Synthesis of Equation-Free Control Structures for Dissipative

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Synthesis of equation-free control structures for dissipative distributed parameter systems using proper orthogonal decomposition and discrete empirical interpolation methods Manda Yang, and Antonios Armaou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02322 • Publication Date (Web): 31 Jul 2017 Downloaded from http://pubs.acs.org on August 1, 2017

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Industrial & Engineering Chemistry Research

Synthesis of equation-free control structures for dissipative distributed parameter systems using proper orthogonal decomposition and discrete empirical interpolation methods

Manda Yang † and Antonios Armaou ∗ † ‡ , ,

†Department

of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, USA

‡Department

of Mechanical Engineering, Wenzhou University, Zhejiang, China

E-mail: [email protected]

Abstract We describe an equation-free control framework for the regulation of dissipative distributed parameter systems, with emphasis on improving the accuracy of the estimation by using a correction term. This control method is capable of regulating systems that have unknown dynamics but known eect of the control action. The system state and the dynamics are estimated by using the oine observations (snapshots ensemble) and the online continuous measurement of a restricted number of point sensors. First, we construct a reduced order model (ROM) with unknown terms using Galerkin/proper orthogonal decomposition (POD). Then the state of the ROM is estimated by a static observer with the information from the state sensors; and the mapping between the dynamics of the system and velocity sensors are generated using a similar approach. Discrete empirical interpolation method (DEIM) is employed to determine the sensor 1

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locations. To improve the accuracy of the estimation, a correction term is updated consistently. The proposed equation free control framework is illustrated through a diusion-reaction process and the performance of the proposed method is evaluated by simulation.

Introduction Distributed parameter systems (DPS) exist in many chemical and material industry processes.

This kind of systems shows spatial variation because of the existence of diusion,

1

convection and reaction; examples include plasma enhanced chemical vapor deposition , plasma etching reactors

2

3

and reaction in porous catalyst particles .

Due to the motiva-

tion to improve product quality and increase economic prot, controlling these processes is signicantly important. A standard way to control these processes is to construct reduced order models (ROMs) via the Galerkin method

4,5

and then design observers and controllers based on the ROMs,

given the fact that the behavior of most of the above chemical and material industry processes can be captured by nite dimensional systems

6,7

. The prerequisite basis functions are

generated by proper orthogonal decomposition (POD)

8,9

. Compared with other methods,

basis functions constructed by POD can better represent the dominant behavior of previous observation

8,10

.

Having constructed ROMs, controllers can be designed based on the ROMs so that less computational eort is required.

Many controller design techniques have been applied to

DPSs, including model predictive control methods

13

11

, feedback linearization

and the back stepping techniques

14

12

, the Lyapunov-based

. To implement the controller and relax the

requirement on the sensors, Luenberger-based observers have been widely used to estimate the state of the systems

15,16

. However, most of the control techniques and observer design

methods require a priori knowledge of the mathematical model of the system; meanwhile the unmodeled dynamics and model uncertainty always exist in processes. Motivated by this

2


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issue, signicant eorts have been devoted to systems with uncertainty control

20,21

techniques

.

and data-driven

Some of the data-driven/equation-free methods include divide-and-conquer

22,23

ral networks

1719

(such as memory based local modeling

3135

24

), subspace identication

2530

, autoregressive moving average with exogenous inputs (ARMAX)

model-free control based on ultra-local model used in many areas

41,42

45

3639

and

. While these methods have been widely

, the realization of them is limited by disadvantages including the

high computational cost In previous work

20,40

, neu-

43,44

and the demanding requirement on training sets.

, we have described another equation-free method to control systems

when the knowledge of the governing law is not available or complete and the actuator eect is. This method is motivated by the feature of discrete empirical interpolation method (DEIM)

46

: the selection of the interpolation indices can limit the growth of the error of the

approximation by DEIM. To take advantage of this feature, sensor locations in this method are determined by the interpolation indices. about sensor network designs

4750

While there have been many contributions

, this approach has the advantage that no information

about the governing equation is required. With the continuous measurement of these sensors, the state in the ROM is estimated by a static observer. In the dynamics estimation part, the mapping from the outputs onto the projection of the dynamics is generated using a similar approach as the static observer.

By using this information, the explicit governing

law becomes superuous. In this manuscript, we try to improve the accuracy of the estimation of the dynamics by using the information of estimated state. A correction term is applied to compensate the error brought by the static part. The dierence between the expected value of the state and the actual value is used to update the correction term. The modied version is applied to a diusion reaction process. The assumption of the availability of the constant regulation term in the previous work is relaxed. We organize the remainder of this manuscript as follows. First, we present the techniques including POD, DEIM and the original equation-free control method. The limitation of the

3



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original equation-free control method is also discussed.

Page 4 of 54

Then, we give the algorithm of

the modied equation-free control method. Finally, both the original equation-free control method and the modied one are applied to a diusion reaction process.

Preliminaries System Description We consider a process described by the following semi-linear PDE:

∂x = L(z)x + g(x) + b(z)u = f (x) + b(z)u, ∂t where

f (x) = L(z)x + g(x),

(1)

subject to the boundary condition

h(x,

∂x ) = 0 on Γ ∂z

(2)

and the initial condition

x(z, 0) = x0 (z) where

x∈R

coordinate; function.

is the state variable,

L(z)

refers to time and

z ∈ Ωz ⊂ R3

is an unknown linear spatial operator and

u ∈ Rs×1

ulated inputs.

t

(3)

denotes manipulated variables, where

b(z) ∈ R1×s

spatially and is known.

h

s

g(x)

is an unknown nonlinear

denotes the number of manip-

describes how the manipulated variables is a function of

x

represents the spatial

u

control the system

and its spatial derivative.

