Kv = valve flow gain, M g = grip mass, Pd = valve driver pole, Kc = load cell stisfness, Gv = equivalent valve leakage, Mf=fiame mass (equiv.), Ga. = equivalent ...
Robust Control for High-Performance Materials Testing F. N. Bailey, J. C. Cockburn, and A. Dee Materials testing requires the design of control systems which can faithfully reproduce normal specimen operating environments over a range of specimen parameters. Root locus based robust design techniques were used in the design of a robust digital controller for a high performance variable amplitude fatigue test. Evaluation of the resulting controller on a prototype laboratory materials testing system shows that it meets design performance goals of 0.25% specimen load accuracy while providing robustness to specimen compliance over a 10to 1 range. This performance significantly exceeds that obtained with traditional hand tuned controllers.
Materials Testing Materials testing is an important part of much of modem manufacturing. To help ensure product reliability the manufacturer subjects critical product components to controlled tests using environment simulators designed to reproduce the operating environments of the components. Since each test must faithfully reproduce the operating environment of the specific component, the environment simulator or testing machine must be carefully controlled to maintain the appropriate test loads on the component over a possibly wide range of test conditions. Moreover, since component characteristics may vary for a variety of reasons such as material changes caused by aging or wear, manufacturing tolerances, parts interchangeability, etc. the testing machine controller must provide performance robustness over a wide range of component characteristics. In this paper we will report on a project where robust control concepts were used to design a controller for a specific materials test. The testing machine was a prototype laboratory materials testing machine built by MTS Systems, Corp. The specific test of interest was a Flight Spectrum EN. Bailey and J.C. Cockbum are with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455. A . Dee is with MTS Systems Corp., P: 0. Box 24012, Minneapolis, MN 55424.
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Fig. 1. Physical model of the load frame and speciman.
Test. This is a test designed to simulate the mechanical stress loads experienced by certain aircraft components. This paper describes the development of a robust digital controller for a prototype MTS materials testing system running a Flight Spectrum Test. The flight spectrum test involves subjecting a specimen to precisely controlled unidirectional force loads. To accomplish this the specimen is inserted into the grips of a load frame, a load is applied through a servo-valve controlled hydraulic actuator and a load cell measures the actual load applied. The physical structure of the load frame, specimen and associated components is shown in Fig. 1 . For precise control of specimen loads a feedback system is employed to ensure that applied specimen loads follow given load commands. In the flight spectrum test a specimen is subjected to force loads defined by a set of sequentially specified load values that we will call theflight spectrum data set. Inspection of the flight spectrum data set reveals that the data set values vary widely from point to point with a dynamic range of over one hundred to one. Since the flight spectrum data set defines required specimen loads only at discrete points, the individual data points must be interpolated to obtain a smooth load command signal. Traditionally this is done using a haversine smoothing curve at frequency fd. This interpolated command signal will be called the jlight spectrum command signal.
The flight spectrum test specifications require that the loads on the specimen track the loads in the flight spectrum data set to a given accuracy (here taken as 0.25% of the full scale values) measured at the flight spectrum data set peaks. (A peak in the flight spectrum data set is defined to be apoint where the derivative of the flight spectrum command signal changes sign.) In addition no additional load peaks may be generated by the control system. Thus the relevant control problem involves the design of a controller for the testing machinehpecimen process that will cause the actual specimen loads to track the flight spectrum command signal with the required accuracy and without generating additional load peaks. Moreover, since material parameters may vary throughout the test or between specimens the controller is to be robust to a range of specimen stiffness parameters. From the control system design viewpoint a materials testing machine system can be considered a servo-mechanism in which we want to have the load waveform track the reference signal. The control system design approach to be employed involves the following four steps: 1) modeling of the controlled process, 2) modeling of the test specifications, 3) selection of an appropriate control system structure and the necessary control algorithms and 4) evaluation of the design on the MTS Laboratory Testing Machine System. These steps will be used to organize the remainder of this paper.
