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QFT robust control design and MU analysis for a solar orbitaltransfer vehicleCitation for published version (APA):Hoekstra, D. (2004). QFT robust control design and MU analysis for a solar orbital transfer vehicle. (DCTrapporten; Vol. 2004.043). Eindhoven: Technische Universiteit Eindhoven.

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QFT Robust Control Design and MU analysis for a

Solar Orbital Transfer Vehicle

Dynamics and Control Technology Group

Department of Mechanical Engineering Eindhoven University of Technology

The Netherlands

Report 2004.43 Eindhoven, april2004

Name: Douwe Hoekstra Email: [email protected] Student: 422286 Code: 4W609

Supervisor: M.Steinbuch

Introduction

This report is part of a project by ESA (European Space Agency), aimed at designing a satellite system. This satellite, the Solar Orbital Transfer Vehicle (SOTV), has solar collectors that have to remain aimed at the sun while rotating the earth. Furthermore, for the moment, the final design of the SOTV has not been completed yet snd several parameters of the design (e.g. masses) are only known within certain limits. The satellite has to follow a trajectory around the earth, this requires the possibility to perform small corrections to this trajectory using thrusters attached to the SOTV. Controllers that perform these corrections have to be designed based on the dynamics of the SOTV. Since the SOTV dynamics is only known to lie in between certain bounds, the Quantitative Feedback Theory (QFT) can be applied to design controllers for all possible dynamics configurations. This is explained in more detail in part 1 of this report.

In part 2 the designed controller from the QFT design method is compared with controllers resulting fi-om a H, design procedure and a LMI design procedure (these designs were done by RBmi Drai, ~ c o l e des Mines de Paris) by means of a MU robustness analyis.

This report is an extension of the report of Jos A.C. Meesters (DCT 2003.74), "Quantitative Feedback Theory applied to a Solar Orbital Transfer Vehicle". The QFT design and MU robustness analysis were supervised by Maarten Steinbuch.

This report consists of two parts:

1 . WP 31 Technical Note: QFT Robust Control Design Pages 3-39

2. MU robustness analysis Pages 40-48

(version December 5 2003)

ESA Contract 16885/02/NL/LvH . WP 31 Technical No te

1 Scope ........................................................................................................................................ 5 Reference Documents .................................................................................................................. 5

..................................................................................................... 2 Control Problem Definition 6 3 SISO loopshaping design ......................................................................................................... 8

................................................................................................... 4 MIMO interaction Analysis 11 ............................................................................................................................. 5 QFT design 12

........................................................................ 5 . i Generation of the templates and bounds 12 5.2 Loop shaping .................................................................................................................. 15

........................................................................... 5.3 Controller Time domain Performance 17 ..................................................................................................... 6 Conclusions & Discussion 18

Appendix 1 Influence of parameters on Plant FRF (x direction) ................................................... 19 Appendix 2 QFT Controllers C,, C, and C , ................................................................................... 23

........................................................................................................... Appendix 3 QFT principles 31

ESA Contract 16885/02/NL/LvH - WP 31 Technical Note

1 Scope

The present document has been prepared by Eindhoven University of Technology, Department of Mechanical Engineering, Control Systems Technology Group, in the frame of the ESA Contract: AOCS for Large Flimsy Appendages as a technical output of WP 3 1. The purpose of this note is to describe the loopshaping and QFT control design approach used to solve the robust control problem defined in the WP 30 [pl].

The results presented here for the 1 :3 scale SOTV application in concentrators axes; the obtained insights will be used in the combined design effort WP 33.

The main issues addressed in the document are:

a recall of the robust control design problem a presentation of a loopshaping design, and its robustness analysis, including a M~M~'/interaction assessment generation of QFT templates and a QFT loopshaping robust performance SISO~ control design and evaluation assessment on the proslcons of the design method

For the design we will use MATLAB version 6.5.0.180913a (R13), MATLAB QFT toolbox version 2.5 [R4] and SOTV constructor files v5.

Reference Documents

Robust Control Problem Formulation CMA-ARMINES, June 4th 2003, WP 30 TN, version 2 revised SOTV: Full and Reduced Models (documented Matlab code) CMA-ARMINES, June 2003, addendum to WP 30 TN M. Steinbuch, M.L. Norg, Advanced Motion Control: an industrial perspective, European J, of Control, 4,278-293, (1998) Craig Borghesani, Yossi Chait and Oded Yaniv, "Users guide for the QFT Frequency Domain Control Toolbox". Constantine H. Houpis, Steven J. Rasmussen, "Quantitative Feedback Theory, Fundamentals and applications", 1999. Isaac M. Horowitz, "Quantitative Feedback Theory", 1992. Gene F. Franklin, J. David Powell, Abbas Emami-Naeini, "Feedback control of dynamic systems", fourth edition, 2002.

