**4. Model-based controller design**

In a typical AO system, the requirement of changing the mirror shape within a predefined time *T* from an initial deformation △*w*<sup>∗</sup> <sup>1</sup> to a final deformation △*w*<sup>∗</sup> <sup>2</sup> is derived from the 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 control loop when designing the AO loop controller.

For model-based shape control of deformable mirrors, the control scheme shown in Figure 6 can be used. It consists of a deformable mirror with multiple inputs *u* = *u*<sup>1</sup> *u*<sup>2</sup> ... *uM* 

<sup>3</sup> In Equation (20) and for the following analysis, it is assumed that the eigenfunctions *Wk* (*r*, *θ*) are normalized with respect to the *L*<sup>2</sup> scalar product �*Wk* , *Wk* � = 1.

(force actuators), multiple outputs *y* = *y*<sup>1</sup> *y*<sup>2</sup> ... *yM* (position sensors), and an underlying shape controller in a two-degree-of-freedom control structure. The control structure contains an online trajectory generator Σ<sup>∗</sup> *traj*, a static and a dynamic model-based feedforward controller <sup>Σ</sup>−<sup>1</sup> *stat* and <sup>Σ</sup>−<sup>1</sup> *dyn*, and a model-based feedback controller <sup>Σ</sup>*crtl*.

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Modeling and Control of Deformable Membrane Mirrors

*traj*,

*traj*

**Figure 6.** Two-degree-of-freedom structure for shape control of deformable mirrors modeled as a multi-input multi-ouput (MIMO) system with co-located non-contacting force actuators and position sensors consisting of a trajectory generator Σ<sup>∗</sup>

**Figure 7.** Illustration of a general input signal △*w*∗(*t*) and the resulting output signal *yd* (*t*) of the trajectory generator <sup>Σ</sup><sup>∗</sup>

There are several ways to generate the output signal *yd* based on the step input △*w*∗, e.g. finite impuls response (FIR) filters, sigmoid basis functions, or segmented polynomials. In [20] and [21] a detailed introduction to this topic is given with respect to control of deformable mirrors. A more general and comprehensive discussion on trajectory generation can be found in [45]. Here, a polynomial ansatz function is used to generate the output signal

Depending on the required smoothness of *yd*, the reference trajectory *yd* is described by

<sup>2</sup> − △*w*<sup>∗</sup>

<sup>1</sup> + (△*w*<sup>∗</sup>

*stat* and <sup>Σ</sup>−<sup>1</sup>

*dyn*, and a feedback controller Σ*ctrl* .

<sup>1</sup> )*κ*(*<sup>t</sup>* − *<sup>t</sup>*0), *<sup>t</sup>* ∈ [*t*0, *<sup>t</sup>*1] , *<sup>t</sup>*<sup>1</sup> = *<sup>t</sup>*<sup>0</sup> + *<sup>T</sup>*, (21)

a static and a dynamic model-based feedforward controller Σ−<sup>1</sup>

within a fixed transition time *T* = *t*<sup>2</sup> − *t*1.

*yd* for simplicity.

Σ∗

*traj* : *yd*(*t*) = △*w*<sup>∗</sup>

There are two control objectives adressed by the control structure shown in Figure 6. One is achieving good tracking performance along a spatio-temporal trajectory *yd* describing the evolution of the mirror shape from an initial shape △*w*<sup>∗</sup> <sup>1</sup> to a new commanded mirror shape △*w*<sup>∗</sup> <sup>2</sup> in a predefined time *T*. The second control objective is the stabilization of the deformable mirror shell by feedback control Σ*ctrl* along the generated trajectory *yd* and in its final position △*w*<sup>∗</sup> <sup>2</sup> with respect to external disturbances and model uncertainties.

