Pitch Rate CAS Design Project Washington University in St. Louis MAE 433 Control Systems Bob Rowe 4.4.7
Design Project Part 2 This is the second part of an ongoing project to design a control and stability system for pitch control of an aircraft. This second part of the design project will cover the following areas: 4. Establishing the system configuration and identifying the actuator 5. Obtaining a model of the process, the actuator, and the sensor 6. Describing a controller and selecting parameters to meet the performance specifications 7. Optimizing the parameters and analyzing the performance 8. Repeating these steps if the performance is unacceptable In part three, the last part of the design project, a prototype will be built and tested from the results of parts one and two. Design Goals As a refresher, it is fitting to go over our design goals and a few of the key variables associated with our problem (see Figure 1a). A summary of the control goals follows: 1. Dead beat pitch response to precision tracking with t r 1. 5s 2. Steady state error of less than 5% 3. Phugoid damping. 4 4. Short Period damping. 35 We will consider an aircraft flying at an altitude of 4, ft at a velocity of 774 ft/s. The aircraft will be modeled during constant, steady flight. The aircraft will be examined mostly in the x-z plane where moments are about the y axis. This can be done assuming that the mass distributions around the x and z axes are symmetric. Figure 1a: Key variables in describing control of aircraft
System Configuration, Actuator, Sensor, and Process Model The results from part one of the design project have shown a need to improve the phugoid and short period responses. Let us see how employing pitch rate feed back to our system will improve the response. Pitch rate feed back should give us more control of the phugoid and short period responses and help us meet our design goals. To provide feed back and control of our system we will need a sensor and an actuator. For the sensor we will use is a gyroscopic pitch rate sensor which can be modeled with a transfer function equal to one. For the actuator we will use a hydraulic elevator as commonly used in aircraft. We will model the elevator as a lag transfer function with a time constant of 1/2 seconds. Thus our plant transfer function becomes the following: G P G actuator G aircraft e q u e e To account for the fact that a negative moment is created for a positive elevator deflection it is necessary to apply a phase reversal by multiply the actuator state variable by negative one. This yields the following relation: e x a The state equation and output equation for our new state variable X a, the actuator state variable, are as follows: 1 x a x y x e a a 1 u e We can now add this new state equation to our existing state equations by augmenting the state matrices. From this we will attain the following newly formed matrices. vt x q x a
3.e 3 8.3979e 5 F 2.1e 4 3.186e 1 3.19e 1 7.8174e 1 3.22e 1 1. 1. 4.29e 1 1.7453e 2 4.589e 4 2.246e 2 2 G 2 ue H 18 / e J As in part one of the design project the previous matrices belong to our matrix state equations that are expressed as follows: x F x Gu y H x Ju Uncompensated Control System - G c = K p With the current model of our system we are not meeting the design goals. This fact can be seen by examining the root locus and the response to a step input. Let s take a look at the unit step response of our system in figures 1 and 2.
Figure 1: Unit step response of uncompensated system Figure 2: Unit step response of uncompensated system (note: larger time scale)
In figure 1 we can see that the response to a unit step input reaches amplitude of one and then starts to decrease in amplitude. In figure 2, where we are viewing the response on a much larger time scale, we can see that the amplitude quickly drops from an amplitude of one and begins to oscillate around zero. The amplitude continues to oscillate until the response settles at zero. The system at this point yields unsatisfactory results. The response does not meet our design goals because it is not dead beat, does not have the right final output, and thus does not meet our goal of steady state error being less than five percent! The root locus, Figure 3, doubles as a pole-zero plot because we can identify the location of all of the system s poles and zeros. Examining and manipulating the root locus proved futile in fully meeting our design goals. We need to find a way to make our system have a dead beat response, assume a steady state error of less than 5%, and make the damping fall within design constraints. Let us try to do this by adding a PI compensator to our system. Figure 3: Root locus of uncompensated system PI Compensated System G c = K p (S+Z)/S By adding a PI compensator we are effectively adding another pole and zero to our system. The form of a PI compensator is as follows: K p S Z S
The value of K p (the gain) and Z (zero location) are to be chosen. These values can be manipulated in such a fashion to design the system response to meet the specified design goals. The PI compensator was designed by manipulating the root locus using SISO tool in Matlab. When values of K p and Z were chosen the step response was examined to see if it met the design goals. Figures 4 and 5 show the results of adding the PI compensator to our system. Figure 4: PI compensated root locus plot (left) Figure 4 shows the root locus plot of the PI compensated system. The zero added can be seen slightly to the left of the imaginary axis and the pole added lies at the origin. Manipulating this root locus plot resulted in choosing a gain value K p = 6.5 and a zero location value Z = 3.62. These values resulted in the following PI compensator. S 3. 62 6.5 S The unit step response to our system while utilizing the above compensator results in a better response than what we had without a compensator. The PI compensated unit step response can be seen in Figure 4.
