Observers and Optimal Estimation for PDE Systems: Sensor ......M. A. Demetriou and D. Ucinski, “State estimation of spatially distributed processes using mobile sensing agents”,

IMA September 6, 2017

IMA Workshop on Sensor Location in Distributed Parameter Systems September 6 - 8, 2017 Minneapolis, MN

SENSOR LOCATIO

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q

SENSOR LOCATIO

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q

1

Observers and Optimal Estimation for PDE Systems: Sensor Location Problems

(A Short Overview) John A. Burns

11Interdisciplinary Center for Applied Mathematics Virginia Tech

Blacksburg, Virginia 24061-0531

People and Background Virginia Tech (ICAM) - Jeff Borggaard, Gene Cliff, Terry Herdman, Lizette Zietsman Carnegie Mellon (Chemical Engineering) - Larry Biegler Humboldt University in Berlin (Math) - Carlos Rautenberg MIT (Aeronautics & Astronautics) – Boris Kramer Oklahoma State (Math) - Weiwei Hu United Technologies Corporation (UT Aerospace, Carrier, UTRC) - Trevor Bailey (CCS), Degang Fu (CCS Shanghai), Rui Huang (CCS), Clas Jacobson (UTRC) Worcester Polytechnic Institute (Mechanical Engineering) - Michael Demetriou, Nikolaos Gatsonis

Applications to energy efficient buildings, thermal management systems, battlefield management, homeland security Any spatially dependent system …


Best sensor location depends on problem: system identification, feedback control with incomplete

information, estimation … 2

Old Ideas


Fred E. Thau, “On optimum filtering for a class of linear distributed-parameter systems”, Journal of Basic Engineering, 91 (1969), 173 – 178.

Alain Bensoussan, “Optimization of sensors' location in a distributed filtering problem”, in Stability of stochastic dynamical systems: Proceedings 1972 International Warwick Conference, R. Curtain ed., pp. 62 - 84, 1972, Springer.

Thomas K. Yu and John H. Seinfeld, “Observability and optimal measurement location in linear distributed parameter systems”, International Journal of Control, 18 (1973), 785 – 799.

Ruth F. Curtain, Akira Ichikawa and E. G. Ryan, “Optimal location of point sensors”, in Proc. IFIP Conference on Modelling and Identification of Distributed Parameter Systems, Rome, 1976.

S. Kumar and J. H. Seinfeld, “Optimal location of measurements in tubular reactors”, J. Chemical Engineering Science, 33 (1978), 1507 – 1516.

Sigeru Omatu and John H. Seinfeld, “Distributed Parameter Systems: Theory and Applications”, 1989, Clarendon Press. (Chapter 11)

S. Omatu, S. Koide and T. Soeda, “Optimal sensor location problem for a linear distributed parameter system”, IEEE Trans. on Automatic Control, 23, (1978), 665 – 673.

C. S. Kubrusly and H. Malebranche, “Sensors and controllers location in distributed systems. A survey”, Automatica, 27 (1982), 242 – 244.

3

Basic Idea


( )( ) ( ) ( )z t z tt w= + u DP SYSTEM

( )( ) ( , , ) ( )qy vt t z t t= ε + SENSED OUTPUT

2( , ) ( , ) ( , ) ( , ) ( ) , , 0( )t z t x z t x t x wz t x tg x x tκ∂∂ = ∇ − ⋅∇ + ∈Ω >u

( ) ( , , , ) ( , ) ( )y t c t x z t x d vx Dq tΩ

= ε +∫∫∫

( , , ), ( )( , , , ) .

0, ( )k t x x B

c t xx B

q qq

qε

ε

∈ Ωε = ∉ Ω

( )( )11 ( ) ( ( ))2

1/23/2

1( , , , )(2 )

Tqx t x tqqc t x e

−− − Λ −ε =

π Λ

( , , , ) ( ) ( , , , ) ( ( )) or c t x x c t x xq q tq qεε = δ − ε = δ −

4

Least Squares Estimator (Formal)