#### **4.1. Design of the trajectory generator** Σ<sup>∗</sup> *traj*

The trajectory generator Σ<sup>∗</sup> *traj* generates a continuously differentiable output signal *yd* that reaches its final value △*w*<sup>∗</sup> <sup>2</sup> within a fixed transition time *T*. In Figure 7, the piecewise constant input △*w*<sup>∗</sup> and filtered output signal *yd* are illustrated for a single set-point change. Since the deformable mirror consists of many inputs and outputs, the signals △*w*<sup>∗</sup> and *yd* are vectorial variables containing deformation values either in modal or physical coordinates. The choice of units can be driven by the higher-ranking AO control loop and is only of minor importance for the following section.4

**Figure 5.** Bode diagram of the analytical local transfer functions of a deformable mirror at different co-located actuator/sensor positions (normalized).

<sup>4</sup> A linear forward or backward transformation can be used to transform physical coordinates *w*(*r*, *θ*, *t*) into modal coordinates *fk* (*t*) using the orthonormal eigenfunctions *Wk* (*r*, *θ*) in 13.

10.5772/52726

12 Adaptive Optics

controller <sup>Σ</sup>−<sup>1</sup>

shape △*w*<sup>∗</sup>



positions (normalized).


Phase [°]


0

Amplitude [dB]

final position △*w*<sup>∗</sup>

The trajectory generator Σ<sup>∗</sup>

reaches its final value △*w*<sup>∗</sup>

importance for the following section.4

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

coordinates *fk* (*t*) using the orthonormal eigenfunctions *Wk* (*r*, *θ*) in 13.

(force actuators), multiple outputs *y* =

**4.1. Design of the trajectory generator** Σ<sup>∗</sup>

the evolution of the mirror shape from an initial shape △*w*<sup>∗</sup>

an online trajectory generator Σ<sup>∗</sup>

*stat* and <sup>Σ</sup>−<sup>1</sup>

*y*<sup>1</sup> *y*<sup>2</sup> ... *yM*

shape controller in a two-degree-of-freedom control structure. The control structure contains

*dyn*, and a model-based feedback controller <sup>Σ</sup>*crtl*. There are two control objectives adressed by the control structure shown in Figure 6. One is achieving good tracking performance along a spatio-temporal trajectory *yd* describing

deformable mirror shell by feedback control Σ*ctrl* along the generated trajectory *yd* and in its

*traj*

constant input △*w*<sup>∗</sup> and filtered output signal *yd* are illustrated for a single set-point change. Since the deformable mirror consists of many inputs and outputs, the signals △*w*<sup>∗</sup> and *yd* are vectorial variables containing deformation values either in modal or physical coordinates. The choice of units can be driven by the higher-ranking AO control loop and is only of minor

<sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> -150

Frequency [Hz]

Frequency [Hz]

**Figure 5.** Bode diagram of the analytical local transfer functions of a deformable mirror at different co-located actuator/sensor

<sup>4</sup> A linear forward or backward transformation can be used to transform physical coordinates *w*(*r*, *θ*, *t*) into modal

100 101 102 103

<sup>2</sup> in a predefined time *T*. The second control objective is the stabilization of the

<sup>2</sup> with respect to external disturbances and model uncertainties.

*traj*, a static and a dynamic model-based feedforward

*traj* generates a continuously differentiable output signal *yd* that

<sup>2</sup> within a fixed transition time *T*. In Figure 7, the piecewise

(position sensors), and an underlying

<sup>1</sup> to a new commanded mirror

**Figure 6.** Two-degree-of-freedom structure for shape control of deformable mirrors modeled as a multi-input multi-ouput (MIMO) system with co-located non-contacting force actuators and position sensors consisting of a trajectory generator Σ<sup>∗</sup> *traj*, a static and a dynamic model-based feedforward controller Σ−<sup>1</sup> *stat* and <sup>Σ</sup>−<sup>1</sup> *dyn*, and a feedback controller Σ*ctrl* .

**Figure 7.** Illustration of a general input signal △*w*∗(*t*) and the resulting output signal *yd* (*t*) of the trajectory generator <sup>Σ</sup><sup>∗</sup> *traj* within a fixed transition time *T* = *t*<sup>2</sup> − *t*1.

