Rocket tracking and decoupling eigenstructure control law

Rocket Tracking and Decoupling Eigenstructure Control Law

P. Oliva

W. C. Leite Filho

Instituto de Aeronáutica e Espaço (IAE) - Divisão de Sistemas Espaciais, 12228-904, São José dos Campos, SP – Brasil

[e-mail]

A control law was designed for a satellite launcher ( rocket ) vehicle using eigenstructure assignment in order that the vehicle tracks a reference attitude and also to decouple the yaw response from roll and pitch manoeuvres and to decouple the pitch response from roll and yaw manoeuvres. The design was based on a complete linear coupled model obtained from the complete vehicle non linear model by linearization at each trajectory point. After all, the design was assessed with the vehicle time varying non-linear model showing a good performance and robustness. The used design method is explained and a case study for the Brazilian satellite launcher ( VLS Rocket ) is reported.

Keywords: Attitude Control, Decoupling, Eigenstructure Assignment.

Introduction

In this work the eigenstructure assignment method is applied to a satellite launcher vehicle model following the guidelines given in D’Azzo & Houpis (1988), Lewis (1992) and Lewis & Stevens (1992). An interesting method is also given in Blackelock (1991) , and other useful references for decoupling using eigenstructure are given in Shapiro et al. (1983). In all these references the design is applied to an aircraft model with an input actuating in two states, for example aileron input influencing both roll and yaw responses. In the studied model each input influences only its related model, for example roll input influences only roll responses. However the coupling of the responses is due to the inertial terms as also due to the non linear terms. In order to design the control law it is necessary to have the vehicle parameters as a function of the flight time as also the trajectory parameters (roll attitude, pitch attitude and yaw attitude) as a function of the flight time. The main objective of the design is to obtain decoupling in yaw, when there is some roll and pitch manoeuvre and decoupling in pitch, when there is some yaw and roll manoeuvre. The decoupling in roll, when there is some yaw and pitch manoeuvre is not the main concern of the design, however it can be also achieved. It is good to use a control law that decouples the modes, because otherwise, one needs to design the vehicle trajectory in a way that the vehicle performs one manoeuvre at time, that is the vehicle will not do combined manoeuvres, what certainly deteriorates the performance.

Vehicle Mathematical Model

To design the control law it is necessary to use a complete linear coupled model, which can be obtained from the linearization of the non linear equations of the vehicle as developed in several well known references as McLean (1990) and Greensite (1970) , for example, and here given by

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

In figure 1 there is a view of the flight vehicle studied in this work. It is a solid rocket engine satellite launcher, with four stages. So there is no thrust control. The figure shows the nozzles without deflection ( at the right ), with a deflection angle ( at the centre ) and the plan view of the four nozzles of the first stage ( at the left ).

The following notation was used in equations (1) to (8) for the vehicle mathematical model :

Note that the equation for the variation of linear speed in the x-body axis is not used, since it is not necessary because there is no thrust control. It is also useful to say that in the problem addressed the vehicle trajectory is designed in a way that the yaw attitude angle y does not reach 90° or -90° , and so equation (6) can be used without any problem.

The linear coupled model can be obtained as also described in Lewis & Stevens ( 1992 ). This model has been obtained from the linearization of these non linear equations for the states v₀ , w₀ , p₀ , q₀ , r₀ , f ₀, q ₀ , y ₀ , and for a specific steady-state velocity U₀. The linearization procedure is very well explained in Greensite ( 1970 ). The linear model can be represented as follows

(9)

with the A and B matrices given by

(10)

(11)

These matrices were obtained by linearizing the non-linear model as described in Oliva & Leite (1998), and the coefficients of these matrices are given by

where the parameters are:

Cn_a

derivative of normal aerodynamic force with respect to angle of attack ( a )

Cl_p

derivative of roll damping moment with respect to roll rate ( p )

Cm_q

derivative of pitch or yaw (damping) moment with respect to pitch-rate,q or (yaw-rate, r)

dynamic pressure

S_ref

vehicle reference area

D_re

vehicle reference diameter

m

vehicle mass

d₁

control moment in roll arm

I_xx , I_yy , I_zz

moments of inertia with respect to the x, y and z axis, respectively

la_x

aerodynamic moment arm

lc_x

control moment in pitch and in yaw arm

F_E

thrust force

F_coy ,F_coz

y and z axis coriolis force

M_coy ,M_coz

y and z axis coriolis moment

Theoretical Background

The vehicle linearized model can be described by equation (9), where A is a ( n x n ) matrix, B is a ( n x e ) matrix, u is a ( e x 1 ) vector, and x is a ( n x 1 ) vector, with the output given by

