4 Navigation & Docking System
5.1 Closed-Loop Control
The closed-loop problem in presence of disturbances is represented in Figure 5.1.
Such a feedback loop consists of a plant G(s), a stabilising controller K(s), a reference signal r, sensor noise ν, input perturbations (actuators and environment) di, and output disturbancesd.
Analysing the system can be done computing the transfer function between the different inputs and outputs. The following equations, which provide the contribution of all inputs
Chapter 5. Control
+
G
+
K
r
_u u
g +d
id
y
+ +
+ +
n
Figure 5.1 – Closed-loop control system.
on outputs, are valid for internally stable closed-loop systems [298]:
y = To(r − n) + GSidi+ Sod (5.1a)
r − y = So(r − d) + Ton − GSidi (5.1b)
u = KSo(r − n) − KSod − Tidi (5.1c)
ug = KSo(r − n) − KSod + Sidi (5.1d)
The input loop gain (or loop transfer function)Li and output loop gainLo are defined as:
Li = KG (5.2a)
Lo = GK (5.2b)
Note that for Single-Input Single-Output (SISO), Li= Lo. The input/output sensitivity functionsSi andSo, as well as the input/output complementary sensitivity functions Ti and To, are defined as:
Si = (1 + Li)−1 (5.3a)
So = (1 + Lo)−1 (5.3b)
Ti=1 − Si= Li(1 + Li)−1 (5.3c)
To=1 − So= Lo(1 + Lo)−1 (5.3d)
The input sensitivity is the transfer function between di and ug and represents the closed-loop sensitivity to input perturbations. Similarly, the output sensitivity is the transfer function betweend and y and provides information on the closed-loop sensitivity to output disturbances.
The analysis ofSi,So,Ti,andTo, provides valuable insight on the expected closed-loop
5.1. Closed-Loop Control
performance.
To simplify the notation,S ≡ So, T ≡ To, and L ≡ Lo. The subscript index i will be specifically used to denote input functions.
The closed-loop response in terms of error is given by (5.1b):
e ≡ r − y = S(r − d) + T n − GSidi (5.4)
To obtain a null error, S and T would have to be null. This is, of course, impossible asT + S = 1. Good command tracking and disturbances rejection can be obtained for S ≈ 0, i.e T ≈ 1. On the other hand, to limit noise amplification in the feedback, T ≈ 0, i.e. S ≈ 1. A large loop gain L is required for good command tracking and stability but a small loop gain L is necessary to restrain noise amplification. These conflicting requirement can be accommodated as signals involved in a feedback loop have different frequency contents.
Disturbances, di andd, caused by the dynamics or the environment, and the reference signalr typically have a large amplitude at low frequencies and roll-off above a particular frequency. On the other hand, sensor noise ν is generally composed of white noise or high-frequency signals. The different nature between these input signals allows satisfying competing requirements at the same time. Thus, requiring a large loop gain L at low frequencies for tracking and disturbances rejection should not amplify sensor noises, or at least limit its amplification. The loop gain can then roll-off above a particular frequency thus limiting high-frequency noise amplification.
To prevent noise amplification at the plant input, the termKS is of paramount importance as it multiplies the noise signal directly (see (5.1c) and (5.1d)). KS should thus be close to one (0 dB) at low frequencies, to have control authority, and roll-off at high frequencies where sensors noise is important.
Typical shapes forS and T and KS are shown in Figure 5.2. Note that the abscissa will often be referred to as the frequency although it is expressed in terms of angular rate.
On the ordinate are the singular values ofS and T which provide more information than the eigenvalues. S, T , and KS being computed at the plant output are composed of as many channels as there are outputs.
5.1.1 Singular Values
The Singular Values Decomposition (SVD) of a matrixA ∈ Cn×m is:
A = U ΣV∗ (5.5)
Chapter 5. Control
Figure 5.2 – Sensitivity, complementary sensitivity, and controller times sensitivity functions.
where * denotes the complex conjugate transpose. The matricesU ∈ Cn×nandV ∈ Cm×m are unitary and contain respectively the input and output singular vector. Σ∈ Rn×m with k = min(n, m) singular values σi ordered in decreasing order along the main diagonal.