There are several ways to generate the output signal *yd* based on the step input △*w*∗, e.g. finite impuls response (FIR) filters, sigmoid basis functions, or segmented polynomials. In [20] and [21] a detailed introduction to this topic is given with respect to control of deformable mirrors. A more general and comprehensive discussion on trajectory generation can be found in [45]. Here, a polynomial ansatz function is used to generate the output signal *yd* for simplicity.

Depending on the required smoothness of *yd*, the reference trajectory *yd* is described by

$$\Delta^\*\_{\text{traj}}: \quad y\_d(t) = \triangle w\_1^\* + (\triangle w\_2^\* - \triangle w\_1^\*) \kappa(t - t\_0), \quad t \in [t\_0, t\_1], \ t\_1 = t\_0 + T,\tag{21}$$

$$\kappa(t) = \begin{cases} 0 & \text{if } t \le (t\_1 - t\_0) \\ \frac{(2n+1)!}{n!(t\_1 - t\_0)^{2n+1}} \sum\_{i=0}^{n} \frac{(-1)^{n-i}}{i!(n-i)!(2n-i+1)} (t\_1 - t\_0)^i t^{2n-i+1} \\ & \text{if } 0 \le t \le (t\_1 - t\_0) \\ 1 & \text{if } t \ge (t\_1 - t\_0). \end{cases} \tag{22}$$


static command *ustat*

system dynamics <sup>Σ</sup>−<sup>1</sup>

displacement [µm]

<sup>1</sup> measured

0 5 10 15 20 time [ms]

without feedforward control showing a settling time above 5-10 ms

for a practical implementation in existing systems.

Σ−<sup>1</sup>

Afterwards, the final feedforward command *ud* can be computed as

is implemented at this stage and only the trajectory generator Σ<sup>∗</sup>

therein in case of zero dynamics in the differential equation (19).

*stat* : *<sup>u</sup>stat*

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

identified


displacement [µm]

**Figure 8.** Measured step responses of the first and second eigenfunctions of an Alpao DM88 deformable membrane mirror

consider only the static components of the inverse system dynamics (23) and compute a

*<sup>m</sup>* (*t*) = <sup>1</sup>

Regarding the computational complexity, in [20] is shown that the computational demands of feedforward control scale quadratically with the number of system inputs and outputs. However, since the control signal only needs to be computed with the frequency of the outer AO-loop (e.g. 1 kHz in modern astronomical AO systems), there are currently no burdens

In Figure 8 and Figure 9 measurement results of a setpoint change without and with feedforward control for an Alpao deformable mirror with 88 voice coil actuators is shown. In the experiment, the first two eigenfunctions of the deformable mirror are excited by a modal step input (spatially distributed step command for all 88 actuators at the same time) and a model-based feedforward control input *ud* (spatially distributed time-varying signal for all 88 actuators) based on a polynomial of degree *n* = 3 and identified modal damping and eigenfrequencies in (24). Clearly, overshoot and settling time of the feedforward scheme show considerable improvements to a pure step command. Thereby, no feedback control

Feedforward control of deformable membrane mirrors has also been demonstrated at the P45 adaptive secondary prototype of the Large Binocular Telescope adaptive secondary mirror in [21] with similar results. In addition, a flatness based feedforward control is proposed

In order to further improve the system response to feedforward control and in order to add additional disturbance rejection capabilities to the DM control scheme, a model-based

*dyn* are used to generate the control command *u* = *ud* (see Figure 6).

*bj ω*2 *<sup>j</sup> fj*(*t*) 

0 5 10 15 20 time [ms]

Modeling and Control of Deformable Membrane Mirrors

*ud*(*t*) = *udyn*(*t*) + *ustat*(*t*). (26)

. (25)

*traj* and the inverse modal

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measured identified

The transition time *T* must be chosen shorter than the higher-ranking AO loop cycle time but long enough to comply with input constraints of the force actuators. The shorter the transition time *T* is chosen, the higher the required input forces *u* will be in order to drive the mirror from one deformation to another one. The required smoothness of the trajectory *yd* depends on the parameter *n* and should be chosen *n* ≥ 3 in order to be able to apply the model-based feedforward control scheme presented in the next section.