Γ

is the process

boundary. We assume 2 sets of restricted numbers of point sensors are available: state sensors and velocity sensors. The outputs corresponding to the state sensors are:

Z δ(z − zi )xdz (i = 1, 2, · · · , ks )

yi = Ωz

4


(4)

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and the outputs corresponding to the velocity sensors are

Z δ(z − zi )xdz ˙ (i = ks + 1, · · · , ks + kv )

yi =

(5)

Ωz

where

x˙

denotes the time derivative of

denotes the location of

ith

x, δ(·)

represents the Dirac delta function, and

zi

sensor. The control objective is to maintain the state variable at

a desired steady state.

Assumption 1

The relative degree of the output is 1

Assumption 2

The velocity sensors sample the state evolution of a point quickly and accu-

rately enough so that

Remark 1

x˙

51 .

can be evaluated with negligible noise.

The velocity sensor can be either the same sensor as the state sensor or a dif-

ferent type of sensor. In both cases, they may be collocated. We assume the velocity sensors can sample frequently enough and have knowledge of time in order to be able to estimate the time derivative of state change at that point.

Method of Weighted Residuals To reduce the computational cost, the method of weighted residuals (MWR)

52

is employed

to project the distributed parameter system into a nite dimensional functional subspace. MWR approximates the state by a superposition of predetermined basis functions

{φi }

multiplied by time-dependent coecients,

x(z) =

n X

ci (t)φi (z),

(6)

i=1

where the time-dependent coecients

ci

are called modes.

Then a set of ordinary dierential equations (ODEs) is generated by projecting the gov-

5



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erning equation into a subspace spanned by predetermined weighting functions

n Z X i=1

n Z X c˙i φi ϕj dz = (

Ωz

i=1

Z ϕj Lφi dz)ci +

Ωz

Z n X ϕj g( ci φi )dz + (

Ωz

.

ϕj b(z)dz)u,

Ωz

i=1

{ϕj }

(7)

∀j = 1, 2, · · · , n. This step forces the projection of the residual of the governing equation (Eq. 1)

r=

∂x − L(z)x + g(x) + b(z)u ∂t

in the subspace spanned by the weighting functions to be zero. In this equation-free control framework, we apply this technique not only to reduce the computational cost, but also to facilitate the accuracy improvement of the dynamics estimation. The weighting functions to the basis functions

Remark 2

{φi }.

{ϕj } are constructed so that the residual r is orthogonal

This method is also called the Galerkin method

53

.

In highly dissipative PDE systems, it's observed that a nite number of modes

can capture the dominant behavior of the original system

54 .

By taking advantage of this

observation, the projection of the original system in a low order subspace can generate a ROM with enough accuracy for control purposes.

Proper Orthogonal Decomposition The eciency of Galerkin's method hinges on the basis functions. Since the knowledge of the model is not available, we employ proper orthogonal decomposition (POD)

8,9

to generate

them after sampling the process and creating a training set. Compared with any other set of basis functions of the same size, POD can better capture the dominant trend of a system in the sense that the

L2

norm of the dierence between

x

and

P

i ci φi will be minimized for

the given training set. Also, this technique will later be combined with DEIM to estimate system dynamics in this work. A brief introduction of POD is presented for completeness.

6


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First, POD requires a set of spatial proles of the system response collected experimentally or via numerical simulation. Each prole

xi

{xi (z)}

at

t = ti

be

is called a snapshot.

The POD basis functions are determined by solving an optimization problem:

max

where

h·i

p (p, p),

denotes average,

where

p

and

q

(·, ·)

h(xi , φ)2 iN i=1 kφk2 L2 -inner

represents

(p, q) =

product:

R Ωz

p · q dz

and

kpk =

represent two snapshots. The solution of this optimization problem

is equivalent to the solution of the following eigenvalue problem of the covariance matrix

AT A AT Av = λv where tors

A = [x1 , x2 , . . . , xN ] ∈ RM ×N

{vi }

and

xi

(8)

is discretized using

M

grid points. the eigenvec-

are sorted so that the corresponding eigenvalues in the covariance matrix (Eq. 8)

are in descending order. Then the basis functions are linear combination of snapshots:

1

Φ = AV Λ− 2

where

Λ = diag(λ1 , · · · , λN )

represents the eigenvectors of

(9)

contains the eigenvalues of Eq.8 ,

AT A and Φ = [φ1 , φ2 , · · · , φN ].

V = [v1 , v2 , · · · , vN ] ∈ RN ×N

This result can also be obtained

from singular value decomposition (SVD):

A = ΦΣV T

where

Σ = Λ1/2 .

According to corollary 2.4.7 in Matrix Computations

55

, for a matrix

A

with rank n:

A=

n X

1/2

λi φi viT

i=1 Therefore, only

n basis functions in φ need to be retained and the basis functions correspond-

7



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ing to zero (and small) eigenvalues can be truncated at a small loss of accuracy. Since these basis functions are usually contaminated with signicant round-o error this step actually robusties POD

Remark 3

56

.