The largest singular valueσ1 is commonly referred to as ¯σ. The smallest, σkis denoted
¯σ.
Singular values, as opposed to eigenvalues, can be computed for any matrix. Furthermore, eigenvalues can lead to wrong interpretations [286]. For the two-input/two-output plant G
the eigenvalues are degenerated and null, which means that for any input, the output should be zero. However, for the input vector [0 1]T, the output is [100 0]T. Eigenvalues provide a measure of the gain only when the output and input are in the same direction.
TheU and V matrices of G are:
The inputV and output U matrices give the direction so that an amplification σi of the
5.1. Closed-Loop Control
amplification whereas the last columns give the direction of minimum amplification. It can be directly seen that an input vector in the direction [0 1]T provides an output in the direction [1 0]T, amplified byσ1 = 100.
5.1.2 Bandwidth
The bandwidth defines the frequency range over which the control is effective. It can be derived from the sensitivity function as:
σ(S(jω¯ B)) =−3 dB (5.8)
The plant complementary sensitivity T relates the plant output y and the reference r: y = T r. Control is thus effective as long as T 0 dB. This leads to an alternate bandwidth definition:
σ(T (jω¯ BT)) =−3 dB (5.9)
Finally, the crossover frequency ωc is defined as the frequency at whichL changes sign:
σ(L(jω¯ c)) = 0 dB (5.10)
For systems with Phase Margin (PM)<90 deg [286, p. 39]:
ωB < ωc < ωBT (5.11)
In the interval [ωB, ωBT], S and T are likely to be larger than 0 dB and will then amplify noises and disturbances in this frequency band, without increasing tracking performances.
Acting on T and S bandwidth allows controlling this specific frequency band where performances may be degraded.
Note that in the case PM=90 deg, then ωB = ωc = ωBT.
5.1.3 Margins
Phase Margin (PM) and Gain Margin (GM) are used to measure how much gain and phase can be included in the feedback before it becomes unstable. Classically, SISO PM and GM are efficiently represented using Nyquist’s plot, as shown in Figure 5.3 [286].
ω180 defines the frequency at which the phase changes sign. The crossover frequencyωc
Chapter 5. Control
Im
Re 1− 1
GM 1
PM
L(jω180)
L(jωc)
L(jω)
Figure 5.3 – Nyquist plot of the loop gain L.
becoming unstable:
τ = PM
ωc (5.12)
For MIMO systems, an alternate definition can be used [299, p. 118]. For a perturbed feedback system, it can be shown that the system remains stable if it does not have poles in the half-left plane and if:
!
¯σ(1 + L) > ¯σ(Δa(s)) ∀s ∈ DR
¯σ(1 + L−1) > ¯σ(Δm(s)) ∀s ∈ DR
(5.13)
where Δa and Δm are additive and multiplicative uncertainties, and DRis the standard Nyquist contour, encircling the right-half plane with R sufficiently large.
Defining α = minω ¯σ(1 + L), then:
GM1+L=
"
1
1 + α, 1 1− α
$
(5.14a)
P M1+L=±2 sin−1α 2
(5.14b)
These are margins to additive uncertainties.
5.1. Closed-Loop Control
which are margins to multiplicative uncertainties.
The total gain and phase margins are defined as:
GM = GM1+L∪ GM1+L−1 (5.16a)
P M = P M1+L∪ P M1+L−1 (5.16b)
Note that using this definition the PM is bounded by 60 deg. The negative GM is often referred to as the “gain reduction” margin.
It is possible to compute guaranteed minimum values on the PM and GM using the sensitivity function. DefiningMS = max
ω |S(jω)|, the following expressions are true [286, p.