#### **4.2. Design of the model-based feedforward controller** Σ−<sup>1</sup> *stat* **and** <sup>Σ</sup>−<sup>1</sup> *dyn*

The basic idea of model-based feedforward control is the application of a desired input signal *ud*(*t*) such that the system response *y*(*t*) follows a desired output behavior *yd*(*t*) (see Figure 6). The transient deformation of the deformable mirror is described by the planned output *yd*(*t*) and leads to a new steady shape △*w*<sup>∗</sup> <sup>2</sup> using an inverse model of the system dynamics of the DM. The design of the desired response *yd*(*t*) is performed in the on-line trajectory generator Σ<sup>∗</sup> *traj* described in Section 4.1.

As shown in Figure 6, the feedforward control concept is seperated into a static and a dynamic component <sup>Σ</sup>−<sup>1</sup> *stat* and <sup>Σ</sup>−<sup>1</sup> *dyn*. The feedforward component <sup>Σ</sup>−<sup>1</sup> *dyn* contains the dynamic inverse of the relevant mirror dynamics and is based on the modal system dynamics (19) reading

$$f\_{\vec{j}}(t) + \underbrace{\frac{\lambda\_d + \kappa\_d \mathfrak{f}\_{\vec{j}}^4}{\rho h}}\_{2d\_{\vec{j}}\omega\_{\vec{j}}} f\_{\vec{j}}(t) + \underbrace{\frac{D\mathfrak{f}\_{\vec{j}}^4}{\rho h}}\_{\omega\_{\vec{j}}^2} f\_{\vec{j}}(t) = \underbrace{\frac{1}{\rho h} \sum\_{m=1}^M W\_{\vec{j}}(r\_m, \theta\_m)}\_{b\_{\vec{j}}} u\_m(t) \,. \tag{23}$$

By transforming the desired output behavior *yd* into modal coordinates *fj*(*t*) with relation (13), the left hand side of Equation (23) is fully defined by the desired modal output behavior *fk*(*t*) and the first two time derivatives of *fk*(*t*). Thus, the dynamic part of the required input command *ud* achieving the planned output behavior *fk*(*t*) can be computed as

$$
\Sigma\_{dyn}^{-1}: \quad u\_m^{dyn}(t) = \frac{1}{b\_j} \left( \vec{f}\_j(t) + 2d\_j \omega\_j \dot{f}\_j(t) + \omega\_j^2 f\_j(t) \right) \tag{24}
$$

for a limited number of *j* = 1, . . . , *N* dynamic eigenmodes. Since viscous and Rayleigh damping typically increase for higher order eigenmodes, it is not necessary to consider all eigenmodes in the dynamic feedforward component <sup>Σ</sup>−<sup>1</sup> *dyn*. Instead, it is sufficient to

with

**Figure 8.** Measured step responses of the first and second eigenfunctions of an Alpao DM88 deformable membrane mirror without feedforward control showing a settling time above 5-10 ms

consider only the static components of the inverse system dynamics (23) and compute a static command *ustat <sup>m</sup>* (*t*) as

$$
\Sigma\_{\rm stat}^{-1} : \quad \mu\_m^{\rm stat}(t) = \frac{1}{b\_j} \left( \omega\_j^2 f\_j(t) \right). \tag{25}
$$

Afterwards, the final feedforward command *ud* can be computed as

14 Adaptive Optics

generator Σ<sup>∗</sup>

reading

dynamic component <sup>Σ</sup>−<sup>1</sup>

¨ *fj*(*t*) +

*κ*(*t*) =

*yd*(*t*) and leads to a new steady shape △*w*<sup>∗</sup>

*traj* described in Section 4.1.

*stat* and <sup>Σ</sup>−<sup>1</sup>

*λ<sup>d</sup>* + *κdβ*<sup>4</sup> *j*

˙ *fj*(*t*) +

*ρh* � �� � 2*djω<sup>j</sup>*

Σ−<sup>1</sup>

*dyn* : *<sup>u</sup>dyn*

all eigenmodes in the dynamic feedforward component <sup>Σ</sup>−<sup>1</sup>

  0 if *t* ≤ (*t*<sup>1</sup> − *t*0) (2*n*+1)! *n*!(*t*1−*t*0)<sup>2</sup>*n*+<sup>1</sup>

1 if *t* ≥ (*t*<sup>1</sup> − *t*0).

model-based feedforward control scheme presented in the next section.