Since the proposed method applies to system without the knowledge of the gov-

erning law, the snapshots have to be collected from experiments. The proposed approach is also applicable to systems for which a black box simulator is available; we employ these high delity simulations to obtain the snapshots. In the simulation results shown later, the snapshots are generated by using the latter case. For illustration purposes, we have access to the governing equation. But we pretend to not know it and regulate the system without using the information of the governing law.

Remark 4

We use POD instead of SVD because when

requires more resources than POD to compute

M

is large, SVD is slower and

Φ.

Discrete Empirical Interpolation Method After we construct a reduced order model for the distributed parameter system, we need to estimate the state of ROM for controller design. To achieve this goal, we propose a datadriven control method by modifying discrete empirical interpolation method (DEIM)

46

. In

this section, we briey introduce DEIM for completeness. Note that DEIM provides us with basis functions that are discretized in space. DEIM has been used to mitigate the computational burden associated with the nonlinear term in ordinary dierential equations (ODEs) systems generated by POD-Galerkin method

5759

. Since the basis functions generated by POD are orthogonal and self adjoint,

the ROM in Eq. 7 can be discretized to

c˙j =

n X i=1

Dj,i ci + ϕTj g(

n X

ci φi ) + Bj u,

j = 1, 2, · · · , n

i=1

8


(10)

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where

j th

Dj,i

row of

and

Bj

j th

denote the

B = ΦT bz ∈ Rn×s ,

row,

ith

column element of

respectively (where

can be seen that compared with the linear part

bz

Lx,

is the spatially discretized

Ψ = [ψ1 ψ2 . . . ψk ] ∈ RM ×k

this manuscript) for

g(x)

and the

b(z)).

evaluating the nonlinear term

computationally intensive. DEIM rst calculate snapshots of basis functions

D = ΦT LΦ ∈ Rn×n

It

g(x)

is

g(x), then generate discretized

(which are called nonlinear basis functions in

by using the POD algorithm so that

g(x)

can be approximated by

a superposition of nonlinear basis functions

g ≈ Ψ(P T Ψ)−1 P T g

where

P = [~ep1 , ~ep2 , . . . , ~epk ] ∈ RM ×k . ~epi

is the

pi th

(11)

column of identity matrix

refers to the size of it. DEIM algorithm is used to determine the subscripts

pi ,

IM ,

where

M

which is also

called interpolation indices. Here, matrices

P T g requires the nonlinear term g to be evaluated at k spatial grid points and other Ψ

and

P

are predetermined. Please note that

of total spatial grid points

M.

k

is much smaller than the number

Hence the computation cost associated with evaluating

g

is

reduced. The interpolation indices in

Remark 5

P

are linked to sensor locations in our work.

In DEIM, the linear part and nonlinear part are dierentiated because of the

dierence in computational cost in evaluating them:

In the linear term, the spatial and

temporal components are decoupled, so that coecients in the ROM can be predetermined; while the nonlinear term needs to be evaluated at each grid point and at each time step. In the proposed method, the linear part and the nonlinear part are not dierentiated so that the knowledge of the linear operator and the nonlinear function is not required.

At the same

time, choosing sensor locations using nonlinear basis functions can also limit the growth of the error as it does in DEIM.

In summary, DEIM contains the following steps:

9



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•

generate nonlinear snapshot set using

•

apply POD to nonlinear snapshots to get nonlinear basis functions

•

determine interpolation indices by using

g

and snapshot set

{xi } Ψ

Ψ

Data-driven Control Based on DEIM In this section, we review how the equation-free control is used to regulate systems.

The

ow chart of the proposed method is given in Fig. 1. This method makes the following assumptions:

Assumption 3 {fi }

A representative set of snapshots of

x

and a set of snapshots of

respectively, are available before the process operation starts, or

from processing

{fi }

f , {xi }

and

can be obtained

{xi }.

Assumption 4 Assumption 5 (state space partition)

b(z),

The eect of the actuators,

on the state is known.

60 The innite dimensional system of Eq.1 can

be partitioned into a nite dimensional slow (stable or unstable) subsystem and an innite dimensional fast stable subsystem.

Please note that the availability of oine

f

snapshots does not necessarily require the

knowledge of the governing equation. As long as the sampling time corresponding to each or

x˙

can be measured directly,

f

can be calculated by

f=

∂x ∂t

x

− b(z)u.

basis functions construction First, POD or SVD is applied to functions:

{φi }

for

{xi }

and

{ψi }

{xi } for

and

{fi }.

{fi }

respectively to generate 2 sets of basis

To take full advantage of the continuous point

measurement, the modal representation dimension for of state sensors

ks

and the mode size for

f

is

x (i.e.

kv .

10


n) is the same with the number

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determine sensor locations by using DEIM algorithm During system evolution, the continuous measurement from point sensors (both state sensors and velocity sensors) are used for state and dynamics estimation.

The locations of these

sensors are predetermined by applying DEIM algorithm to the basis functions corresponding to

{xi }

and

{fi }.

We choose DEIM to determine sensor locations because it can improve

the observability when the dynamics of the system is unknwon. Take the basis functions for

{xi }

for example: The interpolation indices for each basis

functions are determined inductively using DEIM. Each interpolation index corresponds to the location of a sensor. The rst interpolation index is the position of the peak of the rst basis function. We dene 2 matrices

Φ0 = [φ1 ]

and

of the peak of the rst nonlinear basis function identity matrix. Starting from the second one of residuals

r0 .

