# Two-step MOR for $H_\infty$-robust nonlinear controller design

GAMM Annual Meeting at Magdeburg, March 21, 2024

# Motivation

The Navier-Stokes equations

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

$\nabla \cdot v = 0.$

Control Problem:

• use two small outlets for fluid at the cylinder boundary
• to stabilize the unstable steady state
• with a few point observations in the wake.

# Quadratic Stability

The (uncontrolled) Navier-Stokes equations can be realized as an SDC system $$$\dot x(t) = A(x(t))\, x(t), \quad x(0)=x_0 \in \mathbb R^{n},$$$ with $A\colon \mathbb R^{n}\to \mathbb R^{n\times n}$.

### Theorem

Quadratic Stability (Prop. 1.1, Shamma 2012):
If there exists $X>0 \in \mathbb R^{n\times n}$ s. th. $$$XA(x) + A(x)^TX < 0$$$ along the trajectory $x$, then the system is asymptotically stable.

For $x(t)\in \mathbb R^{n}$, the linear Matrix inequality (LMI) $$$XA(x) + A(x)^TX < 0$$$ has to be checked on an infinite set $\mathcal X\subset \mathbb R^{n}$.

For parametrizations $x(t) = \Phi\rho(t)$, with $\rho(t) \in \mathbb R^{r}$, the LMI $$$XA(\Phi \rho) + A(\Phi \rho)^TX < 0$$$ has to be checked on an infinite set $\mathcal R \subset \mathbb R^{r}$.

# Polytopic LPV System Approximations

### Theorem

Polytopic LPV systems (Apkarian, Gahinet, and Becker 1995):

If $\tilde A(\rho) := A(\Phi\rho)$ is linear, and $\rho(t)\in R\subset \mathbb R^{r}$ with a polytope $R$ of $N$ vertices $\rho^{(i)}$, then quadratic stability holds, if $$$X\tilde A(\rho^{(i)}) + \tilde A(\rho^{(i)})^TX < 0$$$ at the vertices $\rho^{(i)}$, for $i=1,\dotsc,N$.

Thus, for polytopic LPV systems, we need to solve an $$$N\cdot n$$$ dimensional LMI to establish stability.

The direct way (like hinfgs in Matlab) uses the bounding box for $\rho$ and solves a $2^{r+1}\cdot n$ dimensional LMI.

For basic POD (see, e.g., (Hashemi and Werner 2011))

$$$\dot {\hat x} (t) = \hat A(\hat x(t))\, \hat x(t), \quad \hat x(0)=\hat x_0 \in \mathbb R^{k}$$$

the LMI to sizes reduces $2^{k+1}\cdot k$.

However, already for $k=10$, the LMI size is $20\ 480$ despite the low accuracy of the model.

Our approach: two level reduction

1. Reduce the state-space to a moderate dimension $k_x$
2. Parametrize the coefficient with very few dimensions $k_r$.

Then, the system reads $$$\dot {\tilde x} (t) = \tilde A(\rho(\tilde x(t)))\, \tilde x(t), \quad \tilde x(0)=\tilde x_0 \in \mathbb R^{k_x}, \quad \rho(\tilde x(t)) \in \mathbb R^{k_r},$$$ and for $k_x=36$ and $k_r=6$, the LMI size is $254$ while accuracy is good.

Illustration of model accuracy via the limit cycles for

1. the full order model (top)
2. the POD reduction with 10 modes (middle)
3. our two-layer approach (bottom row)

## How about further reduction?

Critical factor is $2^{k_r+1}$: the number of vertices of the bounding box for $\mathcal R \supset \rho$.

Consider a polytope $\mathcal P$ of less vertices that encloses $\rho$

• potentially, we could reduce $2^{k_r + 1} \leftarrow k_r +1$
• however, the vertices of such a simplex will be far off the actual values
• so that no feasible solutions for the LMIs may be found
• we use a multiobjective optimization to find a polytope $\mathcal P$
1. of less vertices
2. at less extremal coordinates

3D case illustration

• set of trajectory values
• enclosing bounding box
• enclosing polytope

Note the extremal values in the polytope

# Controller Design

Using the hinfgs routine from the Robust Control Toolbox

• very basic implementation of the LMI solves
• likely to be deprecated
• little support for general polytopes $\mathcal P$
• closed-loop simulations only with bounding boxes

Example case of $k_x=36$ and $k_r=6$.

1. bounding box – LMI size: $4608 = 2^7\cdot 36$
2. optimized polytope – LMI size: $720 = 20\cdot 36$

We compare the hinfgs runtime against achieved performance $\gamma$ of the controller.

• The polytope representation is generally and significantly faster
• the bounding box achieves a better $H_\infty$ performance

# Conclusion

• Promising two-layer reduction for $H_\infty$ robust gain scheduling for nonlinear systems

• Major issue – solving the LMIs

• theoretical complexity
• unsufficient implementations

## Future Work

• combine model order reduction and controller design in polytopes (see contribution by Yongho Kim)

• call on more recent implementations like in LPVcore

• do the system theory (PhD student wanted)

## References

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.
Das, Amritam, and Jan Heiland. 2023. “Low-Order Linear Parameter Varying Approximations for Nonlinear Controller Design for Flows.” arXiv. https://doi.org/10.48550/arXiv.2311.05305.

## References ctd.

Hashemi, Seyed Mahdi, and Herbert Werner. 2011. “Observer-Based LPV Control of a Nonlinear PDE.” In 50th IEEE Conference on Decision and Control, 2010–15. IEEE. https://doi.org/10.1109/CDC.2011.6161232.
Shamma, Jeff S. 2012. “An Overview of LPV Systems.” In Control of Linear Parameter Varying Systems with Applications, edited by Javad Mohammadpour and Carsten W. Scherer, 3–26. Springer. https://doi.org/10.1007/978-1-4614-1833-7_1.