Maglev Controller Design

Transcription

1 Maglev Controller Design By: Joseph Elliott Nathan Maher Nathan Mullins For: Dr. Paul Ro MAE 435 Principles of Automatic Controls Due: May 1, 2009

2 NJN Control Technologies 5906 Wolf Dale Ct. Suite 1 Raleigh, NC April 30, 2009 Dr. Paul Ro Chief Engineer Magnetech Inc. 42 Broughton Dr. Raleigh, NC Dear Dr. Ro: After reviewing the proposal and design guidelines for the control system on Magnetech s new maglev train, we have come up with a suitable solution. We explored many different possible controller designs, including critically damped gain, PID, and dual pole-zero controllers. In our opinion, a controller of the following form is the most suitable given the design guidelines: 4 s s s 2 + 6s This controller provides the best control action possible for the design criteria. The time response of this controller has a quick settling time of ~1 second and a steady-state error of zero due to the introduction of a pure integrator. There is no overshoot with this controller design. The controller also uses less coil current than the maximum stated in the design guideline. The frequency response of this controller is also superior. It has a closed-loop bandwidth from input to output of over 6.5 Hz, above the recommended 2 Hz. The Bode plot from disturbance to output shows that the controller should attenuate all disturbances, with a minimum attenuation of 10dB. This controller also has been shown to be stable for all gains, based on root locus analysis and gain margin. The gain margin is infinite for this system. The phase margin is 90 degrees, above the 30 degree recommended for a system like this. When testing this controller in a closed-loop scenario with a step input and a sinusoidal disturbance of amplitude 1.0 (at the least attenuated frequency in the bandwidth), the response received was within 25% of the desired steady state output at all times. Testing with a step input and step disturbance of amplitude 1.0 gave a response within 20%. The coil current also did not exceed the maximum in either scenario. Please review the attached proposal for more detail involving this controller and other options that were considered but are not recommended. Sincerely, Nathan Maher Senior Staff Engineer Enclosure Joseph Elliott Senior Staff Engineer Nathan Mullins Senior Staff Engineer

3 Table of Contents 1. Introduction Open-Loop Response Critically Damped Proportional Control PID Controllers Custom Controller Design Final Evaluation References Appendix...20

4 Introduction As a contractor for Magenetch, Inc., we were introduced to the problem occurring with the control system on the new maglev train design. The problem was introduced as a need to control the air gap between the train and track. As experts in the field of controls, the associates at NJN Control Technologies considered multiple designs to devise the best controller for the job. This report details the issues posed by the maglev system and the corrections made by various controllers. Open Loop Response The original maglev train system for the air gap can be seen in Figure 1 below. Desired Air Gap R(s) Coil Current Air Gap C(s) Figure 1: Maglev Train System Amplitude This system transfer function is the mathematical approximation of how the maglev train system reacts to a given input. This transfer function is in the LaPlace Domain, and is inherent to the train s physical design (regardless of disturbance). Using a software package, Matlab, it is possible to graph the response of this system to a step input. Below is a graph of time response of the maglev train system with no control action. Time (s) Figure 2: Open Loop Time Response The step input applied to the system was a step input with amplitude of 1.0, beginning at a time of 1.0 second. As you can see from the figure, the response has several negative aspects. First, the steady-state error of the open-loop response is very large. A perfect system would have the steady-state value of amplitude 1.0, while this system has the steady-state value of around 0.4. There also exists a large amount of oscillation in the response. Clearly control action is needed to correct the system s response. 4

5 Critically Damped Proportional Controller For all of the controllers attempted in this report, the same basic system block diagram is utilized. The following is the block diagram of the closed loop system. The controller is represented by Gc(s). Figure 3: Block Diagram of Maglev System with Controller in Place The first option for a controller is a pure proportional control. The proportional control acts like a spring. This proportional controller was selected so that the system will then be critically damped. A critically damped system will not have oscillation in the response. Judging by the root-locus of the system, the resulting gain at critical damping was The root locus graph is shown below. Figure 4: Root Locus of Maglev System The Simulink block diagram used for this controller is shown below. In this case, K= See the Appendix for calculations. 5

