**3. Fundamental solution and modal analysis**

One way to analyze PDE (7) is to discretize the PDE in its spatial coordinates (*r*, *θ*). Thereby, the spatial differential operator ∇<sup>4</sup> and the boundary conditions (9)-(11) can be approximated by finite difference or finite element methods and the PDE is reduced to a finite set of coupled ordinary differential equations (ODE). The Eigenfrequencies and the eigenvectors of the resulting generalized eigenvalue problem correspond to the spatially discretized eigenfunctions of the PDE. Thereby, the spatial discretization has strong influence on the remaining temporal system dynamics and can even mislead to wrong results if performed insufficiently. The typically large set of ordinary differential equations requires high computing power for dynamic simulation and is ill-suited for model-based controller design, in general.

Another way to analyze PDE (7) is a modal transformation based on the eigenfunctions of PDE (7) fulfilling the homogeneous boundary conditions (9)-(11). The modal transformation leads to an infinite set of ordinary differential equations in modal coordinates describing only the temporal evolution of the eigenfunctions of PDE (7). Thereby, all spatial and temporal properties are preserved and a reduced set of the ODEs can be used for dynamic simulation and controller design.

The eigenfuntions *Wk*(*r*, *θ*) of PDE (7) can be derived by separation of variables with

$$\Gamma: \quad w(r, \theta, t) = \sum\_{k=1}^{\infty} W\_k(r, \theta) f\_k(t), \tag{13}$$

10.5772/52726

107

<sup>=</sup> 0 , (16)

http://dx.doi.org/10.5772/52726

Modeling and Control of Deformable Membrane Mirrors

Λ*c*(*βk*)

since the matrix elements are rather extensive.

since the matrix elements are rather extensive.

  *A*1*<sup>k</sup> A*2*<sup>k</sup> A*3*<sup>k</sup> A*4*<sup>k</sup>*

to infinitely many eigenvalues *β<sup>k</sup>* related to the eigenfunction *Wk*(*r*, *θ*).

 

<sup>=</sup> 0, <sup>Λ</sup>*s*(*βk*)

where Λ*c*(*β*) and Λ*s*(*β*) are [4 × 4] matrices containing linear combinations of Bessel functions depending on the eigenvalue *β<sup>k</sup>* for the cosine and sine parts of *Wk*(*r*, *θ*). A computation of matrices Λ*<sup>c</sup>* and Λ*<sup>s</sup>* should be performed using computer algebra software

where Λ*c*(*β*) and Λ*s*(*β*) are [4 × 4] matrices containing linear combinations of Bessel functions depending on the eigenvalue *β<sup>k</sup>* for the cosine and sine parts of *Wk*(*r*, *θ*). A computation of matrices Λ*<sup>c</sup>* and Λ*<sup>s</sup>* should be performed using computer algebra software

In order to compute a non-trivial solution of the homogeneous equations (16), the parameter *β<sup>k</sup>* needs to determined such that Λ*<sup>c</sup>* and Λ*<sup>s</sup>* contain a non-trivial kernel. This can be achieved by numerically searching for zeros in the determinant of the matrices Λ*<sup>c</sup>* and Λ*<sup>s</sup>* and leads

In Figure 4, the consine parts of the first 21 analytical eigenfunctions *Wk*(*r*, *θ*), *k* = 1, . . . , 21 of a deformable mirror with clamped inner radius and free outer radius are shown based on the calculations given before. The eigenfunctions *Wk*(*r*, *θ*) and eigenvalues *β<sup>k</sup>* were compared with a detailed finite element model of the same mirror and show excellent compliance.

**Figure 4.** First 21 eigenfunctions of a deformable mirror model (7) with clamped edge boundary conditions (9) at the inner

radius *r*<sup>1</sup> and free edge boundary conditions (11) at the outer radius *r*<sup>2</sup> ordered by increasing eigenfrequency.

  *B*1*k B*2*k B*3*k B*4*k*  

where the eigenfunctions *Wk*(*r*, *θ*) describe the spatial characteristics and the modal coefficients *fk*(*t*) describe the time-varying amplitude. At the same time, equation (13) can be used to perform a transformation from modal coordiantes *fk*(*t*) to physcial coordinates *w*(*r*, *θ*, *t*) and back, which will be needed for feedback controller design in Section 4.3.

