CN4227R Project 1 Group 06

Group members:

TIMOTHY YEO YU JINA0101826NYEO JUN WENA0100521H

Submission Date:

Question 1 The objective of experiment 1 is to find the extreme positive parameters for the uncertain process models given the nominal model, and uncertainty weighting function, as shown below.

The 2 uncertain processes assigned to the group are b) and f) as shown below, where and n are parameters to be found.

From the multiplicative uncertainty description,

represents the either the process or and m is given a value of 1 as this gives the worst possible perturbation which allows the extreme positive parameters to be obtained.Therefore,

As and have been given, the objective of this question is to use trial and error to find the maximum positive value of the unknown parameter that when substituted into will result in the value of being less than or equal to the magnitude of for the entire frequency range.To do this, the magnitude of is first plotted for a frequency range of 0 to 100 as shown in figure 1 below.

Figure 1: Plot of the uncertainty weight, Next, the graph of various values of in was plotted in figure 2. For , the initial guess for the value of was 0.04 with increments of 0.01 until any point of graph of exceeded that of the uncertainty weight. The value of at this point was found to be 1.23. At this point, the value of is at its maximum possible value and increasing any further will result in exceeding at the region where the arrow is pointing. A close-up of this region can be found in figure 3. The code required to produce this graph can be found in appendix A1.

Figure 2: Plot of and with varying values of in

Figure 3: Close-up of the arrowed region in figure 2The graph of various values of n in for can be found in figure 4. For , the initial guess for the value of was 1 with increments of 1 until any point of graph of exceeded that of the uncertainty weight. The value of at this point was found to be 89. At this point, the value of is at its maximum possible value and increasing any further will result in exceeding at the region at which the arrow is pointing in figure 4. A close-up of this region can be found in figure 5. The code required to produce this graph can be found in appendix A2.

Figure 4: Plot of and with varying values of in

Figure 5: Close-up of the arrowed region in figure 4

Question 2a)This question requires a PI controller, to be designed based on the Cohen-Coon tuning method and to compare the performance of this controller with 2 other controllers, and designed based on the ITAE performance index for load and set-point respectively. The equations for and is shown below.

Cohen-Coon tuning will be based on the nominal model, which is shown below.

Cohen-Coon tuning makes use of the nominal model parameters to design the controller via the following equations.

Where,

By substituting in the appropriate values, and are found to be 0.471 and 5.803 respectively. This results in the PI controller as shown below.

Next, the performance of the controllers is compared to each other for a unit step change in the set-point using Simulink. A simple feedback loop is designed in Simulink as shown in figure 6. Figure 6: Simple Feedback loop to simulate unit step set-point changes for and The resulting graph is plotted in matlab via the code found in Appendix A3 and is shown in figure 7.

Figure 7: Graph showing the reponses for the 3 different controllers to a unit step set-point change

To determine which controller has the fastest servo response, 2 criteria will be looked at, 1) time taken for the curves to cross the set point and 2) time taken to reach the first peak. Just by looking at the graph in figure 7, it is obvious that the Cohen-Coon design gives the fastest servo response. Table 1 below shows the specific values for the 2 criteria mentioned above.Table 1: 1) Time taken for curves to cross set-point and 2) time taken for curves to reach the first peak for the 3 different controllers.CriteriaController

C1 (Load)C2 (Set-Point)C3(Cohen-Coon)

Time taken to reach first peak/s16.319.615.85

Time taken to cross set-point /s12.6516.5512.05

From the 2 criteria mentioned above, it can be seen from table 1 that the shortest times to achieve these criteria belongs to , the controller designed using the Cohen-Coon tuning method. Therefore has the fastest servo response while has the slowest servo response.However, the faster speed of servo response for is not indicative of good performance while the slower speed of servo response for is not indicative of poor performance. There are other factors that determine performance such as peak related criteria. An example of peak related criteria is the maximum overshoot, the higher the overshoot, the worse the performance and vice versa. Another example of peak related criteria is the decay ratio which is linked to the oscillatory nature of the curve. A small decay ratio improves the performance as it reduces the oscillatory nature of the curve while a large decay ratio worsens performance as it increases the oscillatory nature of the curve.

Question 2b)There are 3 equations available to determine the frequency response based robust stability criteria. These equations are different forms of each other and they are shown below. The equation that will be used in this section is equation (3).

Equation 3 states that the amount of tolerable uncertainty must be more than the actual uncertainty in order for a controller to be robustly stable. From this equation, we can also imply that the greater the amount of tolerable certainty a controller is able to provide given a fixed , the more robustly stable a controller is as it can accept a greater actual uncertainty. Figure 8 shows the graphical representation of the equation (3). has more tolerable uncertainty than and the controller is therefore more robustly stable.At the same time, it can be said that has better performance than as it remains closer to the value of 1 over a larger range of frequencies. This is indicative of good set-point tracking which can be in turn be interpreted as a fast servo response. Therefore, the closer remains to the value of 1 for a larger range of frequencies, the faster the servo response is.It should also be noted that for PI controllers, the values of for each of the 3 controllers tends towards 1 when the frequency goes to 0. This means that at the steady state condition, there is no offset which is to be expected when using a PI or PID controller.

Figure 8: Graphical representation of equation (3)Using the code found in appendix A4, a similar graph of was plot for the 3 different controllers. Figure 9 shows this graph.y-intercept=1

Figure 9: graph for the 3 controllersFrom figure 9, it can be seen that the Cohen-Coon controller has the least tolerable uncertainty, and hence is the least robust stable. However, it remains closest to the value of 1 over the largest range of frequencies, giving it the fastest servo response as found in question 2a). Likewise, the ITAE (set-point) designed controller has the largest tolerable uncertainty and therefore is the most robust stable. However, it deviates from 1 faster than the other 2 controllers, giving it the slowest servo response as found in question 2a). The ITAE (load) designed controller is in between these 2 controllers.The evaluation of 2a) is consistent with the frequency response based robust stability criterion as explained above. This also concurs with the theory that robust stability and performance both cannot be achieved at the same time. It is a problem inherent in feedback control and one will be achieved at the expense of the other. Therefore, it is up to the designer to decide which is of greater importance in the process to be controlled and from there decide the parameters for the controller.

Question 2c)Assuming the process/model mismatch is entirely due to uncertainty in the process gain, the process model can be represented by the equation shown below.

Where,

To determine the range of K that each controller can accept, it is important to first note that the controller gain is positive, therefore the process gain must be positive as well. This acts as the lower bound for the process gain. The upper bound of the process gain will be determined from equation (3) in section 2b. Equation (3) states that must be more than for all frequencies. Hence the approach to solving this question revolves around finding the and solving the inequality shown below.

To solve the inequality above, an expression for must be found via the equation shown below.

In this question, the expression for , after substituting in the and defined above, is shown below.

Finally, equating to the final expression to solve for the upper bound of K is shown below.

Table 2 shows (found from Matlab), the calculated upper limits of K and finally, the range of K for which the controller can tolerate based on the frequency response robust stability criteria

Table 2: Values of |T-1|min, Upper bound of K and Range of KControllerC1 (Load)C2 (Set-Point)C3 (Cohen-Coon)

|T-1|min0.5731.000.404

Upper bound of K6.29285.616

Range of K0