6 Figure 5: Simulink Block Diagram for System with Proportional Gain Although this controller eliminated the oscillation, it did not eliminate steady-state error, and thus was unfit for a final solution. The time response graph showing this is below. Note that the final value is around 0.3, much different than the 1.0 step input. Figure 6: Time Response for System with Proportional Gain However, the coil current at this point was used as a reference point to define the maximum coil current. Since higher proportional gains can be used to control this system at large cost (due to the large amount of coil current involved), the problem proposed by Magnetech Inc. stated that the maximum coil current could be four times the critically damped coil current. This set the value of maximum coil current at 4*1.23=

7 PID Controllers One of the easiest ways to correct a system response is to add a Proportional, Derivative, Integral controller, or PID. This is an easy way to correct the system because it is easy to visualize what each portion of the controller does. Adding integral control action will correct the steady-state error of the system, but can contribute to too much oscillation. The derivative action will damp out the system response, but can create some error. For this system, several different PID controllers were examined. Each controller provided different responses and different advantages and disadvantages. Three different figures are used to display the effectiveness of these controllers. First of all, the time response will show the action of the controller with respect to time. This is important in determining steady-state error, oscillation, overshoot, settling and rise times. The following figure shows the time response of the maglev system with four different PID controllers. Figure 7: Time Response for System with PID Control The R-C Bode plot is a graph of the system from input to output, with respect to frequency. Figure 8 shows how the system will react to different frequency inputs. The important thing to note on this plot is when the graph drops below -3dB. The frequency at this point is called the bandwidth of the system and is a cutoff point for a system having good command following. Beyond that point, the system will not follow the input. 7

8 Figure 8: R-C Bode Plots for System with PID Control The Bode plot from D-C, or from disturbance to output shows how well the system will reject a disturbance. Ideally, the magnitude plot from D-C should be below zero for all frequencies. Figure 9: D-C Bode Plots for System with PID Control 8

9 The open loop Bode plot is important for determining the system stability. Gain and phase margin can be gleaned, and the higher the value of each, the stabler the system. Gain margin must be >0, and phase margin must be >30 to satisfy most design criteria. Figure 10: Open Loop Bode Plots for System with PID Control Figure 11: Root Locus for System with PID Control 9

10 Root locus plots visualize the position of the poles and zeroes of both the train system and the controller. Information that can be gathered from these plots includes damping ratio, overshoot, and speed of the response with respect to gain. These figures will aid in the discussion of the four PID controllers explored during this stage of the investigation. The simulink diagram for the PID controllers is shown below: Figure 12: Simulink Block Diagram for System with PID Control PID 1 For the first PID controller, the gains were randomly selected as Proportional=2, Integral=2, Derivative=2. These gains did not work at all for this system, as they produced an unstable response and also required an infinite amount of coil current. The derivative gain in this case was too large and was causing the response to become unrealistic. From Figure 7, one can see that the time response was unacceptable because it had large oscillation and very long settling time. The steady-state error, while better than the proportional only case, was not very good. Judging from the Bode diagram from R-C (Figure 8), this controller would also have very low bandwidth, and poor command following. It drops rapidly below -3dB, and the phase graph is also very inconsistent. The disturbance rejection of this system is okay because the magnitude of the D-C Bode plot is below zero for all frequencies. PID 2 The second PID controller eliminated the derivative gain that was causing the issues (making it PI only). The second controller had Proportional=3 and Integral=6. For this controller, the main issue was the time response. The settling time was very long (5 sec). Also, the R-C Bode plot showed some fluctuation in the magnitude plot. Because of these issues, it was decided to increase the proportional gain and integral gain in order to speed up the system settling time and eliminate the fluctuations in the R-C Bode plot. PID 3 The third PI controller used Proportional=4.92 (the maximum) and Integral=50. This controller improved on the previous design with a low settling time and very high bandwidth. A concern arose with this PID: it had a ~3.4% overshoot, and the maximum allowed is 1.5% according to the design guideline indicating that ξ (damping ratio) must be greater than

