### Eigenstructure Assignment for Control System Design

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Since the vehicle is an unstable plant, one cannot use the open loop transfer functions gain and phase margin as a stability performance parameter, as specified in Van de Vegte The vehicle flying qualities were assessed in terms of tracking performance, decoupling performance and control effort required for each manoeuvre. In order to access the flight control law performance the decoupling responses are reported in figures 3 , 4 and 5 for the vehicle working with the decoupling control law and working without the decoupling control law. In figure 3 the roll-attitude time response for a step manoeuvre of yaw and pitch are reported.

The yaw-attitude time response following a step manoeuvre of pitch and roll are reported in figure 4 and the pitch-attitude following a step manoeuvre of roll and yaw are reported in figure 5. It is clear from these figures how good is the performance of the eigenstructure control law with respect to decoupling the modes. The only disadvantage is with respect to the roll performance decoupling.

However this is not so bad, since the main objective in the VLS case is the decoupling of yaw and pitch, what is completely obtained looking for figures 4 and 5. One can try to reduce, for example for 0. In figure 3 it can be noticed that the maximum bank angle achieved is about 0. In this case the control law without decoupling was considered the same as with decoupling making the coupling gains as zero into the implementation.

## Eigenstructure Assignment for Control System Design

For the yaw case the maximum yaw attitude achieved is about 0. For the pitch case the maximum pitch attitude achieved is about 0. Obviously one can notice how good has been the decoupling between the yaw and pitch planes, that is the main concern in the case of a satellite launcher vehicle. In figure 6 there is the pitch-attitude response for a step manoeuvre of pitch-attitude, in figure 7 there is the yaw-attitude response for a step manoeuvre of yaw-attitude and in figure 8 there is the roll-attitude response for a step manoeuvre of roll-attitude.

## Rocket tracking and decoupling eigenstructure control law

From these figures it is possible to notice that the eigenstructure control law performance concerning tracking is also very good. From this table, as also from the above figures it can be noticed that the responses are about the same in the three channels. From figure 9 it can be noticed that the maximum control required is about 0.

It must be remembered that in the real flight vehicle, step manoeuvres are not performed, in fact figures 14 , 16 and 18 show actual flight manoeuvres applied to this vehicle, which are very smooth, as one can notice. In figures 10 and 11 the control effort required for decoupling is found to be a maximum of 0.

Certainly these values can be considered very low, regarding that the maximum allowed is 3. It is known that for a good performance robustness the minimum singular value must be large at low frequencies, and for a good stability robustness the maximum singular value must be small at high frequencies. In this system,. The low frequency boundary is given in terms of the minimum singular value s min being large, and the high frequency boundary is given in terms of the maximum singular value s max being small. These performance specifications were derived in terms of the maximum s max and minimum s min singular values of the loop gain GK j w.

The boundaries were derived using the H-infinity norm in the frequency domain, and they are plotted in figures 13 and The boundaries for a good design are reported in figures 13 and 14 , as LF for the lower frequency boundary and HF for the high frequency boundary. The high frequency boundary will guarantee good stability due to parameter uncertainties, that can be additive or multiplicative, and also due to unmodeled flexible and vibrational modes. In figure 13 the singular values of the loop gain are plotted.

From this figure it is possible to notice that at low frequencies the minimum singular value of the loop gain is large enough and so a good performance robustness is achieved. At high frequencies the maximum singular value of the loop gain is small enough and so also a good stability robustness performance will be obtained. One can notice that the maximum singular value is not close to the other two singular values minimum in this figure. Certainly the maximum singular value is that corresponding to the roll mode, and the other two corresponding to the pitch and yaw mode, due to the vehicle symmetry.

In figure 14 the singular values plots for the closed loop transfer function is plotted. In this figure one can notice, that the three singular values are very close to each other throughout the frequency range, except at high frequencies, where they are not very close, and so the speed of the responses will be nearly the same in the three channels of the system, that is, pitch, yaw and roll, as showed in figures 6 , 7 and 8. Again the singular value not so close to the other two is the one corresponding to the roll mode, and one can notice in the time history response that the roll response differs a little bit from the yaw and pitch responses.

To assess the performance of this design the obtained gains were used in the vehicle non-linear model, that is, the model described by equations 1 to 8 , which are varying in time, since all the parameters are function of the flight time, that is, the trajectory of the satellite launcher. The gains were maintained fixed during the simulation time, and the responses following simultaneous manoeuvres, that are actual vehicle manoeuvres, are reported in figures 15 , 17 and 19 , with the respective manoeuvres in figures 16 , 18 and In figure 15 there is the roll attitude response following a simultaneous pitch and yaw manoeuvre, showed in figure It can be noticed that the maximum roll attitude is about 0.

In figure 3 the maximum roll attitude achieved was about 0. It was expected a maximum of 0.

## 525.777 - Control System Design Methods

This performance can be improved by making use of gain scheduling. In figure 17 there is the yaw attitude following a simultaneous roll and pitch manoeuvre showed in figure It can be noticed that the maximum yaw attitude was around 0. Here the performance was deteriorated, however even with this loss of performance, the experience shows that 0. Certainly, a better performance can be obtained with gain scheduling.

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In figure 19 there is the pitch attitude following a simultaneous roll and yaw manoeuvre showed in figure In this case the maximum pitch attitude was about 0. In view of this the performance is not so affected, and one can say that 0. Again the use of gain scheduling can improve the controller performance. It can be said that the eigenstructure control law offers a very good performance in all aspects. The non linear responses showed were obtained with fixed gains, so in fact these responses can be improved by using gain scheduling with time, as usual in this kind of system.

The singular value analysis has also shown that the control law robustness with respect to performance and stability is quite good. From table 1 it can be noticed that it is possible to simplify the control law implementation making some of the small gains as zero and maintaining a good performance. From figures 3 and 15 it was noticed that the roll response is the one that presents more sensitivity to coupling effects, this is due to the numerical inaccuracy in the solution of equation 27 for the roll mode, as also it can be improved with a new choice of the non zero component of the associated eigenvector, as explained there.

However, this fact is perfectly acceptable since the main objective was the decoupling of yaw and pitch responses, which was satisfactory obtained. Email or Username Forgot your username? Password Forgot your password? Keep me signed in. No SPIE account? Create an account Institutional Access:. The alert successfully saved. The alert did not successfully save.

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Liu and Patton offer a uniquely integrated introduction to eigenstructure assignment theory and techniques for multi-input multi-output control system design. Features include: Introduction to the Eigenstructure Assignment Toolbox for use with MATLAB examples available via the Internet providing engineers with a powerful set of tools for the design of multivariable systems Broad coverage including the principle of eigenstructure assignment, basic, insensitive, robust and multiobjective eigenstructure assignment for multirate sampled-data systems, descriptor systems and fault detection systems Description of the majority of known eigenstructure assignment methods for both state and output feedback control offering the reader a concise reference Combination of time-domain and frequency-domain performance specifications for robust control design Postgraduates and researchers studying control engineering will appreciate the combination of mathematical theory and practical issues.

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