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THE RCBODE PLOT AND ADVANCED COMPENSATOR STRUCTURES FOR HIGH PERFORMANCE TWO STAGE ACTUATION AND OTHER APPLICATIONS

William C. Messner Data Storage Systems Center Carnegie Mellon University

5000 Forbes Avenue Pittsburgh, 15213-3890

bmessner@andrew.cmu.edu v. 412.268.2510 f. 412.268.334

29 April 2011

Abstract The RCBode plot is a visualization method that depicts closed-loop robust performance on the Bode plot of the controller as allowable and forbidden regions. Intersections between the controller frequency response and the forbidden regions indicate frequency regions where the robust performance criterion are not satisfied. A significant aspect of the RCBode plot is that the nominal models, the performance weighting function, and uncertainty weighting function of the robust performance criteria can be represented by measured frequency response data alone�—realizable transfer functions are not needed. This proposal is directed to applications of the RCBode to

Design of high performance controllers for two-stage actuator systems

Controller synthesis on the fly during the HDD certification process

We will extend the loop-shaping technique using loop shaping to the design of controllers for dual-input/single-output systems found in HDD two-stage actuator systems using sensitivity function decoupling. Exploiting the fact that measured frequency response data and measured disturbance spectrum data are sufficient for constructing the RCBode plot, we will automate the loop shaping design process to enable optimal controller synthesis during the HDD certification process. In both cases, we will employ new compensator structures (e.g. asymmetric lead compensator and complex proportional-integral-lead compensator) that use complex zeros and complex poles for decoupling the phase contribution of a compensator from its shape at a particular frequency. We will also examine the utility of using these compensators for the construction of weighting functions for and -synthesis.

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The boundary functions of the magnitude plot of the RBode plot are obtained by defining using the identity that and then rewriting the robust

performance criterion as

(4)

where the dependence of the functions on variable has been suppressed to simplify notation. Solving the resulting quadratic inequality at each frequency by assuming that the magnitude as a variable while is known determines the magnitude boundary functions. Rewriting the robust performance criterion again

(5)

and solving the resulting trigonometric inequality at each frequency assuming that is a variable while the magnitude is known defines the boundary functions for the forbidden regions on the phase plot. Figure 1.a shows a RBode plot for which intersections between the gray forbidden regions and the open loop frequency response exist, and therefore robust performance is not achieved. Figure 1.b. shows an RBode plot where the intersection have been eliminated, and therefore robust performance is achieved. The design procedure is to shape the loop by selection of component compensators of the controller to avoid the forbidden regions.

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Solving the resulting quadratic inequality at each frequency by assuming that the magnitude as a variable while is known determines the magnitude boundary functions for the Bode plot of

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and solving the resulting trigonometric inequality at each frequency assuming that is a variable while the magnitude is known defines the boundary functions for the forbidden regions on the phase plot for . Figure 3.a shows a RBode plot for which intersections between the gray forbidden regions and the open loop frequency response exist, and therefore robust performance is not achieved. Figure 3.b. shows an RBode plot where the intersection have been eliminated, and therefore robust performance is achieved. Again the design procedure is to shape the loop by selection of component compensators of the controller

to avoid the forbidden regions. The big advantage of the RCBode plot is that multiple plant frequency responses can be considered simultaneously, thus addressing structured uncertainties by sampling the set of possible frequency responses. The forbidden regions on the RCBode plot are the unions of the forbidden regions for each individual plant frequency response, had it been plotted separately. (See Figure 4.) A design procedure employing the RCBode that has been successfully applied to SISO systems is the following.

Step 1. Determine the one or more frequency responses of nominal models , the performance weighting function , and an uncertainty weighting function .

Step 2. Design an initial controller which stabilizes the all of the with positive gain margins and positive phase margins. Let and go to Step 4.

Step 3. Let n . Cascade the compensator with to eliminate intersections occurring at the lowest frequency. The compensator should have the property that at lower frequencies where intersections have already been eliminated to ensure that no new intersections are introduced over frequency regions that have already been addressed.

Step 4. Plot the RCBode plot using , , the updated , and the multiple . If there are no intersections, the design is complete done. Otherwise go to Step 3 to

iteratively eliminate the intersections by proceeding from lowest to highest frequency. Comparison with other methods. Automated design tools for robust control system based on , -synthesis, and Linear Matrix Inequalities (LMI) have been widely used to assure both stability and performance in closed loop systems subject to plant uncertainties. >Hirata et al. (1992); Shim et al. (2003); Du et al. (2008)]. A drawback of theses automated methods is that they require a realizable

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Proposed Research Approaches There will be two aspects of this research: (1) application of the RCBode plot to design of high performance controllers for track follow during two-stage actuation, and (2) application of the RCBode plot to controller synthesis on the fly during the HDD certification process.