( ) ( ) ( ) ( )Tt W tw w t = δτ − τ ( ) ( ) ( ) ( )Tt V tv v t = δτ − τ

*( ) ( )M Wt t= *( ) ( )N Vt t=

[ ]( ) ( ) ( ) ( ) ( , , ) ( )e ez (t) z t t y t t z tq= + − ε u Kalman Filter

[ ] [ ]* 1( ) ( ) ( , , ) ( )Nt t qt t −= Π ε

[ ] [ ]2( ) ( , , , ) ( ) ( , , , )ez t z t Trace t Traq ce t q − ε = Π = Π ε u u

[ ] [ ] [ ]* 1*( ) ( ) ( ) ( ) ( ) ( ) ( , , ) ( ) ( , , ) ( ) ( )t t t t t t tqN t tq Mε ε−Π = Π + Π −Π Π + u u

[ ]( ) [ ]( )(0) (0) (0) (0) (0)T

z z z z Π = − −

[ ]2

0 0( , , ) ( ) ( , , , ) ( , , , )f fT T

eJ z s z s ds Trace s dsq q q ε = − ε = Π ε ∫ ∫ u u uTotal Error

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An “Optimal Sensor Location Problem”


[ ] [ ] [ ]( )* 1* *

0( ) ( ) (0) ( ) ( ) ( ) ( ) ( , , ) ( ) ( , , ) ( ) ( )

tqt S t S t S t s s s t s t s S t sM qN dsε ε− Π = Π + − −Π Π −

∫

( )( ) ( , ) tS t S t e= = uu( ) ( , , , )t t q εΠ = Π u

*( , ; , ) ( ) ( ) (0) ( )G t S t Sq tεΠ = Π − Πu

[ ] [ ] [ ]( )* 1 *

0( ) ( ) ( ) ( , , ) ( ) ( , , ) ( ) ( )

tS t s s s t s t s S t s dsq N qM ε ε− − − − Π Π −

∫

( , ; , ) 0qG εΠ =u

subject to

[ ]0

min ( , ; , ) ( , ; , )f

q

T

QJ Trace s dsq qε ε

∈Π = Π∫u u


[ ]( ) [ ]( )(0) (0) (0) (0) (0)T

z z z z Π = − −

6

An “LQG Optimal Sensor Location Problem”


( )( ) ( , ) tS t S t e= = uu( , , )q εΠ = Π u

( , ; , ) 0qG εΠ =usubject to

[ ]min ( , , , ) ( ; , )q Q

J Traceq qε ε∈

Π = Πu u

[ ] [ ] [ ]* 1*0 ( ) ( ) ( , ) ( , ) qN Mq ε ε−= Π + Π −Π Π + u u

( )( ) ( ) ( )z t z tt w= + u

( ) ( , ) ( ( ))y t z t v tq= ε +

[ ] [ ] [ ]* 1*(, ; , ) ( ) ( ) ( , ) ( , ) G q qN Mq ε ε ε−Π = Π +Π −Π Π + u u u

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Example


DOOR OPEN DOOR CLOSED

SENSOR LOCATION

optq

SOLVING THE OPTIMAL SENSOR LOCATION PROBLEM

SENSOR LOCATION

q

SENSOR LOCATION

q

8

SENSOR LOCATION

optq

Example


DOOR CLOSED

SENSOR LOCATION

q SENSOR LOCATION

optq

( ( ))optqTrace Π

22% REDUCTION IN ESTIMATION ERROR

( ( ))Tr ce qa Π

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Theoretical Basis (Hard Stuff)



[ ]( ) [ ]( )(0) (0) (0) (0) (0)T

z z z z Π = − −

Infinite Dimensional Filtering

S. G. Tzafestas and J. M. Nightinggale, “Optimal filtering, smoothing and prediction in linear distributed parameter systems”, Proc. lEE, 115 (1968), 1207 - 1212..

P. L. Falb, “Infinite-dimensional Filtering: The Kalmann - Bucy filter in Hilbert space”, Information and Control, 11 (1967), 102 - 137.

A. Bensoussan, L’identification et le filtrage, Ph.D. Thesis, I.R.I.A., 1969.

R. F. Curtain, “Infinite-Dimensional Filtering”, SIAM J. Control, 13 (1975), 89 - 104.

J. S. Meditch, “Least squares .filtering and smoothing for linear distributed parameter systems, Automatica (1971), 315 - 322.

