5.4 Controller Synthesis and Performance Evaluation
With the weighting filters of Section 5.3, a controller is designed using the D-K iteration procedure in the µ-Analysis and Synthesis Toolbox in MATLAB, [3]. It should be noted that the exact shape of the weighting functions that specify the performance is determined in an iterative manner, such that the maximum performance is achieved for the selected robustness specifications. It turned out that numerical conditioning of the plant by means of scaling or balancing of the generalized plant, which is often required, is not needed for the AVIS and numerical reliable results are obtained. Since the D-K iteration scheme is not convex, global convergence cannot be guaranteed.
Consequently, the scheme was used for different initial controllers. Nevertheless, the resulting controller did not show any dependance on the initial controller choice. Convergence of the D-K iteration scheme was achieved in 5 steps.
The µ-controller which is obtained using the D-K iteration scheme, consist of four, 90th order controllers. The frequency response functions of these controllers are depicted in Figure 5.9.
Because of the high order, these controllers cannot directly be implemented. Therefore, balanced truncation as described in [22] is used to obtain four, 40thorder controllers. These controllers are also depicted in Figure 5.9. It should be noted that the difference between the 90thorder and the 40thcontrollers cannot be distinguished in the frequency plots.
For the reduced order controller, the robust performance goal is verified. Recall that robust performance is achieved if
kµ∆a(N (jω))k∞≤ 1 (5.11)
The frequency response function of the structured singular value, µ∆a(N (jω)), is shown in Fig-ure 5.10. From this figFig-ure it can easily be seen that the structFig-ured singular value is smaller than 1, for all frequencies. The H∞-norm of µ∆a(N (s)), obtained with the reduced order controller is equal to
kµ∆a(N (s))k∞= 0.97, (5.12)
which means that robust performance is achieved.
In Figure 5.11, the frequency response function of each entry of the sensitivity function, based on the true plant, is depicted. From this figure, it can be concluded that disturbance rejection of almost 20 [dB] is achieved around the resonances of the logical axes. From the cross-terms of the sensitivity function it can be concluded that an interaction between the disturbance of one axis on the output of the other axis is introduced. Nevertheless, it should be noted that the interaction is very small.
Subsequently, the reduced order controller is implemented with a sample frequency of 1 [kHz], using the XPC-Target Toolbox in Matlab. In Figure 5.12, the velocity of the payload is depicted.
The black line shows the velocity if the controller is active, whereas the grey line represent the uncontrolled velocity of the payload. It can be concluded that the controlled velocity of the payload is significantly smaller than the uncontrolled situation. Moreover, the frequency content of the signal is determined and depicted in Figure 5.13. It can clearly be observed that the frequency content of the output signals is significantly lower between 2 an 8 [Hz] if the controller is active.
This means that the disturbances are rejected according to specification.
Summarizing, it can be concluded that the objective to design a multivariable controller, which achieves robust performance for two axes of the AVIS using µ-synthesis, is successfully accom-plished.
10−1 100 101 102 103
Figure 5.9: Full and Reduced Controller
10
−110
010
110
210
348 5.4. CONTROLLER SYNTHESIS AND PERFORMANCE EVALUATION
10−1 100 101 102 103
-20 -15 -10 -5 0 5
Magnitude[dB]
Frequency [Hz]
(a) S33
10−1 100 101 102 103
-100 -90 -80 -70 -60 -50 -40 -30 -20
Magnitude[dB]
Frequency [Hz]
(b) S34
10−1 100 101 102 103
-70 -60 -50 -40 -30 -20 -10
Magnitude[dB]
Frequency [Hz]
(c) S43
10−1 100 101 102 103
-20 -15 -10 -5 0 5
Magnitude[dB]
Frequency [Hz]
(d) S44
Figure 5.11: Sensitivity
5. ROBUST CONTROL OF THE AVIS 49
i i
i i
i i
×10−5
0 1 2 3 4 5 6 7 8 9 10
-5 0 5
×10−5
0 1 2 3 4 5 6 7 8 9 10
-2 0 2
˙ φ[rad/s]
time [s]
Controller OFF Controller ON
˙z[m/s]
time [s]
Figure 5.12: Velocity of the payload
10−1 100 101 102 103
-220 -200 -180 -160 -140 -120 -100
Magnitude[dB]
Frequency [Hz]
(a) Frequency contents ˙z
10−1 100 101 102 103
-220 -200 -180 -160 -140 -120 -100
Controller OFF Controller ON
Magnitude[dB]
Frequency [Hz]
(b) Frequency contents ˙φ
Figure 5.13: Implementation results
50 5.5. DISCUSSION
5.5 Discussion
In this chapter the application of the µ-theory to two axes of the AVIS, i.e. the translation along the z-axis and the rotation around the x-axis, has been presented.
