**5. Conclusion**

Control of deformable mirrors becomes relevant when the mirror dynamics are slower or equivalent with the dynamics of optical disturbances that shall be compensated by the deformable mirror. Active shape control of deformable mirrors can thereby increase the mirrors bandwidth and make it suitable for the AO task, again. When designing a model-based shape controller for a deformable mirror in a two-degree-of-freedom control structure, there are three components to be considered (see Figure 6): First, a trajectory generator Σ*traj* providing continuous reference trajectories *yd* for fast setpoint changes of the DM shape. Second, a static and dynamic feedforward controller <sup>Σ</sup>−<sup>1</sup> *stat* and <sup>Σ</sup>−<sup>1</sup> *dyn* generating control commands for driving the mirror along the precomputed trajectory based on inverse system dynamics. Finally, a feedback controller Σ*ctrl* responsible for compensation of model errors in the feedforward part and rejection of external disturbances.

In this chapter, model-based design steps for all three components were shown supported by experimental and simulation results. The key for decentralized controller design is the right choice of weighting matrices *Q* and *R* in the LQR framework. Followed by a loop transfer recovery approach, the state feedback controller can be transformed into an ouput feedback controller that can be implemented as FIR filters on existing hardware.

Future astronomical optical telescopes, e.g. the European Extremely Large Telescope (E-ELT) or the Giant Magellan Telescope (GMT), will include deformable mirrors with many thousand actuators and position sensors. Due to the inherent slow dynamics of the large deformable mirror shells, active shape control of these elements is inevitable. Model-based shape control in a two-degree-of-freedom structure can greatly improve the performance of these elements and should be considered for comparible AO systems with high performance requirements, also.