0272- 1708/92/$03.000 1992IEEE April 1992
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Modeling the Process In this section we develop a mathematical model of the controlled process. The process is chosen as the load framekpecimen system with the current amplifier, load cell and load cell amplifier included. This process has a (current amplifier) voltage input and a (load cell amplifier) voltage output. The materials testing system shown in Fig. 2 includes the following major components: 1) load frame with servo-valve controlled hydraulic actuator; 2) prototype transputer based digital controller with 16 bit A/D and DIA convertem; 3) load cell (force transducer) with 9 7 . 8 6 ~ 1 0N~ = 20 mV; 4) load cell amplifier with gain of 500; 5) three pole Butterworth low pass filter anti-aliasing filter (AAF) withf, = 200 Hz; 6) current amplifier with gain = 25e-4 AN. Although the controlled variable of interest is in fact the specimen force f & from an instrumentation and controller design point of view it is more convenient to consider the controlled variable in the testing machine system as the load cell amplifier output e, as shown in Fig. 2. While e, = Gc(s)fs,the transfer function Gc(s)is almost constant over the frequency range of interest here. Generalized Circuit Model. A generalized circuit model for the load frame/specimen system is shown in Fig. 3. Two important points about this model should be noted. First, we are assuming a first order model for the servo valve dynamics. This is a simplification of the rather complex nonlinear behavior of the actual valve. Second, we are assuming that eFG,$ with Gc constant. This is in fact true only to the extent that&=K& with K2 constant. However, in Fig. 3 we see that the force & is in fact divided between Kc and Mg. Moreover this force division is frequency dependent with K2(s) =
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Fig. 2. Closed loop materials testing system.
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Fig. 3. Generalized circuit model of load frame, specimen and components. State variables, inputs, and outputs: ei = current ampl6er input, vg = velocity of upper grip, qs = valve flow,& =force in load cell, pa = actuatorpressure, vf = velocity of loadframe head, va = actuator piston velocity, ec = load cell ampl6er voltage,& =force in specimen, ff = force in frame. Parameters: Kd = valve driver gain, Ks = specimen stifiess. Kv = valve flow gain, M g = grip mass, Pd = valve driver pole, Kc = load cell stisfness, Gv = equivalent valve leakage, Mf=fiame mass (equiv.),Ga = equivalent actuator leakage, Kf =frame stiffness, Ca = equivalentfluid compliance, Bf=frame damping, Aa = actuator area, G, = load celVamplijier gain, Ma = actuator mass, Gi = current amplij?er gain, B, = actuator damping.
Mathematical Model. The next step in building the model is the conversion of the symbolic generalized circuit model into a mathematical model. Using standard techniques [4] an eight order state variable model can be obtained directly from K c ( Mf 's + Bf s + K f ) the general( M g K f + M g K c + M f K c ) s 2 + B f K c s + K f K c I , , i z e d circuit \-I
While Kz(jo) has a DC gain of one, it clearly has a frequency dependence with higher fresuency gain values depending on the associated circuit parameters. A frequency response plot of the magnitude of the force divider ratio Kz(s) shows that with existing parameters IKz(j0)l remains within 0.25% of unity only f o r 6 3 5 Hz.This frequency dependence of Kz(s) shows that for any given system parameters there is always an upper limit to the fresuencies where the load cell will accurately read the force in the specimen and thus where the model developed here will be valid
PROCESS
using the state variables and parameters defined in Fig. 3. The determination of the appropriate model parameters proceeds in three steps: 1) the determination of preliminary parameter values from component descriptions, 2) the measurement of actual system behavior and 3) the adjustment of preliminary parameter values to fit predicted system behavior to measured behavior. Here fitting was accomplished mainly through adjustment of Pd, Ge. and K p The values given in the Appendix have been adjusted to obtain a close fit to the experimental data.
The process model has eight poles and two zeros. Arealpoleats--5,threelightlydamped complex pole pairs at s&j2OOO, sdj4000, s=+j12000and complex zeros at s=kj2500 are associated with the hydro-mechanical structure of the load frame, specimen and actuator. An additional pole at s = -Pd = -720 arises from the single pole valve driver model assumed in Fig. 3. The complex pole-zero pair near s=3$2000 and the complex poles near s&j12000 represent an approximation of the resonant structure of the load frame components. Experimental data (see Fig. 4) shows that the actual resonances are similar but somewhat more complex.
Experimental Data. Actual frequency response data for the process with AAF included was collected by injecting random noise at the summing junction input es (see Fig. 2) and analyzing the transfer function from ei to ed using a Tektronix 2630 Signal Analyzer.
