Identification of a dynamical model of a thermally actuated deformable mirror

Abstract

Using the subspace identification technique, we identify a finite dimensional, dynamical model of a recently developed prototype of a thermally actuated deformable mirror (TADM). The main advantage of the identified model over the models described by partial differential equations is its low complexity and low dimensionality. Consequently, the identified model can be easily used for high-performance feedback or feed-forward control. , a TADM has been used to correct static wavefront aberrations. In the above cited paper, a static (steady-state) model of the TADM has been identified and, on the basis of this model, a control action for the TADM has been derived as the solution of a constrained least-squares problem. However, this control strategy requires that the time between two consecutive control iterations is approximately equal to the settling time (or the rise time) of the TADM. Consequently, this wavefront correction strategy is relatively slow and its performance might be additionally degraded in the case of time-varying wavefront aberrations.To achieve fast correction of both static and timevarying wavefront aberrations, the time between control iterations has to be significantly smaller than the TADM's settling time. In such cases, a dynamical model of a TADM has to be developed to accurately correct wavefront aberrations [9,10]. Once this dynamical model has been obtained, model-based control strategies [11,12] can be employed to maximize the performance of the wavefront correction. Apart from the control perspective, a dynamical model of a TADM is important because it can be used to simulate the dynamical behavior of the AO system before the real system has been built.A dynamical model of a TADM must meet two requirements. First, it must accurately capture the TADM's dynamics. Second, to be used for control, it must be relatively simple [13,14] and preferably low dimensional (i.e., it must have a relatively small number of states). However, the dynamics of TADMs are governed by the thermoelastic system of partial differential equations (PDEs) [15]. Furthermore, in the case of the TADMs that have been proposed in [3,7], the thermoelastic system of PDEs must be coupled with the biharmonic plate equation [16]. The dynamical model based on these PDEs is infinite-dimensional and as such is too complex to be used for control. To apply the model-based control strategies of [11,12], a more compact, finite-dimensional model must be developed. One way to develop such a model would be to discretize the system of PDEs and corresponding boundary conditions using the finite element method (FEM) [17]. However, the FEM can be applied only if all physical parameters of the TADM are known. Furthermore, the FEM model is usually high dimensional and thus is still relatively complex to be used for control.In this Letter, we follow another way of model building that is based on system identification techniques [18]. Accordingly, from experimental data, we identify a loworder, state-space model of a recently developed TADM...

In situ correction of mirror surface to reduce round-trip losses in Fabry–Perot cavities

“…However, since the correction depends on defects on the mirror surface, which do not change over time, the optimal correction must be computed only once and does not need to be adjusted in real-time. To speed up the searches it is possible to take advantage of the measurable time response of the correction [11] and implement model predictive control systems [12].…”

Predictive control of thermally induced wavefront aberrations

“…In the AO system, the control sampling period is T = 2s. This value is chosen on the basis of the step response analysis of the TADM (previously characterized in [26]). This sampling period divides the time t into k discrete time instants, that is, t = kT .…”

“…In [26], it has been experimentally demonstrated that TADM can produce variation of essentially the first 9 Zernike coefficients (omitting the piston). We therefore focus on correction of these 9 Zernike coefficients which will be represented by a vector:…”

“…In [26] it has been demonstrated that the state-space model of the 4 th order (the dimension of the state x(k)) is able to relatively accurately predict the dynamical behavior of the TADM. However, in order to achieve even better prediction accuracy in this paper we are using the identified model of the 10 th order.…”

“…However, in order to achieve even better prediction accuracy in this paper we are using the identified model of the 10 th order. The input of the TADM is v = r 2 3 , where r 3 is a voltage applied to the 3 rd actuator of the TADM (for more details consult [26]). For simplicity, we actuate only one channel of the TADM (the generalization for all 19 actuators of the TADM is straightforward).…”