A data-driven Koopman model predictivecontrol framework for nonlinear flows

Hassan Arbabi, Milan Korda and Igor Mezic ∗

June 6, 2018

Abstract

The Koopman operator theory is an increasingly popular formalism of dynami-cal systems theory which enables analysis and prediction of the nonlinear dynamicsfrom measurement data. Building on the recent development of the Koopman modelpredictive control framework [1], we propose a methodology for closed-loop feedbackcontrol of nonlinear flows in a fully data-driven and model-free manner. In the firststep, we compute a Koopman-linear representation of the control system using a varia-tion of the extended dynamic mode decomposition algorithm and then we apply modelpredictive control to the constructed linear model. Our methodology handles both full-state and sparse measurement; in the latter case, it incorporates the delay-embeddingof the available data into the identification and control processes. We illustrate theapplication of this methodology on the periodic Burgers’ equation and the boundarycontrol of a cavity flow governed by the two-dimensional incompressible Navier-Stokesequations1. In both examples the proposed methodology is successful in accomplishingthe control tasks with sub-millisecond computation time required for evaluation of thecontrol input in closed-loop, thereby allowing for a real-time deployment.

Keywords: Flow control, Koopman operator theory, Feedback control, Dynamic modedecomposition, Model predictive control

1 Introduction

Flow control is one of the central topics in fluid mechanics with an enormous impact onother fields of engineering and applied science. Its wide range of applications includes, justto name a few, reduction of aerodynamic drag on vehicles and aircrafts, mixing enhancementin combustion and chemical processes, suppression of instabilities to avoid structural fatigue,lift increase for wind turbines, and design of biomedical devices. To emphasize the impact of

∗The authors are with the department of Mechanical Engineering, University of California, Santa Barbara,CA, 93106, USA. harbabi, milan.korda, mezic@engineering.ucsb.edu

1The MATLAB implementation of the Koopman-MPC framework for the examples is available at https://github.com/arbabiha/KoopmanMPC_for_flowcontrol.

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flow control, it is worth noting that discovery of efficient flow control techniques for reductionof drag on ships and cars can result in mitigation of yearly CO2 emission by millions of tonsand annual savings of billions of dollars in the global shipping industry [2, 3].

Despite all the interest and continuous effort, the flow control still poses a dauntingchallenge to our theoretical understanding and computational resources. The main sourceof difficulty is the combination of high-dimensionality and nonlinearity of fluid phenomenawhich results in computational or experimental models which are too complex and costlyto control using well-developed strategies of modern control theory. The recent advances innumerical computation has led to partial success with active control of flows using modelsbased on Navier-Stokes equations [2, 4–6]; however, these methods suffer from two majorshortcomings: first, nonlinear models obtained from Navier-Stokes are still high-dimensionaland computationally costly, thereby not allowing for fast implementation of nonlinear andcomputationally complex control techniques such as nonlinear model predictive control, andsecond, linear models used with LQR/LQG or adjoint-based controllers often rely on locallinearization around equilibria or a trajectory of the flow which makes them valid only locally,and may result in suboptimal or even unstable control performance.

An alternative approach that has gained traction in the last two decades is identificationof relatively low-dimensional flow models from data provided by numerical simulations or ex-periments. Some of the data-driven methods combine the measurement data with underlyingphysical model to identify models of the system. The major examples include construction of(usually autonomous) state space models via Galerkin projection of the Navier-Stokes equa-tions onto the modes obtained by proper orthogonal decomposition (POD) of data [7–9], oridentification of linear input-output systems using balanced POD [10, 11]. There are also afew applications of system identification methods to construct linear input-output modelspurely from data, including the eigensystem realization algorithm [12, 13], as well as sub-space identification and autoregressive models [14, 15]. Utilization of the above techniquesin a variety of problems has shown great promise for low-dimensional modeling and controlof complex flows from data.