P 0 = [ep1 ],

φ1

q = 2,

and

ep1

where

p1

is the index position

denotes the

pth 1

column in an

the interpolation indices are the peaks

To obtain them, rst we solve

(P 0T Φ0 )d = P 0T φq

for

d.

The residual

r0

(12)

is constructed by subtracting the components of previous basis func-

tions from the current basis function

r0 = φq − Φ0 d

The

Φ0

q th

to

(13)

interpolation index is the position of the peak of the residual.

[Φ0 , φq ]

and

P0

[P 0 , ~epi ]

to

and repeat the process for the next

Then we update

q = 3.

is repeated until all the interpolation indices are determined and the nal The nal

P0

is called

Px

indices are dierent for

and

x

Pf

and

f.

P0

This process is obtained.

in next section; the subscripts show that the interpolation Note that the basis functions

φi

have been sorted so that

the corresponding eigenvalues in the covariance matrix (Eq. 8) are in descending order. The

11



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index position

pi

is nonrepeated

46

Page 12 of 54

, i.e. all index positions are dierent.

State & Dynamics estimation A static observer is designed for the system.

By using the basis functions

Φ

and their

corresponding interpolation indices, an output to state mapping that is similar to Eq. 11 can be obtained

x ≈ Φ(PxT Φ)−1 X where

X = [y1 y2 · · · yks ]T

are outputs measured by state sensors, and

(14)

Px

interpolation indices. The state in the ROM can be obtained by multiplying

x

c ≈ c˜ := (PxT Φ)−1 X

Similarly,

f

Pf

is obtained based on

Ψ

Φ:

(15)

(16)

and

Y = [y1+ks − b(z1+ks )u y2+ks − b(z2+ks )u · · · yks +kv − b(zks +kv )u]T

Remark 6

by

can be estimated by using the same approach:

f ≈ f˜ := Ψ(PfT Ψ)−1 Y

where

contains the

(17)

Although Eq. 16 provides an estimate of the nonlinear term of the governing

equation, static observer is used instead of Luenberger-type dynamic observer despite the fact that dynamic observer can dramatically relax the requirement on the number of sensors. The rst reason is the estimate of

f

f

may not be accurate enough to estimate the state since

is unknown. The second reason is the observer gain is not easy to determine when the

governing law is not available.

12


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Controller Based on Feedback Linearization The controller constitutes of 2 parts: the rst part

u1 is designed using feedback linearization

and the second part comes from the estimation of the error of the nonlinearity observer (Eq. 16). For a given set point

x0 ,

the corresponding set point

c0

in the ROM is calculated by

c0 = ΦT x0

We dene the distance

u1

e ∈ Rn

(18)

from the set point as

e = c − c0

(19)

e˙ + Kd e = 0

(20)

is designed so that

where

Kd ∈ Rna ×na

is a diagonal matrix with positive entries and

na

is the number of

actuators. In discretized context, we express the estimated ROM in the following form:

We can obtain

u1

c˜˙ = ΦT f˜ + Bu1

(21)

u1 = B −1 − Kd (c − c0 ) − ΦT f˜

(22)

from

With the estimated state (Eq.15 ) and

f

(Eq. 16), the implementation of

u1

doesn't require

the explicit information of the governing equation.

Limitations of Equation-free Method In previous work

45

, a predetermined term is assumed to be available so that a steady state

is translated to the operation point when the manipulated input is 0. This term not only

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Page 14 of 54

reduces the number of actuators required to regulate the system, but also relaxes the requirement on the accuracy of the estimation. Hence, the number of sensors used for estimation can be reduced. However, this transformation method may not always be feasible, especially when the governing law of the system is unknown. Motivated by this limitation, we try to regulate the system without translating the steady state. In this situation, a more accurate estimation of the term

f

is required, otherwise the error will lead to o set (which will be

demonstrated in the numerical results). Since the accuracy of the estimation hinges on the snapshot ensemble, one way to solve the problem is to enrich the snapshot ensemble by using various initial conditions and operating conditions. However, there is no guarantee that the enriched snapshots can improve the accuracy. Here we propose another approach to reduce the estimation error of

f

by using the information of

x.

Modied equation free method Motivation By properly choosing the sensor locations via DEIM, the state

x can be estimated by using the

information of a restricted number of sensors with enough accuracy; however, the estimation for

f

usually cannot achieve the same level of accuracy. This is because

satisfy the state space partition assumption that

x

does.

f

does not necessarily

This can be demonstrated in a

simple example as follows: We collect 40 snapshots of

x

and 40 snapshots of

f

from the same open loop process

(Fig. 2a) of a diusion reaction process. We apply POD to each snapshot ensemble. The energy captured by rst

n

basis functions for

x

and

f

are compared in Fig. 2b. It can be

seen from this result that more basis functions are required for

f

than

x to capture a certain

amount of energy. Motivated by this observation, we will use the information of the estimate of

f

so that the total number of sensors can be reduced.