Y. Sakawa, “Optimal filtering in linear distributed-parameter systems”, International J. Control, 16 (1972), 115 - 127.

R. F. Curtain, “A survey of infinite-dimensional filtering”, SIAM Review, 17, (1975), 395 – 411.

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G. Da Prato and J. Zabczyk, “Stochastic Equations in Infinite Dimensions”, Cambridge University Press, 2014.

Riccati Differential & Integral Equation


[ ] [ ] [ ]( )* 1* *

0( ) ( ) (0) ( ) ( ) ( ) ( ) ( , , ) ( ) ( , , ) ( ) ( )


∫

R. F. Curtain and A. J. Pritchard, “The infinite-dimensional Riccati equation for systems defined by evolution operators”, SIAM Journal on Control and Optimization, 13 (1976), 951 - 983.


J. S. Gibson, “The Riccati integral equations for optimal control problems on Hilbert spaces”, SIAM Journal on Control and Optimization, 17 (1979), 537 – 565.

Types of solutions: (weak, strong, uniform operator? Meaning of the integral: (weak, strong, Bochner, Pettis, McShane “P”?

J. A. Burns and Carlos Rautenberg, “Solutions and Approximations to the Riccati Integral Equation with Values in a Space of Compact Operators”, SIAM Journal on Control and Optimization, 53 (2015), 2846 – 2877.

Carlos Rautenberg, “A Distributed Parameter Approach to Optimal Filtering and Estimation with Mobile Sensor Networks”, Ph.D. Thesis, Virginia Tech, 2010.

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J. A. Germani , L. Jetto and M. Piccioni, “Galerkin approximation for optimal linear filtering of infinite-dimensional linear systems”, SIAM J. Control and Optimization, 26 (1988), 1287 – 1305.

W. W. Hu, K. Morris and Y. Zhang, “Sensor location in a controlled thermal fluid”, 2016 IEEE 55th CDC, 2259 - 2264.

Some Issues & Challenges

IMA September 6, 2017 12

( )( ) ( ) ( )z t z tt w= + u DP SYSTEM

( )( ) ( , , ) ( )qy vt t z t t= ε + SENSED OUTPUT

T. M. Hintermuller, Carlos N. Rautenberg, M. Mohammadi and M. Kanitsar“Optimal sensor placement: A robust approach”, WIAS Preprint No. 2287, 2016, submitted..

Existence of solutions: (weak, strong, uniform operator?


[ ] [ ] [ ]( )* 1* *

0( ) ( ) (0) ( ) ( ) ( ) ( ) ( , , ) ( ) ( , , ) ( ) ( )


∫

Existence of optimal solutions? “Point measurements”?

Meaning of the integral

Real time computation … especially for mobile sensor systems

Robustness

Mobile Sensor Systems


Why moving sensors? Applications Can improve “observability”

Why now? Applications !! Advances in theory Advances in computing Possible to implement in real time

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Y. Khapalov, “Continuous observability for parabolic system under observations of discrete type”, IEEE Transactions on Automatic Control, 38 (1993), 1388 - 1391.

A. Y. Khapalov, “L ∞ - exact observability of the heat equation with scanning pointwise sensor”, SIAM journal on control and optimization, 32 (1994), 1037 - 1051.

A. G. Butkovskiy and L. M. Pustylnikov, “Mobile Control of Distributed Parameter Systems”, John Wiley, 1987.

M. A. Demetriou and D. Ucinski, “State estimation of spatially distributed processes using mobile sensing agents”, 2011 American Control Conference, 1770 - 1776.

E. RafajŁowicz, “Optimum choice of moving sensor trajectories for distributed-parameter system identification”, Int. J. Control, 32 (1986), 1441 – 1451.

Real Time Mobile Sensor Systems


T. Egorova, N. A. Gatsonis and M. A. Demetriou, “Plume estimation using static and dynamic formations of unmanned aerial vehicles”, 2016 IEEE 55th Conference on Decision and Control, 2270 - 2275..

M. A. Demetriou, “Emulating a mobile spatially distributed sensor by mobile pointwise sensors in state estimation of partial differential equations via spatial interpolation”, 2017 American Control Conference, 3243 - 3248.

Real time solutions!