(12)

with y an ( k x 1 ) vector, C a ( k x n) matrix, E a (t x n) matrix, F a [(k-t) x n ] matrix, and x a ( t x 1 ) vector as the outputs that are going to track the ( t x 1 ) input vector x_ref. . In order to maintain controllability it is necessary that k ³ t . The outputs y_i that must be controlled, i = 1,2,...,t , must be the first t elements in the output vector y. If necessary, the output vector must be reordered, so that, x = E x , contains all the controlled outputs. The objective of the feedback controller is to make the output vector x to track the command input x_refand so in the steady state x = x_ref. In order to design such a tracker an error vector is included into the model as

(13)

and so the composite open loop system is given by

(14)

(15)

with the new matrices

(16)

(17)

(18)

(19)

(20)

so the composite model is given by

(21)

The control law will be given by

(22)

and the closed loop equation will be given by

(23)

The objective is to assign both a closed loop eigenvalue spectrum

(24)

and an associated set of eigenvectors

(25)

which are selected in order to achieve the desired time response characteristics. So the closed loop eigenvectors are related by the equation

(26)

and this equation can be write as

(27)

where v_iis the eigenvector associated with the l _i eigenvalue and

(28)

so that

(29)

where

(30)

and

(31)

For the design case studied the E matrix is given by

(32)

the reference vector is given by

(33)

So the composite state vector will be given by

(34)

It is possible to notice from this model how the vehicle is completely coupled in the three axis. In figure 2 there is a block diagram of the vehicle with the control law. In this figure the x_ref vector represents the reference vector defined by equation (33).

The x_a state vector represents the vector defined by equation (34). The vector of gains G_z , G_y and G_rthat appear in this figure are rows of the K matrix showed in equation (29), and they are the following:

(35)

Flight Control Law Design

In order to design the flight control law it is necessary to solve equation ( 27 ) and then equation ( 29 ) to obtain the feedback gain vector K. To start the design it is necessary to choose the closed loop eigenvalues ( l _i ) . Then for each selected eigenvalue ( l _i ) introduced into equation ( 27 ) we will get 11 equations for 14 unknowns ( v₁ , ... , v₁₁, u₁ , u₂ , u₃ ), so for the solution of this system it is possible to choose arbitrarily three eigenvectors ( v_i) and then to obtain the remaining eigenvectors and also the u_i. Using this degree of freedom one can choose some of the eigenvector components as zero in order to obtain decoupling of the modes.

Eigenvectors Selection For Decoupling

It is necessary to say hat the closed loop eigenvectors must be linearly independent in order to carry on the calculation of V^-1 in equation (29). Also it is clear that if a zero is chosen for an eigenvector component, there is no energy injection into this component.

For the eigenvalue associated with the ( q , q ) mode one can take the following eigenvector

(36)

where the vi’s will be complex numbers, and it is possible to notice that the y and f responses associated with this eigenvector will be decoupled because the corresponding rows were selected as zero. In this equation and from here on the symbol j will mean the complex number, that is, j = Ö -1 . In this case the vector u_i will be given by

(36-a)

Then introducing ( 36 ) and ( 36-a ) with the associated eigenvalue into equation ( 27 ) , that it will be solved to obtain v₁and u₁ . Note that the components relative to the states y and f have been chosen as zero, since the desired feature is that these two sates remain decoupled from the ( q , q ) mode. That is, energy will not be injected into these two components. In this case the third choice was for the component relative to the w state, since the w, q and q states are all relative to the pitch plane motion. So, energy will be injected into this component, what was already expected. The value 1+ j was chosen as a first attempt, if the system does not work well with this choice, then one can go back and try another value for this component. If there are large excursions in the w response, this is probably due to the fact that the used value has introduced too much energy into this component, and it will be necessary to reduce the magnitude of these, say for example 0.5 + 0.5 j , and so on.

Here it is useful to say that although not performed in this work, one can also try to do the design selecting the following components as zero:

(a) p and r component in v₁

(b) p and y components in v₁

(c) r and f components in v₁ .

These choices are due to the fact that it is also desirable to have no coupling with these states.

For the eigenvalue associated with the ( y , r ) mode one can take the following eigenvector

(37)

where the vi’s will be complex numbers, and it is possible to notice that the q and f responses associated with this eigenvector will be decoupled because the corresponding rows were selected as zero.

Here the components relative to the q and f states were selected as zero, since the objective is to have no coupling between the y manoeuvres with q and f . Again the third choice was taken as 1 + j for the v component, since v , r and y are all in the same plane, yaw plane, and it is desired to have the v motion coupled with the y and r motions, as expected in the original flight vehicle dynamics. In this case other possible choices are:

(a) p and q components as zero

(b) p and q components as zero

(c) q and f components as zero.