**4.2. Design of the model-based feedforward controller** Σ−<sup>1</sup>

*n* ∑ *i*=0 (−1)*<sup>n</sup>*−*<sup>i</sup>*

The transition time *T* must be chosen shorter than the higher-ranking AO loop cycle time but long enough to comply with input constraints of the force actuators. The shorter the transition time *T* is chosen, the higher the required input forces *u* will be in order to drive the mirror from one deformation to another one. The required smoothness of the trajectory *yd* depends on the parameter *n* and should be chosen *n* ≥ 3 in order to be able to apply the

The basic idea of model-based feedforward control is the application of a desired input signal *ud*(*t*) such that the system response *y*(*t*) follows a desired output behavior *yd*(*t*) (see Figure 6). The transient deformation of the deformable mirror is described by the planned output

of the DM. The design of the desired response *yd*(*t*) is performed in the on-line trajectory

As shown in Figure 6, the feedforward control concept is seperated into a static and a

inverse of the relevant mirror dynamics and is based on the modal system dynamics (19)

By transforming the desired output behavior *yd* into modal coordinates *fj*(*t*) with relation (13), the left hand side of Equation (23) is fully defined by the desired modal output behavior *fk*(*t*) and the first two time derivatives of *fk*(*t*). Thus, the dynamic part of the required input

for a limited number of *j* = 1, . . . , *N* dynamic eigenmodes. Since viscous and Rayleigh damping typically increase for higher order eigenmodes, it is not necessary to consider

*Dβ*<sup>4</sup> *j ρh* � �� � *ω*2 *j*

command *ud* achieving the planned output behavior *fk*(*t*) can be computed as

*bj* � ¨

*<sup>m</sup>* (*t*) = <sup>1</sup>

*dyn*. The feedforward component <sup>Σ</sup>−<sup>1</sup>

*fj*(*t*) = <sup>1</sup>

*fj*(*t*) + 2*djω<sup>j</sup>* ˙

*ρh*

*M* ∑ *m*=1

*<sup>i</sup>*!(*n*−*i*)!(2*n*−*i*+1)(*t*<sup>1</sup> <sup>−</sup> *<sup>t</sup>*0)*<sup>i</sup>*

*t* 2*n*−*i*+1

(22)

if 0 ≤ *t* ≤ (*t*<sup>1</sup> − *t*0)

*stat* **and** <sup>Σ</sup>−<sup>1</sup>

<sup>2</sup> using an inverse model of the system dynamics

*Wj*(*rm*, *θm*)

� �� � *bj*

*fj*(*t*) + *ω*<sup>2</sup>

*<sup>j</sup> fj*(*t*) �

*dyn*. Instead, it is sufficient to

*dyn*

*dyn* contains the dynamic

*um*(*t*), (23)

(24)

with

$$
\mu\_d(t) = \mu^{dyn}(t) + \mu^{stat}(t). \tag{26}
$$

Regarding the computational complexity, in [20] is shown that the computational demands of feedforward control scale quadratically with the number of system inputs and outputs. However, since the control signal only needs to be computed with the frequency of the outer AO-loop (e.g. 1 kHz in modern astronomical AO systems), there are currently no burdens for a practical implementation in existing systems.

In Figure 8 and Figure 9 measurement results of a setpoint change without and with feedforward control for an Alpao deformable mirror with 88 voice coil actuators is shown. In the experiment, the first two eigenfunctions of the deformable mirror are excited by a modal step input (spatially distributed step command for all 88 actuators at the same time) and a model-based feedforward control input *ud* (spatially distributed time-varying signal for all 88 actuators) based on a polynomial of degree *n* = 3 and identified modal damping and eigenfrequencies in (24). Clearly, overshoot and settling time of the feedforward scheme show considerable improvements to a pure step command. Thereby, no feedback control is implemented at this stage and only the trajectory generator Σ<sup>∗</sup> *traj* and the inverse modal

system dynamics <sup>Σ</sup>−<sup>1</sup> *dyn* are used to generate the control command *u* = *ud* (see Figure 6).