T. Egorova, N. A. Gatsonis and M. A. Demetriou, “Estimation of Gaseous Plume Concentration with an Unmanned Aerial Vehicle”, Journal of Guidance, Control, and Dynamics, 39 (2016), 1314 – 1324.

Luenberger type “suboptimal” observer

Need to develop “theoretical” framework

Regional Estimators / Observers


Why regional estimators? Applications More “practical”

Why now? Applications (AGAIN) Advances in theory Can be used to improve computation time Possible to implement in real time

R. Al-Saphory, “Asymptotic Regional Boundary Observer in Distributed Parameter Systems via Sensors Structures”, Sensors, 2 (2002), 137 - 152

E. Zerrik, L. Badraoui and A. El Jai, “Sensors and regional boundary state reconstruction of parabolic systems”, Sensors & Actuators: A. Physical, 75 (1999), 102 - 117.

R. Al-Saphory and A. El-Jai, “Sensors characterizations for regional boundary detectability in distributed parameter systems”, Sensors & Actuators: A. Physical, 94 (2001), 1 – 10.

R. A. Al-Saphory, N. J. Al-Jawari and A. N. Al-Janabi, “Asymptotic Regional Gradient Full-Order Observer in Distributed Parabolic Systems”, International J. Contemporary Mathematical Science, , 11 (2016), 343 – 358.

R. Al-Saphory and A. El-Jai, “Sensors structures and regional detectability of parabolic distributed systems”, Sensors & Actuators: A. Physical, 90 (2001), 163 - 171.

Regional Estimators: Trivial Example


(0,2) and let =(0,1)Ω = ⊂ ΩR

2

2( , ) ( , ) ( , ), (0,2), 0t xz t x z t x f t x x t∂ ∂∂ ∂= + ∈ >

(0, ) ( ), ( ,0) 0, ( ,2) 0z x x z t z tϕ= = =

( ) ( , ), 0 2y t t qz q= < <Ideal “point sensor”

Estimate ( , ) 0 1 for z t x x< <

2

2( , ) ( , ) ( , ), (0,1), 0e et xz t x z t x f t x x t∂ ∂∂ ∂= + ∈ >

(0,1)(0, ) ( ) | , ( ,0) 0, ( ,1) ( )e e ez x x z t z t y tϕ= = =

( ) ( , ), 01 1 2y t z t q= < <= ( , ) ( , ), 0 1ez t x z t x x≡ ≤ ≤

Regional Estimators / Observers


“Perfect” estimator (idealized) Uses boundary information (makes sense) Unbounded output operator Theoretical issues remain

Computational advantages Only need to “grid” the (smaller) region Can use multi-grid & multi-region methods Demetriou & Gatsonis use similar ideas to achieve real time implementation

How do we put all this together? Theoretical framework (possibly unbounded observations) Understanding relationships between optimal sensor location problems and “sub-optimal locations” with real time algorithms

Computational challenges (real time) For Kalman type optimal sensor location problems For moving sensor systems (with flight dynamics) For robustness

Issues and Challenges

Fast Riccati Solvers: Chandrasekhar


[ ] [ ]** *( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( , ) ( ) t t t t t qt tqΠ = Π +Π −Π Π + u u

* *( ) ( )W MtM t = = = * *( ) ( ) pN tVt IN= = = =


[ ]*( ) ( ) ( , , )t t t q= Π ε

*

0( ) (0) ( )[ ( )]

tt s s dsΠ = Π + ∫

[ ]**( ) ( ) ( ) ( , ) ,t t t t q=

[ ]( )( ) ( ) ( ) ( , ) ( ),t t qt t= − u

(0) 0=

(0) =

Chandrasekhar (PDEs)

Chandrasekhar Equations (ODEs)


[ ]**( ) ( ) ( ) ( , ) ,F t L t L t C qt=

[ ]( )( ) ( ) ( ) ( , ) ( ),L t A F t C t L tq= − u

(0) 0F =

(0)L G=

Chandrasekhar ODEs

Peter L. Falb, “Infinite-dimensional Filtering: The Kalmann - Bucy filter in Hilbert space”, Information and Control, 11 (1967), 102 - 137.