14


x to correct

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Updating Dynamics Estimation In the modied equation free method, we use a correction term to improve the accuracy of

f.

the estimation for

A correction term

∆f ∈ Rna

for

ΦT f

in Eq. 21 is dened as

∆f = ΦT f − ΦT fe

Unfortunately,

∆f

units of time at

cannot be evaluated. We estimate

t = t0 , t1 , · · · , tv , tv+1 , · · · ,

where

∆f

(23)

by

f ∆f

that is updated every

∆t

tv = v∆t.

The estimated ROM becomes:

f + Bu c˜˙ = ΦT f˜ + ∆f

Assuming

f ∆f

is constant in the time interval

(24)

f = ∆f f , the control action [tv , tv+1 ] and ∆f v

in that interval is adjusted accordingly:

gv u = B −1 − Kd (c − c0 ) − ΦT f˜ − ∆f

The state estimated by the observer at accurate, the state during

[tv , tv+1 ]

t = tv

is denoted by

c˜v .

(25)

If the correction term

f ∆f v

is

should obey the rule determined by Eq. 20. Therefore,

e˜v+1,1 = e˜v,1 e−Kd,1 ∆t e˜v+1,2 = e˜v,2 e−Kd,2 ∆t

(26)

. . .

where

e˜i,j

indicates the dierence between

and the objective and

j th

estimated state variable in the ROM at

Kd,i denotes the ith entry on the diagonal of Kd .

the actual estimated distance error of the correction term

e˜v+1

t = ti

The dierence between

and the theoretical distance in Eq. 26 indicates the

f. ∆f

15



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f t = t0 , ∆f

At at

t = tv

during

and

is chosen to be

t = tv+1 .

[tv , tv+1 ]

f 0, at t = tv+1 , ∆f

Page 16 of 54

is updated using the estimated distance

e˜

Take the rst state variable in the ROM as an example, the dynamics

is

f v,1 + e∆,1 + B1 u c˜˙1 = φT1 f˜ + ∆f

(27)

With the control action in Eq. 25, we can obtain

e˜˙1 = −Kd,1 e˜1 + e∆,1

Assumption 6 terval

[tv , tv+1 ],

We assume the variance of the error and

e∆

e∆

(28)

is negligible during each time in-

can be approximated by a constant.

Based on assumption 6, it can be derived that

e∆,1 =

−e−Kd,1 ∆t Kd,1 ef v,1 + Kd,1 e] v+1,1 −K 1 − e d,1 ∆t

(29)

The errors for other state variables are calculated using the same approach.

Then

f ∆f

is

updated to

f f ∆f v+1 = ∆f v + e∆

Remark 7

The information used to correct the estimation of

used to estimate

Remark 8

x.

(30)

f

is from the sensor network

Hence, no additional sensors are required in this updating part.

Since the estimation of the system dynamics in this equation-free method is

associated with POD, the accuracy of the estimation hinges on the quality of the basis function which depends on how the snapshot ensemble is collected. As a result, the number of sensors for both

x

and

f

can be reduced with a better collected snapshot ensemble or more snapshots

to achieve the same level of accuracy.

The modied equation-free control method is summarized in Table 1.

16


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Remark 9 (error bound)

In previous work

kf˜ − f k2 ≤ (1 +

where

g

kf − ΨΨT f k

√

45 , the error bound for

f

is proved to be

T 2M )kv −1 kψ1 k−1 ∞ kf − ΨΨ f )k2

is small based on the assumption 3 that implies the nonlinear function

can be represented by basis functions

Ψ.

The updating step in the proposed method relaxes

this assumption.

Remark 10

In this work, we try to regulate a process of which model information is not

available, control objective is known, and oset is inevitable (all the actuators need be to used to achieve the best regulating performance). Plantwide control

61 can be used by combining the

proposed control method with a system identication algorithm to gather information about the system and thus achieve better performance in terms of economic benet, production eciency and safety.

Numerical Results In this section, we rst show that regulating without shifting the steady state requires a higher level of accuracy of the estimation of the dynamics in the original equation-free method. Then we compare the original equation-free method and the modied equation-free method.

Diusion-reaction Process We consider a diusion reaction process. In this process, an elementary exothermic reaction takes place along the surface of a catalytic rod of length

π.

The concentration of the reactant

is a constant. The only state variable is the temperature, which is aected by the exothermic

17



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reaction and diusion

ρC

62

Page 18 of 54

. The governing equation is:

E ∂T ∂ 2T k3 ls = k1 2 − Ak2 (z)e− RT c0 ∆H + (Tc (z) − T ) ∂t ∂z ∆A

(31)

subject to the boundary condition:

T (0, t) = T0 , T (π, t) = T0

where

T

(32)

denotes temperature.

We dene

x=

T −T0 and Eq. 31 becomes: T0

3 X ∂x ∂ 2x −γ/(1+x) = 2 + βT (z)e + βu ( bi (z)ui − x) ∂t ∂z i=1

subject to

x(0, t) = 0, x(π, t) = 0

βu =

(33)

where

k3 ls =2 ∆AρC

γ=

E =2 RT0

Because of a spatial distribution on the activity of the catalyst, we have

βT (z) = 16[cos(z) + 1]

(34)

Without control, the system evolves to a spatially varying steady state (shown in Fig. 2a). The objective is to force the temperature to the spacial invariant steady state

x = xo . Three cooling jackets are used to regulate the system (Fig.

