H infinity control for launchers

8/9/2019 H infinity control for launchers

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Control of Launchers

Claudia Valeria Nardi, Marc Alomar

5th of February, 2015

Abstract

In this report we synthesize an H ∞ controller for a space launcher of the ARIANE 5 class. We only consider

the control of the pitch angle during the atmospheric ascent trajectory, and limit the study to the rigid launcher

model. In the first part, we develop a state-space representation of the launcher, and perform a frequency and

time domain analysis of the system. We conclude that the open-loop system is unstable. In the second part, we

synthesize a robust controller using the H ∞ technique. We build the augmented model, and test a first set of

weight functions. The closed-loop system is stable, showing good gain and phase margins. We explore the effectof changing the weight function parameters, and study the effects on p erformance and robustness. Finally, the

controller is fine-tuned to get the best compromise between performance and robustness.

1 Rigid Launcher Model

In this section we build two rigid launcher models: one for the time domain simulation, and one for the H ∞ synthesis.To build a linear, time-independent model, we assume that there is a low variation of mass and inertia, that thelaunch vehicule is symmetric and rigid, and that the system is stationary (coefficients are time-independent).

1.1 Time domain simulation model

The state space representation, which describes the launcher’s dynamics through its angular and lateral movement

about the commanded trajectory, is given by:

d

dt

∆θ̇∆θ∆ż

∆ β̇ R∆β R

=

0 A6 A3 0 K 11 0 0 0 00 A1 A2 0 K 20 0 0 −2ξ 0ω0 −ω200 0 0 1 0

∆θ̇∆θ∆ż

∆ β̇ R∆β R

+

0 B1 K 10 0 00 B2 K 2

ω20

0 00 0 0

∆β c∆W

∆β FZ

y =

X ∆α

=

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

0 1 1/V R 0 0

∆θ̇∆θ∆ż

∆ β̇ R

∆β R

+

0 0 00 0 00 0 00 0 00 0 0

0 −1/V R 0

∆β c∆W

∆β FZ

The launcher state vector consists of the pitch angle error, ∆θ, and its time derivative, the derivative of thelateral position error, ∆̇z, the actual thrust deflection, ∆β R, and its time derivative. The output vector, y, containsthe full state vector and the angle of attack ∆α. The inputs of the system are the commanded thrust deflection,∆β c, the wind W, and a thrust deflection offset, ∆β FZ . To estimate the fluid consumption of the actuator, wecalculate the deflection integral, C β . The numerical values of the state-space model are given below, and we considera launcher’s true airspeed V R = 325.34 m/s.

A1 A2 A3 A6 ξ 0 ω0 B1 B2 K 1 K 2

- 31.6 - 0.0467 0.005 1.631 0.63 23.9 - 0.005 0.0467 - 2.514 - 15.6893

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0 1 2 3 4−100

−50

0

50

A o A ( d e g )

0 1 2 3 4−1

0

1

W i n d ( m / s )

0 1 2 3 4−1

0

1

2

D e f l ( d e g )

0 1 2 3 4−100

−50

0

50

A t t ( d e g )

0 1 2 3 4−10

0

10

20

D e f l r a t e ( d e g / s )

0 1 2 3 4−200

−100

0

100

A t t r a t e ( d e g / s )

0 1 2 3 40

2

4

C β

( d e g )

t (s)

0 1 2 3 4−20

0

20

40

D r i f t

( m / s )

t (s)

Figure 1: Time response of the uncontrolled launcher, under a commanded thrust deflection pulse of 1º, 1 s.

Figure 1 shows the response of the uncontrolled launcher to a commanded thrust deflection pulse, of 1 º and 1s.As we can see, the angle of attack diverges: in only 4 seconds the angle of attack has become about - 75º, whichis definitely catastrophic. The launcher attitude, its rate and the drift show a similar behavior, meaning that thelauncher is naturally unstable. Therefore, it is necessary to control it to stabilize the system. The poles of thesystem show the instability,

Poles −15 ± 18.56 i - 1.32 0.05 1.23One of the poles is positive, meaning that the system is unstable. This pole is responsible for the divergence

seen on figure 1.

