Model Order Reduction for Parameter-Varying Systems Xingang Cao Promotors: Wil Schilders Siep Weiland Supervisor: Joseph Maubach Eindhoven University of Technology CASA Day November 1, 216 Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 1 / 25
Introduction Overview 1 Introduction 2 Model Order Reduction 3 Model Order Reduction for Linear Parameter-Varying Systems 4 Conclusions and Future Work Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 2 / 25
Introduction Self Introduction Russia Biography: 199 born in Baotou Mongolia 29-213 BE in Electrical Engineering Tongji University, Shanghai 213-215 MSc. In Systems and Control TU Eindhoven Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 3 / 25
Introduction Modeling & Control of Thermal Effects in Printing Systems Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 4 / 25
Introduction Model Order Reduction in a Broad Field Systems and control, Lyapunov, Truncated Balanced Realization Scientific computing, numerical MOR, Krylov methods, linear algebra, tensors Pade-via-Lanczos and PRIMA Passivity preserving Structure preserving Linear and nonlinear problems Parameterized methods Tensor analysis Large-scale Lyapunov systems Balanced realizations Observability and controllability Gramians Port-Hamiltonian systems Hankel singular values Projection methods Mathematical modeling, behavioral modeling, models via data MOR at operator level Reduced basis methods Karhunen-Loeve expansions Neural networks Vector fitting Behavioral models Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 5 / 25
Model Order Reduction Overview 1 Introduction 2 Model Order Reduction 3 Model Order Reduction for Linear Parameter-Varying Systems 4 Conclusions and Future Work Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 6 / 25
Model Order Reduction What is Model Order Reduction? Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 7 / 25
Model Order Reduction Abstract Problem Formulation Given a dynamical system Σ : {ẋ = f (x, u) y = g(x, u) with x(t) R n, u(t) R m and y(t) R p. Find another dynamical system with x(t) R r and Σ : { x = f ( x, u) ŷ = ĝ( x, u) r n. Constraints: Approximation error small, i.e., Σ Σ ɛ; Preserve characteristics, e.g., stability, dissipativity, etc; Computational efficient for simulation and control design. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 8 / 25
Model Order Reduction Why Model Order Reduction? Realistic models have high complexity (n 1 2 1 9 ), but we want to perform simulations fast and with reliable outcomes; enable model based control design; cope with limited memory storage capacity; perform online tasks (optimization, prediction) efficiently; enable implementation on embedded systems. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 9 / 25
Model Order Reduction How to do Model Order Reduction? Model Reduction by Projection: Signal projection: V : R n R r x V x Residual: R = V x f (V x, u) Residual projection: W R = W V x W f (V x, u) z.15.1.5 -.5 -.1.8.6.1.4.5.2 -.5 y -.1 x Trivial projection: 3D to 2D. Galerkin projection: W = V & V V = I r. Petrov Galerkin projection: W V & W V = I r. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 1 / 25
Model Order Reduction How to Find the Projections? Consider a linear time-invariant (LTI) system { Eẋ = Ax + Bu Σ : y = Cx + Du Take the Laplace transformation L : ẋ(t) sx (s), u(t) U(s) and y(t) Y (s) Σ defines an input-output mapping G(s) = C(sE A) 1 B + D. Expand G(s) = G(s + σ) around a point s, G(σ) = C (s E A) 1 E (s }{{} E A) 1 B }{{} i= M B σ i Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 11 / 25
Model Order Reduction How to Find Projections (Cont d)? Theorem If W and V span the left and right principal subspaces of the approximation error is small. M = (s E A) 1 E, Compute W and V by Krylov subspace methods W := span{w } = span{c, M C,..., (M ) q 1 C } V := span{v } = span{ B, M B,..., M q 2 B} dim(w) = dim(v) = r. Remark For B, C R n 1 (single-input single-output system), r = q 1 = q 2. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 12 / 25
Model Order Reduction for Linear Parameter-Varying Systems Overview 1 Introduction 2 Model Order Reduction 3 Model Order Reduction for Linear Parameter-Varying Systems 4 Conclusions and Future Work Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 13 / 25
