**4.3. Design of the model-based feedback controller** Σ*ctrl*

Feedforward control combined with suitable trajectory generation methods improve the input output behavior of deformable mirrors significantly. However, in case of model uncertainties or external disturbances, the tracking performance shown in Figure 9 can be affected considerably. For this reason, active position feedback is integrated in modern deformable mirrors measuring the local mirror position at each actuator generating an error compensation command.

Due to the spatially distributed mirror dynamics, global instead of local position control of deformable mirrors is the most promising solution for error compensation and disturbance rejection. However, the computational complexity for high order spatial control of the deformable element typically exceeds available computing power. For this reason, existing deformable membrane mirrors for large telescopes incorporate local feedback instead of global feedback control and neglect some of the global dynamics of the deformable mirror [10, 13, 14, 46–48]. As a side effect, dynamic coupling of separately controlled actuators through the deformable membrane can lead to instability of the individually stable loops and draws the need for carefully designing the control parameters of the local feedback loops.

In the following, an advanced control concept for position control of large deformable mirrors is derived based on the detailed dynamical model of the deformable mirror (7), suitable boundary conditions (9)-(12), and its represenation in modal coordinates (23). The presented controller design differs from existing ones since it incorporates a detailed mirror model and comprises a decentralized structure at the same time.

For feedback controller design, the first N relevant modal differential equations (23) are combined in state space form as

$$\begin{aligned} \dot{\mathbf{x}}(t) &= A\mathbf{x}(t) + Bu(t) \\ y(t) &= \mathbf{C}\mathbf{x}(t), \end{aligned} \tag{27}$$

$$\text{with } \mathbf{x}(t) = \begin{bmatrix} f\_1(t) \ f\_2(t) \ \dots \ f\_N(t) \ \dot{f}\_1 \ \dot{f}\_2 \ \dots \ \dot{f}\_N(t) \end{bmatrix}^T \text{ and system matrices}$$

16 Adaptive Optics

loops.

displacement [µm]

time [ms]

feedforward control showing a settling time of 2 ms

implementation is discussed.

compensation command.

0 0.5 1 1.5

time [ms]

measured identified

displacement [µm]

0 5 10 15 20 0 5 10 15 20

**Figure 9.** Measured responses of the first and second eigenfunctions of an Alpao DM88 deformable membrane mirror with

feedback controller is designed in the following section and its sparse structure for local

Feedforward control combined with suitable trajectory generation methods improve the input output behavior of deformable mirrors significantly. However, in case of model uncertainties or external disturbances, the tracking performance shown in Figure 9 can be affected considerably. For this reason, active position feedback is integrated in modern deformable mirrors measuring the local mirror position at each actuator generating an error

Due to the spatially distributed mirror dynamics, global instead of local position control of deformable mirrors is the most promising solution for error compensation and disturbance rejection. However, the computational complexity for high order spatial control of the deformable element typically exceeds available computing power. For this reason, existing deformable membrane mirrors for large telescopes incorporate local feedback instead of global feedback control and neglect some of the global dynamics of the deformable mirror [10, 13, 14, 46–48]. As a side effect, dynamic coupling of separately controlled actuators through the deformable membrane can lead to instability of the individually stable loops and draws the need for carefully designing the control parameters of the local feedback

In the following, an advanced control concept for position control of large deformable mirrors is derived based on the detailed dynamical model of the deformable mirror (7), suitable boundary conditions (9)-(12), and its represenation in modal coordinates (23). The presented controller design differs from existing ones since it incorporates a detailed mirror model and

For feedback controller design, the first N relevant modal differential equations (23) are

*x*˙(*t*) =*Ax*(*t*) + *Bu*(*t*)

*y*(*t*) =*Cx*(*t*), (27)

measured identified

**4.3. Design of the model-based feedback controller** Σ*ctrl*

comprises a decentralized structure at the same time.