In this paper, we present a general and fully data-driven framework for control of nonlin-ear flows based on the Koopman operator theory [16,17]. This theory is an operator-theoreticformalism of classical dynamical systems theory with two key features: first, it allows a scal-able reconstruction of the underlying dynamical system from measurement data, and sec-ond, the models obtained are linear (but possibly high-dimensional) due to the fact that theKoopman operator is a linear operator whether the dynamical system is linear or not. Thelinearity of the Koopman models is especially advantageous since it makes them amenableto the plethora of mature control strategies developed for linear systems. This frameworkfor design of controller, is called Koopman-MPC and follows the work in [1]. In the firststep of our approach, we build a finite-dimensional approximation of the controlled Koopmanoperator from the data, using a variation of the extended dynamic mode decomposition algo-rithm (EDMD) [18], with a particular choice of observables assuring linearity of the resultingapproximation. The ideal data would include measurements on a number of system trajecto-ries with various input sequences. In the second step, we apply the model predictive control(MPC) to these linear models to obtain the desired objectives. The distinguishing feature isthat this framework leads to a linear MPC, solving a convex quadratic programming prob-lem, and thereby enables a rapid solution of the underlying optimization problem, which is

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necessary for real-time deployment. This methodology can be also applied to problems withsparse measurements, i.e., problems with a limited number of instantaneous measurements(e.g. point measurements of the velocity field at several different locations). In that case,delay-embedding of the available measurements (and nonlinear functions thereof) is used toconstruct the Koopman-linear model; the MPC is then applied to the linear system whosestate variable is the delay-embedded vector of measurements.

The outline of this paper is as follows: A brief review of related work is given in section 1.1.Section 2 gives a review of the Koopman operator theory for dynamical systems with input.In section 3, we describe the EDMD algorithm for construction of the Koopman-linear modelfrom measurement data. In section 3.1, we discuss using delay-embedding to constructand control Koopman-linear models from sparse measurements. An overview of the MPCframework is given in section 4. In section 5, we present two numerical examples: theBurgers’ system on a periodic domain and the 2D lid-driven cavity flow. We formulate thecontrol problem for these cases using various objectives and demonstrate our approach forboth full-state and sparse measurements. We summarize the results and discuss the outlookin section 6.

1.1 Review of related work

The Koopman operator formalism of dynamical systems is rooted in the seminal worksof Koopman and Von Neumann in the early 1930s [16,19]. This formalism appeared mostlyin the context of ergodic theory for much of the last century, until in mid 2000’s when theworks in [20, 21] pointed out its potential for rigorous analysis of dynamical systems fromdata. The notion of Koopman mode decomposition (KMD) which is based on the expansionof observable fields in terms of Koopman operator eigenvalues and eigenfunctions was alsointroduced in [21]. KMD was first applied to a complex flow in [22] where its connection withthe DMD numerical algorithm [23] was pointed out. The work in [22] showed the promiseof this viewpoint in extracting the physically relevant flow structures and time-scales fromdata. Following the success of this work, KMD and its numerical implementation throughDMD, has become a popular decomposition for dynamic analysis of nonlinear flows [24–28].

The application of the Koopman operator to data-driven control of high dimensionalsystems is much less developed. The earliest works on generalizing the Koopman operatorapproach to control systems was presented in [29, 30] accompanied with a numerical varia-tion of DMD algorithm [31]. To the best of our knowledge, however, the only applicationfor feedback control of fluid flows are the works in [32, 33]. The work in [32] considered theproblem of flow control using a finite set of input values. For each value of the input, aKoopman-linear model was constructed from the data and the control problem was formu-lated as a switched optimal control and implemented in a receding horizon fashion. Thismethodology was successfully used for tracking reference output signals in Burgers equationand incompressible flow past a cylinder. The work in [33] proposed to remove the restrictionof the input to a finite set, by interpolating between the Koopman-linear systems for eachinput value which led to an improvement of the control performance. In [1] (which this workis based on), a more general extension of the Koopman operator theory to systems withinput was presented, and used to construct linear predictors especially suitable for modelpredictive control, demonstrating the effectiveness of the approach (among other examples)

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on the control of the Korteweg-de Vries PDE. The results in [1] showed superiority of thecontrolled Koopman-linear predictors constructed from data to models obtained by local lin-earization and Carleman’s representation both for prediction and for feedback control. Letus also mention the earlier work in [34] that utilized KMD to construct the normal forms,for dynamics of the flow past an oscillating cylinder; the input forcing appeared as a bilinearterm in the normal forms for this flow. See [35] for application to pulse-based control ofmonotone systems as well as [36] for system identification and [37] for state estimation.

