#  Low-order Linear Parameter Varying Approximations for Nonlinear Controller
Design for Flows

1st Amritam Das  Department of Electrical Engineering
Eindhoven University of Technology
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
am.das@tue.nlORCID:0000-0001-8494-8509 2nd Jan Heiland  Max Planck Institute
for Dynamics of Complex Technical Systems
39106 Magdeburg, Germany
heiland@mpi-magdeburg.mpg.deORCID:0000-0003-0228-8522

###### Abstract

The control of nonlinear large-scale dynamical models such as the
incompressible Navier-Stokes equations is a challenging task. The
computational challenges in the controller design come from both the possibly
large state space and the nonlinear dynamics. A general purpose approach
certainly will resort to numerical linear algebra techniques which can handle
large system sizes or to model order reduction. In this work we propose a two-
folded model reduction approach tailored to nonlinear controller design for
incompressible Navier-Stokes equations and similar PDE models that come with
quadratic nonlinearities. Firstly, we approximate the nonlinear model within
in the class of LPV systems with a very low dimension in the parametrization.
Secondly, we reduce the system size to a moderate number of states. This way,
standard robust LPV theory for nonlinear controller design becomes feasible.
We illustrate the procedure and its potentials by numerical simulations.

###### Index Terms:

Model reduction, LPV systems, Robust control, Nonlinear control

##  I Introduction

We consider general nonlinear, control-affine systems of type

$\dot{x}(t)=f(x(t))+Bu(t),$ (1)

where $x$ is the system’s state with values in $\mathbb{R}^{n}$ with $n$
possibly large, where $u$ is the input with values in $\mathbb{R}^{p}$, and
where $f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$.

The computer-aided controller design for (1) with $n$ large is a challenging
problem with no generic approach being established yet. Commonly used methods
like _backstepping_ [1], feedback linearization [2, Ch. 5.3], or sliding mode
control [3] require structural assumptions and, thus, may not be accessible to
a general computational framework. The both holistic and general approach via
the _HJB_ equations is only feasible for very moderate system sizes or calls
for model order reduction; see, e.g., [4] for a relevant discussion and an
application in fluid flow control. As an alternative to reducing the systems
size, one may consider approximations to the solution of the _HJB_ of lower
complexity. For that, e.g., truncated polynomial expansions [5] are considered
or heuristic approximative solutions via the so called _state-dependent
Riccati equation_ ; see, e.g., [6, 7].

In this work, we propose the embedding of (1) into the class of _linear
parameter-varying_ (LPV) systems of type

$\dot{x}(t)=A(\rho(t))\,x(t)+Bu(t),$ (2)

with $A(\rho(t))\in\mathbb{R}^{n\times n}$ for parameter values
$\rho(t)\in\mathbb{R}^{r}$ by implementing two layers of complexity reduction
so that established theory and algorithms ([8]) for robust LPV controller
design become available.

As laid out in [9], the controller design techniques for LPV systems can be
classified into the categories _polytopic_ , _LFT-based_ , and _gridding_.

A general assumption of all these approaches is that the set of possible
parameter values is bounded. If, in addition, the parameter dependency of the
coefficients is affine-linear, then the theory and the related computations
simplify significantly; see, e.g., [8, 10].

The polytopic approach is seen as the most developed approach with the notable
results from [8, 11] that provide algorithms and theory for a scheduled
$H_{\infty}$-robust controller and that are the base of the hinfgs routine in
the MATLAB _Robust Control Toolbox_.

In a first approximation step, we seek for very low-dimensional encodings of
the states for replacing the nonlinear source terms by a low-dimensional
linear parameter varying (LPV) surrogate. In this step that is directed to
adapt the nonlinearity to a format that is accessible to the LPV theory for
computer-aided controller design, we tailor the approximations for a best
representation of the source terms at very low-dimensions.

We will show that this approach can lead to LPV models that well approximate
the actual dynamics with as much as $r$ dimensions in the parametrization,
with $r$ well less than $10$. Thus, application of standard linear matrix
inequalities (LMI) approaches already comes into reach. Still, this would
require the solution of at least $r+1$ but more likely of about $2^{r}$
coupled LMIs of the size of the original system. Therefore, we propose a
second layer of approximation that is tailored for the low-order LPV
representation to accurately follow the original dynamics with a moderately
sized state space.

This integrated approach to nonlinear controller design via tailored
approximations and established robust LPV theory has not been considered so
far.

