Value For K 1 0 example essay topic

Page 1 Abstract The objective for this project was to give one the opportunity to apply the concepts of modern control systems in a realistic design scenario. The project also provided a chance to gain a better understanding of how closed loop systems and forward loop systems model the dynamics of a space vehicle re-entering the earth's atmosphere. These are the tools that can be utilized in real world engineering projects. The system modeling the space vehicle dynamics was successfully used with specific gain values to produce a stable system built to design specifications. Introduction The design of a winged space vehicle is one that has occupied the National Aeronautics Space Administration throughout the 1950's and 1960's. It was not until the 1970's that NASA put into use the first winged space orbiter.

Set to this backdrop of history, we began our modest endeavor to model the wing dynamics of our spacecraft design. The goal for the final design of the vehicle dynamics system is to modify the system to be stable in a specific range. For this project we were able to find a range of stability for the design problem. One of the objectives was to first model the system with the integral gain K 1 equal to 1 and the proportional gain K 2 equal to 0.

The next objective of this project is to model the vehicle system for specified output parameters. The requirements for this project were to obtain the Bode diagram, gain margin, and phase margin. It is then possible to determine the stability, rise time, peak time, settling time, and percent overshoot of the system by examining the graphs. The second requirement was to stabilize the system by selecting two values of the gain (K 1 and K 2), which will result in a gain crossover frequency of 1 rad / sec with a phase margin of at least 40 degrees and a gain margin of at least 45 db.

Design The project first involved examining the control system with the integral gain K 1 = 1 and the proportional gain K 2 = 0. The second stage of the project involved selecting the values of the integral gain K 1 and the proportional gain K 2 that will give the phase margin of at least 40^0 and the gain margin of at least 45 db. The control diagram that models the space vehicle can be examined at the beginning of the appendix. For the first part of the project, with K 1 set to equal 1 and K 2 set to equal 0, the system was unstable. See Figure A 3 in the appendix for the Bode plot that listed the gain and the phase margin as 0.

The root locus diagram (Figure A 4) showed that one of the poles and one of the zeros were located on the origin (0, 0). When one varies the K values, the pole becomes increasingly unstable as it moves deeper into the right hand side of the graph. The second part of the design involved finding the optimum gain values that produced the proper phase and gain margin. In order to accomplish this, Matlab was used to model a wide range of various gain values. First, K 2 was set to a constant value of 0 and K 1 was tested in the range of -500 to 475.872 in increments of 51.86. See Table B 1 in the appendix for the results of this test.

Every value of K 1 in this range yielded an unstable control system. Next, K 1 was set to a constant value of 0 and K 2 was varied along a range of different values. The range that was used in this case started with -292.976 and ended with 1390.03. The change in each increment of this range was 70.124. If one examines the results listed in Table B 2 several trends become immediately obvious.

The system is unstable for the first four negative values. After increasing to -12.48, the system became stable for the rest of the range. Next one can notice that the gain margin of the system starts at a very low value and then increases to the peak value of 33.554 db listed in this simulation. The gain margin then decreases until it becomes very small at the end of the chosen range. The second simulation run focused on the numbers surrounding the peak of the first run. K 2 was tested in the range from -12.48 to a maximum of 137.52.

The increments changed during this run in steps of 25. Table B 3 shows that a K 2 value of 12.52 produced a gain margin of 46.947 db and an infinite phase margin. The third run focused on the K 2 values around 12.52. This run started at 2.52 and increased in steps of 5 until the maximum K 2 value equaled 22.52. In Table B 4 the results found came close to the gain design requirement of 45 db. The system was modeled again with the integral gain K 1 set to 0 and the proportional gain K 2 set to 17.52.

These K values yielded an infinite phase margin and a gain margin of 44.021 db. This was within 2.2% error of the intended gain margin of 45 db. Figure B 1 through B 10 shows that this system is stable. Conclusions In this project, it was possible to choose the integral gain K 1 and the proportional gain K 2 so that they yielded the engineering specifications for the phase margin and the gain margin. In order for the system to be stable, K 1 must be 0. When K 1 is non-zero, the transfer function had an extra S factor multiplied to the denominator.

When the poles were factored this produced a pole that was located at the imaginary intercept (0, 0) of the pole-zero graph. Thus, a value for K 1 = 0 was needed to remove this instability. The only thing that was left was to find the optimum value for the proportional gain K 2. This was possible by examining various ranges of values.