High Order Approximations for Control& Estimation of PDE Systems

John A. Burns and James CheungInterdisciplinary Center for Applied Mathematics

Virginia TechBlacksburg, Virginia 24061-0531

WORKSHOP ON DYNAMICS, CONTROL AND NUMERICSFOR FRACTIONAL PDEs

December 5 7, 2018San Juan, Puerto Rico

Motivation: From Energy Independence to Homeland Security to National Defense

Mathematical Characteristics Common to All

Air ConditioningThermal + Fluid

+ Mechanical +

Wing FlutterStructure + Fluid

Plume EstimationConvection-Diffusion

+ Flight DynamicsOptimal Sensor LocationThermal + Fluid

Heat ExchangerThermal + Fluid

Mathematical Characteristics Common to All

Air ConditioningThermal + Fluid

+ Mechanical +

Wing FlutterStructure + Fluid

Plume EstimationConvection-Diffusion

+ Flight DynamicsOptimal Sensor LocationThermal + Fluid

Heat ExchangerThermal + Fluid

Modeling, control, estimation & optimizationof multidiscipline systems

People & Problems

Virginia Tech: Jeff Borggaard, Gene Cliff, Terry Herdman, Veronica Gheorghe, Zach Grigorian, James Cheung

Carnegie Mellon: Larry BieglerHumboldt, Berlin: Carlos N. RautenbergOklahoma State: Weiwei HuUT CCS: Trevor Bailey, Qingfan Zeng, Degang Fu, Rui

Huang, Clas JacobsonWPI: Michael Demetriou, Nikolaos Gatsonis

Defense

HomelandSecurity

Energy Independence

PDE Optimization & Control of Dynamic Sensor Systems

computational complexity

Combine physics based models, higher order methods and optimization for real time estimation and control

Wing Flutter: Aeroelasticity

Simplest Linear Flutter Model

( )h t

( ) t - total airfoil cirulation( )t

( )t( ) ( ) ( ) ( ) ( ) ( ) ( ) (t) ( )T

x t h t t t t h t t t =

0 0( ) ( ) ( ) ( ) ( ) ( ) ( )d x t s x t s ds x t s x t s ds u t

dt + + = + + + A D L K G

8,82( ) Uss

s

=D8,8 0=A

non-atomic neutral delay differential equation

weakly singular kernel

U

Simple Aeroelastic Models

J. A. Burns, E. M. Cliff and T. L. Herdman, A State Space Model for an Aeroelastic System, 22nd IEEE Confeence on Decision and Control, (1983), 1074-1077.J. A. Burns, T. L. Herdman and H. Stech, Linear functional differential equations as semigroups on product spaces, SIAM J. Math. Anal. 14 (1983), 98-116.J. A. Burns, E. M. Cliff and R. K. Powers , Modeling and Control of an Aeroelastic System, Fourth IFAC Symposium on Control of Distributed Parameter Systems, UCLA, July 1986, 6-11.

A. V. Balakrishnan, Active Control of Airfoils in Unsteady Aerodynamics, Applied Mathematics and Optimization, 4 (1978), 171-195.

H. Ozbay and J. Turi,, Robust Control of an Aeroelastic System Modeled by a Singular Integro-Differential Equation, Fourth IFAC Symposium on Control of Distributed Parameter Systems, UCLA, July 1986, 2682-2686.J. A. Burns and T. L. Herdman, Neutral Functional Integro- Differential Equations with Weakly Singular Kernels, JMAA, 1990, 1074-1077.

D. Swinney, A Fractional Calculus Model of Aeroelasticity, PhD Thesis, AFIT, Dayton, OH, 1989.R. Bagley, D. Swinney and K. Griffin , Fractional Order Calculus Model ofthe Generalized Theodorsen Functions, J. Aircraft, 30 (1993), 1074-1077.

1003--1005

BIG advances in past 30 years - I. Lasiecka...

