xT we assume by now that our elevator can only go up During this phase the acceleration and the speed are x 0 x 0 When the command is issued the system changes the state and goes to an acceleration state where the new equations are x a Two things may happen the cruise speed is reached then the system has to go in the cruise state where the new equation is x 0 namely constant speed the distance of the elevator from the target is less than or equal to the braking distance and the speed is less then the cruise speed then the 1 EE249 Spring 2003 Homework 4 system has to go directly to the deceleration state This means that the target position was so close to the starting position that the elevator didn t have time to reach the cruise speed If the distance to the destination becomes smaller than or equal to the braking distance while the elevator is cruising the elevator has to decelerate When the speed drops down to the landing speed the elevator has to move with constant speed vl until the target is reached within a tolerance that you can decide and finally has to stop ideally the speed can go to zero in zero time in practice this cannot happen QUESTION 1 describe an hybrid model of the system that I have informally explained QUESTION 2 build such a model using HyVisual and plot position and speed of the elevator to go from position 0 0 to position 10m Acknowledgment I would like to acknowledge Claudio Pinello for helping me in preparing this homework 2
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