combined in state space form as

$$A = \begin{bmatrix} 0 & 0 & 0 & \dots & 0 & 1 & 0 & 0 & \dots & 0\\ \frac{D\theta\_1^4}{\rho h} & 0 & 0 & \dots & 0 & \frac{\lambda\_d + \kappa\_d \beta\_1^4}{\rho h} & 0 & 0 & \dots & 0\\ 0 & 0 & 0 & \dots & 0 & 0 & 1 & 0 & \dots & 0\\ 0 & \frac{D\theta\_2^4}{\rho h} & 0 & \dots & 0 & 0 & \frac{\lambda\_d + \kappa\_d \beta\_2^4}{\rho h} & 0 & \dots & 0\\ 0 & 0 & 0 & \dots & 0 & 0 & 0 & 1 & \dots & 0\\ 0 & 0 & \frac{D\theta\_3^4}{\rho h} & \dots & 0 & 0 & 0 & \frac{\lambda\_d + \kappa\_d \beta\_3^4}{\rho h} & \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \dots & 0 & 0 & 0 & 0 & \dots & 1\\ 0 & 0 & 0 & \dots & \frac{D\theta\_N^4}{\rho h} & 0 & 0 & 0 & \dots & \frac{\lambda\_d + \kappa\_d \beta\_N^4}{\rho h} \end{bmatrix},\tag{28}$$

$$B = \frac{1}{\rho h} \begin{bmatrix} W\_1(r\_1, \theta\_1) & \dots & W\_1(r\_M, \theta\_M) \\ W\_2(r\_1, \theta\_1) & \dots & W\_2(r\_M, \theta\_M) \\ \vdots & \dots & \vdots \\ W\_N(r\_1, \theta\_1) & \dots & W\_N(r\_M, \theta\_M) \\ 0 & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & 0 \end{bmatrix} \tag{29}$$

$$\mathbf{C} = \begin{bmatrix} W\_1(r\_1, \theta\_1) & \dots & W\_N(r\_1, \theta\_1) & 0 \dots \dots 0 \\ W\_1(r\_2, \theta\_2) & \dots & W\_N(r\_2, \theta\_2) & 0 \dots \dots 0 \\ \vdots & \ddots & \vdots & \vdots \ddots \vdots \\ W\_1(r\_M, \theta\_M) & \dots & W\_N(r\_M, \theta\_M) & 0 \dots \dots 0 \end{bmatrix} \dots \tag{30}$$

A global multi-input multi-ouput linear quadratic regulator is designed for controlling the system along the planned trajectory *xd*(*t*) using the optimization criterion

$$J = \int\_0^\infty \left(\mathbf{x}(t) - \mathbf{x}\_d(t)\right)^T \mathcal{Q}\left(\mathbf{x}(t) - \mathbf{x}\_d(t)\right) + u(t)^T \mathbf{R}u(t) \,\mathrm{d}t,\tag{31}$$

with *Q* being a positive definite and *R* being a positive semi-definite matrix. Assuming the pair (*A*, *Q*) is observable, a minimization of (31) can be achieved by the state feedback

$$
\Sigma\_{\rm ctrl} : \quad \mathfrak{u}(t) = -K \left( \mathfrak{x}(t) - \mathfrak{x}\_d(t) \right) \tag{32}
$$

with

$$K = \mathbb{R}^{-1} \mathbb{B}^{T} P.\tag{33}$$

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Modeling and Control of Deformable Membrane Mirrors

100 101 102 103 104

100 101 102 103 104

Frequency [Hz]

**Figure 10.** Bode diagram of the closed loop local transfer functions at different co-located actuator/sensor positions using

In Equation (37), it can be seen that the columns of *Kp* contain information about how much displacement and velocity feedback is required for a certain actuator in vector *u*(*t*). A visualization of entries in *Kp* shown in Figure 11 reveals that only a limited amount of displacement and velocity information around each actuator is needed to compute the feedback signal *u*(*t*). This property can be used to truncate the spatial extension of the

For a deformable mirror with 672 actuators and sensors, the corresponding normalized entries of the state feedback matrix *Kp* are visualized in Figure 11. Clearly, a choice of *c*<sup>1</sup> = 1000 and *c*<sup>2</sup> = 0.1 results in a fully decentralized structure of *Kp*. Comparing the position and velocity feedback entries of *Kp* in Figure 11, dynamic effects of boundary conditions and actuator position can be seen. Consequently, a model-free decentralized control law for all actuators seems unreasonable and the suggested linear quadratic regulator approach should be considered when designing deformable mirror controllers for astronomical telescopes or comparible application areas. Depending on the truncation area, the computational demands of this control concept scale linearly with the number of actuators and show the applicability

Although controller *K* requires full state information, the state feedback controller can be transformed into an output feedback controller using loop transfer recovery (LTR) and results in an output controller Σ*ctrl* as shown in Figure 6. The ouput controller can be implemented on existing hardware as a finite impulse response (FIR) filter for each actuator/sensor pair where the number of filter coefficients is mainly driven by the acceptable approximation

of global LQ - control for shape control of large deformable mirrors in general.

global LQ regulator and leads to a decentralized control scheme.

error of the loop transfer recovery approach.