T. Kailath, “Some Chandrasekhar-type algorithms for quadratic regulators”, 11th IEEE CDC, 1972, 219 – 223..

T. Kailath, “Some new algorithms for recursive estimation in constant linear systems”, IEEE transactions on Information Theory, 19 (1973), 750 – 760..

A. Lindquist, “Optimal filtering of continuous-time stationary processes by means of the backward innovation process”, SIAM J. Control, 12 (1974), 747 - 754.

A. Lindquist, “A new algorithm for optimal filtering of discrete-time stationary processes”, SIAM J. Control, 12 (1974), 736 - 746.

Chandrasekhar Equations (PDEs)


[ ]**( ) ( ) ( ) ( , ) ,t t t t q=

[ ]( )( ) ( ) ( ) ( , ) ( ),t t qt t= − u

(0) 0=

(0) =


Peter L. Falb, “Infinite-dimensional Filtering: The Kalmann - Bucy filter in Hilbert space”, Information and Control, 11 (1967), 102 - 137.

Peter L. Falb and D. Kleinman, “Remarks on the infinite dimensional Riccati equation, IEEE Transactions on Automatic Control, 11 (1966), 534 - 536.

M. Sorine, “Sur le semi-groupe non lineaire associe a l'equation de Riccati”, RR-0167, INRIA Report , 1982.

J. Casti and L. Ljung,, “Some new analytic and computational results for operator Riccati equations”, SIAM J. Control, 13 (1975), 817 – 826.

J. S. Baras and D. G. Lainiotis, “Chandrasekhar algorithms for linear time varying distributed systems”, J. Information Sciences, 17 (1979), 153 – 167..

Chandrasekhar Equations (PDEs)


M. Sorine, “Sur le semi-groupe non lineaire associe a l'equation de Riccati”, RR-0167, INRIA Report , 1982.

J. A. Burns, K. Ito and R. K. Powers, “Chandrasekhar equations and computational algorithms for distributed parameter systems”, 23 rd IEEE CDC, (1984), 262 - 267.

Kevin P. Hulsing, “Methods of Computing Functional Gains for LQR Control of Partial Differential Equations”, Ph. D. Thesis, Virginia Tech, (1999).

K. Ito and R. K. Powers, “Chandrasekhar equations for infinite dimensional systems”, SIAM Journal on Control and Optimization, 25 (1987), 596 - 611.

K. Ito and R. K. Powers, “Chandrasekhar equations for infinite dimensional systems. II. Unbounded input and output case, Journal of differential equations, 75 (1988), 371 - 402.

H. T. Banks and K. Ito, “A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems”, SIAM J. Control and Optimization, 29 (1991), 499 – 515.

[ ]**( ) ( ) ( ) ( , ) ,t t t t q=

[ ]( )( ) ( ) ( ) ( , ) ( ),t t qt t= − u

(0) 0=

(0) =


Future Computing (Parallel & DG on GPUs)


J. L. Steward, A. Aksoy and Z. S. Haddad, “Parallel Direct Solution of the Ensemble Square-Root Kalman Filter Equations with Observation Principal Components”, Journal of Atmospheric and Oceanic Technology, 75 (2017), in press.

Y. Wang, Y. Jung, T. A. Supinie and M. Xue, “A hybrid MPI--OpenMP parallel algorithm and performance analysis for an ensemble square root filter designed for multiscale observations”, Journal of Atmospheric and Oceanic Technology, 30 (2013), 1382 - 1397.

J.-F. Remacle, R. Gandham and T. Warburton, “GPU accelerated spectral finite elements on all-hex meshes”, Journal of Computational Physics , 324 (2016), 246–257.

Past decade dramatic changes and advances in both mathematical foundations and computational science …

[ ]**( ) ( ) ( ) ( , ) ,t t t t q=

[ ]( )( ) ( ) ( ) ( , ) ( ),t t qt t= − u

(0) 0=

(0) =


Future Computing (Parallel & DG 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), 246–257.

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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 for quasi-triangular generalized Lyapunov equations”, J. Parallel Computing, 57 (2016), 212–221.


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Documents

Observers and Optimal Estimation for PDE Systems: Sensor ......M. A. Demetriou and D. Ucinski, “State estimation of spatially distributed processes using mobile sensing agents”,