IFAC Seminar – Data-driven Methods in Control – 2021

Jan Heiland & Peter Benner (MPI Magdeburg)

\[\dot x = f(x) + Bu\]

Control of an inverted pendulum

- 9 degrees of freedom
- but nonlinear controller.

Stabilization of a laminar flow

- 50’000 degrees of freedom
- but linear regulator.

A general approach would include

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

\[ \dot x = [A(x)]\,x + Bu \]

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*\[ u=-[B^*P(x)]\,x. \]

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).

**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.

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

\[ \dot x \approx [A_0+\Sigma \,\rho_k(x)A_k]\, x + Bu \]

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.

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) .

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).

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

**Approximation** of *Navier-Stokes Equations* by *Convolutional Neural Networks*

The *Navier-Stokes* equations

\[ \dot v + (v\cdot \nabla) v- \frac{1}{\Re}\Delta v + \nabla p= f, \]

\[ \nabla \cdot v = 0. \]

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 \[ (v\cdot \nabla) v \approx (\tilde v \cdot \nabla) v = [\sum_{k=1}^r\rho_k(V_k\cdot \nabla)]\,v. \]

Can we do better than POD?

Lee/Carlberg (2019): *MOR of dynamical systems on nonlinear manifolds using deep convolutional autoencoders*

Kim/Choi/Widemann/Zodi (2020): *Efficient nonlinear manifold reduced order model*

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

Learn convolutional kernels to extract relevant features.

While extracting the features, we reduce the dimensions.

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

A number of convolutional layers for feature extraction and reduction

A full linear layer with nonlinear activation for the final encoding \(\rho\in \mathbb R^{r}\)

A linear layer (w/o activation) that expands \(\rho \to \tilde \rho\in \mathbb R^{k}\).

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

interpolated to two pictures with

`63x95`

pixels eachmakes a

`2x63x69`

tensor.

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

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

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

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

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.

Simulation parameters:

- Cylinder wake at \(\Re=40\), time in \([0, 8]\)
`1000`

snapshots/data points- 2D-CNN with 4 layers
`kernelsize, stride = 5, 2`

.`batch_size = 40`

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.

Thank You!

Apkarian, Pierre, Pascal Gahinet, and Greg Becker. 1995. “Self-Scheduled \(H_\infty\) Control of Linear Parameter-Varying Systems: A Design Example.” *Autom.* 31 (9): 1251–61. https://doi.org/10.1016/0005-1098(95)00038-X.

Banks, H. T., B. M. Lewis, and H. T. Tran. 2007. “Nonlinear Feedback Controllers and Compensators: A State-Dependent Riccati Equation Approach.” *Comput. Optim. Appl.* 37 (2): 177–218. https://doi.org/10.1007/s10589-007-9015-2.

Beeler, S. C., H. T. Tran, and H. T. Banks. 2000. “Feedback control methodologies for nonlinear systems.” *J. Optim. Theory Appl.* 107 (1): 1–33.

Benner, Peter, and Jan Heiland. 2018. “Exponential Stability and Stabilization of Extended Linearizations via Continuous Updates of Riccati Based Feedback.” *Internat. J. Robust and Nonlinear Cont.* 28 (4): 1218–32. https://doi.org/10.1002/rnc.3949.