Furthermore, the main difficulty in Kalman filter tuning, is the selection of the spectral density matrix, Q and the covariance matrix, R. According to the probabilistic Kalman Filter theory, the spectral density matrix, Q and the covariance matrix R should be selected according to the level of state and output noises respectively. The smaller the values of Q and R, the faster the convergence of the parameters. However, small values of Q and R also result in more noisy estimates. If the elements of these matrices are too small, instability of the CDEKF algorithm can occur.
The CDEKF is successfully implemented in Matlab and has resulted in a converging parameter set. The comparison of FRFs using the model and parameter-set with FRF measurements has demonstrated that an accurate mathematical description of the AVIS has been obtained.
The µ-theory has been adopted to design a multivariable controller for the AVIS. The major advantage of the µ-theory is that a robust non-conservative controller can be designed in a sys-tematical way for systems in which structured perturbations are present such as the AVIS. The nominal model of the plant and model uncertainties have to be specified as well as the performance and robustness bounds, which have to be selected by means of weighting filters. The remaining optimization problem can efficiently be solved with the algorithms available in the µ-Analysis and Synthesis Toolbox in Matlab, resulting in an optimal controller.
The specification of the nominal model and the model uncertainty is the most crucial part of the controller design. In this thesis, the rigid body model of the AVIS has been used as nominal model. Besides this, FRF measurements have been performed and considered as the true plant.
Nevertheless, the model uncertainty, which has been defined as the difference between the nominal model and the true plant, has appeared to be too large to obtain a controller with satisfactory performance. Therefore, the model uncertainty has been fitted with a sixth order filter using the frsfit -algorithm and included into the nominal model. However, it is very difficult to determine which part of the system should be considered as nominal plant and which part of the system should be considered as model uncertainty. If the model uncertainty is too small, the robust performance goal is not achieved for the implemented controller. If the model uncertainty is too large, this results in a conservative controller.
The µ-theory has successfully been used to developed a 2-DOF multivariable controller for the translation along the z-axis and the rotation around the x-axis. Experimental results have shown that the level of disturbances around the resonance frequency of the AVIS has been reduced with almost 20 [dB] using this controller.
6.2 Recommendations
The analysis of the modelling, identification and 2-DOF multivariable control of the AVIS has not only provided some valuable insights in the vibration isolation problem, but also yields four main recommendations for further research.
• In this thesis, a model of the AVIS has been derived which describes the coupling between the degrees-of-freedom. This model has only been used for the design of a multivariable controller using µ-synthesis. However, it is also very interesting to explore the possibilities to use this model for other controller design methods.
– Firstly, the model can be used to carry out a so-called modal transformation, in order to obtain a decoupled model of the AVIS, for which simple SISO controller design techniques can be used.
– Secondly, since a model of the AVIS is available in which vibrations of the chassis are considered as non-manipulated inputs, a model-based feedforward technique, such as the Computed Torque Control technique, is directly applicable to improve the perfor-mance of the vibration isolation system. However, since the motion of the chassis is needed in order to design a feedforward controller, the experimental set-up has to be modified such that vibrations of the chassis can be measured.
• In this study, a multivariable controller for two axes of the AVIS has been designed using µ-theory and its performance has been evaluated. It should be noted that it is very interesting to compare the multivariable, 2-DOF controller with the iteratively tuned SISO controller, which is used in current industrial applications. In this comparison, not only the performance of the controllers, but also the design procedure should be taken into account.
• Moreover, the specification of the nominal model and the model uncertainty requires further attention. In this thesis, a two degree-of-freedom model based on the rigid body dynamics, in combination with a sixth order fit of the difference between this model and a measured FRF, has been selected as a nominal model for 2-DOF AVIS. The difference of this nominal modal and the FRF has been used as uncertainty model. Another approach is to base the specification of the nominal model and the model uncertainty solely on FRF measurements.