Direct relations to the vast research on LPV systems are given as follows.
Although the typically considered LPV systems are of moderate size (see [9,
Tab. V] that classifies state space dimensions larger than $10$ as _high
dimensional_), the unfavourable increase of complexity with the parameter
dimension has triggered various work on reducing the so-called _scheduling
dimension_ ; see, e.g., [12], [13], [14], or [15] that base on sparse
optimization, _principal component analysis_ (PCA), _auto encoders_ , or
general _deep neural networks_ , respectively.

The idea of using model order reduction in combination with LPV controller
design has been followed by [16] where a PDE system is reduced as a whole
before the treatment as an LPV system. Recently, we have delivered a proof of
concept ([7]) that LPV approximations of nonlinear systems can with low
parameter dimensions work well for nonlinear controller design.

##  II POD for low-dimensional LPV approximations

Under mild conditions (see, e.g., [17]), the system (1) can be brought into
the so-called _state-dependent coefficient_ (SDC) form

$\dot{x}(t)=A(x(t))\,x(t)+Bu(t),$ (3)

with $A(x(t))\in\mathbb{R}^{n\times n}$. We will assume that, in particular,
the map $x\to A(x)\in\mathbb{R}^{n\times n}$ is (affine) linear, i.e. $A(x)$
can be realized as $A(x)=A_{0}+L(x)$ with $L$ linear.

######  Remark II.1.

For some systems like the spatially-discretized Navier-Stokes equations the
SDC representation with affine dependencies is naturally induced by its
structure; see [18]. If the map $x\to A(x)$ is not linear, one can seek to
find an approximation that is linear; see, e.g., [15].

We note that (3) is an LPV representation (2) of (1) with the trivial
parametrization $\rho(t)=x(t)$ and with, in particular, a large parameter
dimension namely $r=n$.

Next we illustrate how a general model order reduction scheme, can provide LPV
approximations with possibly low-dimensional, e.g., $r\ll n$, parameter
domains.

Let $V_{r}\in\mathbb{R}^{n\times r}$ be a POD basis that encodes and decodes
the state $x(t)$ as $\rho(x(t))=V_{r}^{T}x(t)$ and

$x(t)\approx\tilde{x}(t)=V_{r}\rho(x(t))=\sum_{i=1}^{r}v_{i}\rho_{i}(x(t)),$

where $v_{i}\in\mathbb{R}^{n}$ is the $i$-th POD mode.

With this encoding and decoding, the nonlinear term in system (3) can be
approximated as

$\begin{split}A(x(t))\,x(t)\approx A(\tilde{x}(t))\,x(t)=\\\
A(\sum_{i=1}^{r}v_{i}\rho_{i}(t))\,x(t)=[A_{0}+\sum_{i=1}^{r}\rho_{i}(t)L(v_{i})]\,x(t)\end{split}$

which defines an affine-linear low-dimensional LPV approximation

$\tilde{x}(t)=[A_{0}+\sum_{i=1}^{r}\rho_{i}(\tilde{x}(t))L(v_{i})]\,\tilde{x}(t)+Bu(t)$
(4)

to (3) and (1). We set

$A_{i}:=L(v_{i}),\quad i=1,\dotsc,r.$ (5)

In the second step, we now project the LPV approximation to reduced order
coordinates. For that let $V_{k}$ with $k\geq r$ be the POD basis that
includes $V_{r}$ and $\bar{\rho}(t)=V_{k}\tilde{x}(t)$. Then an approximation
to (4) with state dimension $k$ (as opposed to $n$) reads

$\dot{\bar{\rho}}(t)=[\bar{A}_{0}+\sum_{i=1}^{r}\bar{\rho}_{i}(t)\bar{A}_{i}]\,\bar{\rho}(t)+\bar{B}u(t).$
(6)

with $\bar{A}_{i}:=V_{k}^{T}A_{i}V_{k}$, for $i=0,1,\dots,r$ and
$\bar{B}:=V_{k}^{T}B$.

Note that $r$ can be chosen independently of $k$. Thus, a very low-dimensional
approximation can be achieved independently of a possibly larger state space
that can be tailored for the best compromise in terms of size and accuracy.

##  III Controller Design for LPV Systems

The polytopic approach is seen as the most developed approach with the notable
results from [8, 11] that provide algorithms and theory for a scheduled robust
controller and that are the base of the hinfgs routine in the MATLAB _Robust
Control Toolbox_.