Optimization of Real Time Dynamic Sensor Systems

( )( ) ( ) ( )z t z tt = + V

[ ] [ ]( , ) ( , ) ( , ) ( , ) ( , ) ( ) ,( ) , 0tz t x K t x z t x t x z t x g x x tt= + >V

( ( ), )

( ) ( , ) ( , ) ( )jB tw

jy t h t x z t x dx D t

= +

2

2

( )(( ( ), ( ), ( ), ( )) ( ( ), ( ))j j j j j j j j j

d tf x t y t z t t f w tu

dw

tut

= =

( ) ( ( ), ( ), ( ))j j j jt x t y zw t t=

2

2( ) ( ( ), ( ))d t t t

dt=

uwfw

( ) [ ( ) ( ) ( )] ( ) ) )) ( ((e ez t t t z t tt t = + + yV

[ ( ) ( , ( )) ( , ( ))] ( ) ( , ( )) ( ) ( )et t t t z t t t t t= + +

V yw w w

F0max ( ) ( )et TE z t z t =

State estimator (observer)

: Find the observer ( ) ( , ( ), ( )) to mioperator nimizeopt optt t t t=Problem

uwF 2

0( ) ( )

T

eE z s z s ds= ( )F 20 ( ) ( )T eE z s z s ds=

control / estimation and optimization

of acomposite system

or or

( ) ( ( )) ( )( )t z t t= +

y w

Optimization of Real Time Dynamic Sensor Systems

[ ] [ ]( , ) ( , ) ( , ) ( , ) ( , ) ( ) ,( ) , 0tz t x K t x z t x t x z t x g x x tt= + >V

2

2( ) ( ( ), ( ))d t t t

dt= wf uw

( ) [ ( ) ( ) ( )] ( ) ) )) ( ((e ez t t t z t tt t = + + yV

[ ( ) ( , ( )) ( , ( ))] ( () ( , ( ) ) )) (et t t t z t tt t t= + + yw wV w

REAL TIME estimation, control & optimization ?

( ) ( ( ), ( ), ( ))j j j jw t x t y t z t=

IMPOSSIBLE ! MAYBE NOT ?

Egorova, T., Gatsonis, N.A., and Demetriou, M.A., Estimation of Gaseous Plume Concentration with an Unmanned Aerial Vehicle, Vol. 39, No. 6, pp. 1314-1324, Journal of Guidance, Control, and Dynamics, 2016.

Demetriou, M. A., Gatsonis, N.A., and Court, J., A coupled controls-computational fluids approach for the estimation of the concentration from a moving gaseous source in a 2D domain with a Lyapunov-guided sensing aerial vehicle, IEEE Transactions on Control Systems Technology, Vol. 22, 3, pp. 853 867, 2014

( ( ), )

( ) ( , ) ( , ) ( )jB tw

jy t h t x z t x dx D t

= +

Moving Intruder & Sub-Optimal Luenberger Estimator

Intruder with plume (Left) Estimated Plume by a UAVcarrying a noiseless sensor (Right)

Demetriou&

Gatsonis(WPI)

Moving Sensor & Sub-Optimal LuenbergerEstimator with Grid Adaptation

Demetriou & Gatsonis (WPI)

BUT, REAL TIME OPTIMAL ESTIMATION IS IMPOSSIBLE - RIGHT?

MAYBE NOT - WITH HIGHER ORDER (DG hp) METHODSAND

GPU COMPUTING !Adaptive multi-grid / domain decomposition method

Computing (Parallel, DG & Spectral Element on GPUs)

This computation (a 3D heat equation) involved more than 20 million equations

J.-F. Remacle, R. Gandham and T. Warburton, GPU accelerated spectral finite elements on all-hex meshes, Journal of Computational Physics 324(2016), 246257.

and was solved in a few seconds on a single $500 GPU processor by using a DG spectral element method !

M. Kohler and J. Saak, On GPU acceleration of common solvers forquasi-triangular generalized Lyapunov equations, J. Parallel Computing, 57 (2016), 212221.