Feedforward control of deformable membrane mirrors has also been demonstrated at the P45 adaptive secondary prototype of the Large Binocular Telescope adaptive secondary mirror in [21] with similar results. In addition, a flatness based feedforward control is proposed therein in case of zero dynamics in the differential equation (19).

In order to further improve the system response to feedforward control and in order to add additional disturbance rejection capabilities to the DM control scheme, a model-based

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115

, (28)

with *x*(*t*) = �

*A* =

 

*Dβ*<sup>4</sup> 1

> . . . . . . . . . ... . . . . . . . . . . .

*f*1(*t*) *f*2(*t*) ... *fN*(*t*) ˙

<sup>0</sup> *<sup>D</sup>β*<sup>4</sup> 2

0 0 *<sup>D</sup>β*<sup>4</sup>

3

0 0 0 ... *<sup>D</sup>β*<sup>4</sup>

*<sup>B</sup>* <sup>=</sup> <sup>1</sup> *ρh*

*C* =

*J* = �∞

0

 

. .

system along the planned trajectory *xd*(*t*) using the optimization criterion

(*x*(*t*) − *xd*(*t*))

*f*<sup>1</sup> ˙ *f*<sup>2</sup> ... ˙

*<sup>ρ</sup><sup>h</sup>* 0 0 ... 0 *<sup>λ</sup>d*+*κdβ*<sup>4</sup>

*fN*(*t*)

0 0 0 ... 0 1 0 0 ... 0

0 0 0 ... 0 0 1 0 ... 0

0 0 0 ... 0 0 0 1 ... 0

*<sup>ρ</sup><sup>h</sup>* ... 0 0 0 *<sup>λ</sup>d*+*κdβ*<sup>4</sup>

0 0 0 ... 0 0 0 0 ... 1

*W*1(*r*1, *θ*1) ... *W*1(*rM*, *θM*) *W*2(*r*1, *θ*1) ... *W*2(*rM*, *θM*)

. ... .

*WN*(*r*1, *θ*1) ... *WN*(*rM*, *θM*) 0 ... 0

. ... .

0 ... 0

*W*1(*r*1, *θ*1) ... *WN*(*r*1, *θ*1) 0 ... 0 *W*1(*r*2, *θ*2) ... *WN*(*r*2, *θ*2) 0 ... 0

*W*1(*rM*, *θM*) ... *WN*(*rM*, *θM*) 0 ... 0

A global multi-input multi-ouput linear quadratic regulator is designed for controlling the

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

. . . . . ... . . .

. ... .

*<sup>ρ</sup><sup>h</sup>* 0 ... 0 0 *<sup>λ</sup>d*+*κdβ*<sup>4</sup>

*N*

. .

. .

  1

�*<sup>T</sup>* and system matrices

*<sup>ρ</sup><sup>h</sup>* 0 0 ... 0

*<sup>ρ</sup><sup>h</sup>* 0 ... 0

3 *<sup>ρ</sup><sup>h</sup>* ... 0

> 

> >

*<sup>T</sup> <sup>Q</sup>* (*x*(*t*) <sup>−</sup> *xd*(*t*)) <sup>+</sup> *<sup>u</sup>*(*t*)*TRu*(*t*)d*t*, (31)

Σ*ctrl* : *u*(*t*) = −*K* (*x*(*t*) − *xd*(*t*)) (32)

. ... .

Modeling and Control of Deformable Membrane Mirrors

. .

*N ρh*

, (29)

. (30)

 

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

2

*<sup>ρ</sup><sup>h</sup>* 0 0 0 ... *<sup>λ</sup>d*+*κdβ*<sup>4</sup>

. .

. .

**Figure 9.** Measured responses of the first and second eigenfunctions of an Alpao DM88 deformable membrane mirror with feedforward control showing a settling time of 2 ms

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