##  IV Implementation Issues

For the computation of the snapshots that are used to extract the POD basis, a
forward simulation for an example input is performed.

###  IV-A Polytope or Bounding Box

The computed snapshots are then projected to the $\rho$-coordinates in order
to estimate the polytope $W\subset\mathbb{R}^{r}$ that contains $\rho(t)$.

The vertices $w_{i}$ of $W$ are then used to define the controller.

Intuitively, the performance of the controller will be better if the volume
$|W|$ of $W$ is smaller and if the vortices $w_{i}$ are closer to extremal
values of $\rho(t)$. In this respect, the optimal choice of $W$ would be the
convex hull of the snapshots of $\rho$.

On the other hand, the convex hull is likely to have a large number of
vortices, which makes the application of LPV controller design approaches
costly as they, e.g., require the solution of $n$ coupled LMIs where $n$ is
the number of vertices of $W$.

Typically, $W$ is chosen as a bounding box with, accordingly, $2^{r}$
vortices. In order to reduce the volume, the box can be rotated and expressed
in coordinates obtained of the principal components; see, e.g., [13]. In our
affine-linear case (4) with

$\rho(t)=U_{\mathsf{p}c}U_{\mathsf{p}c}^{T}\rho(t)=:U_{\mathsf{p}c}\rho_{\mathsf{p}c}(t)$
(7)

and, accordingly,

$\tilde{x}(t)=V_{r}\rho(t)=V_{r}U_{\mathsf{p}c}\rho_{\mathsf{p}c}(t)=:\sum_{i=1}^{r}v_{{\mathsf{p}c},i}\rho_{{\mathsf{p}c},i}(t)$
(8)

where $U_{\mathsf{p}c}$ is the _principal components_ coordinate
transformation, this reparametrization reads

$\sum_{i=1}^{r}\rho_{i}(t)A_{i}=L(\tilde{x}(t))=\sum_{i=1}^{r}\rho_{{\mathsf{p}c},i}(t)A_{{\mathsf{p}c},i},$
(9)

where $A_{{\mathsf{p}c},i}:=L(v_{{\mathsf{p}c},i})$ and $v_{{\mathsf{p}c},i}$
denoting the $i$-th column of $V_{r}U_{\mathsf{p}c}$, for $i=1,\dotsc,r$. For
a direct retransformation of the system, we can also resort to the relation

$A_{{\mathsf{p}c},i}=\sum_{j=1}^{r}U_{ji}A_{j},$ (10)

where $U_{ji}$ is the $j$-th row entry of the $i$-th column of
$U_{\mathsf{p}c}$.

Adding on these standard ways on defining the parameter domains, in our
examples we use optimization to find a suitable polytope that gives a good
compromise of volume and number of vortices and that underbids the bounding
box in both variables. For this, the following optimization setup was defined
and solved using the built-in methods of computing convex hulls and genetic
optimization in _SciPy_ ; see, e.g., [19].

  * (0.)

Let $V\in\mathbb{R}^{r}$ be (the set of vertices of) the convex hull of given
measurements of $\rho(x(t_{j}))$, for $j=1,\dots,N$ and a given state
trajectory $x$.

  * (i.)

In the $i$-th iteration, extend $V$ by $n_{k}$ vertices
$v^{(i)}_{k}\in\mathbb{R}^{r}$ and compute the convex hull $V^{(i)}$ of
$V\cup\\{v^{(i)}_{1},\dotsc,v^{(i)}_{n_{k}}\\}$

  * (ii.)

Update $v^{(i)}_{k}$ to minimize both the volume and the number of vertices of
$V^{(i)}$.

We report on efficiency (and feasibility) of the computation of the LPV
controller and on its performance for the three approaches of considering

  * •

the bounding box in the original $\rho(t)$ coordinates,

  * •

the bounding box in the PCA coordinates of $\rho(t)$, or

  * •

an optimized polytope of fewer vertices

as the polytope for the parameter variation; see Figure 1 for an illustration
of the bounding box and an optimized polytope.

Figure 1: An illustration of the first three components of $\rho(t)$, of the
bounding box (with 8 vortices), and an optimized polytope of less volume and
with only 5 vortices). Each data object is in three-dimensional space but
projected along the coordinate axes to planes in the two-dimensional space.