Model Order Reduction for Linear Parameter-Varying Systems Linear Parameter-Varying Systems Linear parameter-varying (LPV) system is { E(p)ẋ = A(p)x + B(p)u Σ(p) : y = C(p)x + D(p)u where p(t) : R + P Rµ. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 14 / 25
Model Order Reduction for Linear Parameter-Varying Systems Track the Parameterized Projection Subspaces Problem Track/Estimate the principal subspaces of M(p) = (s E(p) A(p)) 1 E(p) with starting vectors C(p) and B(p). Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 15 / 25
Model Order Reduction for Linear Parameter-Varying Systems Geodesics on Grassmann Manifold Definition Let Y(t) : I S be a smooth, injective curve and Ẏ(t) for all t R. There exists a smooth vector field X : S T S such that Ẏ(t) = X (Y(t)) for all t I. If ẎẎ := ( X X )(Y(t)) =, then Y(t) is a geodesic. On Grassmann manifold, it satisfies Ÿ(t) + Y(t)(Ẏ(t) Ẏ(t)) =, t [, 1]. Given Y() = S and Ẏ() = H, Y(t) = ( ) ( ) cos(σ S Z U t) Z, sin(σ t) S 1 : = Y(1). H thin svd == U Σ V, t [, 1]; Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 16 / 25
Model Order Reduction for Linear Parameter-Varying Systems Data-Driven Subspace Tracking Step 1. Sample parameter p and get {p i } N i=1 ; Step 2. Compute the projections {W } N i=1 and {V }N i=1 ; Step 3. Define ψ i = vec(w i Wi ) and φ i = vec(v i Vi ); Step 4. Assume that ψ i and φ i satisfy Σ ψ : ψ i+1 = A ψ ψ i + B ψ p i+1, Σ φ : φ i+1 = A φ ψ i + B φ p i+1 ; Data Prediction φ 2 φ 1 φ A φ A φ φ 1 φ 2 B φ B φ p 1 p p 1 p 2 Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 17 / 25
Model Order Reduction for Linear Parameter-Varying Systems A Small-scale Example Example Consider the subspaces 1 W(p(t)) = span{, 1 V(p(t)) = span{, cos(p(t)) }, sin(p(t)) cos(p(t) + π 32 ) }. sin(p(t) + π 32 ) where e i, i = 1, 2, 3 R 3 and p(t) [ 1 1, π 4 π 32 ]. Take 2 random examples and apply the data-driven method to obtain a 3rd-order model. When the parameter is varying linearly from 1 1 to, compare the approximated and the real subspace representations. π 4 π 32 Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 18 / 25
Model Order Reduction for Linear Parameter-Varying Systems Approximation of W (p) 2 1.5 W 11 = 1 Wreal Wapprox 1.5 W 12 = Wreal Wapprox 1.5 W 13 = Wreal Wapprox 1.5 -.5 -.5.2.4.6.8 p W 21 = 1 Wreal.5 Wapprox -.5-1.2.4.6.8 p -1.2.4.6.8 p W 22 = cos(p) 1.9.8 Wreal Wapprox.7.2.4.6.8 p -1.2.4.6.8 p W 23 = sin(p).8.6.4.2 Wreal Wapprox.2.4.6.8 p Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 19 / 25
Model Order Reduction for Linear Parameter-Varying Systems Approximation of V (p) 2 1.5 V 11 = 1 Vreal Vapprox 1.5 V 12 = Vreal Vapprox 1.5 V 13 = Vreal Vapprox 1.5 -.5 -.5.2.4.6.8 p V 21 = 1 Vreal.5 Vapprox -.5-1.2.4.6.8 p -1.2.4.6.8 p V 22 = cos(p+π/32) 1 Vreal Vapprox.9.8.7.2.4.6.8 p -1.2.4.6.8 p V 23 = sin(p+π/32).8.6.4.2 Vreal Vapprox.2.4.6.8 p Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 2 / 25
Conclusions and Future Work Overview 1 Introduction 2 Model Order Reduction 3 Model Order Reduction for Linear Parameter-Varying Systems 4 Conclusions and Future Work Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 21 / 25
Conclusions and Future Work Conclusions 1 Data-driven method: allows to track the parameter-varying subspaces; 2 Large-scale problems: off-line cost for data-driven method is high; 3 Data-driven method: faster at online phase than direct method. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 22 / 25
Conclusions and Future Work Future Work Formulate the LPV model order reduction problem as {W (p(t)), V (p(t))} = argmax Trace(W M(p(t))V ) W,V Gr(r, R n ) st. W V = I or W W = I, V V = I, W and V starts with specific vector B and C. where Gr(r, R n ) := {span(a) A R n r, rank(a) = r}. Solve it by optimization on matrix manifold methods. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 23 / 25
Conclusions and Future Work Questions? Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 24 / 25
Conclusions and Future Work References Schilders, Van der Vorst, Rommes. Model Order Reduction: Theory, Research Aspects and Applications. Springer, 28. Absil, Mahony, Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press, 29. Benner, Gugercin, Willcox. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM review, 215. Proctor, Brunton, Kutz. Dynamic Mode Decomposition with Control. SIAM Journal on Applied Dynamical Systems, 216. Xingang Cao (CASA TU/e) CASA Day 216 November November 1, 216 25 / 25