18


(35)

3), which separates the

Page 19 of 54

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domain into 3 regions. In each region, the actuator aects the state uniformly.

b1 (z) =H(z − 0) − H(z − 0.3π) (36)

b2 (z) =H(z − 0.3π) − H(z − 0.5π) b3 (z) =H(z − 0.5π) − H(z − π) where

H(·)

denotes Heaviside step function.

There are 2 sets of point sensors to continuously measure the temperature of the rod at points

zi

where

i = 1, 2, · · · , ks

and the time derivative of temperature at

zi

where

i=

ks + 1, ks + 2, · · · , ks + kv . Z

π

δ(z − zi )x(z)dz,

yi =

∀ 1 ≤ i ≤ ks

(37)

∀ 1 + ks ≤ i ≤ ks + kv

(38)

0

Z

π

δ(z − zi )x(z)dz, ˙

yi = 0

A diagram of the actuators and sensors is given in Fig. 3.

Original Equation-free Method In this section, we regulate the diusion reaction process via the original equation-free method without shifting the steady state.

The snapshots used for generating basis func-

tions are collected from an open-loop process with the following initial condition:

x(t0 ) = 0.3sin(4.3z) − 0.11cos(1.38z) + 1

(39)

40 snapshots are collected and displayed in Fig. 2a. Since

x can be accurately estimated with a small number of sensors, we use 5 state sensors

(more than

99.99% is captured) to estimate the state and vary the number of velocity sensors.

It is found that at least 23 velocity sensors are required. The system response in systems

19



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Page 20 of 54

with 21, 23, and 30 velocity sensors are displayed in Fig. 4a, 4b and 4c, respectively. To better compare the results, we dene the distance from the set point as:

sZ d=

(x − xo )2 dz

(40)

The distance from the set point in Fig. 4a, 4b and 4c are compared in Fig.6. It can be seen that as the number of velocity sensors increase, the system response approaches to the set point. In this case, as the total number of sensors increases, the error of the observer (for the state) also decreases (Fig. 7). Fig. 5a, 5b and 5c give the corresponding control actions. While none of them has chattering, the larger error that results from decreasing the number of sensors leads to actuator saturation.

Original Equation-free Method with Enriched Snapshots Ensemble To show that a better collected set of snapshots can increase the quality of the estimation, in this section, we use 2 dierent initial conditions to collect oine snapshots. The rst initial condition is the same with the last section (Eq. 39), the second initial condition is Eq. 41.

x(t0 ) = cos(z)

(41)

80 snapshots are collected, 40 from the open-loop process with the rst initial condition and the other 40 from the second one. We still use 5 state sensors; yet the number of velocity sensors can be decreased from 23 to 12.

The conclusion can be drawn from the system

response from the closed-loop systems with 10, 12 and 15 velocity sensors (Fig. 8a, 8b and 8c) and the distance from the set point (Fig. 9).

20


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Modied Equation-free Method In this section, we use the correction term to improve the quality of the estimation. The value of the correction term is updated every 0.5 units of time. The same snapshot ensemble as section Original Equation-free Method is used. The simulation results show that the number of velocity sensors can be decreased to 6 in this example. Fig. 10a, 10b, 10c and 10d illustrate the system response in systems with 5, 6, 8, and 10 velocity sensors respectively. The temporal prole of the distance from the set point is depicted in Fig. 11 and the control actions are given in Fig. 12a, 12b and 12c. The overall performance of the controller is better than the one without the updating step. Note that when there are not enough sensors, the error in the correction term may lead to oscillation in the control actions.

sensor noise & model mismatch In this section, we consider the sensor noise and model mismatch.

We use the same 40

snapshots as section Original Equation-free Method and Modied Equation-free Method. These snapshots are collected from a process in which assume in the real process, is given in Fig.

13.

βT

becomes time-variant.

βT

is time-invariant (Eq. 34). We

The spatial temporal prole of

βT

We also assume 2 state sensors and 2 velocity sensors have noise,

specically

yi = yreal (1 + θi ),

and 0.3 for

i = 1, 2, 7, 8.

where

θi

is random uniform noise of amplitude 0.3, 0.1, 0.2

The noise of the rst sensor is displayed in Fig. 14. We also use

5 state sensors as previous sections. Because of space limitation, we only present the result for 6 and 10 velocity sensors. Fig. 15a and 15b depict the system response. Although the state oscillates because of the time-varying coecient and sensor noise, the controller still manages to regulate the system. The distance from the set point and the observer error are give in Fig. 16a, 16b, 17a and 17b. Both the distance and error are smaller in system with 10 velocity sensors. This is expected since the signal-to-noise ratio is higher with 10 sensors than 6 sensors.

21



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Discussion To further increase the quality of the modied equation-free method, there are at least 3 approaches we can investigate:

1. Enrich the snapshot ensemble

2. Increase the frequency of updating the correction term

3. Change the timing of activating the correction term

snapshot ensemble To show that a better collected set of snapshots can increase the quality of the estimation, we use 2 dierent initial conditions to collect o-line snapshots. The two initial conditions are the same with those in section Original Equation-free Method with Enriched Snapshots Ensemble (Eq. 39, Eq. 41). In total, 80 snapshots are collected. Results of systems using dierent number of velocity sensors and dierent snapshot ensembles are compared in Fig. 18 and 19. They are denoted by a; b, in which

b

a refers to the number of velocity sensors and

is the number of initial conditions. Fig. 18 illustrates the temporal prole of the distance

from the set point.