1. Design of high performance controllers for two-stage actuation Until now the RCBode plot has been used only for controller design for SISO systems. The PI proposes to extend the loop-shaping design technique to two-stage actuator systems through the use of sensitivity function separation.

Consider the block diagram of Figure 5 depicting a two-stage actuator system where the relative position of the microactuator is measured and fed back to the input of the controller for the voice coil motor (VCM). It can be easily shown that the total sensitivity function of this system is [Li and Horowitz (2000)]

(6)

where

is the transfer function of the VCM, is the transfer function of the microactuator, is the transfer function of the controller for the VCM, and is the transfer function of the controller microactuator.

The function is the sensitivity function of VCM loop. That is, the closed-loop with the microactuator branch removed.

(7)

Likewise is the sensitivity function of the microactuator loop. That is the closed-loop with the VCM branch removed.

(8)

The sensitivity function of the entire system is thus separable as the product of the two sensitivity functions for which controllers can be designed sequentially.

Figure 5. Block diagram of a two stage actuator system where the relative position of the microactuator is fed back to the input of the VCM.

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Design procedure. The design procedure will be to sequentially design controllers and for to obtain the two sensitivity functions and while accounting for the relative contribution of the VCM and the microatuator as function of frequency. To that end define

(9)

Note that at low frequencies where the VCM response greatly exceeds the microactuator response. At high frequencies where the microactuator response dominates. By a continuity argument, there exists a frequency where

implying .

The function will be used to modify the performance weighting function for the VCM loop and the performance weighting function for the microactuator loop to allocate the contributions of the VCM and the microactuator as a function of frequency.

The proposed procedure is as follows.

Step 1. Let

be the uncertainty weighting function for the VCM dynamics, be the uncertainty weighting function for the microactuator dynamics,

be the performance weighting function for total closed-loop such that performance is achieved if .

Define the VCM sensitivity weighting function as

(10)

where is the inverse of the desired maximum value of the VCM loop

sensitivity function. The function has low pass characteristics accounting for the hand-off frequency. One possible definition of is

. (11)

Using this definition

(12)

Step 2. Design the controller using the RCBode plot and loop shaping method of the RCBode plot background section with the nominal plant , the uncertainty weighting function , and the performance weighting function . Using of Equation 10 implies that

for .

10

Step 3. After has been designed, determine and define

(13)

Step 4. Design the controller using the RCBode plot and loop shaping method of the RCBode plot background section with the nominal plant , the uncertainty weighting function , and the performance weighting function .

Note that applying the RCBode plot method will assure that

(14)

Multiplying both sides of this inequality by gives

(15)

Thus, the total sensitivity function will satisfy the performance bounds.

Significantly, the RCBode plot allows multiple VCM models and multiple microactuator models to be considered at each design stage. Furthermore, the RCBode plot will need to be modified to

account for multiple microactuator performance weighting functions , since will change with different VCM models .

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2. Controller synthesis on the fly during HDD certification The fact that the RCBode plot requires only frequency response data for the model of the plant and

and can be arbitrary non-negative functions of frequency offer the tantalizing possibility that controllers could be synthesized on the fly during HDD certification. The automated procedure would be as follows.

Step 1. Collect frequency response data for a single actuator ( and for a two stage implementation).

Step 2. Use multiple frequency response of the single system to determine one or more nominal models and an uncertainty weighting function . ( and ,

, and for a two stage implementation.) Step 3. Determine the disturbance spectrum and use it to create . Step 4. Apply an automated version of the loop shaping procedure from the RCBode background

section, or apply an automated version of the two-stage loop shaping procedure employing the RCBode plot above.

The key research problem is that of developing the automated versions of the loop shaping procedures for Step 4. At this point the principle investigator�’s understanding of the problem means breaking it down as follows.

a. Develop a method to automatically determine the compensator to stabilize the nominal plant with positive phase margin and positive phase margin using frequency response data only.

b. Develop a method to determine which compensator structure to apply to eliminate the lowest frequency intersection.

In both cases, determining whether gain compensation or phase compensation or a combination is appropriate will require development of rules and heuristics that account for the characteristics the available compensator structures. These compensators include standard structures [Franklin et al. (1998)] and newer ones developed by the principle investigator and others over the last decade can decouple phase contribution from shape at a compensation frequency through the use of complex zeros. [Atsumi et al. (2007), Messner et al. (2007), Messner (2008), Messner (2009)] Figure 6 shows two examples of a couple of the new compensator structures.