The numerical engine behind the system identification part of the framework presentedin this paper is the Extended Dynamic Mode Decomposition (EDMD) algorithm proposedin [18]. Although the original DMD algorithm was invented independent of the Koopmanoperator theory [23], the connection between the two was known from early on [22], andDMD-type algorithms have become the popular methods for computation of the Koopmanoperator spectral properties. Nevertheless, the convergence of DMD algorithms for approx-imation of Koopman operator is just recently established in [38, 39]. For application ofKoopman-MPC to systems with sparse measurements, the EDMD is modified to include thedelay embeddings of instantaneous measurements. Delay embedding is a classic techniquein system identification literature (see, e.g., [40] for a comprehensive reference) and controlas well as in linear and nonlinear time-series analysis (e.g., [41]). In the field of dynamicalsystems, the classical reference is the work of Takens [42] on geometric reconstruction ofnonlinear attractors. The work in [43] suggested the combination of this technique with theDMD algorithm for identification of nonlinear systems and its role in approximation of theKoopman operator was studied in [38, 44]; the use for control, in the Koopman operatorcontext, was described in [1].

2 Koopman operator theory

In this section, we first review the basics of the Koopman operator formalism for au-tonomous dynamical systems and then discuss its extension to systems with input and out-put. We will focus on discrete-time dynamical systems to be consistent with the discrete-time nature of the measurement data, but most of the analysis easily carries over to thecontinuous-time systems. We refer the reader to [17,45] for a more detailed discussion of theKoopman operator basics.

Consider the dynamical system

x+ = T (x), x ∈M (1)

defined on a state space M . We call any function g : M → R an observable of the system,and we note that the set of all observables forms a (typically infinite-dimensional) vectorspace. The Koopman operator, denoted by K, is a linear transformation on this vector spacegiven by

Kg = g T, (2)

where denotes the function composition, i.e., (Kg)(x) = g(T (x)). Informally speaking, theKoopman operator updates the observable g based on the evolution of the trajectories in the

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state space. The key property of the Koopman operator that we exploit in this work is itslinearity, that is, for any two observables g and h, and scalar values α and β, we have

K(αg + βh) = αKg + βKh, (3)

which follows from the definition in eq. (2). We call the observable φ a Koopman eigenfunc-tion associated with Koopman eigenvalue λ ∈ C if it satisfies

Kφ = λφ. (4)

The spectral properties of the Koopman operator can be used to characterize the state spacedynamics; for example, the Koopman eigenvalues determine the stability of the systemand the level sets of certain Koopman eigenfunctions carve out the invariant manifolds andisochrons [46–48]. Moreover, for smooth dynamical systems with simple nonlinear dynamics,e.g., systems that possess hyperbolic fixed points, limit cycles and tori, the evolution ofobservables can be described as a linear expansion in Koopman eigenfunctions [49]. Inthese systems, the spectrum of the Koopman operator consists of only point spectrum (i.e.eigenvalues) which fully describes the evolution of observables;

Kng =∞∑j=0

vjφjλnj . (5)

where vj is called the Koopman mode associated with Koopman eigenvalue-eigenfunctionpair (λj, φj) and it is given by the projection of the observable g onto φj. See [22] for moredetail on Koopman modes, and [49] on the expansion in (5).

The extension of the Koopman operator theory to a controlled system denoted by

x+ = T (x, u), x ∈M, u ∈ U , (6)

requires one to work on the extended state space, which is the Cartesian product of the statespace M and the space of all input sequences `(U) = (u0)∞i=0 | ui ∈ U. We denote theextended state space by S = M × `(U). Now, given an observable g : S → R we can definethe non-autonomous Koopman operator,

(Kg)(x, (ui)∞i=0) = g(T (x, u0), (ui)

∞i=1). (7)

See [1] for more details on this extension. We emphasize that the linear representation ofthe nonlinear system by the Koopman operator is globally valid and generalizes the locallinearization around equilibria [49].