##  V Numerical Example

Figure 2: Illustration of the computational domain and the developed state of
the uncontrolled flow field.

We consider the two-dimensional cylinder wake with control at moderate
Reynolds numbers; see Figure 2. The controls are designed as two outlets at
the cylinder periphery of size $\pi/6$ located at $\pm\pi/3$ through which
fluid can be injected or sucked away. Mathematically, the control is modelled
by parabola-shaped spatial shaped function that is scaled by the scalar
control value. For inclusion in the FEM scheme, these Dirichlet conditions are
relaxed towards Robin-type boundary conditions with a relaxation parameter
$\epsilon=10^{-5}$; see [20] for implementation details.

As the output $y$, we consider averaged velocities in 3 square domains of
observations of size $D^{2}$ located at a distance of $H$ behind the cylinder
symmetrically with respect to the channel middle. Here $D$ denotes the
diameter of the cylinder and $H$ the height of the channel. With the two
components of the velocity, overall, an output $y(t)\in\mathbb{R}^{6}$ is
obtained with the first 3 values corresponding to the stream wise components
and the final 3 values to the lateral components of the velocities.

The corresponding PDE model is spatially discretized by _Taylor-Hood_
quadratic-linear mixed finite elements on a nonuniform grid. From the FEM
model of about 50 000 degrees of freedom, the low-dimensional LPV
approximation is obtained through the following algorithm

  1. 1.

Starting from the steady-state solution, the FEM model is integrated in time
from $t=0$ to $t=5$ with a test input applied to trigger the instabilities.

  2. 2.

From the FEM solution, 417 equispaced snapshots are collected to define the
POD basis.

  3. 3.

With the POD basis, the reduced order LPV approximation as in (6) is computed.

In view of determining $k$ and $r$, i.e., the size of the POD reduced model
and the size of the parameter in the LPV approximation, we note that the
system is chaotic which makes a quantitative decision delicate (as small
perturbations have arbitrarily large effects). That’s why we took the
qualitative view of examining the resulting limit cycles of selected
components in the phase portraits of $y_{2}$ vs. $y_{5}$ and $y_{1}$ vs.
$y_{5}$, see Figures 3–4 for results of different choices of $k$ and $r$ and
Figure 5 for the reference. These phase portraits show the data points, say
$(y_{2}(t_{i}),y_{5}(t_{i}))$, for $500$ time equidistant instances $t_{i}$
from the output $y$ of the corresponding simulations with zero inputs on the
time interval $(0,12)$.

For the following numerical studies we chose the setup $k=36$ and $r=6$ that,
judging from Figure 4 (first line) in comparison to Figure 5 (first line),
seems to well cover at least the range of values in the phase portraits for
the smaller values of $r$.

Figure 3: Phase portraits $y_{2}$ vs. $y_{5}$ and $y_{1}$ vs. $y_{5}$ of the
output $y$ for POD dimension $k=24$ and parameter dimension $r=6,10,15$ on the
time interval $(0,12)$. Figure 4: Phase portraits $y_{2}$ vs. $y_{5}$ and
$y_{1}$ vs. $y_{5}$ of the output $y$ for POD dimension $k=36$ and parameter
dimension $r=6,10,15$ on the time interval $(0,12)$. Figure 5: Phase portrait
of selected components of the output $y$ for the full order FEM model on the
time interval $(0,12)$. Because of the symmetry of the overall setup, the
phase portraits of the remaining components bear no additional information and
are not shown here.

###  V-A Computing the LPV Controller

For the computation of the LPV controller we used the _Matlab Robust Control
Toolbox_[21] in the release 2022b with built-in function hinfgs that computes
an LPV controller with a guaranteed quadratic robustness performance
$\gamma_{*}$. The computational costs of the underlying optimization with
coupled LMI constraints are significant and make larger values of $k$ and $r$
quickly infeasible for numerical studies.

As for the different approaches of defining the enclosing polytope
$W\supseteq\\{\rho(t)\colon t>0\\}$ we made the following observations.

  * •

The bounding box in the original coordinates with $2^{6}=64$ achieves the best
robustness performances $\gamma_{*}$ though with rather high computational
costs.

  * •

The bounding box in the PCA coordinates comes with the same number of vertices
to consider. However, the observed convergence in $\gamma_{*}$ was tediously
slow which led to infeasible numerical costs for lower target values of
$\gamma_{*}$.