' Multi-input multi-output Single-input single-output

ESA Contract 16885/02/NL/LvH - WP 31 Technical No te

2 Control Problem Definition

The performance specifications for the controllers were defined in the reference document [Rl], and can be summarized in the following table

Attit~de poiating 1 0.03 deg ( lo) I 0.03 deg ( lo) random accuracy Frequency domain

Table 1 Performance Specifications

stability criteria

Time domain stability criteria - end of slew manoeuvre Time domain stability criteria - disturbance rejection

The mapping from the time domain specifications from Table 1 to frequency domain criteria is in general a non trivial issue. However, for 2nd order systems, an approximate relation between settling time and required bandwidth is given in [R7, p. 147-1481 as:

bias 6 dB gain

In our case, with an accurary of 0.03 deg after a maximum slew manoeuvre of 180 deg, a choice for closed loop damping <=0.8, and the required settling time t, = 15s this boils down to on=- ln(0.03/180)/(0.8*15) = 0.72 radls = 0.12 Hz. This can be regarded as an approximate bandwith requirement for linear second order systems, for higher order systems as in the system under consideration in this report, it is merely an indication of the bandwith requirement. The mapping of the disturbance rejection specification into frequency domain criteria can be done using spectral information which is not done here.

margin

1 deg maximum deflection

The demands on Gain Margin and Phase Margin can be translated into one stability demand, the Modulus- or Sensitivity Margin [R7, p. 409 called vector margin]. These are related to each other by the following set of equations:

30 deg phase margin (if delay not

where amin = min(al,az). A Sensitivity Margin of 6 dB (amin = 0.5) corresponds to a worst-case GM of 6 dB (a l = 0.5) and a PM of 29" (a2 =0.5). This value will be used for the QFT design procedure.

20 deg phase margin (if included in design model)

15 sec settling time (to remain within +I-0.03 deg)

15 sec stabilisation time (to remain within +I-0.03 deg)

delay included in design model)


The control laws must provide robustness with respect to a certain number of uncertain (or time- varying) parameters that are listed together with thdir uncertainty range in the following table:

SIC CoM I 1 cm SIC inertias 10%

Damping 0.005-0.05 Misalignment 1 0.005 deg

on cent ria tors modes nncertzzinties

Table 2 Uncertain Parameters

Cross-coupling Terms I " 15 kg.mA2

Engine uncertainties

The parametric dispersion was performed using the generic Matlab tool S0TV.m that was developed by CMA in order to be used by the control teams for both the QFT and the H, designs. This tool allows to work, for each position on orbit, in either SC or concentrators frames, using full and reduced models and to change very easily all the parameters listed in Table 2 above using the single function typically invoked in the form: sOTVS e t ( ' Parameter Name , Value ) . This flexible object-oriented program is completely described in the document [R2].

Frequency

Analysis of the effect of the parameters mentioned in Table 2 on the dynamics is shown in Appendix 1 (for the x axis). From this analysis it is clear that the subset of relevant parameter variations consists of the SIC inertias, combined with the linked participation coupling terms, of the flexible modes frequencies and damping and of the concentrator rotation angle 8. In the sequel of the report we will address uncertainties in these parameters only. Notice that, as an additional feature, in this report uncertainties in mode-damping (0.005-0.05) is taken into account, whereas it is taken constant (0.009) in the WP 32 results. The verification of the other specifications will be ultimately assessed during the validation tasks of WP 41, where worst case scenarios identified during the design stages will be defined and systematically tested.

15%

Coupling Coefficients Thrust direction

Since the QFT design method is based on frequency response data, the plant set has to be made finite, by making a choice for a grid in the uncertainty space as well as in the frequency parameter. This is well-known to be a principle drawback of the method, since there is not a guarantee on robustness nor performance in between the chosen grid points, although by taking a dense grid in practice it works satisfactory. In the sequel of the report we will use a grid of the uncertain parameters to do our calculations as follows:

20% 6BeWh,= k 0.5"

& Coef I [-~~0:0.005:0, 0.01:0.01:0.20] 1 p=sotvShiftParticipationFactors ( 0 . 2 0 , ) I

Table 3 Uncertain Parameters grid

Concentrator rot. Angle (theta)

[O:pi/4:pi] s o t = s o t v ( x / 2 , p ) ;


The resulting system dynamics of changing the concentrator rotational angle 9 is symmetrical in the domain [O:x] and [.n:2x]. The combination of the parameter values of Table 3 leaves us with a plant set of approximately 600.000 elements, computational time of the QFT bounds would be a big issue then. However, by smartly eliminating parameter uncertainty combinations that do not produce extreme bounds (e.g. resulting in plants located inside calculated uncertainty bounds) this set has been reduced to 4560 plants.

Before the actual QFT design will be done, a §IS0 loopshaping design will be made (Section 31, including a robustness assessment. A MIMO analysis will be done in Section 4. The QFT robust performance design consists of several design steps which will be discussed in Section 5.