After inserting (13) in the homogeneous PDE (7) with *q*(*r*, *θ*) = 0, an analytical description of the eigefunctions *Wk*(*r*, *θ*, *t*) can be found as described in [23] reading

$$\begin{split} \mathcal{W}\_{k}(r,\theta) &= \left( A\_{1k}I\_{k}(\beta r) + A\_{2k}Y\_{k}(\beta r) + A\_{3k}I\_{k}(\beta r) + A\_{4k}K\_{k}(\beta r) \right) \cos(k\theta) \\ &+ \left( B\_{1k}I\_{k}(\beta r) + B\_{2k}Y\_{k}(\beta r) + B\_{3k}I\_{k}(\beta r) + B\_{4k}K\_{k}(\beta r) \right) \sin(k\theta). \end{split} \tag{14}$$

Here, *Jk*,*Yk*, *Ik*, *Kk* are Bessel functions and modified Bessel functions of first and second kind. The eigenfunctions fulfill the relation

$$
\nabla^4 \mathcal{W}\_k(r, \theta) = \mathcal{J}^4 \mathcal{W}\_k(r, \theta) \tag{15}
$$

and are self-adjoint due to the biharmonic operator ∇4.

After inserting the fundamental solution (14) in the boundary conditions (9)-(11), the free parameters *A*1*k*,..., *A*4*<sup>k</sup>* and *B*1*k*,..., *B*4*<sup>k</sup>* can be found by computing a non-trivial solution of the resulting homogeneous equations

$$
\Lambda\_{\mathfrak{C}}(\beta\_k) \begin{bmatrix} A\_{1k} \\ A\_{2k} \\ A\_{3k} \\ A\_{4k} \end{bmatrix} = 0, \quad \Lambda\_{\mathfrak{s}}(\beta\_k) \begin{bmatrix} B\_{1k} \\ B\_{2k} \\ B\_{3k} \\ B\_{4k} \end{bmatrix} = 0 \,, \tag{16}
$$

where Λ*c*(*β*) and Λ*s*(*β*) are [4 × 4] matrices containing linear combinations of Bessel functions depending on the eigenvalue *β<sup>k</sup>* for the cosine and sine parts of *Wk*(*r*, *θ*). A computation of matrices Λ*<sup>c</sup>* and Λ*<sup>s</sup>* should be performed using computer algebra software since the matrix elements are rather extensive.

8 Adaptive Optics

design, in general.

and controller design.

The eigenfunctions fulfill the relation

the resulting homogeneous equations

and are self-adjoint due to the biharmonic operator ∇4.

**3. Fundamental solution and modal analysis**

One way to analyze PDE (7) is to discretize the PDE in its spatial coordinates (*r*, *θ*). Thereby, the spatial differential operator ∇<sup>4</sup> and the boundary conditions (9)-(11) can be approximated by finite difference or finite element methods and the PDE is reduced to a finite set of coupled ordinary differential equations (ODE). The Eigenfrequencies and the eigenvectors of the resulting generalized eigenvalue problem correspond to the spatially discretized eigenfunctions of the PDE. Thereby, the spatial discretization has strong influence on the remaining temporal system dynamics and can even mislead to wrong results if performed insufficiently. The typically large set of ordinary differential equations requires high computing power for dynamic simulation and is ill-suited for model-based controller

Another way to analyze PDE (7) is a modal transformation based on the eigenfunctions of PDE (7) fulfilling the homogeneous boundary conditions (9)-(11). The modal transformation leads to an infinite set of ordinary differential equations in modal coordinates describing only the temporal evolution of the eigenfunctions of PDE (7). Thereby, all spatial and temporal properties are preserved and a reduced set of the ODEs can be used for dynamic simulation

> ∞ ∑ *k*=1

where the eigenfunctions *Wk*(*r*, *θ*) describe the spatial characteristics and the modal coefficients *fk*(*t*) describe the time-varying amplitude. At the same time, equation (13) can be used to perform a transformation from modal coordiantes *fk*(*t*) to physcial coordinates *w*(*r*, *θ*, *t*) and back, which will be needed for feedback controller design in Section 4.3.