The idea is to identify a set of FRFs for different operating points or input signals. The average value of these FRFs at each frequency can now be selected and fitted to obtain a nominal model. Moreover, the difference between each FRF and nominal model can be determined and the maximum magnitude for all frequencies can be fitted and selected as a worst-case uncertainty model. The advantage of this approach is that a more reliable description the model uncertainty is obtained. Since the level of uncertainty is directly related to the performance of the AVIS, this method should be further explored.
• In this thesis, the µ-theory is applied to two axes of the AVIS. Since the long term goal of the vibration isolation project is to design a six degree-of-freedom controller for the AVIS, the next step for further research is to increase the number of degrees-of-freedom which are subjected to multivariable control. One could for example think of designing a full 6-DOF controller directly, using the µ-theory. However, because of the size of this problem, the trade-off between the conflicting performance and robustness specification will be more complicated as well as the implementation of the resulting controller. Another approach is to design multiple, multivariable controllers for combinations of degrees-of freedom. Never-theless, the selection of the combinations of degrees-of-freedom needs further attention as well.
56 6.2. RECOMMENDATIONS
[1] http://acept.la.asu.edu/PiN/rdg/elmicr/elmicr.shtml.
[2] Design it easy tool. http://www.dct.tue.nl/New/Steinbuch/diettot.zip.
[3] G.J. Balas, J.C. Doyle, K. Glover, A. Packard, and R. Smith. µ-Analysis and Synthesis Toolbox Users’s Guide. The MathWorks, Natick, MA, version 3 edition, 2004.
[4] O.H. Bosgra, H. Kwakernaak, and G. Meinsma. Design methods for control systems. Notes for a disc course, Dutch Institute of Systems and Control, The Netherlands, 2002.
[5] A. Damen and S. Weiland. Robust control. Lecture notes, Eindhoven University of Technol-ogy, Eindhoven, The Netherlands, 2002.
[6] D. de Roover. Motion of a Wafer Stage, A Design Approach for Speeding Up IC Production.
PhD thesis, Delft University of Technology, 1997.
[7] J.C. Doyle, K. Glover an P.P. Khargonekar, and B.A. Francis. State space solutions to standard H2and H∞control problems. IEEE Transactions on automatic control, ac-34(8):831 – 847, 1989.
[8] A. Gelb. Applied Optimal Estimation. The M.I.T. Press, Cambridge, Massachusetts, 1999.
[9] J. Holterman. Vibration Control of High-Precision Machines with Active Structural Elements.
PhD thesis, University of Twente, 2002.
[10] IDE. Active and passive isolation modules, 1998.
[11] J.M. Maciejowski. Multivariable Feedback Design. Addison-Wesley, Wokingham, United King-dom, 1989.
[12] The MathWorks. xPC Target. http://www.mathworks.com/products/xpctarget.
[13] Y. Nakamura, M. Nakayama, K. Masuda, K. Tanaka, M. Yasuda, and T. Fujita. Develop-ment of six-degrees-of-freedom microvibration control system using giant magnetostrictive actuators. Smart Mater. Struct., (9):175 – 185, 2000.
[14] A. Packard and J. C. Doyle. The complex structured singular value. 29(1):71–109, January 1993.
[15] C. Scherer. Theory of robust control. Lecture notes, Delft University of Technology, Delft, The Netherlands, 2001.
[16] S. Skogestad and I. Postlethwaithe. Multivariable feedback control. John Wiley & Sons, Chichester, United Kingdom, 1996.
58 BIBLIOGRAPHY
[17] Vagners J. von Flotow A. Hardham C. Thayer, D. and K. Scribner, editors. Six Axis Vibration Isolation Using Soft Actuators and Multiple Sensors. Proceedings of the 21st Annual AAS Guidance and Control Conference, Februari 1998.
[18] S. Tøffner-Clausen, P. Andersen, and J. Stoustrup. Robust control. Technical report, De-partment of Control Engineering, Institute of Electronic Systems, Aalborg University, 2001.
[19] Marc van de Wal, Gregor van Baars, Frank Sperling, and Okko Bosgra. Multivariable H∞/µ feedback control design for high-precision wafer stage motion. Control Engineering Practice, 10:739 – 755, 2002.