1.2 Synthesis model

In order to calculate the H ∞ controller, we need an appropriate synthesis model. Since we will synthesize a controlleronly for the pitch angle, we have to reduce our MIMO representation, discussed before, to a SISO model with ∆β cas the single input, and ∆θ as output. Therefore, we remove all the disturbances in our synthesis model, and letone single output.

The transfer function of the SISO model is given by the ratio between the pitch angle error and the commanded

thrust deflection,

G(s) = ∆θ

∆β c=

−1436s − 112s5 + 30.16s4 + 571s3 − 22.36s2 − 929.2s + 46.7

The system is of fifth-order. The Bode diagram of the transfer function G(s) is shown in figure 2.If we consider the uncontrolled launcher without actuators, the dynamics of the system is described by

d

dt

∆̇θ∆θ

=

0 A61 0

∆̇θ∆θ

which is the equation of an harmonic oscillator, of frequency ω =√

A6 = 1.3 rad/s. This relation implies that A6is positive, meaning that the center of pressure is above the center of gravity. As a consequence, the launcher is

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−200

−150

−100

−50

0

50

M a g n i t u d e ( d

B )

10−3

10−2

10−1

100

101

102

103

−180

−135

−90

−45

0

P h a s e ( d e g )

Rigid launcher Bode plot

Frequency (rad/s)

Figure 2: Bode diagram of the rigid launcher uncontrolled system.

Figure 3: Standard form of the system.

aerodynamically unstable. This expression allows us to identify that the pole - 1.32 describes the natural dynamicsof the system, while the imaginary poles −15 ± 18.56 i describe the actuator dynamics. The actuators are muchfaster than the natural system, which is required to control the launcher.

2 Augmented system

To synthesize an H ∞ controller, we have to express the system in the standard form. We choose two exogenousinputs, which represent the reference attitude, z1, and the disturbance on the thrust deflection, z2, which represents

a disturbance on the actuator. The exogenous outputs are the attitude error, z1, and the actuator command, z2.The standard form of the system is shown in figure 3. G represents the dynamics of the launcher, and K is thecontroller to synthesize.

The relation between the exogenous inputs and outputs is

z1 = W 11

1 + KGw1 − W 1 G

1 + KGW 3w2

z2 = W 2K

1 + KGw1 − W 2 KG

1 + KGW 3w2

In a matricial form, we can identify the transfer functions T ij ,

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−50

0

50

M a g n i t u d e ( d B )

10−2

100

102

0

45

90

P h a s e ( d e g )

1/W1

Frequency (rad/s)

20

22

24


100

101

−1

0

1

P h a s e ( d e g )

1/W2

Frequency (rad/s)

−50

0

50


10−2

100

102

0

45

90

P h a s e ( d e g )

1/W1W3

Frequency (rad/s)

25.0362

25.0362

25.0362


100

101

−1

0

1

P h a s e ( d e g )

1/W2W3

Frequency (rad/s)

Figure 4: Bode diagram of the filters. These functions bound the transfer functions of the system.

z

1

z2

=

T 11

T 12

T 21 T 22

w1

w2

which relate the exogenous inputs and outputs. The H ∞ controller will minimize these transfer functions. Moreprecisely, the algorithm will solve the sub-optimal H ∞ problem: it will ensure that the H ∞ norm of the transferfunctions T ij (in the one dimensional case, the maximum) is smaller than the constant γ . As we will see in the nextsection, the filters allow us to shape the transfer functions in different frequency domains.

Now, let’s take a look to the weight functions. We consider the following expressions

W 1 = k1s + a1s + b1

, W 2 = k2s + a2s + b2

, W 3 = k3

k1 a1 b1 k2 a2 b2 k3

0.33 3.14 0.0628 0.08 22 22 0.7

The functions W 1 and W 2 are first order filters, whereas W 3 is a constant. As we will see in the next section,the inverse of these functions will bound the transfer functions T ij . Figure 4 shows the four inverse filters 1/W 1,1/W 2, 1/W 1W 3, and 1/W 2W 3. The transfer function 1/W 1 is a high-pass filter, with a magnitude of M LF = 0.061at low frequencies and M HF = 3.03 at high frequencies. In the middle range, the gain increases linearly, as well asthe phase. The cutoff frequency is ω−3dB = 1.04 rad/s. The function 1/W 2 is constant because of the choice of theparameters, a2 = b2. However, if a2 > b2 the function will be similar to 1/W 1. On the contrary, if a2 < b2 the filterwill be low-pass.