3 SISO loopshaping design

In this section we will apply classical control design principles first, being the standard tool in industrial motion systems, see [R3], in order to acquire insight into the design problem at hand. Hereto we will focus on robust stabilization first; as an example we take the x axis. Similar results can be obtained for the y and z axes. As a first step model reduction is done, not because it is really necessary for loopshaping nor QFT design, since they are both working with frequency response function (FRF) data, but solely for limiting the computational time required to generate the FRF plots for all uncertainty data. Comparison of the fiequency response function of the full order model (n=86), as shown in Figure 1 for the x axis, with the FRF of the reduced model (n=8, using fiequency truncation), shows that the difference between the reduced and h l l FRFs is only visible at high frequencies.

Figure 1 Comparison of Bode plots between full and reduced system (x-axis, 8=x/2)


Analysis of the FRFs of the three axes reveals that the robust stabilization design problem boils down to providing sufficient phase lead, since the most relevant observation is that for all uncertain plants phase never crosses -1803 degrees (see also Appendix 1). Henceforth, the cross- over frequency can be put well within the frequency range of the parasitic dynamics, since phase lag is limited due to the co-located nature of the control problem.

Figure 2 Bode plot of the x controller

A standard first order leadllag filter is used providing sufficient phase lead near the cross-over, see Figure 2. For all three axes a similar controller could be used. Application of the controller to the x axis model and perturbing all relevant parameters, the set of 4560 loop gains L=CP (where C is the controller and P the plant model) has been generated. For plotting purposes, a number of 360 FRFs out of the set of 4560 of the open-loop plants L are shown in Figure 3. It can be seen that although cross-over changes, there does not seem to be any stability problem, since phase is always good. A further proof for this can be obtained if we analyse the closed-loop transfer fimctions, of which the sensitivity function s=(I+L)-' is the most important. In Figure 4 the set of FRFs of the sensitivity S is shown for the x axis, corresponding to the open-loop FRFs of Figure 3. Realizing that the peak value of S equals inversely to the shortest distance to the critical point (-1,O) in the Nyquist plot, we can observe that for all plants the peak is below 6dB, meaning 0.5 distance to the critical point, which shows the nice robustness properties of the design. Also it can be seen from Figure 3 that the required bandwidth of at least 0.12 Hz has been realized, meaning that the time-domain specifications also are met.

fk*360, k=0,1,.. depending on the plotting in Matlab


For completeness, the Frequency Response Functions corresponding to the most extreme parameter variations are shown in Appendix 2 for the three directions x, y and z using the final QFT controllers fi-om Section 5.

frequency [Hz]

Figure 3 Open loop for x for a set of 360 plants

Figure 4 Sensitivity for x, for a set of 360 plants


4 MIMO interaction Analysis

Before we can assume that further synthesis and analysis can be done using SISO approaches it is important to access the amount of interaction present. Among the many interaction tools possible, we use the following procedure:

first design SISO controllers based on the individual SISO plants (i.e. the diagonal plants P(i,i) of the 3x3 plant P we have in this study), see previous section, then apply these 3 SISO controllers to the diagonal plant (Pdiag) and to the MlMO plant P calculate the closed-loop sensitivity function for both cases: s ~ ~ ~ ~ = ( I + c P ~ ~ ~ ~ ) - ' , s=(I+ CP)-', and check the differences if the diagonal entries of the sensitivity function are unchanged, then the plant can be seen as decoupled.

The result for 24 plants (most extreme parameter variations for theta=[O n/2 n]) is shown in Figure 5. Clearly the diagonal entries (containing 2 plots each!) do not show a significant difference between the SISO and MIMO based sensitivity functions Sdiag and S respectively. Hence, we can conclude that further design can be done using a SISO approach.

Figure 5 Sensitivities for MIMO and 3x SISO models

The most extreme MIMO interaction occurs around theta=n/2.


5 QFT design

Before going into details on QFT, first the procedure that will be followed is pointed out: for a chosen grid of frequencies, and a chosen grid of plant parameters, plant FRFs are calculated. For each frequency this results in so-called plant templates (one template results for all possible combinations of plant parameters from the gridded parameter space). One of the plants (FRFs) is chosen as the nominal plant. For each template, i.e. for each frequency, the specification(s) at that frequency idare checked by moving around with the temphie in the gairihase p h e (Wkhols chart). This results in so-called bounds associated with the chosen nominal plant. In fact, we have enlarged the specification bound by the size of the plant template, to make sure that if the nominal plant does fulfil the bound, that the perturbed plants fulfil the original specification. Taking into account the bounds, the loop shaping process can now be done using the nominal plant. Summarizing: QFT boils down to converting the family of plants into one nominal plant, and a family of bounds for which the nominal plant must hold. Loop shaping techniques can then be used straightforward. See also [R4]-[R6] and Appendix 3 for further reading.

5.1 Generation of the templates and bounds

We use the four most relevant parameters to span up the parameter space. The templates will not be very smooth because the variations of the SOTV-model do not represent coefficients in the numerator or denominator of the transfer function, but are far more hidden in the dynamics. One still needs a nice (i.e. 'closed') template, and therefore it is important to make a good choice for the grid in parameter and frequency space. Figure 6 shows a template (i.e. for one specific frequency) where the edges are not good enough. Performing the calculations again, but now with some more plants, gives the result of Figure 7. As can be seen from this figure, there are actually calculated too many plants, that result in inner points of the template. This results in a larger calculation time, but one is now sure that the edges are smooth, and thus the right bounds are being calculated.