Heat Exchanger: Basic Vapor Compression System

1

2

3

2

2

( ) ( )

( ) ( )

( ) ( ) -des

f

v t P tv t V tv t C t

=

=

=

- compressor input power (kw)- compressor volume flow rate (m / s) venturi area valve (m )

1 ( )v tCOMPRESSOREXPANSIONVALVE

2 ( )v t

3( )v t

CONDENSER

EVAPORATOR

outQ

inQ "hot space""cool space"

MO

TOR

1( ) voltageu t =

1( ) ( )a aH w t v t=

1( ) ( ) ( )a a a aw t uA w t B t= +

actu

ator

dyn

amic

s

Heat

Exc

hang

ers (

HX)

"cold fluid"

DYNAMIC MODELINGA CHALLENGE

WELL-POSEDNESS OF CLOSEDCYCLE IS NOT OBVIOUS

ODE1 PDE1 ODE2 PDE2 ODE1: Closed Cycle

( , ) ( , )t s t st s =

0s =s L= s

2( , ) ( )yL tt =2( , ) ( ( ))t s t sy L = +

2( ,0) ( )yt t L =

21( )) (yu t Lt = 2 ( )c t= 2 ( )c t L=

0s =s L= s

1 1 1( )( ) ( )w t a w t b u t= +

11( )) (cy wt t=

( , ) ( , )t s t st s =

1( ,0) ( )v tt =

1( )t

11( () )v yt t=

1( )w t

2 ( , )( ) Lv t t=

1( , ) ( )t s v t s = +

2 2 2 () ) )( (t a t tb u= +

22 ( )) (cy t t=

22 ( () )u vt t=1 ( )c w t L=

ODE1 PDE1 ODE2 PDE2 ODE1: Closed Cycle

2 2 1( ) ( ) ( )t a t b c w t L= +

1 1 2( ) ( ) ( )w t a w t b c t L= + ( )( )( )

w tt

t

=

x

1 1 2

2 2 1

0 0( ) ( ) ( )( )

0 0( ) ( ) ( )a b cw t w t w t L

x ta b ct t t L

= = +

AGAIN, a delay differential equation

1 2( )( ) : ( ) ( ,0) , (0)D H L

= =

xx

1 2

( )( )( )L

= + F F

x

( )1

2 1 12 2

( )( ) ( )

( ,0) ( ,0)Z L L L Lw

= =

zx

1 2( ) ( ) ( )t t t L= + F Fx x x

ODE1 PDE1 ODE2 PDE2 ODE1: Closed Cycle

1( , ) ( )t s c w t s= + 2( , ) ( ( ))t s c t L s= +

Example shows that these type of systems are rarelyself-adjoint and in fact can be highly non-normal.

1 1 2

2 2 1

0 0( ) ( ) ( )( )

0 0( ) ( ) ( )a b cw t w t w t L

x ta b ct t t L

= = +

This system is closely related to a delay differential equationNumerical methods developed for delay systems offer insightMuch more difficult if the ODE systems are vector systemsSteady state behavior of closed cycle cannot be implied by linking steady state solutions of each component

Interest in high order "dual convergent" approximationschemes for optimal estimation and control.

A Simple (Linear) Counter Flow Heat Exchanger

( )2

1 1 11 2 11 12

( , ) ( , ) ( , ) ( , ) ( , ) , ( ( ),0) ,T t x T t x T t x h T t x T t x T tt x

tx

g = + =

1v

( )2

2 2 22 1 22 22

( , ) ( , ) ( , ) ( , ) ( , ) , ( , ) 0,T t x T t x T t x h T t x T t x T t Lt xx

= + + =

2v

11

( , ) 0T t Lx

=

22

( ,0) 0T tx

=

actuator dynamicspossible controls - disturbances

( )u t ( )g t( ) ( ) ( )a aw t w t u t= +A B

( ) ( )ag t w t= H

( ) ( ) ( )a aw t w t u t= +A B

A Simple (Linear) Counter Flow Heat Exchanger

( )2

1 1 11 2 11 12

( , ) ( , ) ( , ) ( , ) ( , ) , ( ( ),0) ,T t x T t x T t x h T t x T t x T tt x

tx

g = + =

1v

( )2

2 2 22 1 22 22

( , ) ( , ) ( , ) ( , ) ( , ) , ( , ) 0,T t x T t x T t x h T t x T t x T t Lt xx