We can see that the oscillation in the system response is mitigated (

the amplitudes of the peaks at t=2.5 decrease). The errors of the observers are compared in Fig. 19. The errors in systems with enhanced snapshot ensemble are signicantly smaller. The error of the observer decreases and the system oscillates less as the number of velocity sensors increases except the case of

8

velocity sensors

Despite the dierence between the standard POD and the proposed equation free method, we are optimistic that the research results on improved snapshot sampling, such as Smith's work

63

and Graham's work

64

, might be extended to this equation free method.

22


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frequency of updating The second approach is to improve the accuracy of the correction term by varying the frequency of updating. Here we present the results for updating the correction term every 0.25 units of time. The results for updating every 0.5 units of time with 5 and 10 velocity sensors are denoted by lf5 and lf 10 in Fig. 21-20c, where lf means lower frequency. The distances from the set point are compared in Fig. 21 and Fig. 20a-20c depict the control actions. Similar to the rst approach (section snapshot ensemble ), increasing the updating frequency can also improve the accuracy of the estimation of the dynamics and mitigate the oscillation of the control actions. The eect of the frequency of updating is specially obvious with fewer sensors. However, this approach does not reduce much observer error. Although the information used in the correction term updating are measured continuously, the correction term is not updated continuously due to the computational cost. On the other hand, the updating frequency should be high enough to satisfy assumption 6. A more advanced way to deal with it is to identify criteria to determine the updating frequency. The research about varying the snapshot acquisition frequency in adaptive proper orthogonal decomposition

65

might be useful.

activation timing for the correction term The third approach is to activate the correction term after the transient period. The idea is the system in the fast stable subspace has not decayed to zero in the transient period. Hence, the large error in the ROM in the transient period will enter the correction term and may make the estimation worse before the correction term converges, which explains the peak in Fig. 11. In this section, the correction term is not activated until results in section

t = 0.5.

The

with 5 and 10 velocity sensors are denoted by m5 and m10 in Fig.

22-23c. Fig. 22 depicts the distance from the set point and Fig 23a-23c present the control actions. It can be observed that the performance of the controller is improved, especially when the number of velocity sensors is increased to 10, where the peak in Fig.22 disappears.

23



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The only exception is the control action in the system with 6 velocity sensors has a larger amplitude of oscillation. The waiting time should be long enough for the fast dynamics to relax, which depends on the ratio of the energy captured by the basis functions in the ROM. On the other hand, this waiting time cannot be too long since the equation free method without the updating step is not able to stabilize the system with a small number of sensors during this waiting period. To better present the improvement of above ideas, the amplitudes of the peak of the distance from the set point (Fig. 18, 21 and 22) are compared in table 2. It can be seen that except changing activation timing when using 6 sensors, all the 3 approaches either eliminates the oscillation or reduces the amplitude of the peak.

Conclusion In this manuscript, a modied version of equation-free control method is proposed and successfully applied to a diusion reaction process.

The assumption made in translating

the steady state in the previous work is relaxed. The result illustrates that the method is robust in the presence of noise and model mismatch. Three approaches to further improve the performance of the proposed method are discussed.

Acknowledgement Financial support from the National Science foundation, CMMI award #13-00322 and Provincial thousand talents introduction program, Zhejiang China is gratefully acknowledged

24


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(59) Henneron, T.; Clénet, S.; Université, L. E. P. Model Order Reduction of Non-Linear Magnetostatic Problems Based on POD and DEI Methods. IEEE TRANSACTIONS

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A Tutorial. Nonlinear Dynamics

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(64) Graham, M.; Kevrekidis, I. Alternative approaches to the Karhunen-Loeve decomposition for model reduction and data analysis. Computers & chemical engineering

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32


Page 33 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


offline Basis functions

snapshots

Sensor locations

online process

State measurement

State estimation controller

Velocity measurement

Dynamics estimation

Figure 1: ow chart of equation free control framework operations and online block diagram.

33



1 0.9

energy

5

x

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 54

0.7

0 4 2

t

0.5 1

2 0 0

f(x) x

2

3

4

5

mode size n

z

Figure 2a: system response of the open-

Figure 2b: energy captured by rst

loop process.

sis functions.

34


n

ba-

Page 35 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


u1

y1

u2

y2

u3

y3

…

yk

Figure 3: diagram of actuators (separate cooling jackets) and point sensors.

35



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 4a: system response

Figure 4b: system response

Figure 4c: system response

of the closed-loop process



with 5 plus 21 sensors.



36


Page 36 of 54

Page 37 of 54

-1

0

-3 0

5

10

3

-2 21 23 30

-3 0

5

t Figure 5a:

comparison of

10

control action u3

21 23 30

control action u2

3

control action u1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 1 0 -1 0

control

5

10

t

t Figure 5b:

21 23 30

comparison of

Figure 5c:

action

the control action of the 3rd

the control action of the 1st

the

actuator.

2nd actuator.

of

the

37


actuator.

comparison of

1.5

Page 38 of 54

0.3 21 23 30

1

21 23 30

0.2

error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

distance from set point


0.5

0 0

0.1

5

0 0

10

t

5

10

t

Figure 6: temporal prole of the distance

Figure 7: comparison of the error of the

of the state from the set point.

observer.