Figure 6 Template with not smooth enough edges

Figure 7 Template with smooth edges

Choice of frequencies. Since the number of calculated bounds and templates is typically much smaller than the frequency grid chosen for the calculation of the nominal plant FRF the question is what frequencies to take? One option is to make a set of extreme plants (parameters at their boundaries, Appendix 2) and for that set calculate the poles and zeros. The used approach however is one of trial- and error, choosing first a random set of frequency points (at least containing some of the mode frequencies), and then judge fiom the resulting QFT Nichols plot


whether points should be added. Later on, when the design is finished, it might be usehl to choose other frequency points and check whether the new bounds are still fulfilled with the same controller. This has however not been done in this report.

Table 5 Chosen frequency grid for QFT design

Some compi

Figure 8 Some of the templates of the SOTV. The QFT toolbox works with frequencies in radls instead of [Hz], the shown frequencies in Hz are: [0.3 0.8 1.9576 4.6792 31.8311

In Figure 9 some example bounds are shown for the x-axis, note that the shape of the bounds doesn't depend on any initial controller whatsoever. Plotting all the bounds is of no use because then nothing is visible anymore. Notice that the bounds have continuous lines on the top, meaning the final open loop design has to stay above the indicated line, and dotted lines on the bottom, meaning the open loop design has to stay below the indicated line. The bounds have been calculated using the sensitivity margin of 6 dB only, as this is the most important demand.


Figure 9 Some of the bounds of the SOTV

5.2 Loop shaping

Now that the bounds are computed, one can begin to shape the open loop frequency response L. For robustness, the Gain and Phase margin specifications are met when choosing a Sensitivity Margin of 6dB as discussed in Section 2. The controllers derived in Section 3 will be used as initial controllers. As a starting point for the design process this seems a logical choice, however controller design can be used as an initial controller for the QFT design procedure. This includes e.g. a unity controller but also the results from a LMI or H, design process can be used. This process will be investigated in more detail in WP33, the "Mixed QFT/LMI/H, Design & Analysis".

As expected, for all three directions the controllers turn out to be fulfilling the QFT design requirements (as can be verified for each axis by looking at the controllers in the QFT design environment), meaning that the resulting open loop response is outside the corresponding bounds. For noise-cancellation purposes a first order Lowpass filter is added and to reduce the steady- state error, integral action is added. For the x-axis, several bounds corresponding with Figure 9 are shown with the open-loop response of the nominal plant:


Figure 10 Open-loop response of the nominal Plant in x direction

The design procedure has been repeated for y and z direction, the following list shows all three resulting controllers:

(format: Controller=Gain*Integral action*Lead/Lag*LowPass filter) Clearly, the relevant frequency range is higher for y and z, and phase lead in a wider frequency range is necessary. Since specifications in time domain are met quite easily, a design with low bandwidth, i.e. below the parasitic dynamics, would possibly be favourable for y and z.

For a complete overview of the shape of these controllers and the resulting responses, the reader is referred to Appendix 2. For convenience, a table with some of the controller's specifications has been added:


Table 6 Controller specifications

5.3 Controlier Time domain Performance

The resulting controllers all show a settling time to a step input of approximately 7 seconds (Appendix 2), a sensitivity margin of 6 dB was incorporated in the QFT design procedure. The design thus fulfils the robust performance specifications. Preliminary simulations using the CMA Simulator are provided in the figure below for two engine firings. For a detailed description of the CMA simulator the reader is referred to [WP32] appendix 2.

time (sec.)

Figure 11 Time domain CMA simulation tool [ W 3 2 appendix 21


6 ConcIusions & Discussion

The present document described the main features of the robust control design conducted within the classical loopshaping techniques and QFT design for WP 3 1. The results presented show that simple lead filters already show stabilizing behaviour for all plant models. An analysis of the multivariable features of the SOTV model showed that in the current configuration, interaction is relatively small, so that SISO designs can be applied for the MIMO plant. In the QFT design procedure, the simple lead filters are used as a starting point for the loop shaping procedure and are extended with an integral- and Iowpass filter part. The resulting controllers (all of 3rd order) fulfil the demand of a settling time of 15 secs and have good time domain performance in the f d l simulation model.

The three major drawbacks of the QFT (and FRF loop shaping techniques) are: the fact that gridding in uncertainty and frequency space has to be used, the dependence of the final controller result on the loop shaping skills of the engineer, the use of the Nichols chart is not helpful with respect to dependency on frequency.

However, the major benefits of QFT are: low (fixed) order controllers result, the designer has maximum control of the design process, thereby making it possible to understand the problem difficulty and the problem result, there is no conservatism with respect to the robustness issue.

