Acceptable Control Analysis

As mentioned earlier, one of the key insights derived from this course is the limitation to achievable closed-loop performance due to intrinsic system properties. Once the students had obtained the physical models of the system, they computed a linearized approximation at a steady state operating point and analyzed the controllability properties of the resulting linear system. The inputs and outputs of the system were appropriately scaled before the controllability analysis was carried out. The first metric considered was the Relative Gain Array (RGA) as a function of frequency. For the system configuration employed in this study, the students found that the diagonal RGA elements were very near to 1 at low frequency, suggesting an easily decoupled system. However, as the frequency increased to the bandwidth region, the students discovered that the diagonal RGA values decreased significantly, indicating the importance of multivariable interactions in the bandwidth of interest. Such an insight is particularly valuable at the graduate control level to highlight the limited interpretation of the steady state RGA value. Additional insight is derived from an analysis of the singular values of the system. More specifically, their ratio (the condition number), gives an indication of the sensitivity of the plant to uncertainty. The condition number at low frequencies was small, between 1 and 3. However, it decreases with frequency, implying that the plant is more sensitive to uncertainty at steady state than at higher frequencies. In addition, the low frequency minimum singular value is above 1. This means that adequate control action should be possible; the input moves will be able to move the outputs a sufficient amount to track setpoints. The minimum singular value of the plant is greater than 1 up to the frequency of . This indicates a potential constraint on the controller bandwidth because of high frequency input saturation. Another quantity of interest in control systems in general, and the Four Tank System in particular, is the location and direction of multivariable process zeros. For the operating conditions in this study, the multivariable zeros are found to be at -0.0791 and . The input zero direction corresponding to the Right-Half-Plane (RHP) zero is , and the output direction is . From these directions, one can see that forcing one pump up while the other is forced down causes the system to display inverse response. The presence of the RHP-zero could also be seen in a plot of the RGA, in that the elements of the RGA change sign from frequency to frequency . The lesson that the students will take away from this analysis is that the RHP-zero also limits the controller bandwidth.