Control of Nonlinear & Large-Scale Systems

A general approach would include

• powerful backends (linear algebra / optimization)
• exploitation of general structures
• data-driven surrogate models
• all of it?!

• Under mild conditions, the flow \(f(x)\) can be factorized \[ \dot x = [A(x)]\,x + Bu \] – a state dependent coefficient system – with some \[A\colon \mathbb R^{n} \to \mathbb R^{n\times n}.\]

• Control through a state-dependent state-feedback law \[ u=-[B^*P(x)]\,x. \]

Nonlinear SDRE Feedback

• Set \[ u=-[B^TP(x)]\,x. \]

• with \(P(x)\) as the solution to the state-dependent Riccati equation \[ A(x)^TP + PA(x) - PBB^TP + C^TC=0 \]

• the system \[\dot x = f(x) + Bu \;=[A(x)- BB^TP(x)]\,x\] can be controlled towards an equilibrium; see, e.g., Banks, Lewis, and Tran (2007).

Linear Updates as an Alternative

Theorem Benner and Heiland (2018)

• …

• If \(P_0\) is the Riccati solution for \(x=x_0\)

• and if \(E\) solves the linear equation \[A(x)E + E(A(x_0)-BB^TP_0)=A(x_0)-A(x)\]

• with \(\|E\| \leq \epsilon < 1\),

• then \(u=-B^TP_0(I+E)^{-1}\) stabilizes the system.

\[ \DeclareMathOperator{\spann}{span} \DeclareMathOperator{\Re}{Re} \]

The linear parameter varying (LPV) representation/approximation \[ \dot x = f(x) + Bu = [\tilde A(\rho(x))]\,x + Bu \approx [A_0+\Sigma \,\rho_k(x)A_k]\, x + Bu \] with affine parameter dependency can be exploited for designing nonlinear controller through scheduling.

Scheduling of \(H_\infty\) Controllers

• If \(\rho(x)\in \mathbb R^{k}\) can be confined to a bounded polygon,

• there is globally stabilizing \(H_\infty\) controller

• that can be computed

• through solving \(k\) coupled LMI in the size of the state dimension;

see Apkarian, Gahinet, and Becker (1995) .

Series Expansion of SDRE Solution

For \(A(x)=\sum_{k=1}^r\rho_k(x)A_k\), the solution \(P\) to the SDRE \[ A(x)^TP + PA(x) - PBB^TP + C^TC=0 \] can be expanded in a series \[ P(x) = P_0 + \sum_{|\alpha| > 0}\rho(x)^{(\alpha)}P_{\alpha} \] where \(P_0\) solves a Riccati equation and \(P_\alpha\) solve Lyapunov (linear!) equations;

see Beeler, Tran, and Banks (2000).

We see

Manifold opportunities if only \(k\) was small.

• Let \(v\) be the velocity solution and let \[ V = \begin{bmatrix} V_1 & V_2 & \dotsm & V_r \end{bmatrix} \] be a, say, POD basis with \[v(t) \approx VV^Tv(t)=:\tilde v(t),\]

• then \[\rho(v(t)) = V^Tv(t)\] is a parametrization.

• And with \[\tilde v = VV^Tv = V\rho = \sum_{k=1}^rV_k\rho_k,\]

• the NSE has the low-dimensional LPV representation via \[ (v\cdot \nabla) v \approx (\tilde v \cdot \nabla) v = [\sum_{k=1}^r\rho_k(V_k\cdot \nabla)]\,v. \]

Question

Can we do better than POD?

Convolution Autoencoders for NSE

1. Consider solution snapshots \(v(t_k)\) as pictures.

2. Learn convolutional kernels to extract relevant features.

3. While extracting the features, we reduce the dimensions.

4. Encode \(v(t_k)\) in a low-dimensional \(\rho_k\).

Our Example Architecture Implementation

Input:

• Velocity snapshots \(v_i\) of an FEM simulation with \[n=50'000\] degrees of freedom

• interpolated to two pictures with 63x95 pixels each

• makes a 2x63x69 tensor.

Training for minimizing:

\[ \| v_i - VW\rho(v_i)\|^2_M \] which includes

1. the POD modes \(V\in \mathbb R^{n\times k}\),

2. a learned weight matrix \(W\in \mathbb R^{k\times r}\colon \rho \mapsto \tilde \rho\),

3. the mass matrix \(M\) of the FEM discretization.

Going PINN

Outlook: the induced low-dimensional affine-linear LPV representation of the convection \[\| (v_i\cdot \nabla)v_i - (VW\rho_i \cdot \nabla )v_i\|^2_{M^{-1}}\] as the target of the optimization.

Implementation issues:

• Include FEM operators while
• maintaining the backward mode of the training.

Results

… and Outlook

• LPV with affine-linear dependencies are attractive if only \(k\) is small.

• Proof of concept that CNN can improve POD at very low dimensions.

• Next: Include the parametrized convection in the training.

• Outlook: Use for nonlinear controller design.