For the eigenvalue associated with the (f , p ) mode one can take the following eigenvector

(38)

where the vi’s will be complex numbers, and it is possible to notice that the y and f responses associated with this eigenvector will be decoupled because the corresponding rows were selected as zero.

In this case notice that the components relative to the states q and y was taken as zero, since it is the objective to have no coupling between a f manoeuvre and the q and y responses. In this case the value 1 + j was taken for the p component, since p and f are both in the same plane, roll plane. In this case other possible choices that can be studied are the following:

(a) r and q components as zero

(b) r and q components as zero

(c) q and y components as zero.

For the eigenvalue associated with the ( w , v ) mode one can take the following eigenvector

(39)

where the vi’s will be complex numbers, and it is possible to notice that the q and y responses associated with this eigenvector will be decoupled because the corresponding rows were selected as zero.

Notice that taking the components associated with q and y as zero there will be no energy input into these components when there is manoeuvres that affect the w and v states, that is, angle of attack and sideslip respectively. The value 0.5 + j 0.5 was taken for the f component in order to put a low energy value into these mode. Again, if this choice was not satisfactory, one must return and carry on the calculations with a new choice. For this case other possible choices are the following:

(a) q and f components as zero

(b) y and f components as zero

(c) q and r components as zero.

Of course the eigenvector v₂will be the complex conjugate of v₁, as also the v₄ will be the complex conjugate of v₃and so v₆ the complex conjugate of v₅ and v₈the complex conjugate of v₇ .

The eigenvector associated with the eigenvalue corresponding to e_q can be taken as

(40)

again it is possible to notice the decoupling objective in this choice. Note that the components associated with y and f was chosen as zero, since the objective is no coupling between the pitch plane and the other two planes, yaw and roll. The value 1 was taken for the q component, since it is in the same plane as e_q .

The eigenvector associated with the eigenvalue corresponding to e_y can be taken as

(41)

again it is possible to notice the decoupling objective in this choice. Here note that the zero was allocated to the components associated with q and f since the objective is zero coupling between the yaw plane and the other two planes, roll and pitch. The value 1 was taken for the y component, since it is in the same plane as e_y .

Finally the eigenvector associated with the eigenvalue corresponding to e_f can be taken as

(42)

again it is possible to notice the decoupling objective in this choice. Obviously the components relative to q and y are chosen as zero in order to decouple the roll plane from these two other planes. The value 1 was allocated to the component associated with f because it is in the same plane as e_f .

With this choice of eigenvectors it is then possible to solve equation ( 27 ) for each v_iand u_iand then obtain the vectors U and V for the calculation of the feedback gain K.

Case Study

Taking the data of the VLS example vehicle the following matrices, ( flight conditions at 8 seconds of flight ) can be used

(43)

(44)

and taking the following closed loop eigenvalues,

l q _,q = -1.7± j 1.9 l y _,r = -1.9± j 1.8 l f _,p = -1.3 ± j 1.2 l _w,v= -0.0221 ± j 0.0006 l _eq = -4.4 l _ey = -4.0 l _ef = -25

with the associated eigenvectors given by ( 36 - 42 ) it is possible to solve equation ( 27 ) for each eigenvector in order to obtain U and V. After this it is possible to obtain the feedback gain K for the system. To choose the closed loop eigenvalues a simplified pre-design was carried out as described in Oliva & Leite (1998). This simplified design is carried with three decoupled models, that is:

a model for the pitch plane with the states : w , q , q and e_q

a model for the yaw plane with the states : v , r , y and e_y

a model for the roll plane with the states : p , f and e_f

This design can be performed by any applicable method to the problem, in order to obtain suitable closed loop poles, and then one can use these closed loop poles in the eigenstructure design. In Oliva & Leite ( 1998 ) this pre-design was performed by the linear quadratic ( LQ ) method. So, this pre-design only takes into account the tracking performance, but not the decoupling performance.

Control Law Performance

The gains obtained by the eigenstructure design are reported in table 1. In this table, column 1 is the G_z vector, column 2 is the G_y vector and column 3 is the G_r vector, as in figure 2. From this table it can be noticed that some of the gains are very small, and can be dropped out in the control law implementation, which simplifies the implementation work. This simplification does not deteriorate the system performance.

It is clear how the gains of the eigenstructure design will be able to offer a good performance with respect to control effort and control rate effort from the magnitudes of these gains, since a gain magnitude lower than 3 was expected, based on past designs and the actual gains used in the VLS design, that works without decoupling control law. In table 2 the gain margin ( GM ) and phase margin ( PM ) for the closed loop transfer functions are reported.