38


Page 39 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8a: closed-loop pro-

Figure 8b: closed-loop pro-

Figure 8c: closed-loop pro-

cess response when using 5



plus 10 sensors.

plus 12 sensors.

plus 15 sensors.

39


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



1.5

10 12 15

1

0.5

0 0

10

20

t Figure 9: temporal prole of the distance of the state from the set point.

40


Page 40 of 54

Page 41 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 10a: nondimensionalized temper-

Figure 10b: nondimensionalized temper-

ature prole in space and time of the


closed-loop process when using 5 plus 5


sensors.

sensors.

Figure 10c: nondimensionalized temper-

Figure 10d: nondimensionalized temper-





sensors.

sensors.

41


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



1.5

5 6 8 10

1

0.5

0 0

5

10

t Figure 11: temporal prole of the distance of the state from the set point.

42


Page 42 of 54

Page 43 of 54

2

0

-2 0

5

3

-1

-2

5 6 8 10

-3 0

10

5

t Figure 12a:

comparison of

10

control action u3

5 6 8 10

control action u2

-0.5

control action u1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2 1 0 -1 0

control

5

10

t

t Figure 12b:

5 6 8 10

comparison of

Figure 12c:

action



the

actuator.

2nd actuator.

of

the

43


actuator.

comparison of


0.3

noise

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 54

0

-0.3 0

10

20

30

40

50

t Figure 14:

Figure 13: spatial temporal temporal prole of

temporal prole of the rst

sensor noise.

βT .

44


Page 45 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure

15a:

spatiotemporal

prole

of

Figure 15b: system response of the closed-

the nondimensionized temperature for the

loop process with 5 plus 10 sensors in


presence of sensor noise and model mis-

sensors in presence of sensor noise and

match.

model mismatch.

45



1.5


1.5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1

0.5

0 0

Page 46 of 54

10

20

30

40

1

0.5

0 0

50

t

10

20

30

40

50

t

Figure 16a: distance of the state from the

Figure 16b: distance of the state from the

set point in system with 5 plus 6 sensors

set point in system with 5 plus 10 sensors

in presence of sensor noise and model mis-

in presence of sensor noise and model mis-

match.

match.

46


Page 47 of 54

0.3

0.2

0.2

error

0.3

error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

0 0

0.1

10

20

30

40

0 0

50

10

t Figure 17a:

20

30

40

50

t

error of observer in system

Figure 17b:

error of observer in system

with 5 plus 6 sensors in presence of sensor

with 5 plus 10 sensors in presence of sen-

noise and model mismatch.

sor noise and model mismatch.

47


1.5 1

Page 48 of 54

0.4

10;1 5;2 6;2 8;2 10;2

10;1 5;2 6;2 8;2 10;2

0.3

error

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



0.5

0.2 0.1

0 0

5

0 0

10

t

5

10

t


Figure 19: error of the observer.

of the state from the set point.

48


Page 49 of 54

2

0

-2 0

5

-1

3 lf5 lf10 5 6 8 10

control action u3

lf5 lf10 5 6 8 10

control action u2

-0.5

control action u1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-2

-3 0

10

5

t Figure 20a:

comparison of

10

lf5 lf10 5 6 8 10

2 1 0 -1 0

t

5

10

t

Figure 20b:

comparison of

Figure 20c:


the

action


actuator.

2nd actuator.

control

of

the

49


actuator.

comparison of

1.5


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



lf5 lf10 5 6 8 10

1

0.5

0 0

5

10

Page 50 of 54

1.5

m5 m10 5 6 8 10

1

0.5

0 0

t

5

10

t



of the state from the set point with dier-

of the state from the set point with dier-

ent frequencies of updating.

ent activation timing.

50


0 m5 m10 5 6 8 10

0

-3 0

5

control action u2

3

control action u1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-1

3

m5 m10 5 6 8 10

control action u3

Page 51 of 54

-2

-3 0

10

5

t Figure 23a:

comparison of

10

m5 m10 5 6 8 10

2 1 0 -1 0

t

5

10

t

Figure 23b:

comparison of

Figure 23c:


the

action


actuator.

2nd actuator.

control

of

the

51


actuator.

comparison of


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 54

Table 1: algorithm of equation-free method oine

x

and

f

1

collect snapshots of

2

generate basis functions for

3

determine sensor location by applying DEIM to basis functions

4

calculate the matrices for output-state mapping and output-dynamics mapping

5

design controller

6

set

6

continuous measurement of state sensors

Eq. 15

7

continuous measurement of velocity

Eq. 16

8

determine control action

x

and

f

using POD

Eq. 9 Eq. 15-16

f =0 ∆f 0 online

9

update correction term

u f ∆f

→x sensors → f

Eq. 25 for

T

Φ f

using estimated

x

52


Eq. 29-30

Page 53 of 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: the amplitude of the peak of the distance from the set point modied number of sensors

5

10

0.760

0.594

snapshots ensemble mehtod

frequency of updating activation timing

further improvement 5

6

0.333

0.276

N/A

N/A

0.65

0.648

0.627

0.546

0.677

0.815

0.497

N/A

53


8

10


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Page 54 of 54

process

snapshots

controller

Basis functions

State estimator

Dynamics estimator

Sensor location

State measurement

Velocity measurement


Synthesis of Equation-Free Control Structures for Dissipative

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