Abstract:

The performance requirements for this project lay out exactly what the design model will do. In addition to the qualitative descriptions given earlier this section will define the quantitative and measurable conditions the system will be expected to meet or achieve. First is a definition of the different characteristics that are used to evaluate control systems. Then it is necessary to define the different types of systems that arise from different values of the aforementioned characteristics. Once this has been accomplished the performance requirements for each possible application will be described. Finally it will be possible to present the desired specifications that our design model will meet or exceed.

Definition of Characteristics:

Rise Time (t_r) -The time it takes the system to reach the vicinity of its new set point following a change in the control signal. This parameter can be interpreted as being how quickly the system reacts to a change. A very short rise time implies that the system can respond very quickly to a change in the control signal. Conversely a long rise time implies a slow response. In general the rise time must be set with the expected speed of disturbances well known. A system with a fast rise time may respond erratically to small variations in the control signal caused by noise. A slow rise time would not react as strongly to these rapid fluctuations. However if the control signal is expected to change rapidly a fast rise time is necessary in order to track the control signal.

Settling Time (t_s) - The time it takes for system transients to decay. This parameter can be thought of as how long it takes the system to settle down after a change in the control signal. A long settling time means that the system will continue to wiggle around its new set point for an extended period of time. A short settling time means that the system response will wiggle when it reaches its new set point, but straightens out quickly. For a system that cannot tolerate any oscillation in its output the settling time needs to be kept to a minimum. However if the oscillations do not perversely affect the desired system performance a long settling time is acceptable.

Overshoot (M_P) - The maximum amount the system overshoots its final value divided by its final value, expressed as a percentage. When the system passes through its new set point, the amount by which it “misses” is termed the overshoot. When considering how much offset a system can withstand it is important to take into account such things as physical limitations. If the system is operating close to a point of failure then any overshoot will push the system past the limit. Also if a system has multiple states of control it is imperative that any overshoot does not falsely switch the control systems modes. An example of this would be a control system that engages another set of gears when the process needs to be sped up, and if incorrectly engaged the process of disengaging the gears is more time consuming that simply slowing the process down.

Peak Time (t_p) - The time that it takes the system to reach the maximum overshoot point. This term is somewhat dominated by the rise time. But it is nonetheless helpful in understanding at which point the system begins to approach the steady state value.

Damping Ratio- Reflects the level of damping as a fraction of the critical damping value. The damping ratio is related to the settling time as it controls how strongly the system response is allowed to respond to a change in input. A damping ratio of zero describes a situation in which the system output does not decline at all, but instead oscillates perpetually. A damping ratio of one, that is equal to the critical damping value, allows the system to approach the desired output but prevents it from having any overshoot.

Natural Frequency- The frequency at which an undamped system will oscillate. For a system with a damping ratio of zero the output will oscillate at the natural frequency. Furthermore any oscillations in the output signal, even if the system is damped, will be at the natural frequency. A point of concern is that an input of the natural frequency will cause unstable system behavior.

Figure 1: Generic System Output with Characteristics Labeled

Definition of System Types:

Undamped- The initial disturbance produces a harmonic response that continues indefinitely. This type of system is defined by its settling time of infinity and it’s damping ratio of zero. The output is not forced to decay and so it oscillates perpetually at the natural frequency.

Underdamped- The displacement overshoots the steady state value initially, and then eventually decays to the steady state value. This system is characterized by a non zero overshoot, an existent settling time, and a non zero damping ratio. The output after some measure of time closely approximates the desired output.

Critically Damped- An exponential rise occurs to approach the steady s tate value without overshoot. This system is characterized by zero overshoot, the critical damping ratio, and does not have a need for settling time. The main parameter that defines this system is the rise time.

Overdamped- The system approaches the steady state value without overshoot, but at a slower rate. This system is identical to a critically damped system, except that it has a longer rise time.

Figure 2: Generic Systems of Various Types

Possible Design Application Performance Requirements:

Airplane Propeller Application

General- In this application the system needs to maintain synchronization between the propellers on an airplane in order to prevent “beats” that negatively affect the crew and passengers.

Rise Time- The rise time necessarily needs to be several times than the average fluctuation in wind speed. The reasoning for this is that if the wind turbine is still responding to a gust of wind that happened in the past, when a new gust of wind hits the system will start to chase that value. As this process repeats the output is constantly chasing the correct value but never reaching it. A constraint on the lower limit of the rise time is that it cannot respond strongly to errant gust of wind that are not indicative of a general increase or decrease in wind speed. In this case the system would be reacting to insignificant events. Another criterion that needs to be considered when engineering this parameter is whether the system has any settling time. If there is a settling time then this parameter loses some of its importance. In that the time to steady state would consist of both the rise time and the settling time. However if the system has no settling time then this parameter is of high importance. The main factor that needs to be considered when calculating the needed value for this application is the typical weather conditions in the area of operation.

Value- The propeller needs to be within 5% of the desired speed in 0.5 seconds or less.

Settling Time- When calculating this value it is necessary to take into account the maximum deviation in frequency that the propellers are allowed. This value is determined by the limit of the human ears powers of perception. Professor Kohne informed this group that 20 beats per minute or less is an acceptable margin of error. Therefore if during the settling time the system output has a frequency deviation of 20 hertz or less it can be longer. In other words a band of 40 hertz around the desired frequency may be considered the target area. If the settling time has periods where the output is greater than 20 hertz different then it needs to be as short as possible.

Value- 1.5 seconds (approximately three times the rise time)

Overshoot- The changing environment is of critical importance for the definition of this parameter. If the master propeller is hit by a gust of wind that speeds it up slightly, the slave propeller will respond and overshoot the master speed. Say it is an unfortunate moment and another gust strikes the master propeller. Again the slave will respond and overshoot. If several gusts proceed in this fashion it is conceivable that the slave propeller will be severely out of sync with the master for a period of time. The beats will be quite audible at this time and the system will have failed. Therefore the overshoot needs to be minimized to a fraction of 20 hertz to prevent this scenario from occurring.

Value- 10 Hertz

Our Design Performance Goals:

General- The design model that will be constructed is intended to synchronize the rotation, of both frequency and phase, of two DC motors. Experiments will be conducted that subject the motors to different initial conditions and changing environmental conditions.

Rise Time- 5% Frequency deviation within 4 seconds

Settling Time- Frequency synchronization within 8 seconds

Overshoot- Frequency overshoot of 10 hertz or less

System Type- Underdamped or critically damped.

Environmental Constraints- The design must be able to operate at the low temperatures of high altitude and also function under conditions of high vibration.

Cost- The cost of purchasing components must be less than $100, and the total labor time for construction must be less than 60 man hours.

Ethical Considerations-Since this system is intended to control the propulsion method for an airplane it needs to be completely safe. It is unacceptable for the system not to have a fail safe manual method, or to routinely fail during operation. Any design flaws need to be completely addressed.