[20] N. v.d. Wouw. Multibody dynamics. Lecture notes, Eindhoven University of Technology, Eindhoven, The Netherlands, 2003.
[21] M.S. Whorton and J.T. Eldridge. Damping mechanisms for microgravity vibration isolation.
Technical report, NASA, 1998.
[22] P. Wortelboer. Frequency-weighted Balanced Reduction of Closed-loop Mechanical Servo-systems: Theory and Tools. PhD thesis, Delft University of Technology, 1994.
[23] G. Zames. Feedback and optimal sensitivity: Model reference transformations, multiplicative seminors, and approximate invereses. IEEE Transactions on automatic control, ac-23:301 – 320, 1981.
[24] A. Lugunga Ya Zenga. Iterative siso feedback design for an active vibration isolation sys-tem. Traineeship report: Dct 2005.10, Eindhoven University of Technology, Eindhoven, The Netherlands, 2005.
[25] K. Zhou, J.C. Doyle, and K. Glover. Robust and Optimal Control. Prentice hall, Upper Saddle River, New Jersey, 1996.
Technical information of the AVIS
A.1 Technical specification of the AVIS
Description : Active Vibration Isolation System
Type : TCN 25 LM10
IDE Drawing No. : 289.000 LM10
Customer : TUE
TCN-25 Serial No. : 39-01-002 Controller Serial No. : 39-01-126
Description Number
TCN 25 Isolator # 1 1
TCN 25 Isolator # 2 1
TCN 25 Isolator # 3 1
TCN 25 Isolator # 4 1
Mechano Pneumatic Leveling Valve Type MV-3 3
Horizontal Force Motors LM 10 4
Vertical Force Motors LM 10 4
Velocity Sensors 6
Sensor Cable 5m 3
Actuator Cable 5m 4
Power Cord Euro 1
A.2 Linear Force Actuator
The actuators which are used for active vibration isolation are linear force motors. These actuators operate in parallel with the passive isolation system. The linear force actuators are composed of two parts, a stator and a translator as depicted in figure Figure A.1. The stator consists of two permanent magnets, which are placed on a metal strip. The translator contains a coil, that acts like an electromagnet when a current is supplied.
60 A.3. GEOPHONE
N NS
S
T r a n s l a t o r
S t a t o r
s
x
Figure A.1: Linear force motor
The force equation is derived using Faraday’s law of induction, F = −dφ
dxi, (A.1)
where φ is the electromagnetic coupled flux, x the position of the coil and i the current through the coil. The expression for the electromagnetic coupled flux is given by
φ = ˆφ cosπx s
, (A.2)
where ˆφ is the amplitude of the flux and s is the distance between the permanent magnets.
Combination of (A.1) and (A.2) results in:
F = φπˆ
s sinπx s
i. (A.3)
A.3 Geophone
The sensors that are used for the active isolation system are geophones. Geophones are inertial sensors based on the production of a voltage in a coil when a magnet is passed through it. The relation between the velocity of the geophone, ˙xg, and the induced voltage in the coil, Vg, is given by:
Hg(s) = Vg
˙
xg = −G s2
s2+ 2ξgωgs + ω2g (A.4) with ωgthe natural frequency of the seismometer, ξg the damping ratio and G = Bl the transduc-tion constant, where B is the magnetic field density generated by the permanent magnet and l is the length of the coil. Above the suspension frequency the geophone behaves as a velocity sensor, below it as a jerk sensor. Since this behavior for frequencies below the suspension frequency is not desired, the geophones are pre-filtered with an analog filter (stretcher), in order to obtain velocity information. The transfer function between velocity of the payload and the geophone voltage for resulting combination of geophone and stretcher is depicted in Figure A.2(a). The velocity of the payload is obtained using a calibrated laser-doppler set-up.