11 PID 4 Our final PI controller is a variation of PID 3, lowering the Integral gain slightly to 30. This reduces the % overshoot to an acceptable level. The proportional gain was also reduced to meet the maximum coil current specification. The time response of this controller displays its fast rise time and settling time (~1.25 s) to a steady state error of zero. The R-C Bode plot indicates this controller provides good command following well beyond the stated bandwidth of 2 Hz, out to ~7 Hz. The D-C Bode plot exhibits good disturbance attenuation; a minimum attenuation of -10 db is achieved. Looking at the open loop Bode plot, the phase margin is determined to be ~90, well above the 30 minimum required for stability. This plot has an infinite gain margin, which is invalid in determining stability. PID Conclusion Through experimentation, it was determined that the maximum coil current tracks with the proportional gain in a PI controller. To reduce coil current, the proportional gain can be reduced, but this will sacrifice performance throughout the system. Increasing integral gain reduces settling time but can lead to increased oscillation. Custom Controller Design In this section, we took our knowledge of the system s response to PID control and applied it to a more customizable control solution in which two poles and two zeroes can be used in the controller. As in the section above, graphs of each system characteristic are made which include data from each controller. Figure 13: Time Responses for System with Custom Control 11

12 Figure 14: R-C Bode Plots for System with Custom Control Figure 15: Open Loop Bode Plots for System with Custom Control 12

13 Figure 16: D-C Bode Plots for System with Custom Control Figure 17: Root Locus of System with Custom Control 13

14 Controller 1 Controller 1 is defined by the transfer function below: 4.34 s s + 11 Equation 1 This controller was designed to direct the root locus (Figure 17) to the real axis quickly by adding a zero on the real axis. This function has good damping at low gain values. Due to this function s lack of a pure integrator, the time response (Figure 13) for this system revealed a large steady state error, rendering this function useless. It also has a bandwidth of zero as shown in Figure 14. Controller 2 Controller 2 was then made with integral action to mitigate steady state error. Equation 2 4 s s s Looking at the time response of the system, it does not rise close enough to unity right away, leading to a long settling time of 2.75s. The R-C Bode plot (Figure 14) magnitude comes close to the -3 db limit at ~0.5 Hz, and the phase plot shows some oscillation of about 10 degrees. We sought to improve the R-C Bode response and the settling time. The disturbance rejection was acceptable as evidenced by Figure 16. Controller 3 This controller improves on controller 2 while maintaining a similar root locus pattern. This method improves the time response and R-C Bode somewhat, but not enough. The response of this system was not fast enough. Equation s s 2 + 8s Controller 4 We decided to move the zeroes off of the real axis for controller 4, and saw increased time response performance and a flatter R-C Bode plot. By moving the zeroes of the controller closer to the poles of the system, the response sped up. Equation 4 4 s s s 2 + 5s Controller 5 This is our final controller. The performance of this controller is the best we have to offer given the coil current restriction. The transfer function of this controller is given below: Equation 5 4 s s s 2 + 6s 14

15 A compilation of the critical system plots for Controller 5 is shown below: Figure 18: Controller 5 Diagram Compilation Controller 5 has a very good time response, with no overshoot. By moving the pole farther out on the real axis, the overshoot was decreased. The settling time is worse than controller 4, but the difference is very small. The closeness of the zeroes to the system poles allows for a response very close to the input. The R-C Bode plot magnitude crosses the -3dB threshold at ~6.52 Hz well beyond the 2 Hz required for a maglev system. While there is a sharp peak in the disturbance rejection as shown in Figures 16 and 18, this peak does not go above -9 db, so an ample attenuation of the disturbance is achieved. The gain margin for this system is infinity, and the phase margin is ~90 degrees, as determined by the open loop Bode plot in Figure 15. The following figure shows the block diagram in Simulink for this controller. 15