Type of compensation Phase compensation Gain compensation Gain and phase compensation Standard structures

Lead Lag

Constant gain Low pass High pass Resonant filter Notch filter

PI PI lead PID

Newer structures with complex zeros

Complex lead Complex lag Asymmetric complex lead Asymmetric complex lag

Phase adjustable resonant filter

Phase adjustable notch filter Complex proportional-

integral-lead (CPIL) As side benefit of this project, we will work with our ASTC colleagues to examine the utility of the new compensator structures for the construction of weighting functions for and -synthesis by (1) providing them with MATLAB functions for these compensators and/or (2) using them ourselves

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Preliminary algorithm development. It is important to note that if an intersection with a forbidden region occurs on the magnitude plot at a particular frequency, then an intersection must occur on the phase plot at the same frequency. Likewise, if an intersection with a forbidden region occurs on the phase plot at a particular frequency, then an intersection must occur on the magnitude plot at the same frequency. If a forbidden region extends from - to + on the magnitude plot and a forbidden regione extends from - to + on the phase plot at the same frequency, then no compensation can eliminate the intersection. This case will occur if and only if both and at that frequency. The weighting functions must be changes. Some rules are readily apparent

If a forbidden region extends from - to + on the phase plot, then no change in the phase alone will eliminate the intersection. Gain compensation is required. (Figure 3.a. at 120, 240, 360, 3000, 3500, 4000, 6000, and 104 Hz.)

o If the forbidden region on the gain plot extends from - dB to a finite dB value, then positive gain (in dB) must be applied. (Figure 3.a. at 120, 240, 360 Hz.) If the forbidden region is narrow, resonant filger compensation is suitable.

o If the forbidden region on the gain plot extends from a finite dB to + dB to value, then negative gain (in dB) must be applied. (Figure 3.a. at 3000, 3500, 4000, 6000, and 104 Hz .) If the forbidden region is narrow, then notch filter compensation is suitable.

If a forbidden region extends from - to + on the magnitude plot, then no change in the gain alone will eliminate the intersection. Phase compensation is required.

Phase compensation is preferable to gain compensation, because it requires less actuator effort. Thus phase compensation is chosen for the region 800-1000 Hz on Figure 3.a.

When both positive and negative phase compensation are possible: o It is preferable to choose positive phase compensation when magnitude forbidden region

extends to - dB. o It is preferable to choose positive phase compensation when magnitude forbidden region

extends to + dB. How to automatically choose the parameters of the appropriate compensators to apply the correct amount of gain or phase compensation over the correct frequency range will require the largest amount of effort of this project.

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REFERENCES 1. Atsumi, T. and Messner, W. (2010a). Optimization of Head-Positioning Control in a Hard Disk Drive

Using the RBode Plot. Proceedings of the 2010 Workshop on Advanced Motion Control, 542 �– 547. 2. Atsumi, T. and W. Messner, W. (2010b). Loop-Shaping Controller Design with the RBode Plot for

Hard Disk Drives. Proceedings of the 2010 American Controls Conference, 2659 �– 2664. 3. Atsumi, T. and Messner, W. (2010c). Modified Bode Plots for Head-Positioning Control in Hard

Disk Drives with Structured and Unstructured Uncertainties. Proceedings of the 2010 IFAC Conference on Mechatronics.

4. Atsumi, T., Okuyama, A., and Kobayashi, M. (2007). Track-Following Control Using Resonant Filter in Hard Disk Drives. IEEE/ASME Transactions on Mechatronics, 12(4), 472̽479.

5. Doyle, J., Francis, B., and Tannenbaum, A. (1992). Feedback Control Theory. Macmillan Publishing Company,bNew York.

6. Du, C., Xie, L., Zhang, J., and Guo, G. (2008). Disturbance Rejection for a Data Storage System via Sensitivity Loop Shaping and Adaptive Nonlinear Compensation. IEEE/ASME Transactions on Mechatronics, 13(5), 493̽501.

7. Franklin, G.F., Powell, J.D., and Workman, M. (1998). Digital Control of Dynamic Systems. Addison Wesley Longman, Menlo Park, California.

8. The Institute of Electrics Engineers of Japan, Technical Committee for Novel Nanoscale Servo Control (2007). Benchmark Problem, Ver. 3.1. URL http://mizugaki.iis.u-tokyo.ac.jp:80/nss/.

9. Li, Y. and Horowitz, H. (2000). Track-following controller design of MEMS based dual-stage servos in magnetic hard disk drives. Proceedings of the 2000 International Conference on Robotics and Automatio, 953-958.