From Figure A.2(a) it can be concluded that the stretcher doesn’t compensate the geophone characteristics very well. Therefore, an additional, digital filter is designed to compensate the deviating behavior. This filter is based on a fit of the geophone-stretcher combination, shown in Figure A.2(b). The frequency characteristics of the digital filter are shown in Figure A.2(c),
and the combination of the geophone, stretcher and digital filter is depicted in Figure A.2(d). It can be concluded that the output of the combination of the digital compensation, stretcher and geophone, is now proportional to the velocity of the payload in the desired frequency range.
FRF Geophone and analog geophone compensation
Frequency [Hz]
FRF digital geophone compensation
Frequency [Hz]
(c) FRF Digital Geophone Filter
FRF Geophone, agc and dgc
Frequency [Hz]
(d) FRF geophone-stretcher-digital filter combina-tion
Figure A.2: Digital Geophone-Stretcher Compensation
A.4 Actuator and Sensor Matrix
Actuator Matrix
The AVIS is a 6-DOF vibration isolation system. Since there are eight individual actuators and six logical axis, an actuator matrix is constructed, which relates the individual actuator signals to the actuator signals for each logical axis. The relation is given by:
a = A · u, (A.5)
where
u =
ux uy uz uφ uθ uψ T
, (A.6)
is the vector of actuator signals for each logical axis and a =
ay1 az1 ax2 az2 ay3 az3 ax4 az4 T
, (A.7)
62 A.4. ACTUATOR AND SENSOR MATRIX
is the vector with the individual actuator signals. For the construction of the actuator matrix, A, the polarity of the actuators is determined. The results are presented in Table A.1. Using the polarity of the actuators and the kinematic model of the payload, see Figure A.3, the actuator matrix becomes:
0.0000 −0.5000 −0.0000 0.0000 −0.0000 −0.6485 0.0974 0.1233 −0.2500 −0.6667 0.5263 −0.0000
−0.5000 0.0000 −0.0000 0.0000 −0.0000 −0.5119 0.0974 −0.1233 −0.2500 0.6667 0.5263 −0.0000 0 0.5000 −0.0000 0.0000 −0.0000 −0.6485
−0.0974 −0.1233 −0.2500 0.6667 −0.5263 −0.0000 0.5000 0.0000 −0.0000 0.0000 −0.0000 −0.5119
−0.0974 0.1233 −0.2500 −0.6667 −0.5263 0.0000
Table A.1: Measured direction of motion for a positive DC signal
Actuator Direction Actuator Direction
ay1 y− az1 z−
Figure A.3: Position of the Isolators
Sensor Matrix
In the same way as the actuator-matrix is built, a sensor-matrix is constructed. The polarity of the sensors is given in Table A.2. The relation between the individual sensors signals and the sensor signals for each logical axis is given by:
˙
is the velocity vector along the logical axis and s =
sy1 sz1 sx2 sz2 sy3 sz3 T
, (A.11)
is the vector with individual sensor signals. The sensor matrix, S, is given by:
S =
−0.3947 0 1.0000 0.1158 −0.3947 −0.1158
0.5000 0.1467 0 −0.1467 −0.5000 0
0 −0.5000 0 −0.0000 0 −0.5000
0 −1.3333 0 1.3333 0 0
0 0 0 1.0526 0 −1.0526
1.0526 0 0 0 1.0526 0
(A.12)
Table A.2: Direction of motion for a positive geophone signal
Sensor Direction Sensor Direction
sy1 y− sz1 z+
sx2 x− sz2 z+
sy3 y+ sz3 z+
64 A.4. ACTUATOR AND SENSOR MATRIX
Linearized Model of the AVIS
In this appendix, the entries of the linearized dynamic model of the AVIS are presented. The linearized dynamics of the AVIS are given by:
M (η˜ m)¨q + ˜C(ηb, ηp) ˙q + ˜K(ηk, ηp)q = ˜T (ητ, ηp)u (B.1)
m) ∈ R6×6 is the inertia-matrix. In contrast with the nonlinear inertia matrix, the entries of the linearized inertia-matrix are independent of the generalized coordinates and only depend on the mass and inertia of the system.