= + + =

2v

11

( , ) 0T t Lx

=

22

( ,0) 0T tx

=

actuator dynamicspossible controls - disturbances

( )u t ( )g t( ) ( ) ( )a aw t w t u t= +A B ( ) ( ) ( )a aw t w t u t= +A B

( ) ( )ag t w t r= H

A Simple (Linear) Counter Flow Heat Exchanger

CHANGE OF VARIABLES 1 1( , ) ( , ) ( )t x T t x g t = 2 2( , ) ( , )t x T t x =

( )2

1 1 111 2 1 1 12

( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( )t x t x t x h t x t x uF w t B tt xx

= + + + 1

v

( )2

2 2 22 1 2 222

( , ) ( , ) ( , ) ( , ) ( , ) (t)t x t x t x h t x t x F wt xx

= + + + 2

v

11 1

( , )( ,0) 0t Ltx

= =

222

( ,0) ( , ) 0t t Lx

= =

1 2 11 2[ ]a a a a a aF F B = + = + = H H A H H B

( ) ( ) ( )a aw t w t u t= +A B

STATE OF THE ART IN INDUSTRY: FINITE VOLUME METHOD ONSIMPLIFIED STEADY STATE MODELS

CHALLENGES: NUMERICAL SCHEMES FOR CONVECTION-DIFFUSION SYSTEMS (& FULL NONLINEAR CONSERVATION LAWS) THAT ARE:

(a) VALID ACROSS RANGE OF OPERATIONS (e.g., LOW FLOW)(b) SUITABLE FOR OPTIMIZATION & CONTROL DESIGN(c) HIGH ORDER & SUITABLE FOR GPU COMPUTING

Combined FE-FV Approximations

( )jP Basic Idea : Approximate the selfadjoint operators using finite elements

P. Deuring, R. Eymard, and M. Mildner, L2-stability independent of diffusion for a finite element - finite volume discretization of a linear convection-diffusion equation, SIAM Journal on Numerical Analysis, 53 (2015), 508--526

( )2

1 1 111 2 1 1 12

( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( )t x t x t x h t x t x uF w t B tt xx

= + + + 1

v 11 1( , )( ,0) 0t Ltx

= =

( )2

2 2 22 1 2 222

( , ) ( , ) ( , ) ( , ) ( , ) (t)t x t x t x h t x t x F wt xx

= + + + 2

v 222

( ,0) ( , ) 0t t Lx

= =

( ) ( ) ( )a aw t w t u t= +A B

J. A. Burns and L. Zietsman, Control of a Thermal Fluid Heat Exchanger with Actuator Dynamics, in Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, December, 2016, 31313136.

( )jH v and the non - selfadjoint operators using a upwind finite volume scheme.

FVM IS LOW ORDER USE HIGHER ORDER DG or hp SCHEMES?

2

2( )j j

xP

=The differential operators

are elliptic (and selfadjoint)( )j j xv vH

=The differential operators

are NOT selfadjoint

Combined FE-FV Approximations( )jH v on the non - selfadjoint "hyperbolic" operators Focus

Use "higher order" (and dual convergent) DG & " " type schemeshp

- K. lto and F. Kappel, A Uniformly Differentiable Approximation Scheme for Delay Systems Using Splines, Applied Mathematics and Optimization, 23 (1991), 217--262 .

Related to an idea introduced by Ito and Kappel in 1991 (for delay equations)

- J. S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms,Analysis, and Applications. Springer Science & Business Media, 2007- J. S. Hesthaven. Numerical Methods for Conservation Laws:From Analysis to Algorithms, SIAM, 2018.- J. Cheung, Overcoming Geometric Limitations in the Finite Element Method by Means of Polynomial Extension: Application to SecondOrder Elliptic Boundary Value and Interface Problems, PhD Thesis, Florida State University, 2018.