To our opinion such techniques should always be used as a starting point for design, in order to serve as a benchmark, and to provide insight. In the next phase, if one would like to explore the limits of performance, one might use the QFT designs as initial controllers for H, or a structured singular value based p-analysis and synthesis. Moreover, a thorough robustness assessment (of any controller) can be performed by doing this p-analysis, using the LFT uncertain plant fi-om WP32; this counteracts the gridding drawback of the QFT design method. In the same way, it is possible to use the controllers from optimization techniques and plot them in a QFT Nichols chart for comparison. This coupling of SISO loopshaping design and optimization based design will be the subject of WP 3 3.


Appendix 1 Influence of parameters on Plant FRF (x direction)

Figure 12 Influence of propellant mass

Figure 13 Influence of damping


Figure 14 Influence of Centre of Gravity

Figure 15 Influence of modes (frequency shift)


F i ~ u r e 16 Influence of inertias

Figure 17 Influence of thrust direction


Figure 18 Influence of variation of theta (0-2pi)


Appendix 2 QFT Controllers C,, C, and C, The FRFs in this appendix all result from taking the boundaries of the parameter space in all their combinations, hence resulting in 16 plants. All other descriptions lie in-between and these are thus representative to all plants in the parameter space.

Figure 19 Plants, x axis


Figure 20 Controller C,

Figure 21 Open loop x-axis


Figure 22 Sensitivity function x-axis

Figure 23 Step response of closed-loop system x-axis


Figure 24 Plants y-axis



Figure 26 Open loop y-axis

Figure 27 Sensitivity function y-axis


Figure 28 Step response of closed-loop system y-axis

Figure 29 Plants z-axis



Figure 31 Open loop z-axis


Figure 32 Sensitivity function z-axis

Figure 33 Step response of closed-loop system z-axis


Appendix 3 QFT principles

This section is taken fiom [J.A.C. Meesters: "Quantitative Feedback Theory applied to a Solar Orbital Transfer Vehicle", TUIe DCT report 2003.74, Eindhoven, The Netherlands]. The frequency unit is taken to be radls in this appendix only. Before explaining the main design steps within Quantitative Feedback Theory (QFT), first the usage of Nichols charts is explained. On the horizontal axis the frequency is displayed, on the vertical axis the logarithm of the amplitude is displayed. In the chart itself there are some lines

Figure 34 Example of the nichols chart

The lines are lines of constant closed-loop response (i.e. the complementary sensitivity T=1-S) concerning amplitude and phase. The ones for constant magnitude are labelled, the ones for constant phase all originate at phase -180 and magnitude OdB, while ending vertically downwards. Constant magnitude lines are called M-lines, whereas constant phase lines are called N-lines. Designing a system with the use of loopshaping techniques is time consuming when one designs in the bode-domain, without the use of a computer. For instance when a controller is added with gain 3 dB and phase +90 degrees, the open-loop FRF has to be recalculated for every frequency point. In the Nichols Chart one just has to move the curve upwards with 3 dB and rightwards with 90 degrees. The closed-loop bandwidth can be read from the crossing point between the M-line that indicates 0.707 dB (closed loop) and the open-loop FRF. When a resonance is present, its closed-loop peak magnitude can be read from the M-circle which has the highest amplitude and which is tangent to the open-loop FW. The following example plant is used:

P = k

with k=5, a=0.5 b=5. s2 + a . s + b '


The FRF of the plant is plotted on a Nichols Chart in Figure 35. Just for convenience the Bode plot of the above system is also depicted in Figure 36. As expected the Nichols plot (NC) starts at phase zero and magnitude approximately

20 log,. (i) = 20 log,, ($1 = OdB , and ends up at phase -1 80 and magnitude minus infinity. The

blue dot indicates the maximum peak (open loop). One disadvantage of the NC is that the frequency is not plotted along. It is retrievable however by tracking along the plot with the mouse (when using a computer). Doing so, the blue dot is indicated at w=2.21 radlsec, gain-13.1 dB, phase=-83.5 degrees. Looking at the Bode plot, this information is confirmed.

Figure 35 nichols representation of P

Figure 36 Bode plot representation of P


The QFT methodology makes use of these Nichols charts. The main steps are QFT can be summared as follows. For a chosen grid of frequencies, and a chosen grid of plant parameters, plant FRFs are calculated. For each frequency this results in so-called plant templates (one template results for all possible combinations of plant parameters from the gridded parameter space). One of the plants (FRFs) is chosen as the nominal plant. For each template, i.e. for each frequency, the specification(s) at that frequency, is checked by moving around with the template in the gainlphase piane (Nichois chart). Tnis resiilis in so-called Sowids associated with the chosen nominal plant. in fact, we have enlarged the specification Sound by the size of the plant template, to make sure that if the nominal plant does fulfill the bound, that the perturbed plants fulfill the original specification. Taking into account the bounds, the loop shaping process can now be done using the nominal plant. Summarizing: QFT boils down to converting the family of plants into one nominal plant, and a family of bounds for which the nominal plant must hold. Loop shaping techniques can then be used straightforward.