16 Figure 19: Simulink Block Diagram for System with Lead-Lag Control Evaluation of Final Controller When designing this controller, it was possible to get a very good response from the system while still staying in the allowable control current range to keep the cost of the system down. In this section, the different goals listed in Part C of the design guidelines will be examined with respect to the chosen controller-train system. First of all, the root locus of the system shows that it will never go unstable. This is important for a situation like a Maglev train where safety is of great concern and control of the gap between the train and track must be maintained. The time response of the system has a very fast rise time. The system is not very oscillatory and has no overshoot. The time response also shows that the system settles quickly, within about one second, and thanks to the addition of a pure integrator, there is no steady state error in our design. Judging by the R-C Bode diagram for the system, we can see that this controller allows for a bandwidth of around 6.5 Hz. This means that the system will be capable of tracking a wide range of input frequencies very well. Taking a look at the D-C Bode plot, we see that our controller has adequate disturbance rejection characteristics. The Bode plot for the system never amplifies the system disturbance and always maintains an attenuation of at least 9 db over all frequencies. The open loop Bode plot was also examined to determine gain and phase margins. The gain margin of this system with the chosen controller is infinite, (as displayed in the root locus). The phase margin of the system was determined to be about 90 degrees, well above the 30 degree design guideline for most systems. To demonstrate the effectiveness of our controller, different disturbances were input. We explored the effects of both step and sinusoidal disturbances on the time response. 16

17 Figure 20: System Response to Step Disturbance with Controller 5 Figure 20 shows the time response of our system to a step input of 1. This should simulate driving the train to twice its standard height. Due to our control action, the train remained within 20% of its nominal height. Figure 21: System Response to Sinusoidal Disturbance with Controller 5 17

18 In Figure 21, we see the effects of a sinusoidal disturbance of magnitude 1 at the frequency least attenuated by our controller. Controller 5 reduces the effect of this disturbance, maintaining a position within 25% of the nominal value. Having decided on our controller, there are some things that could have been improved. The damping ratio of our system is much smaller than suggested which, if increased, could eliminate the small amount of oscillation seen in our system. This would potentially slow the system down but with some more manipulation of the poles and zeros a controller could possibly have been developed that would be as fast as our system but with less oscillation. The D-C Bode diagram shows that our system has good disturbance rejection, though this could have been improved given more time. We would also have liked to try a feed forward disturbance rejection setup to entirely, or nearly entirely, eliminate the influence of any disturbance on the train s height. The cost was a major limitation during the design of this controller. If the control current could have been raised, the controller would have responded better and been faster at damping out disturbances. Given the limitations, the controller does a very good job of controlling the system by keeping it stable and eliminating the effect of disturbances quickly. Another limitation of the system was the restriction on pole-zero cancellation. The design guidelines stated that the zeroes of the controller could not be placed within 2.0 rad/s of the poles of the system. In general, it was found that the closer the zeroes of the controller got to those poles, the better the response of the system became. The system response became faster and tracked the input better. By placing the poles as close as possible to the zeroes without breaking the restriction, we were able to produce the best response. Without this restriction, it would have been possible to produce a faster and more accurate response. 18

19 References Franklin, Gene F., J. David Powell, and Abbas Emami-Naeini. Feedback Control of Dynamic Systems. 5th ed. Upper Saddle River, NJ: Pearson Prentice Hall, Ro, Paul I. "Design Project for MAE 435." to the author Ro, Paul I. "Lecture notes for MAE 435." to the author

20 Appendix This appendix holds the calculations for the critically damped case. Based on Figure 3 in the body of the report, the system transfer function is: Equation 6 Adding a critically damped gain K, the closed loop system transfer function becomes: Equation 7 The closed-loop characteristic equation is: Equation 8 The standard second order equation has the form: Equation 9 Where ωn is the natural frequency, and ζ is the damping ratio. Thus, we can equate: Equation 10 In a critically damped case, the damping ratio ζ=1. Knowing this, it is possible to solve for gain K thusly: Equation 11 Since there cannot be negative gain, the correct critically damped gain is K= The limit on coil current is 4*K=