  * •

Using an optimized polytope of $20$ vertices, each iteration in the hinfgs
computation was sped up by a factor of $3$ which well compensated for an
overall slower convergence. However the slower convergence even led to
stagnation so that the best achievable values of $\gamma_{*}$ were larger than
that of the bounding box approach.

The results on the performance of hinfgs for the different setups and for
different levels of $\gamma_{*}$ are displayed in the chart of Fig. 6. The
conducted experiments suggest that a compact representation of $W$ (in the
sense that its vertices are evenly distributed in space or that the ratio of
surface over volume of $W$ is rather small) is beneficial for convergence in
$\gamma_{*}$. Apparently, the vertices of the optimized polytope are further
apart (see Fig. 1) which may explain the slower convergence and earlier
stagnation. An explanation for the minor performance of the PCA coordinate
transformation may call on the interpretation of the PCA concentrating the
variance of the data in the leading principal components. This may result in
vertices that widely distributed in one dimension and closely located in
another. Nonetheless, the strong (in this case negative) effect supports the
idea that a transformation of the $\rho$ coordinates has the potential of
improving the performance of hinfgs.

Figure 6: Achieved $\gamma_{*}$ versus CPU time (values of three consecutive
simulations) for the bounding box, the PCA transformed coordinates, and the
optimized vertices setup. The left most values correspond to the best achieved
$\gamma_{*}$ for which the iteration stagnates.

###  V-B Nonlinear Closed Loop Simulations

Since the robust control toolbox in _Matlab_ does not provide the
functionality to evaluate the LPV controller within a general
polytope111Basically, _Matlab_ has no built-in function to compute
_barycentric coordinates_ in a convex polytope in dimensions higher than
$r=3$., we considered closed loop simulations with the controller obtained
through the bounding box approach. Also the built-in simulation routines only
support predefined parameter trajectories so that the closed-loop system was
set up manually and simulated with the time integrator ode15s.

As the result, this nonlinear controller did well stabilize the nonlinear
system (with $r=6$ that it was built upon) as illustrated in Fig. 7. Also,
this controller proved a certain robustness by performing similarly well for
the model with parameter dimension $r=15$ which is similar but also shows
different dynamical patterns; cp. Fig. 4(first row vs. third row).

Figure 7: The open loop output of the model for $r=6$ and the closed loop for
LPV controller designed for the $r=6$ model applied to this model and used in
the larger ($r=15$) model.

###  V-C Remarks on Nonlinear Systems as LPV Systems

Although the embedding of a nonlinear system via $A(\rho(t))=A(\rho(x(t))$
into the class of LPV systems is readily covered by the available LPV theory,
it comes with practical consequences in the controller definition. Firstly, as
the trajectory of $\rho$ is not preset but defined through the state
trajectory $x$, the containing polytope (or bounding box) has to be estimated
from state estimations or approximations.

Secondly, the effect of the feedback is ambivalent. While a functioning
controller will stabilize the system around the working point and, thus,
prevent $x$ and $\rho(x)$ from attaining extreme values, short-term
perturbations may lead to overshoots which can drive the system out of the
estimated region.

While, practically, such an overshoot can often be compensated, the computed
controller will fail immediately as the parameter update for the controller is
no more well-defined. In our experiments we observed these critical overshoots
when considering nonzero initial values or discontinuous disturbance signals.
Therefore, in the presented simulations, we started from the zero initial
conditions and applied a disturbance that smoothly faded out after $t=2$.

###  V-D Code Availability

The LPV system data (ready for import in Matlab) and the scripts that were
used to obtain the presented numerical results are available for immediate
reproduction from doi:10.5281/zenodo.10073483 under a CC-BY license.

##  VI Conclusions and Outlook

In this work, we have employed a two-level model order reduction approach so
that, eventually, established LPV theory and algorithms become available for
nonlinear controller design for general nonlinear systems like the
incompressible Navier-Stokes equations.

In a numerical experiment, we illustrated the potential and feasibility of the
combined approach and identified pitfalls of the approach as well as limits in
the existing functionality of standard control systems software.

Notably, although the theory applies and although favourable properties of
tailored polytopes have been illustrated, routines for LPV controller
synthesis basically only allow for bounding boxes as enclosing polytopes.
Thus, a future work will concern interpolation of controllers in polytopes
using, e.g., the formulas provided in [22].

Another immediate future development could concern the solution of large-scale
linear matrix inequalities in the context of LPV controller design.

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