Frequency [Hz]



decentralized LQR control (normalized).


Phase [°]


0


Amplitude [dB]

0

Actuator/Sensor 21 Actuator/Sensor 110 Actuator/Sensor 537

The matrix *P* is the symmetric, positive definite solution of the matrix Ricatti equation

$$A^T P + PA - PBR^{-1}B^T P + Q = 0\tag{34}$$

and can be computed efficiently in modern mathematical computing languages (e.g. Matlab). By choosing *Q* = *CTc*1*C* and *R* = *Ic*2, the optimization criterion for a deformable mirror with *M* inputs and *M* outputs simplifies to

$$J = \int\_0^\infty \sum\_{m=1}^M \left( c\_1 y\_m^2(t) + c\_2 u\_m^2(t) \right) \mathrm{d}t \tag{35}$$

and a decentralized controller can be computed via (33) and (34). The particular choice of weight matrices *Q* and *R* is the essential step for a decentralized controller in the LQR framework. The coefficients *c*<sup>1</sup> and *c*<sup>2</sup> are used to tune the closed loop disturbance rejection and robustness until certain settling time and gain margins are achieved.

Since the controller *K* requires modal signals *x*(*t*), a feedback controller *Kp* in physical coordinates can be computed performing an inverse modal transformation Γ−<sup>1</sup> from (13) reading

$$K\_p = \Gamma^{-1} K.\tag{36}$$

with the feedback law

$$\mu(t) = -K\_p \left( \left[ \mathbf{x}(t) \ \dot{\mathbf{x}}(t) \right]^T - \left[ \mathbf{x}\_d(t) \ \dot{\mathbf{x}}\_d(t) \right]^T \right). \tag{37}$$

For comparison, the closed loop transfer functions at selected co-located actuator/sensor locations with active LQ feedback are shown in Figure 10. In comparison to the open loop transfer functions in Figure 5, all relevant low and high frequency resonances are fully damped in the closed loop case and the resulting bandwidth of the deformable mirror exceeds 1 kHz. In order to tune the resulting performance for existing deformable mirrors, a variation of parameters *c*<sup>1</sup> and *c*<sup>2</sup> can be performed easily in practice.

18 Adaptive Optics

*M* inputs and *M* outputs simplifies to

*J* = ∞

*u*(*t*) = −*Kp*

variation of parameters *c*<sup>1</sup> and *c*<sup>2</sup> can be performed easily in practice.

*x*(*t*) *x*˙(*t*)

*<sup>T</sup>* −

For comparison, the closed loop transfer functions at selected co-located actuator/sensor locations with active LQ feedback are shown in Figure 10. In comparison to the open loop transfer functions in Figure 5, all relevant low and high frequency resonances are fully damped in the closed loop case and the resulting bandwidth of the deformable mirror exceeds 1 kHz. In order to tune the resulting performance for existing deformable mirrors, a

*xd*(*t*) *x*˙*d*(*t*)

0

and robustness until certain settling time and gain margins are achieved.

*M* ∑ *m*=1 *c*1*y*<sup>2</sup>

*<sup>K</sup>* = *<sup>R</sup>*−1*BTP* . (33)

*Kp* <sup>=</sup> <sup>Γ</sup>−1*<sup>K</sup>* . (36)

*<sup>T</sup>*

. (37)

d*t* (35)

*<sup>A</sup>TP* + *PA* − *PBR*−1*BTP* + *<sup>Q</sup>* = <sup>0</sup> (34)