After inserting (13) in the homogeneous PDE (7) with *q*(*r*, *θ*) = 0, an analytical description

*Wk*(*r*, *θ*) = (*A*1*<sup>k</sup> Jk*(*βr*) + *A*2*kYk*(*βr*) + *A*3*<sup>k</sup> Ik*(*βr*) + *A*4*kKk*(*βr*)) cos(*kθ*)

Here, *Jk*,*Yk*, *Ik*, *Kk* are Bessel functions and modified Bessel functions of first and second kind.

After inserting the fundamental solution (14) in the boundary conditions (9)-(11), the free parameters *A*1*k*,..., *A*4*<sup>k</sup>* and *B*1*k*,..., *B*4*<sup>k</sup>* can be found by computing a non-trivial solution of

+ (*B*1*<sup>k</sup> Jk*(*βr*) + *B*2*kYk*(*βr*) + *B*3*<sup>k</sup> Ik*(*βr*) + *B*4*kKk*(*βr*)) sin(*kθ*). (14)

∇<sup>4</sup>*Wk*(*r*, *θ*) = *β*<sup>4</sup>*Wk*(*r*, *θ*) (15)

*Wk*(*r*, *θ*)*fk*(*t*), (13)

The eigenfuntions *Wk*(*r*, *θ*) of PDE (7) can be derived by separation of variables with

Γ : *w*(*r*, *θ*, *t*) =

of the eigefunctions *Wk*(*r*, *θ*, *t*) can be found as described in [23] reading

where Λ*c*(*β*) and Λ*s*(*β*) are [4 × 4] matrices containing linear combinations of Bessel functions depending on the eigenvalue *β<sup>k</sup>* for the cosine and sine parts of *Wk*(*r*, *θ*). A computation of matrices Λ*<sup>c</sup>* and Λ*<sup>s</sup>* should be performed using computer algebra software since the matrix elements are rather extensive.

In order to compute a non-trivial solution of the homogeneous equations (16), the parameter *β<sup>k</sup>* needs to determined such that Λ*<sup>c</sup>* and Λ*<sup>s</sup>* contain a non-trivial kernel. This can be achieved by numerically searching for zeros in the determinant of the matrices Λ*<sup>c</sup>* and Λ*<sup>s</sup>* and leads to infinitely many eigenvalues *β<sup>k</sup>* related to the eigenfunction *Wk*(*r*, *θ*).

In Figure 4, the consine parts of the first 21 analytical eigenfunctions *Wk*(*r*, *θ*), *k* = 1, . . . , 21 of a deformable mirror with clamped inner radius and free outer radius are shown based on the calculations given before. The eigenfunctions *Wk*(*r*, *θ*) and eigenvalues *β<sup>k</sup>* were compared with a detailed finite element model of the same mirror and show excellent compliance.

**Figure 4.** First 21 eigenfunctions of a deformable mirror model (7) with clamped edge boundary conditions (9) at the inner radius *r*<sup>1</sup> and free edge boundary conditions (11) at the outer radius *r*<sup>2</sup> ordered by increasing eigenfrequency.

In the following, a modal decomposition of PDE (7) is used to analyze the mirror dynamics and to derive model-based feedforward control commands in modal coordinates. The modal decomposition of (7) is performed by inserting relation (13) in (7) and using property (15) yielding

10.5772/52726

109

*um*(*τ*)d*τ*, (20)

http://dx.doi.org/10.5772/52726

*<sup>k</sup>*)<sup>2</sup> <sup>−</sup> <sup>4</sup>*Dρhβ*<sup>4</sup>

<sup>2</sup> is derived from the

*u*<sup>1</sup> *u*<sup>2</sup> ... *uM*

*k* .

*<sup>w</sup>*(*r*, *<sup>θ</sup>*, *<sup>t</sup>*) =

− *<sup>t</sup>*

<sup>2</sup>*ρ<sup>h</sup>* (*λd*+*κdβ*<sup>4</sup>

be used for every actuator/sensor pair of the DM.

control loop when designing the AO loop controller.