Determination of the templates

Parametric uncertainty

Suppose there is a plant P:

To give an idea about the size of uncertainty in this system, the bode plot is given in Figure 37 for the following two sets of parameters: [k,a,b]=[10,1,20] and [1,5,30]. All other possible plants lie between these two lines because the preceding two sets of parameters belong to the minimum and maximum plant, concerning magnitude. As can be seen from the figure, the variation of the plants is frequency dependent. In the high and low frequency range there is no difference in phase. Also in the high frequency band there is less variation in magnitude than anywhere else. One basic step in QFT is determining the templates. In a template the variation of the plant is plotted for just one frequency. Take for example w = lradlsec . It is seen from the bode plot that the amplitude varies between -9 and -44 dB, and the phase is between -13 and -48 degrees. There are several ways of plotting such a template. In the Nichols Chart the template is a sort of curved parallelogram instead of a cube (as in the parameter space), as can be seen in Figure 38. This figure indeed has its corners at (-48, -9) being upper-left and (-13,-44) being lower-right.


Figure 37 Bode plot of maximum and minimum plant

Figure 38 Template of P at frequency 1 radlsec.

These kinds of templates now need to be made for an appropriate frequency grid. Which frequencies this frequency grid has to contain is system dependent. When for instance a resonance is present, more frequency points around this resonance should be taken than in the other areas of the relevant frequency range. Furthermore, a good upper and lower bound for the frequency range are the frequencies where phase variations no longer occur. At that point the templates become very thin (a vertical line). In the example under consideration, no resonances

occur, Figure


so one can work with an equally spaced frequency grid. Some templates are shown in 39.

Figure 39 Templates for certain frequencies

This figure indeed shows that there is no phase variation at high and low frequencies. The templates start at phase zero (low frequencies) and end at phase -180 (high frequencies) as expected. Furthermore one can derive an expression for the spread in gain of a phase-invariant template (so high and low frequencies). It is clear that the following holds:

So for s->O this becomes: Pm, = - 1 ' m i , - = -44dB amax bmax 5.30

- ' m a - And Pma, - 10 --= -6dB, and so !EL = 1- 44 - -61 = 38dB

amin . bmin 1 .20 Pmin

Pmax - 'ma* And in the high frequency range: - - - - - 20dB 'min kmin

These results are also clearly visible in the figure. One should always be careful that the templates are closed (if the uncertainty permits it), otherwise errors in the software may occur. The QFT Toolbox [R4] makes its own templates when one is not carefully enough. Varying the right parameters at the same time can close a template. In the example under consideration there are three parameters. It is stressed that it is not


necessary that all possible variations take place before a closed template appears. In practice that would also take too much time. Below is a figure showing the variations that were used to form closed templates. It clearly shows four variations. At variation one for instance gain k is being held at 10 and b at 20, while parameter a is being varied from 1 to 5. Notice that the variation arrows in Figure 40 span the whole parameter space. Taking more than these four variations would only result in points inside the contour of the template (that is already plotted) and thus does not bring along any extra information.

Figure 40 Parameter space

In case of non-parametric uncertainty one usually defines a mean FRF, and then builds such templates so that they include the most outlying FRF fiom the mean FRF. An elliptic template is usually used for this purpose. Because little is assumed to be known about the FRF and its system, the trick of phase invariance at high and low frequency ranges cannot be applied anymore. So indeed the choice for an appropriate frequency range is even more difficult. The templates at other frequencies will generally have the same form, and hence so will the bounds. One can also choose to vary the elliptical shape of the template in the frequency band. Then the bounds will not have the same form.

Generation of the bounds

The next step in QFT is to translate the templates into bounds on the Nichols Chart. For this it is necessary to choose a nominal plant. Which of the plants is chosen to be the nominal one is irrelevant, it can even be one that is outside the parameterspace. For convenience however the maximum plant is chosen to be the nominal one in this subsection. As mentioned before, when one is dealing with a non-parametric uncertainty the nominal plant is already chosen (the mean). For now we only look at parametric uncertainty. The nominal plant corresponds in each template to a certain point. In this case that is the upper left point. From demands on stability and for instance disturbance rejection certain circles and lines will appear (M-lines). One such circle is shown in Figure 41. Together with this circle a template is shown. Obviously, this template has the wrong location because some of the plants are situated inside the circle. These are marked by the black area. When the template is moved in such a way that its upper left point is tangent to the circle (simply by multiplying with a gain less than zero), one (in its mind) takes a pencil and puts its point through the upper left point of the template. Had one chosen the nominal plant to be the lower right point of the template, one had to put the point through the lower right point of the template when the upper left point was tangent to the circle. When this point is made, one moves


the template vertically (simply by a gain bigger than one) in such a way that the location of the template concerning phase will be the same. Now the lower left comer will be tangent to the circle. Again put a pencil through the upper left comer (when this one was chosen to be the nominal plant). Now there are two points, belonging to one phase location of the template. The next step is to take some other phase locations and do the same trick. Be careful not to trespass the circle with any of the points on the template. When changing the phase, other points on the template will be tangent to the circle. In this way one gets several points on the Nichols Chart (two points for every phase location). When these points are connected by a curve, the bound belonging to the frequency of that template is fiiiished. Next one has to go through the same routine for all the other templates. The result is shown in Figure 42. The bounds also become isomorphic when looking at high and low frequencies. This is because in that region the templates are straight lines.