M =˜
66 B.2. ENTRIES OF THE DAMPING MATRIX
B.2 Entries of the Damping Matrix
C(η˜
b, η
p) ∈ R6×6 is the matrix which contains the damping forces. The entries of the inertia matrix are a function of the damping parameters and the position of the isolators to the center of mass of the AVIS.
C˜12= C˜13= C˜14= C˜21= C˜23= C˜25=
C˜31= C˜32= C˜36= C˜41= C˜52= C˜63= 0 (B.8)
C˜11= b11+ b41+ b31+ b21 (B.9) C˜15= −b11pz− b41pz− b31pz− b21pz (B.10) C˜16= −b11py− b41py+ b21py+ b31py (B.11) C˜22= b32+ b42+ b12+ b22 (B.12) C˜24= b12pz+ b42pz+ b32pz+ b22pz (B.13) C˜26= b12px− b42px− b32px+ b22px (B.14) C˜33= b13+ b23+ b33+ b43 (B.15) C˜34= b13py− b23py− b33py+ b43py (B.16) C˜35= −b13px− b23px+ b33px+ b43px (B.17) C˜42= b12pz+ b42pz+ b32pz+ b22pz (B.18) C˜43= b13py− b23py− b33py+ b43py (B.19) C˜44= b32p2z+ b33p2y+ b12p2z+ b42p2z+ b43p2y+ b13p2y+ b22p2z+ b23p2y (B.20) C˜45= −b33pypx+ b43pypx+ b23pypx− b13pypx (B.21) C˜46= −b32pxpz− b42pxpz+ b12pxpz+ b22pxpz (B.22) C˜51= −b41pz− b31pz− b21pz− b11pz (B.23) C˜53= b33px+ b43px− b13px− b23px (B.24) C˜54= b43pxpy− b13pxpy+ b23pxpy− b33pxpy (B.25) C˜55= b41p2z+ b43p2x+ b33p2x+ b13p2x+ b11p2z+ b31p2z+ b21p2z+ b23p2x (B.26) C˜56= b41pypz− b31pypz− b21pypz+ b11pypz (B.27) C˜61= −b11py− b41py+ b21py+ b31py (B.28) C˜62= b12px− b42px− b32px+ b22px (B.29) C˜64= −b32pxpz− b42pxpz+ b12pxpz+ b22pxpz (B.30) C˜65= −b31pzpy+ b41pzpy− b21pypz+ b11pypz (B.31) C˜66= b41p2y+ b11p2y+ b12p2x+ b42p2x+ b22p2x+ b21p2y+ b31p2y+ b32p2x (B.32)
B.3 Entries of the Stiffness Matrix
K(η˜ k, ηp) ∈ R6×6is the matrix which contains the spring forces. The entries of the stiffness matrix depend on the stiffness parameters and the position of the isolators to the center of mass of the AVIS.
K˜12= K˜13= K˜14= K˜21= K˜23= K˜25=
K˜31= K˜32= K˜36= K˜41= K˜52= K˜63= 0 (B.33)
K˜11= k21+ k11+ k41+ k31 (B.34) K˜15= −k31pz− k41pz− k21pz− k11pz (B.35) K˜16= k21py− k11py+ k31py− k41py (B.36) K˜22= k32+ k22+ k42+ k12 (B.37) K˜24= k32pz+ k42pz+ k22pz+ k12pz (B.38) K˜26= k12px− k42px− k32px+ k22px (B.39) K˜33= k13+ k23+ k33+ k43 (B.40) K˜34= k13py− k23py− k33py+ k43py (B.41) K˜35= −k13px− k23px+ k33px+ k43px (B.42) K˜42= k32pz+ k42pz+ k22pz+ k12pz (B.43) K˜43= k13py− k23py− k33py+ k43py (B.44) K˜44= k43p2y+ k23p2y+ k13p2y+ k22p2z+ k42p2z+ k33p2y+ k32p2z+ k12p2z (B.45) K˜45= −k33pxpy+ k43pxpy− k13pxpy+ k23pxpy (B.46) K˜46= −k42pzpx− k32pxpz+ k22pxpz+ k12pxpz (B.47) K˜51= −k31pz− k41pz− k11pz− k21pz (B.48) K˜53= k43px+ k33px− k13px− k23px (B.49) K˜54= k43pxpy+ k23pxpy− k13pxpy− k33pypx (B.50) K˜55= k41p2z+ k33p2x+ k13p2x+ k21p2z+ k11p2z+ k23p2x+ k31p2z+ k43p2x (B.51) K˜56= −k31pypz+ k11pzpy+ k41pzpy− k21pypz (B.52) K˜61= k21py− k11py+ k31py− k41py (B.53) K˜62= k12px− k42px− k32px+ k22px (B.54) K˜64= −k42pzpx+ k12pzpx− k32pzpx+ k22pzpx (B.55) K˜65= k11pypz− k21pypz+ k41pypz− k31pypz (B.56) K˜66= k31p2y+ k22p2x+ k12p2x+ k32p2x+ k42p2x+ k41p2y+ k21p2y+ k11p2y (B.57)
68 B.4. ACTUATOR MATRIX