**4. Model-based controller design**

time *T* from an initial deformation △*w*<sup>∗</sup>

respect to the *L*<sup>2</sup> scalar product �*Wk* , *Wk* � = 1.

with

*gk*(*t* − *τ*) = *e*

*t*

∞ ∑ *k*=1

*<sup>k</sup>*+*ζ<sup>k</sup>* ) <sup>−</sup> *<sup>e</sup>*

*M* ∑ *m*=1

1 *ζk*

− *<sup>t</sup>*

<sup>2</sup>*ρ<sup>h</sup>* (*λd*+*κdβ*<sup>4</sup>

Equation (20) can be used to compute the actuator influence functions for any input signal *um*(*τ*), *m* = 1... *M*. Thereby, not only the resulting static deformation of the mirror can be computed, but also the transient motion of the plate for time-varying input forces.<sup>3</sup> Additionally, relation (20) can be used to compute the frequency responses of a deformable mirror at different actuator locations as shown in Figure 5 for a deformable mirror with clamped inner edge, free outer edge, and co-located force actuators and position sensors. The infinite sum of eigenfunctions *Wk*(*r*, *θ*) in Equation (20) can be approximated with a finite *k* using only a limited number of eigenfunctions. The right number of eigenfunctions can either be driven by a sufficient static coverage of mirror deformations with linear combinations of *k* eigenfunctions *Wk*(*r*, *θ*), or by considering a certain number of eigenvalues *β<sup>k</sup>* in order to sufficiently describe the mirror dynamics up to a certain eigenfrequency.

Obviously, the frequency responses shown in Figure 5 reveal essential variations of local mirror dynamics depending on the co-located actuator/sensor position. From a control point of view, this behavior illustrates the difficulty of designing a decentralized controller that can

In a typical AO system, the requirement of changing the mirror shape within a predefined

higher-ranking AO control loop (see Figure 6). This control loop runs at a fixed cycle time and sends mirror deformations △*w*∗(*t*) to the shape controller in order to correct for measured optical disturbances in the AO system (not shown in Figure 6). The step inputs △*w*∗(*t*) for the model-based shape controller in Figure 6 are in general ill-suited for model-based feedforward control and need to be filtered beforehand. The transition time *T* for changing the mirror shape from its initial shape △*w*<sup>1</sup> to its new desired shape △*w*<sup>2</sup> is assumed to be smaller than the cycle time of the higher-ranking AO control loop. This requirement assures that the closed loop mirror dynamics can be neglected with respect to the higher-ranking AO

For model-based shape control of deformable mirrors, the control scheme shown in Figure

<sup>3</sup> In Equation (20) and for the following analysis, it is assumed that the eigenfunctions *Wk* (*r*, *θ*) are normalized with

6 can be used. It consists of a deformable mirror with multiple inputs *u* =

<sup>1</sup> to a final deformation △*w*<sup>∗</sup>

*Wk*(*r*, *θ*) *Wk*(*rm*, *θm*)*gk*(*t* − *τ*)

*<sup>k</sup>*−*ζ<sup>k</sup>* ), *<sup>ζ</sup><sup>k</sup>* <sup>=</sup>

(*λ<sup>d</sup>* + *κdβ*<sup>4</sup>

Modeling and Control of Deformable Membrane Mirrors

 *G*(*r*,*θ*,*rm*,*θm*,*t*−*τ*)

*τ*=0

$$\sum\_{k=1}^{\infty} \left( D\beta\_k^4 f\_k(t) + (\lambda\_d + \kappa\_d \beta^4) f\_k(t) + \rho h f\_k(t) \right) W\_k(r, \theta) = \sum\_{m=1}^{M} \frac{u\_m(t)}{r\_m} \delta(r - r\_m) \delta(\theta - \theta\_m). \tag{17}$$