Figure 41 Generating bounds


Figure 42 Bounds

For disturbance rejection also bounds can be generated. However for these disturbance rejection demands normally there are no circles available but lines. Therefore the bounds will also be lines. The QFT toolbox provides eleven types of bounds that can be generated. It is pointed out that stability requirements like Gain Margin (GM) and Phase Margin (PM) are easily indicated on the Nichols Chart. The main results are listed below:

with y representing an M-line. So by requiring a PM or GM, the M-line to stay out of can be calculated directly. This shows again another advantage of the Nichols Chart.

In this report the only bounds of concern are those of robust stability. Once all bounds have been plotted on the Nichols Chart, one can throw away the bounds that are not dominant (of course only at the same frequency). At this point one has reached the situation where instead of all the plant FRFs plotted on the Nichols Chart, one has now one nominal transmission, and some bounds to stay out of.

Loop shaping

After this, one is free to shape the open-loop FRF as close to the bounds as possible, taking attention to the complexity of the resulting controller.


In the QFT toolbox there is a file called Zpshapem. With this file one can design the desired open-loop in the Nichols Chart using 8 filter elements. All the eight elements are listed below.

Real pole

Real zero

Complex pole

Complex zero

Super Pd

Integrator (n>O)

or differentiator (n<O)

Lead or lag

Notch

Loop shaping in the Nichols Chart is now possible filling in the numbers for each parameter required by each element, or more advanced by selecting an element, and then moving the FRF line to the location where one wants it to be, with use of the mouse.

MU robustness analysis

MU analysis toolbox version: 3.0.7 (R13) MATLAB Version 6.5.0.180913a (R13)

Remark: The MU robustness analysis has been performed on four candidate controllers resulting fiom reports [WP3 1 J and [WP32]. The analysis has been performed using the mentioned MU analysis toolbox, the theory related to using these tools is not explained in this report and considered to be known. For more information about using the MU robustness analysis, the reader is referred to [MU].

For the MU analysis to take place the dynamic system should be put into the following LFT form:

Figure 1. w] pag 161

Figure 2. C represents the (.

Where

i %I 0 0

A = 0 so, 0 0 0 432

5 ) system matrices as explained

u = ( 4 Y = ( Y ) P =

nd K the controller under inspection.

To obtain the system matrix M simply take out all w to z connections and eliminate the y and u connections by using the relation u=K.y. Another (automated) approach is to use the Mu analysis toolbox command s y s i c .

The C dynamics can be presented in state space notation as follows:

Leading to a so-called SYSTEM notation:

The nominal plant (e.g. the "mean" plant without parameter variations) is represented by the SYSTEM description

And has a frequency response function:

Frequency (tk)

Figure 3. A nominal Plant Frequency Response (for theta=%)

The result of the MU analysis is a calculation of a upper and lower bound for the MU values over a frequency domain of choice. The MU analysis will be calculated for the same frequency domain as can be seen in figure (3). As candidate controllers, the H,, the Hmred6upg3, the LMI

controller and the QFT controller have been selected. The MU analysis will take place for only the x-direction and only for variations in the InertiafCoupling Coefficients since a LFT description of the SOTV has already been constructed for the LMI controller design process. Variations in theta are also considered though on a limited scale. For various angles of theta, LFT descriptions have been constructed for the MU analysis to take place.

Results

MU bounds [real perturbations] 0.12,

- - theta 0

theta 30 theta 60 theta 90 theta 120 theta 180

Frequency (Hz)

Ha (notice different y-axis scaling) MU bounds [real perturbations]

0 0 5 1

0 10'

Frequency (Hz) 1 ; for

LMI Figure 4. MU bounds for the Different Controllerr

theta 0 -- theta 30

theta 60 theta 90 theta 120 theta 180

MU bounds [real perturbations]

Frequency (Hz)

0 05

0 045

0 04

0 035

- 0 0 3 - a 2 0 025 a

Z 0 0 2 -

Hmred6upg3 MU bounds [real perturbatlons]

- - theta 120

0 035 theta I80

m

-

-

-

-

i I

e h I t

F I

Frequency (Hz)

QFT InertiaICoupling coefficients variation

The calculated MU values never cross the +I border indicating instability. No plants exist that could force this considered closed loop system into instability for all presented controllers. Note that the lower bound can not be calculated accurately (thus is zero) because of singularity issues involved with calculating this bound using only real valued parameters in the A block (as is the case). For more details on this issue see [MU] pag. 204 and further. The highest (though still very low) mu upper bound values can be found for theta angles of 90 and 120 degrees. The behaviour of the closed loop systems near the peak values of the mu analysis will be investigated in more depth.