The µ-theory provides powerful tools to design a robust controller for systems in which structured perturbations are present. In a systematical way, the nominal model of the plant and model uncertainties can be specified and performance and robustness bounds can be selected by means of weighting filters. The remaining optimization problem can efficiently be solved with the algo-rithms available in the µ-Analysis and Synthesis Toolbox in MATLAB, [3], resulting in an optimal controller. The µ-synthesis procedure using D-K iteration has proven to work well for the vibra-tion isolavibra-tion problem. Experimental results have shown that a controller has been developed for which the level of disturbances around the frequency of the resonance peak can be reduced with almost 20 [dB].
The major advantage of the µ-theory is that only the weighting filters have to be selected in order to design a controller. The robustness filters follow directly from the model uncertainty, whereas the weighting filters that specify the performance can be selected according to the designer’s insight.
In this chapter, the weighting filters have been selected such that disturbance rejection around the resonance frequency of the passive system has been achieved. However, if an instrument is mounted on the AVIS, which is extremely sensitive around another frequency, the weighting filters can easily be adapted, directly resulting in a controller that is suitable for the new application.
The specification of the nominal model and the model uncertainty needs further attention. In this chapter, a two degree-of-freedom model, based on the rigid body dynamics, has been selected as a nominal model for 2-DOF AVIS. 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.
Using this method, it is very difficult to determine the order of the fit. Since FRFs are in fact an approximation of the true plant’s behavior as well, it is not clear which part of the FRF should be considered as nominal model and which part of the FRF should be treated as model uncertainty.
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. Moreover, this method can also be used to design a controller which is robust for varying payloads. For this application a set of FRFs have to identified for different payloads.
In this chapter the µ-theory is applied to two axes of the AVIS. The next step is design a 6-DOF multivariable controller for the AVIS. The most difficult aspect in designing a 6-DOF controller will be the size of problem. The trade-off between the conflicting performance and robustness spec-ification will be more complicated, since the design of 144 weighting filters is involved. Therefore, a systematic way to design these weighting filters should be developed using a priori knowledge of the system and the model uncertainty. Moreover, the implementation of the 6-DOF controller will be very difficult, since the order of the controller is equal to the order of the generalized plant. For the 2-DOF controller, this was already a problem and controller reduction techniques have been applied. However, this problem can simply be solved using a PC with a more powerful processor,
such a Pentium III.
Furthermore, it should be noted that it is very interesting to compare the multivariable, 2-DOF controller with a well-tuned, SISO controller. In this comparison, not only the performance of the controllers, but also the design procedure should be taken into account.
52 5.5. DISCUSSION
Conclusions and Recommendations
In this thesis the modelling, identification and multivariable control of an Active Vibration Isola-tion System have been discussed. The vibraIsola-tion isolaIsola-tion system that has been considered in this study, the AVIS, has been developed at IDE Engineering and is capable of six degree-of-freedom vibration isolation using air mounts in combination with linear motors and geophones. This re-search can be considered as a first step in order to accomplish the long term goal of the vibration isolation project, to design a six degree-of-freedom multivariable controller for the AVIS. The conclusions of this study are presented below followed by recommendations for further research.
6.1 Conclusions
The Newton-Euler approach provides a straightforward method to develop a multivariable model for multi-body systems, such as the AVIS. For the six degree-of-freedom model of the AVIS, the payload and chassis of the AVIS have been considered as rigid bodies, whereas the isolator modules are modelled as three orthogonal spring-damper combinations, complemented with two actuators working in horizontal and vertical direction. The vibrations of the chassis of the AVIS have been considered as non-manipulated inputs, yielding a highly complex nonlinear dynamic model of the AVIS. Moreover, a linear model has been obtained for which the forces generated by the motion of the chassis have been omitted.
The parameters of the six degree-of-freedom model of the AVIS have been identified using a Continuous-Discrete Extended Kalman (CDEKF). However, the CDEKF cannot straightforwardly be used for the identification of the parameters of the 6-DOF model of the AVIS. There are two reasons for this. Firstly, because of the large amount of parameters which have to be identified, the convergence of the parameters is very slow. Therefore, the number of parameters which are identified with the filter should be minimized. Parameters which are accurately known beforehand or parameters which can be estimated otherwise should not be estimated by the filter. Moreover, the CDEKF can be used iteratively in order to achieve a faster convergence of the parameters.
Secondly, because the geophone voltage is not proportional to the velocity of the payload for low frequencies, the measurement noise can not be considered as a zero-mean Gaussian white noise process, which is supposed by the probabilistic Kalman Filter theory. Consequently, pre-filtering of the measurement data with a high-pass filter is needed.