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In Fig. 4 the experimental Bode plot of process plus AAF is compared with a Bode plot generated from the state variable model augmented by a model of the AAF. Note that there is quite good agreement in magnitude up to about 500 Hz. and good agreement in phase up to about 100 Hz. In general there is good qualitative agreement over the entire frequency range. The phase error developing beyond 100 Hz. is probably due to the simplicity of the valve model used above. Since the controller bandwidth will be limited to 200 Hz, this deviation is not expected to cause problems in the controller design. However, a more complex valve model may be required to design higher bandwidth controllers. Reduced Order Model. In many situations it is desirable to have reduced order models that represent all significant low frequency properties of the process needed in preliminary design studies. To accomplish this we can simplify the generalized circuit model shown in Fig. 3 by ignoring the load frame and load cell dynamics. It is then relatively easy to calculate the transfer function for the reduced order process model as
0 7 7 -
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Fig. 4. Experimental and theoretical frequency response of load frame and specimen (steel specimen).
while the complex roots move out parallel to the j o a x i s but remain in the left half plane. More generally, GI K , Kv K5 Hz and its performance degradation at fd =10 Hz is an expected result of this fact.
70
Conclusions Overall the project has successfully demonstrated that digital controllers for high accuracy materials testing can be analytically designed. Moreover, these controllers can be designed to obtain significant robustness in specimen compliances. The main limit to obtaining controllers with additional compliance robustness was found to be sensor noise. The increasing loop gain required to obtain additional robustness produces additional output noise which can invalidate the flight spectrum test by generating spurious peaks and/or erroneous levels in the specimen load. Since robustness is necessarily obtained through the use of control loop gain and bandwidth, techniques for the design compensators which use these scarce resources with maximum efficiency are important tools in the development of robust controllers for high accuracy materials testing.
Appendix Parameter values of the prototype materials testing machine model (see Fig. 3) are given below in SI units. The Ks values are for standard test specimens. Kd = 8.89 m N K, = 3 . 9 ~ 1 0 ”m3/m Pd = 720 rads Gv= 1 . 9 ~ 1 0m3/MPa -~ Ga=O C, = 0.135~ 10-6 m3/MPa Ba = 1 . 7 5 ~ 1 0N~d m A, = 4 . 8 8 ~ 1 0 m2 -~ Ma = 22.67 k K, = 258x10 N/m (Steel)
8
Ks = 8 9 ~ 1 N/m 0 ~ (Aluminum) Mg = 9.07 kg Kc = 9 . 6 3 ~ 1 0N/m ~ Mf=36.27 k K f = 2 . 5 7 ~ 1 0 N/m Bf= 1 . 9 3 ~ 1 N 0 ~d m Gc = 1O/97.86x1O3 V/N Gi = 2 5 ~ 1 0 - ~ / V 1 0N
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References [ I ] I. Horowitz, Feedback System Synthesis. New York: Academic, 1963. [2] J.C. Cockbum, “Root locus design of robust controllers for linear systems,” M.S. thesis, Dept. of EE, Univ. of Minn., Oct. 1989.
[3] G.F. Franklin and J.D. Powell, Digital Control of Dynamic Systems. Reading, MA: Addison-Wes-
ley, 1980. [4] T. Kailath, Linear Systems. Englewood Cliffs, NI: Prentice-Hall. 1980.
Fred Bailey received the B.S.
degree in electrical engineering from Purdue University. Following graduation he served in the United States Navy as a Pilot and Electronics Officer. He then returned to graduate school, receiving the M.S. and Ph.D. degrees from The University of Michigan. He has been with the Department of Electrical Engineering at the University of Minnesota since 1964 where he is presently Professor of Electrical Engineering with joint appointments in Control Sciences and Computer Sciences. He has also served as technical consultant and in-house course instructor in digital control and signal processing for numerous industrial firms. His research interests include control theory and its applications with emphasis on the control of mechanical motion and computer aided design of control systems. Juan Carlos Cockburn received the B.S. degree in electrical engineering from the Universidad Nacional de Ingenieria in Lima, Peru, un 1984 and the M.S. degree in electrical engineering from the University of of Minnesota in 1990. Since 1988 he has held several teaching and research assistant positions in the Department of Electrical Engineering of the University of Minnesota where he is currently a Ph.D. candidate. His research interests include theory and application of robust control techniques, computer aided design of control systems, and digital control.
i
Arthur M. Dee received the B.A. degree in biological sciences from Indiana University, the B.S. degree in systems engineering from Wright State University in 1976, and the M.S. degree in systems science from Michigan State University in 1978. Since 1984 he has worked as a softwarekontrols engineer at MTS Systems Corporation developing parallel processor based control systems for automated servo-hydraulic material testing machines. His current research interests are focused on the application of intelligent control to material characterization processes.
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