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Figure 5: (Left) Bode plot of the synthesized controller. (Right) Magnitude of the different transfer functions, withrespect to the weights W .

3 First H ∞ Controller Synthesis

3.1 H ∞ controller synthesis

We have used the augmented model to synthesize the H ∞ controller. At the end of the iteration, the value of γ is1.09. The controller is of 7th order, with the following poles,

Poles - 126 - 22 - 13.9± i 19.1 - 10.6 - 0.078 - 0.063

It is not difficult to understand why the controller is of 7th order. We have two filters of 1st order, and a systemof 5th order, which leads to a controller of 7th order altogether. All the poles are negative, which guarantees thatthe system is now stable. As we can see on figure 5, the controller is a low-pass filter of bandwidth 2000 rad/s(frequency at - 3 dB), which also attenuates the system at 1 rad/s. Notice that this frequency corresponds to thenatural frequency of the launcher: the controller avoids exciting this mode. It is interesting to examine the Bodeplots of the transfer functions S , KS , GS , and T (fig. 5). In all cases, the magnitude is bounded by the filterW i, and the following expressions are satisfied, which are a consequence of the previous analysis of the transferfunctions,

|S | ≤ γ/W 1|KS | ≤ γ/W 2

|GS

| ≤γ/W 1W 3

|T | ≤ γ/W 2W 3These expressions show the interest of H ∞ control: thanks to the weight functions W i, we can shape the

closed-loop transfer functions of the system. This technique is in contrast with the classical frequential and modalapproach, where we can just place the poles of the system. The difficulty of H ∞ control resides in choosing theappropriate weights. The choice requires a good understanding of the relation between closed-loop transfer functionsand performance - robustness criteria.

Let’s consider the first Bode plot, which represents the sensitivity function as compared to γ/W1 (fig. 5, left).The H ∞ algorithm ensures that the modulus of S is smaller than γ/W1 , for any frequency, and we can see thatthis relation is satisfied. By choosing an appropriate weight function, we can shape the closed-loop transfer functionS. The shape of the filter depends on the performance objectives. In this case, the sensitivity function relates the

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Figure 6: Bode plot (left) and Nichols plot (right) of the controller rigid launcher, compared to the plant.

commanded attitude, r, with respect to the error. In general we are interested in a low steady-state error, which justifies the choice of a high-pass filter. It is interesting to see that, at high frequencies, the controller doesn’t playany role on the sensitivity function. Since G → 0, S → 1, for any K . In this case, S → 1 when ω > 20 rad/s.

The behavior of S at mid-frequencies requires some attention. In the range 1.2 < ω 0 dB). The sensitivity function also relates the measurement noise and the output:when |S | < 0 dB, the noise is attenuated, but when |S | > 0 dB the noise is amplified by feedback. Therefore, weshould take special attention to noise disturbances in the range 1.2 < ω < 20 rad/s. We can also notice on thefigure that, on this frequency range, the magnitude of S is closest to the boundary γ /W 1.

3.2 Frequency domain assessment

The analysis of the Nichols plot of the controlled rigid launcher confirms that the closed-loop system is stable (fig.6). Since the open-loop plant has one unstable pole, the Nichols plot has to intersect the -180° axis twice to bestable in closed loop. We can see that this is the case for the open-loop plant with the controller, KG, as opposedto the plant G.

LF Gain Margin HF Gain Margin Phase Margin Delay

- 6.9 dB @ 1 rad/s 12.6 dB @ 10 rad/s 31º @ 2.9 rad/s 0.19 s

The gain and phase margins, shown on the table above, measure the robustness of the system. The gain marginrepresents the factor by which KG can increase before becoming unstable. This factor accounts for uncertaintiesin the model. Since the model is never going to be completely accurate, it is necessary to account for a significantgain margin. In the same fashion, the phase margin represents how much lag you can add before the closed loop

becomes unstable. The delay is particularly important for the discretization of the controller: in general, we require∆ > 2T s, where T s is the sampling period.

