**1. Introduction**

The use of deformable mirrors (DMs) in adaptive optics (AO) systems allows for compensation of various external and internal optical disturbances during image aquisition. For example, an astronomical telescope equipped with a fast deformable secondary mirror can compensate for atmospheric disturbances and wind shake of the telescope structure resulting in higher image resolution [1–4]. In microscopy, deformable mirrors allow to correct for aberrations caused by local variations of the refractive index of observed specimen. Especially confocal and multi-photon microscopes particularly benefit from the improved resolution for visualization of cellular structures and subcellular processes [5, 6]. In addition, results of applied adaptive optics for detection of eye diseases and in vitro retinal imaging on the cellular level show promising examination and treatment opportunities [7–9].

In many AO systems, the deformable mirror is assumed to have negligible dynamical characteristics in comparison to the dynamic disturbances compensated by the deformable mirror. Unfortunately, this assumption is not always valid and active shape control of deformable mirrors must be employed to enhance the dynamic properties of the deformable mirror. For example, adaptive secondary mirrors for the Multi Mirror Telescope (MMT), the Large Binocular Telescope (LBT), and the Very Large Telescope (VLT) with diameters around 1 m have their first natural resonant frequencies below 10 Hz. In order to be able to use these systems for compensation of atmospherical disturbances with typical frequencies up to 100 Hz, active shape control is employed pushing the bandwith of these DMs to 1 kHz [10–13].

With up to 1170 voice coil actuators and co-located capacitive position sensors, the new generation of continuous face-sheet deformable mirrors requires fast and precise shape control. Thereby, the main idea for robust control of the mirror surface is the use of distributed voice coil actuators in combination with local position sensing by capacitive

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Ruppel; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Ruppel; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Ruppel, licensee InTech. This is an open access chapter distributed under the terms of the Creative

sensors. The use of voice coil actuators allows for large stroke and exact positioning of the mirror while it is floating in a magnetic field. By changing the spatial properties of the magnetic field, the mirror shell can be deformed into a desired shape to compensate for optical aberrations measured by the AO system. Thereby, a low-contact bearing combined with little intrinsic damping of the mirror shell draws the need for adequate damping in closed loop operation. For the MMT deformable mirror, this problem is tackled by a 40 *µ*m air-gap between the mirror shell and a reference plate behind the shell. The induced viscous damping is sufficient to operate the mirror within the designed specifications [11]. However, recent deformable mirrors for the LBT and VLT shall operate within a larger air-gap to provide more stroke. As a side effect, the requirement for a larger air-gap reduces the natural viscous damping and electronic damping is needed to achieve the same control bandwidth [14]. Therefore, the LBT and VLT type deformable secondary mirrors are controlled by local PD-control for each actuator/sensor pair in combination with feedforward force compensation [15]. Each actuator is driven by a dedicated local position controller running at 40-70 kHz. The shape command for the mirror unit is generated by a higher level wavefront control loop running at about 1 kHz. For small set-point changes of the mirror shell, this control concept is well-suited and has proven to be applicable in practice [16].

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Modeling and Control of Deformable Membrane Mirrors

computed in the feedback loop frequency (40-70 kHz for the LBT and VLT type DMs), but only when a new set-point is commanded (about 1 kHz for the LBT and VLT type DMs). Since typical deformable mirrors do not significantly change their dynamical behavior over time, model-based time-invariant feedforward control can efficiently reduce load on implemented feedback controllers. Studies on model-based feedforward control of large deformable mirrors show that either with poorly tuned feedback control or even without feedback control, high speed and high precision deformations of deformable mirrors can be achieved [20, 21].<sup>1</sup> The required dynamical model of the DM can be identified based on internal position measurements of excited DM actuators. In [22], a practical identification procedure for small scale membrane DMs with static interferometric measurements and dynamic measurements from a laser vibrometer are presented, additionally. It is shown that together with an identified dynamical model, feedforward control can be employed for small scale membrane DMs also, significantly improving the settling time of the membrane

In the following, shape control of small and large deformable membrane mirrors by model-based feedforward and feedback control in a two-degree-of-freedom structure are described in a generalized framework. For this purpose, a scalable physical model of deformable membrane mirrors is derived based on force and momentum equilibriums of a differential plate element in Section 2. A series solution of the resulting homogeneous partial differential equation (PDE) is used to derive the modal coordinates of the inhomogeneous PDE including external actuator forces. This series solution can be employed to analyze and simulate the spatio-temporal behaviour of deformable mirrors and allows the design of a model-based shape controller in Section 4. The controller design section is devided into three parts. In Section 4.1, a trajectory generator for computation of differentiable reference trajectories describing the transient set-point change of the deformable mirror surface is described. In Section 4.2, a static and a dynamic feedforward controller are deduced from the inverse system dynamics of the mirror model, aftwards. Finally, a feedback controller is designed in Section 4.3 as a linear quadratic regulator based on a reduced dynamical mirror model in modal coordinates. After transforming the modal feedback controller into physical coordinates, its decentralized structure is implemented and its usability for future

