Memorandum

To: Al Schmelzer

**From: Nooryusmiza Yusoff**

Re: Simulation of Control Systems

Date: May 13,1997

__PART A__: Hardware

A schematic diagram of the two-tank system is enclosed in the appendix (Figure A1). However, no data were taken during the experiment. The figure was used only to help us visualize the simulation problems in Part B and C.

__PART B__: Simulation of Control System

A pressure transducer was used to measure the water level in Tank 2 in terms of DC voltage signal. Its output, b'(t), was determined to be a second order ODE with respect to the water level in the tank. Its transfer function is shown below:

where K = 0.5 volt/in., t = 0.04 min and z = 0.6

The response of the transducer output, b'(t), to the step change in water level of 5 inches was plotted in Figure 1. The settling time to5% was determined to be 0.2 min. The response to the sinusoidal forcing function, h'(t) = 5 sin (wt) was also investigated. Two values of w (10 and 0.1), were used to examine how the response would be with (t = 0.04 min) and without (t = 0 min) the measurement dynamics. From both Figures 2 and 3, it can be seen that the measurement dynamics delay the response, b'(t), by f = -0.52 for w = 10 and f = -0.005 for w = 0.1. Generally, if the f is small enough, the responses, with or without measurement dynamics, will be similar. Thus, it is not necessary to include the measurement dynamics in such a case as with f = -0.005. However, if the f is large, such as the response with w = 10, the measurement dynamic factor must be included to obtain an accurate response.

The response of the water level, h'(t), to different values of t_{2} (2 and 0.2 min) was investigated. Its transfer function is shown below:

where K_{2} = 2 in./GPM and q'(t) = 1 GPM.

The dynamic and non-dynamic responses of the water level to a unit step increase in q'(t) were plotted on Figures 4, 5 and 6. From Figure 4, as expected, the response with smaller t (0.2 min) reached the steady state level (H = 2 inches) faster than that with larger t (2 min). t represents the time constant of the process - the smaller the t, the faster the response. From Figure 5 and 6, the water level (with t = 0.2 min) reached the steady state level in about 1.5 min but it took about 10 min for t = 2 min. Hence, for smaller t, the measurement dynamics can be neglected if the data would be taken after 1.5 min. However, for larger t, it is not advisable to neglect the measurement dynamics unless the data would be taken after 10 min of the start-up. In general, the measurement dynamics can be neglected if the difference between the response of including the dynamics and neglecting it is small like the case with low t.

__PART C__: Dynamic Response of a Controlled System__ __

This part examined the responses (water level in Tank 2) of the two-tank system to proportional control (P), proportional integral control (PI) and no control. The responses were plotted on the same graph on Figure 7. It turned out that the PI control is the best among the three controls because the response (magnified 100x) decreased quickly to the original steady state level after the system was perturbed. However, the response to the PI control was a little oscillatory, which can make the system less stable for higher value of K_{c} or lower value of t_{I}. As expected, the no control system was the worst because the response increased to a new steady state level (H = 2 ft) but never returned to the original level. Although the response of the P control system was *similar* to that of the no control system, its final result was somewhat better. The water level of the P control system only deviates about 0.3 feet from its original level compare to 2 feet for the no control system. Hence the PI control system was determined to be extremely effective and the best among the three to offset the unexpected perturbation.