Optimal LQG Control

a very small value of k) to avoid singularities if the subspace of DM actuator commands having no effect on σ2 _{is not a priori known. A full derivation of the}

reconstructor can be found in Ellerbroek (2002).

2.2.2 Optimal LQG Control

As shown in Section 2.2.1 and Figure 1.10, classical AO control is limited due to the discrete time delay between measurement and correction. LQG control improves upon classical control methods by making a better prediction of the turbulent phase at the time-step that the correction is to be applied. Equation 2.16 shows how φcor

k+1 which is applied at time [kT, (k + 1)T ) is computed from uk, which is

itself computed from φres

k−1 at time [(k − 1)T, kT ), demonstrating the two-frame

delay shown in Figure 1.10. LQG control instead calculates uk from a prediction

of the turbulent phase φtur

k+1 at time [kT, (k + 1)T ), i.e in the future with respect

to uk.

Starting from the fact that the correction applied by the DM should match the turbulent phase profile so that they cancel each other out and so minimise the residual phase, we can rewrite Equation 2.16 as,

ulq_k = (NtN)−1Ntφtur_k+1, (2.23)

where ulq

k is the optimal full information feed-forward control and (NtN)−1Nt is

the pseudo-inverse of the DM influence matrix, assuming with no loss of generality that Nt_{N is invertible (Kulcsár et al., 2012). This shows that if we know φ}tur

k+1, i.e.

in the full information regime, we can calculate the optimal commands to correct this incident turbulent phase profile. In reality however we don’t generally know what the future turbulent phase will be and so we need to predict it based on past

2.2.2. Optimal LQG Control

measurements, giving rise to,

uopt_k = (NtN)−1Ntφbtur_k+1|k, (2.24)

where uopt

k are optimum DM commands based on the prediction of the turbulent

phase φbtur

k+1|k and the xk+1|k notation represents the variable x at time-step k + 1

given previous measurements at time-step k. This predicted turbulent phase is approximated in the simplest case by the vector valued auto-regressive model, AR(1), which models the turbulent phase as,

φtur_k+1= Aturφtur_k + vk, (2.25)

where Atur _{is a diagonal matrix which enables the adjustment of the cut-off fre-}

quency for each turbulent mode according to priors (Kulcsár et al., 2012) and is given in the Zernike basis by,

aii= exp  −0.3(n(i) + 1)V T D  , (2.26)

where n(i) is the radial order of the i−th Zernike mode, V is the wind speed norm,

T is the sampling period and D is the telescope diameter.

The AR(1) model shown in Equation 2.25 shows how the turbulent phase from one time frame depends on the turbulent phase from the previous frame, however it doesn’t include any actual prior measurements. Applying a Kalman filter to the AR(1) model produces a prediction which then depends on the delay- and control- free measurements, Sk = {s0, ..., sk}, with sk = yk+1+ DNuk−1, resulting in a

prediction of the form,

b

2.2.2. Optimal LQG Control

where Lk is the Kalman gain (Kulcsár et al., 2012). Due to the stationary model

used and because the estimate is to be used for infinite horizon LQG (Kulcsár et al., 2012), the time-varying Kalman gain Lk can be replaced, with no loss of

optimality, by the steady state counterpart,

L∞= AturΣ∞Dt(DΣ∞Dt+ Σw)−1, (2.28)

where Σw is the covariance matrix of the measurement noise in Equation 2.19 and

Σ∞ is the solution to the discrete algebraic Riccati equation (DARE),

Σ∞= AturΣ∞(Atur)t+ Σv− AturΣ∞Dt(DΣ∞Dt+ Σw)−1DΣ∞(Atur)t, (2.29)

where Σv is the covariance matrix of the noise term in Equation 2.25. A solution to

Equation 2.29 can be found by using a DARE solver included in some mathematical programming languages, e.g MATLAB, python/scipy.

The next step is to combine Equations 2.16, 2.19, and φres

k−1= φturk−1− φcork−1 to give,

yk= Dφturk−1− DN uk−2+ wk, (2.30)

which can be substituted into Equation 2.27 via the delay- and control-free mea- surements sk= yk+1+ DNuk−1 resulting in,

b

φtur_k+1|k+1= Aturφbtur_k|k+ L_∞(y_k+1− Dφbtur_k|k+ DNu_k−1), (2.31)

or equivalently,

b

2.2.2. Optimal LQG Control

This result is then used to calculate the predicted turbulent phase at the next step,

b

φtur_k+1|k, by,

b

φtur_k+1|k = Aturφbtur_k|k (2.33)

The equations forφbtur

k|kandφbturk+1|kare then simply the non-trivial part of the Kalman

filter in predictor form adjusted to the state space model given by,

xk+1 = Axk+ Buk+ ξk and yk = Cxk+ wk, (2.34) where, xk =           φtur_k φtur_k−1 uk−1 uk−2           , A=           Atur 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0           , B =           0 0 I 0           , ξk =  vt_k 0 0 0 t , C=  0 D 0 −DN  . (2.35)

And so the predicted turbulent phase is then given by,

ˆxk+1|k = Aˆxk|k−1+ Buk+ L∞(yk−ˆyk|k−1), (2.36)

where ˆyk|k−1is the best estimate of the model output given Sk−1 and is given by,

ˆyk|k−1= C ˆxk|k−1, (2.37)

where L∞ is as given in Equation 2.28.

The state space model demonstrated thus far in Equations 2.34, 2.35, and 2.36 is not however the only model that can be used to describe and then predict the state of the system. Smaller state vectors, xk, can also be used (Kulcsár et al., 2012)

and as the vector given in Equation 2.34 includes two occurrences of u and φtur _it