Deformable membrane mirrors with non-contacting actuators are often represented by static models approximating the nonreactive deformation of the mirror surface. Thereby, both Kirchoff and van Kármán theory is used to describe plate deformations smaller than the plate thickness [23–26] and deformations close to the thickness of the mirror plate [27–29], respectively. Additionally, finite element methods are used to model deformable mirrors, in particular for the development large deformable secondary mirrors in astronomy [30–36]. In order to describe actuator influence functions, radially symmetric Gauss functions or splines are commonly used [2, 37], also. A more detailed static analysis is performed in [38] and [39] for a circularly clamped deformable mirror using a Kirchoff plate model. Additionally, in

<sup>1</sup> In this context, feedback control is mainly needed to account for model uncertainties and to reject external

mirror.

deformable mirrors is discussed.

disturbances (e.g. mechanical vibrations, wind loads).

**2. Mirror modeling**

For high speed deformations over large amplitudes (e.g. chopping of deformable secondary mirrors), disadvantages of local PD-control must be considered. First of all, there is a shape-dependent stiffness and damping variation of the mirror shell. In particular, each deformation of the shell requires a specific amount of external force by the distributed actuators [11]. If local control instead of global control is used for position control of the shell, then robust and subsequently conservative controller design is necessary for all local control loops. Secondly, interaction of neighboring actuators has to be studied carefully. The control loop gains have to be chosen such that only little interaction with neighboring control loops is caused. Otherwise, the local control concept can lead to instability of the shell.

In order to further investigate practical concepts for DM shape control, there have been studies on MIMO optimal feedback control [17–19]. Certainly, the closed loop performance of a global optimal feedback control concept is superior to local PD-control. But still, the computational load of a global MIMO controller may not be suitable for large deformable mirrors with more than thousand actuators. Only in [17], the circular symmetry of the mirror shell is used to reduce the controllers complexity. Thereby, it is shown that symmetry can effectively reduce computational loads without loosing control performance. Although, even when symmetry is fully exploited, the computational effort for a global feedback control concept of future deformable mirrors is considerably high.

The most difficult task for control of deformable mirrors clearly is not the stabilization of the shell in a static shape, but changing the mirrors deflection in a predefined time and maintaining system stability. Instead of using a pure feedback controller for this task, model-based feedforward control concepts are proposed as in [20, 21]. In the past, the concept of model-based feedforward control has successfully been applied to various classes of dynamical systems. This technique is widely used in control practice as an extension of a feedback control loop to separately design tracking performance by the feedforward part and closed-loop stability and robustness by the feedback part. By using model-based feedforward control for shape control a deformable mirror, the feedback control loop is only needed to stabilize the shell in steady state and along precomputed trajectories. Thereby, the feedback loop can be designed to achieve high disturbance rejection. The computational load for feedforward control is comparably low because the feedforward signals may not be computed in the feedback loop frequency (40-70 kHz for the LBT and VLT type DMs), but only when a new set-point is commanded (about 1 kHz for the LBT and VLT type DMs).

Since typical deformable mirrors do not significantly change their dynamical behavior over time, model-based time-invariant feedforward control can efficiently reduce load on implemented feedback controllers. Studies on model-based feedforward control of large deformable mirrors show that either with poorly tuned feedback control or even without feedback control, high speed and high precision deformations of deformable mirrors can be achieved [20, 21].<sup>1</sup> The required dynamical model of the DM can be identified based on internal position measurements of excited DM actuators. In [22], a practical identification procedure for small scale membrane DMs with static interferometric measurements and dynamic measurements from a laser vibrometer are presented, additionally. It is shown that together with an identified dynamical model, feedforward control can be employed for small scale membrane DMs also, significantly improving the settling time of the membrane mirror.

In the following, shape control of small and large deformable membrane mirrors by model-based feedforward and feedback control in a two-degree-of-freedom structure are described in a generalized framework. For this purpose, a scalable physical model of deformable membrane mirrors is derived based on force and momentum equilibriums of a differential plate element in Section 2. A series solution of the resulting homogeneous partial differential equation (PDE) is used to derive the modal coordinates of the inhomogeneous PDE including external actuator forces. This series solution can be employed to analyze and simulate the spatio-temporal behaviour of deformable mirrors and allows the design of a model-based shape controller in Section 4. The controller design section is devided into three parts. In Section 4.1, a trajectory generator for computation of differentiable reference trajectories describing the transient set-point change of the deformable mirror surface is described. In Section 4.2, a static and a dynamic feedforward controller are deduced from the inverse system dynamics of the mirror model, aftwards. Finally, a feedback controller is designed in Section 4.3 as a linear quadratic regulator based on a reduced dynamical mirror model in modal coordinates. After transforming the modal feedback controller into physical coordinates, its decentralized structure is implemented and its usability for future deformable mirrors is discussed.