( )2

1 1 111 2 1 1 12

( , ) ( , ) ( , ) ( , ) ( , ) ( ) ( )t x t x t x h t x t x uF w t B tt xx

= + + + 1

v 11 1( , )( ,0) 0t Ltx

= =

( )2

2 2 22 1 2 222

( , ) ( , ) ( , ) ( , ) ( , ) (t)t x t x t x h t x t x F wt xx

= + + + 2

v 222

( ,0) ( , ) 0t t Lx

= =

( ) ( ) ( )a aw t w t u t= +A B2

2( )j j

xP

=The differential operators

are elliptic (and selfadjoint)( )j j xv vH

=The differential operators

are NOT selfadjoint

A Higher Order FE - C0 Galerkin Scheme

( , ) ( , ) 0 1 0u t x u t x x tt x

= < < >

, , 1v ( ,0) 0u t = (0, ) ( )u x x=

0x

11x0x = Nx=1kx kx 1kx +

0 ( )h x

( )kh x

( )Nh x

11, ( , ]0, otherwis

)e

( k kkx x x

x

=

{ }( ) : 1,2,..,N kW span kh N= = { }( ) : 1,2,..,N kZ span k N= =

1Nx =

Finite Volume - Upwind Scheme

( , ) ( , ) 0 1 0u t x u t x x tt x

= < < >

, , 1v ( ,0)u t = (0, ) ( )u x x=

( ) ( )N N NFV z (t) z t =

1( ) ( )k

NN

kk

z (t a x) t=

=

Dual convergence holds, but low order /slow

For optimization; we want a "higher order" and dual convergent scheme...

NZ

- K. lto and F. Kappel, A Uniformly Differentiable Approximation Scheme for Delay Systems Using Splines, Applied Mathematics and Optimization, 23 (1991), 217--262 .

Take advantage of the Ito-Kappel idea ...

Two Spline Schemes (FE)

( , ) ( , ) 0 1 0u t x u t x x tt x

= < < >

, , 1v ( ,0)u t = (0, ) ( )u x x=

1( ) ( )k

NN

kk

w (t b x) ht=

=

( ) ( )N N NFE FEw (t) w t =

... dual convergence fails

"standard" FE scheme

:N NFE Z W

NW

NW

Two Spline Schemes (IK)

( , ) ( , ) 0 1 0u t x u t x x tt x

= < < >

, , 1v ( ,0)u t = (0, ) ( )u x x=

( ) ( )N N NCG FV w (t) w t =

1( ) ( )k

NN

kk

w (t b x) ht=

=

:N NFV Z Z

1( )( )

NN N

kk

khw t b Wx( ) t=

= 1

() )(N

N Nx k

kkh(t t xw ) b Z

=

=

NZ

NW

Higher order AND dual convergence holds

System Matrices

.5 0 0 0

.5 .5 0 00 .5 .5

00 0 .5 .5

NQ

=

1The matrix representation of is given byN

1

1 0 0 01 1 0 0

0 1 10

0 0 1 1

NFV

vKx

=

1

2

( )( )

( )

( )

N

N

b tb t

b t

b t

=

1( ) ( )k

NN

kk

w (t b x) ht=

=

1 :N N NW Z

Approximating System

( ) ( )N N N NCG CGb (t) H b t =

1[ ]N N NCG FVH K Q=

* 0 0 0* * 0 0* * *

0* * * *

=

triaObse ngulrve ... ar formM. Kohler and J. Saak, On GPU acceleration of common solvers for

quasi-triangular generalized Lyapunov equations, J. Parallel Computing, 57 (2016), 212221.

Goal - Higher Order FE C0G Scheme

2

2( , ) ( , ) ( , )u t x u t x u t x

t xx =

v ( , )( ,0) , 0u t Lu t

x = =

2

2( )

xP

=Finite Element for ( )

xvH v

=Higher Order "DG" and " " FE for hp

1

( ) ( )kN

Nk

ku (t u x) ht

=

= (0, ) ( )u x xinitial data

a comparison of 4 methods:

01) Standard FV method for the case =

( )00

& ii

vvi