3.3 Time domain simulation

Figure 7 shows the time domain simulation of the controlled system. In the simulation we have added a noisymeasurement of θ , to account for the sensor noise. As we can see, the launcher is stable and doesn’t diverge, evenin the case of strong wind gusts.

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Figure 7: Time domain simulation, under an offset deflection (left) and wind gusts (right).

Frequency Analysis Time Domain Analysisγ HF G [dB, LF GM HF GM PM Delay θmax AoAmax C βmax żmax

103 rad/s] [dB] [dB] [º] [s] [º] [º] [º] [m/s]Nominal 1.09 4.2 - 6.9 12.6 31.3 0.19 - 0.87 - 5.01 156 35.16

k2/2 0.92 - 6.2 - 7.53 9.82 32.4 0.18 - 0.75 - 5.02 187 35.082k2 1.51 4.7 - 6.0 13.5 28.5 0.20 - 1.1 - 5.22 113 35.32

4a2, k2/4 1.02 16 - 7.25 15.8 33.7 0.20 - 0.83 - 5.05 194 35.13a2/2, 2k2 1.2 - 5.8 - 6.48 10.1 28.5 0.18 - 0.98 - 5.15 114 35.2

k3/2 1.05 2.74 - 6.94 12.8 32.2 0.20 - 1.03 - 5.16 148 35.362k3 1.21 6.73 - 6.94 12.1 29.6 0.17 - 0.70 - 4.99 175 34.98

Table 1: Summary of the performance and robustness parameters of the different controllers. The simulation isperformed from t = 0 to t = 100 s, with zero guidance input, no thrust deflection offset, and a wind gradient andgust profile. In all the cases, the maximum thrust deflection was β max −4º.

4 Controller Settings

In this section we explore the effect of the weight functions on the synthesized controller. We focus the study onthe W 2 and W 3 filters. In the following, we will consider one parameter at a time, and see how it changes theperformance of the controller. In each case, we first motivate, in a theoretical point of view, the expected effects,and then check the simulation results. A summary of the simulation results is found in table 1.

4.1 H ∞ rigid motion controller - Sensitivity study

4.1.1 Study of W 2 - Gain k2

Let’s focus on the function T , which sets the bandwidth of the controller, ω−3dB. If we increase the gain k2, themagnitude of 1/W 2W 3 will decrease, pushing down the transfer function T . As a consequence, the bandwidthwill decrease (i.e. the function T crosses the -3 dB point before). We expect a system that will respond slowerto the perturbations. At the same time, it will increase the gain margin, i.e. the system will be more robust touncertainties. In terms of fluid consumption, the value of C β will be smaller: the gain of the controller will besmaller.

Figure 8 shows that our predictions are correct. When k2 increases, the gain of the controller at low frequenciesdecreases, but it increases at high frequencies. We can see that the HF gain margin is higher, and that the T

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function is pushed down. Looking at the synthesis table (table 1), we can see that for 2k2 the fluid consumption ismuch lower than in the nominal case. However, the maximum attitude and angle of attack are bigger, because thesystem doesn’t react as fast to the external disturbances.

4.1.2 Study of W 2 - Pulsation a2 and b2

Now, let’s consider the case when a2 > b2. In this case, the filter 1/W 2 will behave like a high-pass filter. As aconsequence, the maximum value of KS , at high frequencies, will increase. As a consequence, the bandwidth of the controller will increase, and it will react much faster to the disturbances. We also expect the gain margins toincrease (remember that the value of K S at high frequencies is related to the robustness, the higher the better).

As we can see in figures 9 and 10, the function W 2 strongly affects the gain at high frequencies. When a2 > b2,the gain increases when ω > 20 rad/s, with respect to the nominal case. We can see significant changes in thetransfer function for KS , which mimic the behavior of the controller gain. The synthesis table shows clearly thesituation: when a2 > b2 the gain margins increase. rHowever, the value of C β is excessive (194, compared to thenominal value of 156). This is the consequence of increasing the bandwidth of the controller.