It can be noticed that a good gain margin ad phase margin were achieved with this design. As a rule of thumb, Stevens & Lewis ( 1992), a phase margin greater than 60° is required to obtain a good loop transient response, and this should be accompanied by a gain margin at least greater than 6 dB, and if possible greater than 15 dB. Regard the phase margin obtained as infinity for the pitch and yaw channels, that is quite good and is reported in Wolovich ( 1994 ) in the Loop Goals Chapter. 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 ( 1990).

Flying Qualities Performance

The vehicle flying qualities were assessed in terms of tracking performance, decoupling performance and control effort required for each manoeuvre.

Decoupling Performance

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. The roll decoupling can be improved with a redesign of the control law, by carrying on a new choice for the eigenvector component taken as 1 + j . One can try to reduce, for example for 0.5 + 0.5 j, and so on, until a desired performance is achieved.

In figure 3 it can be noticed that the maximum bank angle achieved is about 0.05° for the control law with decoupling and about 0.02° for the control law without decoupling. In this case the control law without decoupling was considered the same as with decoupling making the coupling gains as zero into the implementation. For the yaw case the maximum yaw attitude achieved is about 0.06° for the case without decoupling and 0.0025° for the case with decoupling. For the pitch case the maximum pitch attitude achieved is about 0.06° for the case without decoupling and 0.002° for the case with decoupling. 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.

Tracking Performance

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. From these figures it is possible to notice that the eigenstructure control law performance concerning tracking is also very good.

From these figures the following table can be made.

From this table, as also from the above figures it can be noticed that the responses are about the same in the three channels. The performance parameters are very close to those given by the traditional control law design without decoupling, as reported in Oliva & Leite (1998). In the above table the settling time is obtained when the response reaches ± 2% of the final value.

Control Effort

The control effort required to perform a pitch reference step manoeuvre of 1° is showed for the three channels in figures 9, 10 and 11.

From figure 9 it can be noticed that the maximum control required is about 0.6° for a very severe manoeuvre ( step ), while the maximum control effort allowed for this system is 3.5° . 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 0.2° in yaw and 0.05° in roll. The maximum control rate effort in this manoeuvre was found to be 3°/sec , which also is a very good value, since it is for a step manoeuvre. Certainly these values can be considered very low, regarding that the maximum allowed is 3.5° , and the case is for a step manoeuvre.

Singular Value Analysis

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. These frequency domain performance specifications were derived in Stevens & Lewis (1992) considering the general control system of figure 12 . In this system,

r is the command input s is the error signal u is the plant input d is the disturbance input z is the system output n is the measurement noise

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(jw). Selecting s_min(GK(jw))>>1 for w £ w_d where d(s) and r(s) are appreciable, the characteristic of performance robustness is guaranteed. By the other side, selecting s_max(GK(jw ))<<1 for w ³ w_n will guarantee the performance with respect to measurement noise. The boundaries were derived using the H-infinity norm in the frequency domain, and they are plotted in figures 13 and 14. The complete derivation is described in Stevens & Lewis (1992), Chapter 6. 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.

Non Linear Performance

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 ²⁰.

In figure 15 there is the roll attitude response following a simultaneous pitch and yaw manoeuvre, showed in ^{figure 16}. It can be noticed that the maximum roll attitude is about 0.5° . In figure 3 the maximum roll attitude achieved was about 0.05° for an input of 1° in yaw and 1° in pitch. It was expected a maximum of 0.4° , here ( 5% of the maximum input ), and so the performance is not so bad, just 0.1° above. 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 18}. It can be noticed that the maximum yaw attitude was around 0.1° , when in the linear case was around 0.0025° ( 0.25% ), so the expect value was about 0.03° . Here the performance was deteriorated, however even with this loss of performance, the experience shows that 0.1° is a very low coupling for a manoeuvre of 12° in roll and -8° in pitch. Certainly, a better performance can be obtained with gain scheduling.

In ^{figure 19} there is the pitch attitude following a simultaneous roll and yaw manoeuvre showed in ^{figure 20}. In this case the maximum pitch attitude was about 0.05° , while in the linear case was about 0.002° ( 0.2% ), and so the expected value here would be 0.03° . In view of this the performance is not so affected, and one can say that 0.05° is a very good value for a 12° in roll and 4° in yaw. Again the use of gain scheduling can improve the controller performance.

Comments and Conclusions

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 design was performed with the help of the MATHEMATICA (1996) software for the solution of equation ( 27 ) and with the help of MATLAB (1987) for the control law gains calculation, system simulation and system performance analysis. The design procedure is very tedious due to the necessary work to solve equation (27) using MATHEMATICA. 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.

Manuscript received: April 1999, Technical Editor: Paulo Eigi Miyagi.

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Publication Dates

• Publication in this collection
  18 Dec 2000
• Date of issue
  2000

History

• Received
  Apr 1999