B.4 Actuator Matrix
T (η˜
τ, η
p) ∈ R6×8 represents the matrix which relates the manipulated inputs to the external applied forces and torques on the system. The entries of this matrix depend on the actuator constants and the position of the isolators to the center of mass of the AVIS.
T =˜
0 0 τ3 0 0 0 τ7 0
τ1 0 0 0 τ5 0 0 0
0 τ2 0 τ4 0 τ 6 0 τ8
τ1pz τ2py 0 −τ4py τ5pz −τ6py 0 τ8py 0 −τ2px −τ3pz −τ4px 0 τ 6px −τ7pz τ8px τ1px 0 τ3py 0 −τ5px 0 −τ7py 0
(B.58)
FRFs of the six degree-of-freedom
model
70
Figure C.1: FRFs first output
10−2 10−1 100 101 102 103
Figure C.2: FRFs second output
72
Figure C.3: FRFs third output
10−2 10−1 100 101 102 103
Figure C.4: FRFs fourth output
74
Figure C.5: FRFs fifth output
10−2 10−1 100 101 102 103
Figure C.6: FRFs sixth output
76
SISO Control
Figure D.1: SISO controller design xtrans
78
Figure D.2: SISO controller design ytrans
10−1 100 101 102 103
Figure D.3: SISO controller design yrot
80
Figure D.4: SISO controller design zrot
Identification of the 2-DOF AVIS model
0 100 200 300 400 500 600 700 800 9001000 200
0 100 200 300 400 500 600 700 800 9001000 8
0 100 200 300 400 500 600 700 800 9001000 300
Figure E.1: Identification results 2-Dof AVIS model
82
At the end of this thesis I would like to express my gratitude to many people who have contributed to my research in different ways.
First of all, I would like to thank my supervisor prof. dr. Henk Nijmeijer. His enthusiasm, many ideas and confidence encouraged me to push myself forward and motivated me with new challenges that made my project an ongoing and fascinating research. Furthermore, I would like to thank Ines Lopez for her close support, her useful comments and encouragements during this study that helped me to improve the contents of this thesis.
A special word or thanks goes to Ron Hensen and Sander Kerssemakers for their support and their contribution from a practical point of view to the work presented in this thesis. Their advises and technical expertise have further developed my understanding on vibration isolation. Furthermore, I would like to thank Roland Kappel and Han Hartgers from IDE Engineering who have shown interest in my work.
I am also indebted to all the people of the DCT-lab. In particular, I would like to thank the supporting staff of the Dynamics and Control Technology laboratory, Harry van de Loo, Peter Hamels, Rob van den Berg and Sjef Garenfeld, for their support and their advise on many practical issues concerning the modification of the control system of the AVIS and for creating a pleasant environment to work. Moreover, I want to thank my colleague students for the good atmosphere in the DCT-lab and the interesting and clarifying discussions on Kalman filtering, robust control and non-scientific subjects.
At last, but not the least, I am very grateful to my parents and my brother for their love and support during all my years of studies.