MU bounds [real pertubations deta~led region]

theta 30 theta 60 theta 90 theta 120 theta 180

0 08

Frequency (Hz)

H, (notice different y-axis scaling) MU bounds [real perturbatlons, detaded reglon]

theta 60 theta 90 theta 120

0 035 theta I80

Frequency (Hz)

LMI Figure 5. MU bounds for the Different Controllers f

MU bounds [real pertubat~ons, detailed region] 0 05

1 theta 0 /

theta 60 theta 90 theta I20 theta 180

Frequency (Hz)

H,red6upg3 MU bounds [real perturbations, detailed region]

theta 60

theta 120

0 1 i , I

10"' 1 oO lo0 ' 10"' Frequency (Hz)

QFT Inertia/Conpling coefficients variation, detailed region

The lower bounds are again zero. A trick to investigate the behaviour of the lower bound is to use very small complex perturbations of dynamically resembling systems. For more information see [MU] pag. 206 and further. It comes down to replacing the original dynamic system of Figure 1 with the following:

Which leads to a modified A block in which each varying parameter 6, is replaced by a parameter of structure 6R + a2&. By varying the parameter a the influence of the complex parameter 6c can

be varied and it can be proven that as a approaches zero, the calculated MU bounds converge to the original MU bounds of the system corresponding to figure 1. To investigate the peak value near 1.4 Hz, the MU bounds are again calculated for a frequency region around this frequency. a2 Values are mentioned near the associated figures.

Mu bounds [small complex perturbatlons 1 %I

theta 30 theta 60 theta 90 theta 120 rhera 180

0 08

Mu bounds [small complex pelturbatlons 1 %I

theta 30 theta 60

0 04 theta 99 theta 120

0 035 theta 180

Frequency (Hz)

Hm (notice different y-axis scaling) Mu bounds [small complex perturbatlons 1 %]


0 035 theta 180

Frequency (Hz)

Hmred6upg3 Mu bounds [small complex perturbations 1 %I

-

theta 60 theta 90 theta 120

LMI Frequency (Hz)

QFT Figure 7. MU bounds for the Different Controllers for Inertidcoupling coefficients variation, small complex perturbations to investigate lower bound behaviour (1% variation in a').

Mu bounds [small complex perturbat~ons 5 %I


0 08

Frequency (Hz)

Hm (notice different y-axis scaling) Mu bounds [small complex perturbat~ons 5 %I

0 05


Frequency (Hz)

Mu bounds [small complex perturbations 5 %I

Frequency (Hz)

Hmred6upg3 Mu bounds [small complex pertuI3atlons 5 %I

0 05


Frequency (Hz)

LMI I QFT Figure 8. MU bounds for the Different Controllers for InertidCoupling coefficients variation, small complex perturbations to investigate lower bound behaviour (5% variation in a').

As a result, the higher the complex influence the higher and wider the peak MU upper bound. The lower bound ascends rapidly towards the upper bound, with larger complex influence, indicating that the lower bound for the pure real case can most likely also be found near the real upper bounds. As the maximum value of the lower bound indicates the most unstable closed loop dynamics the MU toolbox could find, the resulting dynamic system is a worst case scenario for the presented controllers. The associated worst case InertidCoupling Coefficients for the various controllers and lower bounds presented in figure (8) are presented in figure (9):

Worst Case plants for mrylng theta, small complex perturbailon 5 %I 10

Modesh~R w2 k 7 Modeshlflw3

5 -

-20 0 20 40 60 80 100 120 140 160 180

Theta (degrees)

Worst Case plants for mrylng theta, small complex perturbailon 5 %I

-40 0 20 40 60 80 100 120 140 160 180

Theta (degrees)

LMI Figure 9. InertidCoupling coefficients variations inc

L

lic

Worst Case plants for mrying theta, small complex perturbation 5 %]

-40 0 20 40 60 80 100 120 140 160 1%

Theta (degrees)

Hmred6upg3 Worst Case plants for mlying theta, small complex perturbation 5 %I

Theta (degrees)

QFT ating the worst case systems for a 5% complex perturbation.

The resulting worst case parameters for the H, controller differs from the other controllers, that are fairly consistent in their resulting parameters. Note that most parameters lie beyond the physical boundaries of 4 20% variation. The resulting worst case parameters for the 1% complex variation is not presented because the lower bounds do not have a distinct peaking, the lower bound for the pure real case is non- existent and thus does not produce a worst case system.

References

[WP3 11 WP 3 1 Technical Note: QFT Robust Control Design [WP32] WP 32 Technical Note: LMI / H, Robust Control Design [Ref2_TNWP32] Addendum to [WP32] [MU] MATLAB p-Analysis and Synthesis Toolbox, Gary J. Balas, John C.

Doyle, Keith Glover, Andy Packard, Roy Smith, User's Guide ver. 3

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