An adaptation of the tool MINIMATOR for the control of Partial - - PowerPoint PPT Presentation

An adaptation of the tool MINIMATOR for the control of Partial Differential Equations using interpolating model order reduction techniques

Adrien Le Coënt 1, Florian De Vuyst 1, Christian Rey 2, Ludovic Chamoin 2, Laurent Fribourg 3

Cargèse 2014 2nd International Workshop on Simulation at the System Level

• 1. CMLA Centre de Mathématiques et de Leurs Applications
• 2. LMT-Cachan Laboratoire de Mécanique et Technologie
• 3. LSV Laboratoire de Spécification et Vérification

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Introduction

Framework

Goal : control the evolution of an operating system with the help of actuators Framework of the switched control systems : one selects the working modes of the system over time, every mode is described by differential equations (ODEs or PDEs) Application to transient heat transfer :

Stability control Reachabiity control

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Introduction

A switched control system

Described by the differential equation : ˙ u(t) = fp(t)(u(t)) (1) u(t) ∈ Rn : state variable (temperature field) p(t) ∈ {1, ..., N} : control variable (mode to select) Idea : impose the right p(t) such that u verifies some properties (stability, reachability...)

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Classical approach of the control theory

Very useful for on-line control

Aim : research of the solution of the dual minimisation problem Stability control :

Lyapunov method, late 19th

Optimal control :

Hamilton-Jacobi-Bellmann, 1950 Lev Pontryagin, 1956 Extension to PDEs : J.L.Lions, 1968

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Approach elaborated at the LSV

Off-line control synthesis

MINIMATOR : code developed at the LSV [L.Fribourg and R.Soulat, 2013]

Based on the invariant sets theory Permits the synthesis of state-dependant controllers (correct-by-design) Operational for ODEs Based on a technique of decomposition of the state-space into local regions where the control is uniform (for a given mode)

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction

4

Conclusions and future work

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction

4

Conclusions and future work

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MINIMATOR

Principle (for the stability) :

Let R be a region of the state space one wants to control One looks for a pattern (a sequence of control modes) which sends R in itself If no pattern is found, R is divided into smaller regions and one looks for patterns which send these sub-regions in R

Underlying ideas :

Temporal discretization Measures carried out at the end of every pattern Linearity of the equations which allows the use of zonotopes (matrices) to represent the regions of the state-space

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MINIMATOR

Example : Schematic representation of the boxes R and their images

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MINIMATOR

Properties [L.Fribourg and R.Soulat, 2013] :

The convergence properties are proved under limited hypotheses (contracting modes, verified in practice) Computational cost at most in O(2ndNk) n : dimension of the state-space d : maximal length of decomposition N : number of modes k : maximal length of the patterns

Consequence : dimension of the state-space very limited (< 10 in practice)

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction

4

Conclusions and future work

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MINIMATOR and PDEs

The case of the heat equation

PDE problem to control (boundary control) :            ˙ u(x, t) = λ ρcv ∂2 ∂x2 u(x, t) (x, t) ∈ Ω × (0, T) ∂ ∂nu(x, t) = ϕp(t) (x, t) ∈ Σ × (0, T) u(x, 0) = u0(x) x ∈ Ω (2) Weak discrete form writing then classic FE discretization :

• M ˙

u + Ku = bp u(x, 0) = u0(x) x ∈ Ω (3)

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MINIMATOR and PDEs

Retained system

Described by two modes (sub-systems) : Robin BC on external boundaries Inhomogeneous Neumann BC on internal boundaries

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MINIMATOR and PDEs

1D finite differences (few dofs) [L.Fribourg, E.Goubault, S.Mohamed, S.Putot, R.Soulat, 2014] Direct implementation of a finite element problem in MINIMATOR :

Dimension of the state space = number of dof

⇒ Intractable problem for large-scale systems ⇒ Necessity of a model order reduction (MOR)

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers

4

Conclusions and future work

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers

4

Conclusions and future work

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Model order reduction

u(x, t) =

nr

• k=1

ak(t)ϕk(x) avec nr < 10 Spectral decomposition Kϕi = λMϕi i ∈ {1, ..., nddl} (4) POD (snapshot method, see [L.Cordier, M.Bergmann, 2003]) SVD of the (normalized) snapshot matrix A : USV T = M1/2A ∈ Rnddl×nsnap Computation of the basis (M norm) : [ϕ1...ϕnsnap] = (M1/2T)−1U

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Model Order Reduction

Firsts eigen modes : Firsts POD modes computed on the first subsystem : Firsts POD modes computed on the second subsystem :

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Model Order Reduction

Spectrum associated to the eigen-modes and the POD modes :

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Model Order reduction

Establishment of the reduced model : M ˙ u + Ku = bp ⇒

nr

• k=1

M ˙ akϕk +

nr

• k=1

Kakϕk = bp ⇒

nr

• k=1

˙ akϕ⊤

i Mϕk + nr

• k=1

akϕ⊤

i Kϕk = ϕ⊤ i bp

i ∈ {1, ..., nr} ⇒ Mr ˙ a + Kra = Bp Equivalent writing : state-space representation ˙ a(t) = Ara(t) + Brp(t) Ar is the state matrix and Br is the input (or control) matrix.