The matrix *P* is the symmetric, positive definite solution of the matrix Ricatti equation

and can be computed efficiently in modern mathematical computing languages (e.g. Matlab). By choosing *Q* = *CTc*1*C* and *R* = *Ic*2, the optimization criterion for a deformable mirror with

and a decentralized controller can be computed via (33) and (34). The particular choice of weight matrices *Q* and *R* is the essential step for a decentralized controller in the LQR framework. The coefficients *c*<sup>1</sup> and *c*<sup>2</sup> are used to tune the closed loop disturbance rejection

Since the controller *K* requires modal signals *x*(*t*), a feedback controller *Kp* in physical coordinates can be computed performing an inverse modal transformation Γ−<sup>1</sup> from (13)

*<sup>m</sup>*(*t*) + *<sup>c</sup>*2*u*<sup>2</sup>

*<sup>m</sup>*(*t*) 

with

reading

with the feedback law

**Figure 10.** Bode diagram of the closed loop local transfer functions at different co-located actuator/sensor positions using decentralized LQR control (normalized).

In Equation (37), it can be seen that the columns of *Kp* contain information about how much displacement and velocity feedback is required for a certain actuator in vector *u*(*t*). A visualization of entries in *Kp* shown in Figure 11 reveals that only a limited amount of displacement and velocity information around each actuator is needed to compute the feedback signal *u*(*t*). This property can be used to truncate the spatial extension of the global LQ regulator and leads to a decentralized control scheme.

For a deformable mirror with 672 actuators and sensors, the corresponding normalized entries of the state feedback matrix *Kp* are visualized in Figure 11. Clearly, a choice of *c*<sup>1</sup> = 1000 and *c*<sup>2</sup> = 0.1 results in a fully decentralized structure of *Kp*. Comparing the position and velocity feedback entries of *Kp* in Figure 11, dynamic effects of boundary conditions and actuator position can be seen. Consequently, a model-free decentralized control law for all actuators seems unreasonable and the suggested linear quadratic regulator approach should be considered when designing deformable mirror controllers for astronomical telescopes or comparible application areas. Depending on the truncation area, the computational demands of this control concept scale linearly with the number of actuators and show the applicability of global LQ - control for shape control of large deformable mirrors in general.

Although controller *K* requires full state information, the state feedback controller can be transformed into an output feedback controller using loop transfer recovery (LTR) and results in an output controller Σ*ctrl* as shown in Figure 6. The ouput controller can be implemented on existing hardware as a finite impulse response (FIR) filter for each actuator/sensor pair where the number of filter coefficients is mainly driven by the acceptable approximation error of the loop transfer recovery approach.

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*dyn* generating

*stat* and <sup>Σ</sup>−<sup>1</sup>

Modeling and Control of Deformable Membrane Mirrors

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

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

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

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

The author would like to acknowledge the support by INAF and the Osservatorio Astrofisico di Arcetri in particular, providing finite element data of the LBT ASM and detailed insight into current deformable mirror control schemes. This work was supported by the DFG under

Carl Zeiss AG, Corporate Research and Technology, Optronics Systems, Jena, Germany

[1] Edward P. Wallner. Optimal wave-front correction using slope measurements. *J. Opt.*

[3] B.W. Frazier and R.K. Tyson. Robust control of an adaptive optics system. In *System Theory, 2002. Proceedings of the Thirty-Fourth Southeastern Symposium on*, pages 293–296,

[4] H. M. Martin, G. Brusa Zappellini, B. Cuerden, S. M. Miller, A. Riccardi, and B. K. Smith. Deformable secondary mirrors for the LBT adaptive optics system. In Domenico Ellerbroek, Brent L.; Bonaccini Calia, editor, *Advances in Adaptive Optics II*, volume 6272

DM shape. Second, a static and dynamic feedforward controller <sup>Σ</sup>−<sup>1</sup>

errors in the feedforward part and rejection of external disturbances.

controller that can be implemented as FIR filters on existing hardware.

grant SA-847/10-1 and OS-111/29-1 and by the Carl Zeiss AG.

[2] R.K. Tyson. *Adaptive optics engineering handbook*. CRC, 2000.

requirements, also.

**Author details** Thomas Ruppel

**References**

2002.

*Soc. Am.*, 73(12):1771–1776, 1983.

of *Proc. SPIE*, page 0U, July 2006.

**Acknowledgements**

**Figure 11.** Vislization of the decentralized spatial structure of the feedback matrix *Kp* for selected co-located actuator/sensor pairs.