A multiplication with any eigenfunction *Wj* and integration over domain *S* on both sides gives

$$\begin{aligned} \left\| \iint \mathcal{W}\_{\vec{j}}(\mathbf{r}, \theta) \sum\_{k=1}^{\infty} \left( D\beta\_{k}^{4} f\_{k}(t) + (\lambda\_{d} + \kappa\_{d}\beta\_{k}^{4}) \dot{f}\_{k}(t) + \rho h \ddot{f}\_{k}(t) \right) \mathcal{W}\_{k}(\mathbf{r}, \theta) \mathrm{d}S = \\ \iint \mathcal{W}\_{\vec{j}}(\mathbf{r}, \theta) \sum\_{m=1}^{M} \frac{u\_{m}(t)}{r\_{m}} \delta(r - r\_{m}) \delta(\theta - \theta\_{m}) \mathrm{d}S. \end{aligned} \tag{18}$$

Changing order of summation and integration, using the orthonormality property of the eigenfunctions *Wk*(*r*, *θ*), and considering the sifting property of the Dirac delta function *δ* on the right hand side of Equation (18) leads to an infinite set of second order ordinary differential equations

$$\dot{f}\_{\dot{f}}(t) + \frac{\lambda\_d + \kappa\_d \beta\_{\dot{f}}^4}{\rho h} \dot{f}\_{\dot{f}}(t) + \frac{D\beta\_{\dot{f}}^4}{\rho h} f\_{\dot{f}}(t) = \frac{1}{\rho h} \sum\_{m=1}^{M} \mathcal{W}\_{\dot{f}}(r\_{m\nu}\theta\_{\mathcal{W}}) u\_m(t), \tag{19}$$

$$f\_{\dot{f}}(0) = 0, \quad \dot{f}\_{\dot{f}}(0) = 0, \quad \qquad \qquad \dot{j} \in \mathbb{N}.$$

For each *j*, the resulting ODE (19) describes the temporal evolution of the corresponding eigenfunction *Wj*(*r*, *θ*) by the modal coefficient *fj*(*t*). For simplicity, homogeneous initial conditions are assumed describing a flat and steady mirror surface.

Since there is no coupling between different modal coefficients *fk*(*t*) and *fj*(*t*), *j* �= *k*, the ODEs (19) can be used for dynamic simulation of a mirror surface described by a specific eigenfunction *Wj*(*r*, *θ*) or a linear combination of eigenfunctions. Any physically relevant mirror deformation can be decomposed into a linear combination of the orthonormal eigenfunctions *Wk*(*r*, *θ*) and all relevant dynamic properties of the deformable mirror can be covered with this approach. The decoupled description of the mirror dynamics in modal coordinates (19) allows for simplified system analysis and is the foundation for the following controller design.

Based on the derived normalized eigenfunctions *Wk*(*r*, *θ*), an analytical solution of the biharmonic equation (7) with initial conditions *w*(*r*, *θ*, 0) = 0 and *w*˙(*r*, *θ*, 0) = 0 can be derived in spectral form reading

$$w(r,\theta,t) = \underbrace{\int\_{\tau=0}^{t} \sum\_{k=1}^{\infty} \sum\_{m=1}^{M} \frac{1}{\mathcal{I}\_{k}} \mathcal{W}\_{k}(r,\theta) \, \mathcal{W}\_{k}(r\_{m},\theta\_{m}) \mathcal{g}\_{k}(t-\tau) \, u\_{m}(\tau) \, \text{d}\tau}\_{\mathcal{G}(r\theta,r\_{m}\theta\_{m}t-\tau)} \tag{20}$$

with

10 Adaptive Optics

yielding

∞ ∑ *k*=1 *Dβ*<sup>4</sup>

gives

*Wj*(*r*, *θ*)

differential equations

controller design.

in spectral form reading

¨ *fj*(*t*) +

∞ ∑ *k*=1 *Dβ*<sup>4</sup>

*S*

*<sup>k</sup> fk*(*t*)+(*λ<sup>d</sup>* <sup>+</sup> *<sup>κ</sup>dβ*4) ˙

In the following, a modal decomposition of PDE (7) is used to analyze the mirror dynamics and to derive model-based feedforward control commands in modal coordinates. The modal decomposition of (7) is performed by inserting relation (13) in (7) and using property (15)

A multiplication with any eigenfunction *Wj* and integration over domain *S* on both sides

*fk*(*t*)+*ρh* ¨

*S*

Changing order of summation and integration, using the orthonormality property of the eigenfunctions *Wk*(*r*, *θ*), and considering the sifting property of the Dirac delta function *δ* on the right hand side of Equation (18) leads to an infinite set of second order ordinary