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Model Order Reduction

Implementation of the reduced basis in MINIMATOR : Synthesis of controllers in reasonable computation time Remaining difficulty : relating the decomposition coefficients (i.e. the controlled state variables) to global state conditions u(x, t) =

nr

• k=1

ak(t)ϕk(x)

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers

4

Conclusions and future work

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Interpolating Model Order Reduction

Idea : establishing a basis {ψl(x)}nc

l=1, linear combination of the

eigen/POD modes, such that : ψl(xj) =

nmodes

• m=1

bl

mϕm(xj) = δjl

(5) The interpolation points correspond to the sensors. Equivalent writing in state-space representation : ˙ a(t) = Ara(t) + Brp(t) v(t) = Cra(t) with v the output and Cr = I in the case we interpolate all the sensors.

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Interpolating Model Order Reduction

Interpolation points chosen a priori :

Figure 1 : Map of the interpolation points and simulation of the system.

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Optimization of the sensors locations

First idea : One notes J m the m-order interpolation : J mu(x) =

m

• i=1

u(xi)ψi(x) (6) Research of the best m + 1-order interpolation : xm+1 = argmin

xm+1∈Ω

sup

t∈Sτ

sup

x∈Ωh

|u(x, t) − J m+1u(x)(t)| (7) avec : Sτ = {kτ}k∈[1,...,nsnap] Adaptation of the Empirical Interpolation Method (EIM, [Y.Maday, N.C.Nguyen et al., 2007])

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Optimization of the sensors locations

Results obtained with the interpolating basis built with the eigen modes :

First interpolated point in green (a given quantity of interest) Optimal locations of the sensors : red → black

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Optimization of the sensors locations

Results obtained with the interpolating basis built with 6 eigen modes :

Interpolating modes :

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Optimization of the sensors locations

Results obtained with the interpolating basis built with POD modes :

Map of the optimal locations and decrease of the error :

ǫ = u − J nc(u)∞ u∞

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Optimization of the sensors locations

Results obtained with the interpolating basis built with 6 POD modes :

Interpolating modes :

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Optimization of the sensors locations

Results obtained with the interpolating basis built with POD modes :

Reconstruction error (on the last snapshot) :

Computation of the basis coefficients : {arec} = [ψ]−1{T EF} Reconstruction : {T rec} = [ψ]{arec} Computation of the nodal errors : ǫi =

• T EF

i

− T rec

i

• T EF∞

i ∈ {1, ..., nddl}

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Optimization of the sensors locations

Results obtained with the interpolating basis built with POD modes :

Reconstruction error (on the last snapshot) : Figure 2 : Reconstruction error on the subsystem 1 (left) and the subsystem 2 (right).

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Optimization of the sensors locations

Results obtained with the interpolating basis built with POD modes :

Error between the finite element solution and the reduced order solution for a pattern of length 1 : Figure 3 : Error map obtained after application of the mode 1 during τ (left) and the mode 2 during τ (right).

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers

4

Conclusions and future work

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Optimization of the sensors locations

Results obtained with the interpolating basis built with POD modes : application to stability control

Stabilizing in R = [15, 20]6 R decomposed in 26 = 64 zonotopes Simulation with a uniform temperature field equal to 20◦C for initial condition :

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Optimization of the sensors locations

Results obtained with the interpolating basis built with POD modes : application to reachability control

Start zonotope : R1 = [5, 15]6 Arrival zonotope : R2 = [15, 20] × [0, 25]5 Simulation with a uniform temperature field equal to 5◦C for initial condition :

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Validation of the proposed method

Evaluation of the errors : ǫn = u − J nuL∞ u∞ (8) Comparison with the projection error (Lebesgue constant) : Λn = u − ΠnuL∞ u∞ with Πnu =

n

• i=1

ϕi(u, ϕi)L2 (9)

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Numerical validation of the controller

Stabilization of the system in R = [15, 20]6, simulation of the reduced system (left) and the complete system (right) :

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Outline

1

The tool MINIMATOR

2

MINIMATOR and PDEs

3

Model Order Reduction

4

Conclusions and future work

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Conclusions

The use of reduced bases allows us to synthesize controllers for medium-scale systems. Designing interpolating reduced basis functions allows us to control some quantities of interest. The proposed method leads to reasonably low errors. The obtained controllers are efficient on the tested cases. Remaining work : theoretical error estimation.

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Future work

New procedure : Offline : Synthesis of a correct-by-design controller :

Fine model :

• ˙

u(t) = Au(t) + Bp(t) v(t) = Cu(t) Reduced solution :

• ˙

ˆ u(t) = ˆ Aˆ u(t) + ˆ Bp(t) vr(t) = ˆ Cˆ u(t) with ˆ C = I Synthesis of a ˆ u-dependent controller (decomposition of the space of the ˆ u).

Online : Control of the system

Observations : v Reconstruction of the ˆ u with Kalman filters (two possibilities : Kalman filtering on the fine model then reduction or reduction then Kalman filtering)

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Future work

New procedure :

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Some references

[1] L. Cordier and M. Bergmann. Proper orthogonal decomposition : An

• verview. In von Karman Institute for Fluid Dynamics, Lecture Series

2003-04. LEMTA, France, February 17-21, 2003. [2] L. Fribourg, U. Kühne, and R. Soulat. Minimator : a tool for controller synthesis and computation of minimal invariant sets for linear switched systems, March 2013. [3] L. Fribourg and R. Soulat. Control of Switching Systems by Invariance

• Analysis. Focus series. John Wiley and Sons, ISTE Ltd, 2013.

[4] Y. Maday, N. C. Nguyen, A. T. Patera, and George S.H. Pau. A general, multipurpose interpolation procedure : the magic points. September 2007. [5] Happy birthday Marie ! ! !

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