35]:
According to [300], the margins shall be at least 6 dB and 30 deg for space missions, at the plant output. However, for RVD, and according to ESA, it is preferable to aim for 45 deg margins.
Although such definitions allow a preliminary assessment of controller performance, more advanced techniques can be used to define the robustness of a closed-loop system with respect to a set of bounded uncertainties. This will be further discussed in the next sections.
5.1.4 GNC Loop
The complete GNC loop for the RVD problem is shown in Figure 5.4.
Chapter 5. Control
+
G d
i+
y
r K
- +
EKF
SensorsActuators Wheel Unloading
u
+ + +u
gOL
+
Figure 5.4 – GNC loop with the sensors, actuators, navigation filter and controller.
The blockOL represents the open-loop manoeuvres.
similar structures, with the following exceptions: the reaction wheels unloading controller that is only present in the attitude loop, and the open-loop ΔV s and other feed-forward terms which are only present for the relative position control.
This GNC loop does not include the sloshing dynamic; its coupling to the dynamics is shown in Figure 5.5.
+
+ +
G
S
u u
gy d
iFigure 5.5 – Sloshing dynamics.
Note that the LQR controllers will be tuned without sloshing dynamics, whereas H∞ and μ-synthesis controllers will explicitly account for the sloshing in the generalised plant.
According to Section 3.5.2, to maximise the sloshing perturbation, a filling ratio ofτ = 0.5, corresponding to 1 kg of fuel, will be used. To generate the dynamics (P2P dynamics and Hill’s equations), the chaser dry mass (10 kg), the docking ports locations and orientations, as well as the fuel tank location provided in Section 6.2, are used.
To simplify the discussion, only the case where the target docking port is aligned with its body frame is considered. However, in Chapter 6, controllers for the four different target docking ports orientations will be tested.
5.1. Closed-Loop Control
As each element of the control-loop must be discretised (navigation filters and controllers), using different sample times for each component of the GNC significantly complexifies the software. It has thus been decided to use a single sample time.
The sampling frequency should be ∼ 10 times faster than the fastest mode which needs to be controlled [301]. According to [101], the typical closed-loop bandwidth for RVD are not larger than 0.1 Hz (i.e. 0.6 rad/s). This consideration leads to a sampling frequency of 1 Hz.
The camera of the VBN can provide images up to 14 FPS. With a 10 megapixels sensor, this represents the most significant amount of data that the bus must handle. As the whole GNC loop is sampled at 1 Hz, it has been decided to run the camera at 1 FPS, which will unload the data bus. The large capacity margin on the FPS could be used for other image processing purposes.
5.1.5 Number of Controllers
To satisfy the required accuracy along the approach trajectory, while minimising the fuel consumption, several controllers are necessary with accuracy increasing as range decreases.
The reference trajectory is recalled in Figures 5.6 and 5.7
S2
Figure 5.6 – Trajectory profile for a CubeSat RVD mission.
The relative position control, using the Hill’s equations, will be using different controllers.
For the LQR, a low-bandwidth controller will be used from SK points S2 to S22. From S22 to S23, a medium bandwidth is used. Finally, from S23 to S3 a high bandwidth
Chapter 5. Control
docking port axis
S2
4S3
-15m
-10m -5m
S3
1fly-around VBN+ISL
VBN -2.5m
S3
2V ¯
R¯
Figure 5.7 – Final Approach.
S2 to S23 and a high bandwidth fromS23 to S3.
This is summarised in Table 5.1.
Table 5.1 – Closing: Bandwidth usage.
SK point S2 → S21 S21 → S22 S22→ S23 S23→ S24 S24→ S3
LQR LB LB MB HB HB
H∞ LB LB LB HB HB
For the final approach, the P2P control starts at pointS3. At this point, a low bandwidth controller will be used for the handover between the navigation filters to limit fuel consumption. Once the navigation handover has been performed, a high bandwidth controller will be used for the forced translation until docking.
For absolute attitude control, only one controller is necessary as pointing requirement is constant along the approach.