In the case a2 < b2, the filter 1/W 2 will be low-pass. As opposed to the previous case, it will reduce the gain of SK at high frequencies. We can verify these results in figures 9 and 10. The bandwidth of the controller is smaller,

which translates into a lower consumption controller (i.e. the value of C β is much lower). However, the controlleris less robust: the gain margins decrease.

4.1.3 Study of W 3

Finally, we study the effects of k3. The function SG is bounded by 1/W 1W 3. This function relates the low frequencydisturbances and the command tracking. If we increase k3, the value of SG will decrease. As a result, we shouldhave a better tracking, that is, the system would be less sensitive to the wind gradient, which is of low frequency.

If we take a look to the synthesis table, we can see that the filter with 2k3 produces a controller with the bestperformance values, in terms of wind rejection: it has the smallest value of θmax, AoAmax , and ż .

4.2 H ∞ Controller Settings

As a conclusion to our study of the H ∞ synthesis, we will try to find a controller that satisfies the followingrequirements,

1. Frequency domain: LF gain margin greater than 5 dB, HF gain margin greater than 9 dB, phase margingreater than 25º.

2. Time domain: angle of attack smaller than 5º, and deflection integral smaller than 150º.

If we look back to table 1, we can guess that the best compromise will be met by a combination of a2/2, 2k2, and2k3. The first controller has a very low deflection integral and good margins, but the maximum angle of attack istoo large. By contrast, the latter is the most effective to reduce the maximum AoA, but the value of C β is too high.

We have first tried a combination of both filters. The results were closer to the performance objectives, butthe angle of attack was still too high. To reduce it, we have increased k3, up to 3k3. This controller meets all theperformance requirements, and the performance in the frequency and time domain are shown in figure 12.

k1 a1 b1 k2 a2 b2 k3

0.33 3.14 0.0628 0.16 11 22 2.1

Frequency Analysis Time Domain Analysisγ HF G [dB, LF GM HF GM PM Delay θmax AoAmax C βmax żmax

103 rad/s] [dB] [dB] [º] [s] [º] [º] [º] [m/s]Nominal 1.45 10.3 - 6.66 10 26.3 0.15 - 0.652 - 4.98 144 34.9

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10−3

10−2

10−1

100

101

102

103

−180

−160

−140

−120

−100

−80

−60

−40

−20

0

20

M

a g n i t u d e ( d B )

GS vs γ /W1W3

Frequency (rad/s)

GS, k2 /2

γ /W1W3, k2 /2

GS, 2k2

γ /W1W3, 2k2

Figure 8: Frequency domain behavior of the controller, for different values of k2.

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Figure 9: Frequency domain performance of the controller, when a2 > b2 (left), and when a2 < b2 (right).

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Figure 10: (Continued) Frequency domain performance of the controller, when a2 > b2 (left), and when a2 < b2(right).

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Figure 11: Frequency domain response of the controller, under different k3 values.

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10−3

10−2

10−1

100

101

102

103

104

−10

−5

0

5

10

15

20

25

30


Controller Bode plot

Frequency (rad/s)−185 −180 −175 −170 −165 −160 −155

−15

−10

−5

0

5

6 dB

Nichols plot of the controlled rigid launcher

Open−Loop Phase (deg)

O p e n −

L o o p G a i n ( d B )

a2 /2

0 20 40 60 80 100−5

0

5

A o A ( d e g )

0 20 40 60 80 100−20

0

20

40

W i n d ( m / s )

0 20 40 60 80 100−5

0

5

D e f l ( d e g )

0 20 40 60 80 100−1

−0.5

0

0.5

A t t ( d e g )

0 20 40 60 80 100−10

0

10

D e f l r a t e ( d e g / s )

0 20 40 60 80 100−1

0

1

2

A t t r a t e ( d e g / s )

0 20 40 60 80 1000

50

100

150

C

β (

d e g )

t (s)

0 20 40 60 80 100−20

0

20

40

D r

i f t ( m / s )

t (s)

Figure 12: Characteristics of the synthesized controller. This controller complies with all the specified performancerequirements.

H infinity control for launchers

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