Figure 5: PI compensated unit step response As can be seen in Figure 5 adding a PI compensator dramatically improves the system s response. The response s final value is now somewhere around one. Even when viewing a larger time scale the response s amplitude settles close to one. However, due to the zero the PI compensator did add a substantial amount of overshoot to our system. It can be seen in Figure 5 that the amplitude within the first second reaches a max value around 1.3. The rise time in the response looks ideal because we attain our peak value in under 1 second. This easily meets the design goal of the rise time being less than 1.5 seconds. To meet all of our design goals we should have a dead beat response and no overshoot. We also need to verify our damping and rise time to ensure that it is within our design goals. The system is closing in on what we desire but is still unsatisfactory. PI compensator with a minor loop and closing the loop How then shall we go about changing our response to eliminate the overshoot? We need to change something to create a dead beat response after which we will verify the rest of our design goals. Let us add a minor loop into our system to eliminate the overshoot effect of the zero. Figure 6 shows our current system block diagram with the PI compensator.
1/S Z K p G p Figure 6: PI compensated block diagram This configuration causes our system to have overshoot, yet if we use block diagram algebra we can manipulate our system to a new form. This form is shown in Figure 6 and has the same closed loop poles as does the configuration in Figure 6. 1/S Z K p G p K p Figure 7: PI compensated block diagram with minor loop closed loop Figure 7 shows our PI compensated system with a minor loop. This new configuration will rid our response of overshoot and therefore cause a dead beat response. It is important to note that the closed loop poles in both Figure 6 and Figure 7 are equivalent. This can be seen by looking at single loops in both figures. As you can see, in both instances, there is one loop with a loop gain of G p K p and another with a closed loop gain equal to ZKG P /S. Because both have the same loop gains their poles and zeros are also equivalent. Once the configuration in Figure 7 is attained we have closed the loop. Theory behind closing the loop brings us to the configuration shown in Figure 8.
G 1/S H F Figure 8: Closed loop block diagram For the closed loop system in Figure 8 it can be seen that, u r y also, x Fx G r x y F GHx Gr Also, for the forward path gain K, G goes to K*G x F KGHx KGr And for the feedback path gain K, we let u = r Ky and thus, x F x G r ky x F KGHx Gr y H x Verification of closed loop design G c = K p Z/S Once we have added a minor loop and closed the loop it is time to check and see if our system meets the specified design goals. From the PI compensator root locus we were able to choose values for K p and Z, we will use those values as a starting point in our analysis of the new system. Let us begin by looking at our new system s response to a unit step input. This is shown in Figure 9.
Figure 9: Closed loop response The response seems to be within our design goals. As you can see in Figure 9, the amplitude at 1.5 seconds is.955. This fact meets the design goal of wanting a rise time that is less than 1.5 seconds. Also, the steady state amplitude of the system was right around.97, meeting the design goal that the steady state error must be less than 5%. To make sure that the steady state error was less than 5% the response was plotted on a very large time scale. The amplitude flattened out as expected and never dropped below.95. (I tried to have Matlab place the true rise time and steady state amplitude on the plot but it was buggy and would not do it. Thus, I have shown that the rise time is less than 1.5 seconds and the steady state error is less than 5% in a slightly roundabout way.) Now let us take a look the root locus which can be seen in Figure 1.
Figure 1: Closed Loop Root locus From the root locus in Figure 1 we can verify that the phugoid damping is greater than.4 with an actual value of.553 and that the short period damping is greater than.35 with an actual value of 1.. Conclusion The design goals have been met using a K P = 6.5 and a Z = 3.62. If our design goals were not met we would have chosen different values for the gain and zero to see how our system would change. In this way the design process of control systems can be somewhat iterative. In our case, the gain and zero location we chose worked and we successfully designed a type zero system (characterized by the system s finite error to a step input) that met our specified design goals.
Appendix Matlab m-file % Bob Rowe % Controls Pitch Rate CAS Design Project_part2 clc clear all %Xdot=Fx+Gu %Augmented F matrix F=[-.3,3.186,-32.2,,-.17453; -.83979,-.319,,1,.4589;,,,1,;.21,-.78174,,-.429,.2246;,,,,-2] %Augmented G matrix G=[; ; ; ; 2] %y=hx+ju %H matrix H=[,,,(18/pi),] %J matrix J=[] %Set up SISO sys=ss(f,g,h,j) sisotool(sys) %Set up minorloop using values from SISO kp=6.5 z=3.62 sysgainloop=tf(kp,1) sysminor=feedback(sys,sysgainloop) %Set up PI Compensator using values from SISO num=kp*[1, z] den=[1, ] syscompensator=tf(num,den) %Close the loop sysfp=series(syscompensator,sysminor) syscl=feedback(sysfp,1) %Verify Design requirements figure(1) t=(:.1:5) step(syscl,t) figure(2) rlocus(syscl)