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 2nd International Workshop on Simulation at the System Level Cargèse 2014 1. CMLA Centre de Mathématiques et de Leurs Applications 2. LMT-Cachan Laboratoire de Mécanique et Technologie 3. LSV Laboratoire de Spécification et Vérification Adrien LE COËNT MINIMATOR for PDEs 1 / 42
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 Adrien LE COËNT MINIMATOR for PDEs 2 / 42
Introduction A switched control system Described by the differential equation : u ( t ) = f p ( t ) ( u ( t )) ˙ (1) u ( t ) ∈ R n : 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...) Adrien LE COËNT MINIMATOR for PDEs 3 / 42
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 Adrien LE COËNT MINIMATOR for PDEs 4 / 42
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) Adrien LE COËNT MINIMATOR for PDEs 5 / 42
Outline The tool MINIMATOR 1 MINIMATOR and PDEs 2 Model Order Reduction 3 Conclusions and future work 4 Adrien LE COËNT MINIMATOR for PDEs 6 / 42
Outline The tool MINIMATOR 1 MINIMATOR and PDEs 2 Model Order Reduction 3 Conclusions and future work 4 Adrien LE COËNT MINIMATOR for PDEs 7 / 42
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 Adrien LE COËNT MINIMATOR for PDEs 8 / 42
MINIMATOR Example : Schematic representation of the boxes R and their images Adrien LE COËNT MINIMATOR for PDEs 9 / 42
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 (2 nd N k ) 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) Adrien LE COËNT MINIMATOR for PDEs 10 / 42
Outline The tool MINIMATOR 1 MINIMATOR and PDEs 2 Model Order Reduction 3 Conclusions and future work 4 Adrien LE COËNT MINIMATOR for PDEs 11 / 42
MINIMATOR and PDEs The case of the heat equation PDE problem to control (boundary control) : ∂ 2 λ ˙ u ( x , t ) = ∂ x 2 u ( x , t ) ( x , t ) ∈ Ω × (0 , T ) ρ c v ∂ (2) ∂ n u ( x , t ) = ϕ p ( t ) ( x , t ) ∈ Σ × (0 , T ) u ( x , 0) = u 0 ( x ) x ∈ Ω Weak discrete form writing then classic FE discretization : � M ˙ u + K u = b p (3) u ( x , 0) = u 0 ( x ) x ∈ Ω Adrien LE COËNT MINIMATOR for PDEs 12 / 42
MINIMATOR and PDEs Retained system Described by two modes (sub-systems) : Robin BC on external boundaries Inhomogeneous Neumann BC on internal boundaries Adrien LE COËNT MINIMATOR for PDEs 13 / 42
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) Adrien LE COËNT MINIMATOR for PDEs 14 / 42
Outline The tool MINIMATOR 1 MINIMATOR and PDEs 2 3 Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers Conclusions and future work 4 Adrien LE COËNT MINIMATOR for PDEs 15 / 42
Outline The tool MINIMATOR 1 MINIMATOR and PDEs 2 3 Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers Conclusions and future work 4 Adrien LE COËNT MINIMATOR for PDEs 16 / 42
Model order reduction n r � u ( x , t ) = a k ( t ) ϕ k ( x ) avec n r < 10 k =1 Spectral decomposition K ϕ i = λ M ϕ i i ∈ { 1 , ..., n ddl } (4) POD (snapshot method, see [L.Cordier, M.Bergmann, 2003]) SVD of the (normalized) snapshot matrix A : USV T = M 1 / 2 A ∈ R n ddl × n snap Computation of the basis ( M norm) : [ ϕ 1 ...ϕ n snap ] = ( M 1 / 2 T ) − 1 U Adrien LE COËNT MINIMATOR for PDEs 17 / 42
Model Order Reduction Firsts eigen modes : Firsts POD modes computed on the first subsystem : Firsts POD modes computed on the second subsystem : Adrien LE COËNT MINIMATOR for PDEs 18 / 42
Model Order Reduction Spectrum associated to the eigen-modes and the POD modes : Adrien LE COËNT MINIMATOR for PDEs 19 / 42
Model Order reduction Establishment of the reduced model : M ˙ u + K u = b p n r n r � � ⇒ M ˙ a k ϕ k + Ka k ϕ k = b p k =1 k =1 n r n r � � a k ϕ ⊤ a k ϕ ⊤ i K ϕ k = ϕ ⊤ ⇒ ˙ i M ϕ k + i b p i ∈ { 1 , ..., n r } k =1 k =1 ⇒ M r ˙ a + K r a = B p Equivalent writing : state-space representation a ( t ) = A r a ( t ) + B r p ( t ) ˙ A r is the state matrix and B r is the input (or control) matrix. Adrien LE COËNT MINIMATOR for PDEs 20 / 42
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 n r � u ( x , t ) = a k ( t ) ϕ k ( x ) k =1 Adrien LE COËNT MINIMATOR for PDEs 21 / 42
Outline The tool MINIMATOR 1 MINIMATOR and PDEs 2 3 Model Order Reduction Spectral decomposition and POD Interpolating Model Order Reduction Synthesis of controllers Conclusions and future work 4 Adrien LE COËNT MINIMATOR for PDEs 22 / 42
Interpolating Model Order Reduction Idea : establishing a basis { ψ l ( x ) } n c l =1 , linear combination of the eigen/POD modes, such that : n modes � b l ψ l ( x j ) = m ϕ m ( x j ) = δ jl (5) m =1 The interpolation points correspond to the sensors. Equivalent writing in state-space representation : � ˙ a ( t ) = A r a ( t ) + B r p ( t ) v ( t ) = C r a ( t ) with v the output and C r = I in the case we interpolate all the sensors. Adrien LE COËNT MINIMATOR for PDEs 23 / 42
Interpolating Model Order Reduction Interpolation points chosen a priori : Figure 1 : Map of the interpolation points and simulation of the system. Adrien LE COËNT MINIMATOR for PDEs 24 / 42
Optimization of the sensors locations First idea : One notes J m the m -order interpolation : m � J m u ( x ) = u ( x i ) ψ i ( x ) (6) i =1 Research of the best m + 1-order interpolation : | u ( x , t ) − J m +1 u ( x )( t ) | x m +1 = argmin sup sup (7) x m +1 ∈ Ω x ∈ Ω h t ∈S τ avec : S τ = { k τ } k ∈ [1 ,..., n snap ] Adaptation of the Empirical Interpolation Method (EIM, [Y.Maday, N.C.Nguyen et al., 2007]) Adrien LE COËNT MINIMATOR for PDEs 25 / 42
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 Adrien LE COËNT MINIMATOR for PDEs 26 / 42
Optimization of the sensors locations Results obtained with the interpolating basis built with 6 eigen modes : Interpolating modes : Adrien LE COËNT MINIMATOR for PDEs 27 / 42
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 n c ( u ) � ∞ � u � ∞ Adrien LE COËNT MINIMATOR for PDEs 28 / 42
Optimization of the sensors locations Results obtained with the interpolating basis built with 6 POD modes : Interpolating modes : Adrien LE COËNT MINIMATOR for PDEs 29 / 42
Recommend
More recommend