10. Hirata, M., Liu, K.Z., and Mita, T. (1992). Head positioning control of a hard disk drive using H҄ theory. Proceedings of the 31th IEEE Conference on Decision and Control, 2460̽2461.

11. Messner, W. (2008). Classical Controls Revisited. Proceedings of the 2008 Advanced Motion Controls Conference, Trento, Italy, 26-28 March 2008.

12. Messner, W. (2009). Formulas for Asymmetric Lead and Lag Compensators. Proceedings of the 2009 American Control Conference, 3769-74.

13. Messner, W.C., Bedillion, M.D., Xia, L., and Karns, D.C. (2007). Lead and Lag Compensators with Complex Poles and Zeros. IEEE Control Systems Magazine, 27(1), 44̽54.

14. Shim, D.H., Lee, H.S., and Guo, L. (2003). Mixed Objective Optimization of Track-following Controllers using Linear Matrix Inequalities. Proceedings of the American Control Conference, 4323̽4328.

15. Xia, L. and Messner, W. (2005). Loop Shaping for Robust Performance Using Rbode Plot. Proceedings of the 2005 American Control Conference, 2869̽2874.

16. Xia, L. and Messner, W. (2008). An Improved Version of the RBode Plot. Proceedings of the 2008 American Control Conference, 4940̽4945.

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Resources needed to perform project This project will require the following resources

Personnel 1 graduate student for 1 year 1 faculty for 1 month

Computational MATLAB research license

Resources other than ASTC funding dedicated to perform project

Grants�—none Contracts�—none Other

1 desk top computer (existing) 1 laptop computer (existing)

Resources requested from ASTC funding dedicated to perform project

Funding

Project Period 06/01/2011-05/31/2012

YEAR 1 TOTAL

SALARIES AND WAGES PI Summer - 1 month 13,078 13,078Postdoc 0 0Graduate Student Stipend 28,320 28,320FRINGE BENEFITS 3,793 3,793TOTAL PERSONNEL COSTS 45,190 45,190

OPERATING EXPENSES Technical Supplies & Services 1,000 1,000Travel 2,000 2,000IC Computing Services 290 290TOTAL OPERATING EXPENSES 3,290 3,290

Graduate Tuition 37,950 37,950

TOTAL DIRECT COST 86,430 86,430INDIRECT COST BASE 48,480 48,480INDIRECT COSTS 30,543 30,543TOTAL PROJECT COST 116,973 116,973

Benefits Rate 29.00%Indirect Cost Rate 63.00%

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Expected technical cooperation with sponsor(s): ASTC supplies Realistic MATLAB or Simulink models of one or more HDDs with

Disturbance spectrum data Model uncertainty data Noise data

Sponsor facility utilization�—none Expected students�’ internships�—none

Timeline

Task Y1Q1 Y1Q2 Y1Q3 Y1Q4 Loop shaping using the RCBode plot for the design of high

performance controllers for two-stage actuation

Rewrite RCBode plot function to be more user friendly

Write RCBode commands for multiple microcactuator sensitivity functions to account for multiple VCM frequency responses.

Automate design loop shaping design with the RCBode plot

Implement basic rules

Apply advanced heuristics

Home institution and resources Department of Mechanical Engineering and Data Storage Systems Center Carnegie Mellon University Pittsburgh, PA 15213-3890 Control System Laboratory: Approximately 400 square feet of space containing equipment for fabricating prototype circuits, measuring systems response, and implementing feedback control systems. Computer: Three desktop, 1 laptop Major equipment 2 HP Signal signal analyzers 4 100 MHz digital oscilloscopes dSpace Control Desk Office: Dedicated individual office space. Desk space for graduate students in labs. Desk space for undergraduate students in labs. 1/4 secretarial assistant.

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Biographical sketch William Messner is a professor of mechanical engineering at Carnegie Mellon University. He received his B.S. degree in mathematics from MIT in 1985 and his M.S. and Ph.D. degrees in mechanical engineering from the University of California at Berkeley in 1989 and 1992 respectively. Prof. Messner specializes in application of controls to data storage systems, robotic systems, and microfluidic systems for biological research. During his 2005-6 sabbatical leave he was a Visiting Professor of Medicine at Harvard Medical School where he worked on the rheology of protein networks. With respect to data storage systems, Prof. Messner has worked on controller design techniques (e.g. the PQ Method for two stage actuators), modeling of nonlinear pivot friction, methods for PES generation on lithographically patterned media, and high rotation microactuators for skew compensation, among other things. Prof. Messner is Fellow of the American Association for the Advancement of Science, a Fellow of the ASME, and a Senior Member of IEEE. He is a big fan of Italian cappuccinos and fun number facts.