For each *j*, the resulting ODE (19) describes the temporal evolution of the corresponding eigenfunction *Wj*(*r*, *θ*) by the modal coefficient *fj*(*t*). For simplicity, homogeneous initial

Since there is no coupling between different modal coefficients *fk*(*t*) and *fj*(*t*), *j* �= *k*, the ODEs (19) can be used for dynamic simulation of a mirror surface described by a specific eigenfunction *Wj*(*r*, *θ*) or a linear combination of eigenfunctions. Any physically relevant mirror deformation can be decomposed into a linear combination of the orthonormal eigenfunctions *Wk*(*r*, *θ*) and all relevant dynamic properties of the deformable mirror can be covered with this approach. The decoupled description of the mirror dynamics in modal coordinates (19) allows for simplified system analysis and is the foundation for the following

Based on the derived normalized eigenfunctions *Wk*(*r*, *θ*), an analytical solution of the biharmonic equation (7) with initial conditions *w*(*r*, *θ*, 0) = 0 and *w*˙(*r*, *θ*, 0) = 0 can be derived

*Wk*(*r*, *θ*) =

*fk*(*t*) 

*Wj*(*r*, *θ*)

*ρh*

*M* ∑ *m*=1

*fj*(0) = 0, *j* ∈ **N** .

*M* ∑ *m*=1

*Wk*(*r*, *θ*)d*S* =

*M* ∑ *m*=1

*um*(*t*) *rm*

*um*(*t*) *rm*

*δ*(*r* − *rm*)*δ*(*θ* − *θm*). (17)

*δ*(*r* − *rm*)*δ*(*θ* − *θm*)d*S*.

*Wj*(*rm*, *θm*)*um*(*t*), (19)

(18)

*fk*(*t*) + *ρh* ¨

*<sup>k</sup> fk*(*t*)+(*λ<sup>d</sup>* <sup>+</sup> *<sup>κ</sup>dβ*<sup>4</sup>

*λ<sup>d</sup>* + *κdβ*<sup>4</sup> *j <sup>ρ</sup><sup>h</sup>* ˙

*fj*(*t*) +

conditions are assumed describing a flat and steady mirror surface.

*fj*(0) = 0, ˙

*Dβ*<sup>4</sup> *j <sup>ρ</sup><sup>h</sup> fj*(*t*) = <sup>1</sup>

*fk*(*t*) 

*k*) ˙

$$g\_k(t - \tau) = e^{-\frac{l}{2\mu l} \left(\lambda\_d + \kappa\_d \beta\_k^4 + \zeta\_k\right)} - e^{-\frac{l}{2\mu l} \left(\lambda\_d + \kappa\_d \beta\_k^4 - \zeta\_k\right)}, \quad \zeta\_k = \sqrt{(\lambda\_d + \kappa\_d \beta\_k^4)^2 - 4D\rho h \beta\_k^4}.$$

Equation (20) can be used to compute the actuator influence functions for any input signal *um*(*τ*), *m* = 1... *M*. Thereby, not only the resulting static deformation of the mirror can be computed, but also the transient motion of the plate for time-varying input forces.<sup>3</sup> Additionally, relation (20) can be used to compute the frequency responses of a deformable mirror at different actuator locations as shown in Figure 5 for a deformable mirror with clamped inner edge, free outer edge, and co-located force actuators and position sensors.

The infinite sum of eigenfunctions *Wk*(*r*, *θ*) in Equation (20) can be approximated with a finite *k* using only a limited number of eigenfunctions. The right number of eigenfunctions can either be driven by a sufficient static coverage of mirror deformations with linear combinations of *k* eigenfunctions *Wk*(*r*, *θ*), or by considering a certain number of eigenvalues *β<sup>k</sup>* in order to sufficiently describe the mirror dynamics up to a certain eigenfrequency.

Obviously, the frequency responses shown in Figure 5 reveal essential variations of local mirror dynamics depending on the co-located actuator/sensor position. From a control point of view, this behavior illustrates the difficulty of designing a decentralized controller that can be used for every actuator/sensor pair of the DM.