3 Construction of Koopman-linear system

In this section, we review the construction of the Koopman-linear system as proposedby [1] using the EDMD algorithm [18]. We are looking to approximate the dynamics of thenonlinear flow via a linear time-invariant system such as

z+ = Az +Bu z ∈ Rn, u ∈ Rk,

x = Cz. (8)

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Consider the set of states and inputs of the nonlinear system in the form of

X = [x1, . . . , xK ], X+ = [x+1 , x+

2 , . . . , x+K ], U = [u1, . . . , uK ]. (9)

where x+j = T (xj, uj). Let

g(x) =[g1(x) . . . gm(x)

]>(10)

be a given vector of possibly nonlinear observables. These functions may represent user-specified nonlinear functions of the state as well as physical measurements (i.e., outputs)taken on the dynamical system (or nonlinear functions of such outputs). We are goingto assume that we only have access to values of the observables, and therefore, explicitknowledge of the state variable in (9) is not required. By collecting data on the dynamicalsystem, we can form the lifted snapshot data matrices

Xlift = [g(x1), . . . , g(xK)], X+lift = [g(x+

1 ), . . . , g(x+K)], U = [u1, . . . , uK ]. (11)

These data matrices are the lifted coordinates of the system in the space of observables.Note that, as in [1], we have not lifted U coordinates to preserve the linear dependence ofthe predictor on the original input. The matrices A, B and C are then given by the solutionto the linear least-squares problems

minA,B‖X+

lift − AXlift −BU‖F , minC‖X − CXlift‖F (12)

where ‖ · ‖F denotes the Frobenius norm. The analytical solution to these two problems canbe compactly written as [

A BC 0

]=

[X+

lift

X

] [Xlift

U

]†. (13)

When snapshot matrix Xlift is fat (i.e. number of columns exceeds number of rows), it ismore efficient to compute the matrices by solving the normal equations

V =MG, (14)

with the unknown matrix variable M and given matrices

V =

[X+

lift

X

] [Xlift

U

]>, G =

[Xlift

U

] [Xlift

U

]>.

The solution M to (14) provides the matrices A, B, C through

M =

[A BC 0

].

The matrices A and B describe the linear dynamics of the Koopman-linear state z = g(x).The prediction of the original state x is obtained simply by x = Cz. See [39] for a convergenceanalysis of EDMD for approximation of Koopman operator.

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3.1 Sparse measurements and delay embedding

When the number of observables measured on a dynamical system is insufficient forconstruction of an accurate model, we can use the delay embedding of the observables.Delay embedding (i.e., embedding several consecutive output measurements into a single datapoint) is a classical technique ubiquitous in system identification literature (e.g., [40]) but alsoin the theory of dynamical systems for geometric reconstruction of nonlinear attractors [42].It has also been utilized in the context of Koopman framework in [20,38,50]. The key featureof delay embedding here is that it provides samplings of extra observables to realize theKoopman operator. To be more precise, if we have a sequence of measurements on a singleobservable h at the nd time instants ti, ti+1, . . . , ti+nd−1, we can think of them as samplingof the nd observables [h, Kh, . . . , Knd−1h] at the single time instant ti. Here, we describehow we can incorporate delay-embedding into identification and control of Koopman-linearmodels. The only requirement for identification is that we should have access to at leastnd + 1 sequential time samples on the trajectories where nd is the chosen number of delays.

Let h be the vector of instantaneously measured observables on the dynamical system(e.g., point measurements of the velocity field), and nd be the delay embedding dimension.Consider the state and input matrices described in (9), but now assume that they contain astring of sequential samples with length nd + 1, i.e., for some j, we have

xi+1 = T (xi, ui), i = j, . . . , j + nd − 1. (15)

We can delay embed the measurements on this string to construct a pair of lifted coordinatesin the space of observables,

ζj =

h(xi)...

h(xi+nd−1)ui...

ui+nd−1

, ζ+

j =

h(xi+1)...

h(xi+nd)

ui+1...

ui+nd

, i = j, . . . , j + nd − 1. (16)

It is easy to check that ζ+j = Kζj. By delay-embedding the observations on all sequential

strings of data, we can form the new matrices

X = [ζ1, ζ2, . . . , ζL], Y = [ζ+1 , ζ

+2 , . . . , ζ

+L ]. (17)

We can once again lift the data using a vector of nonlinear user-specified functions g to formthe new lifted matrices,

Xlift = [g(ζ1), g(ζ2), . . . , g(ζL)],

Ylift = [g(ζ+1 ), g(ζ+

2 ), . . . , g(ζ+L )]. (18)

Having Xlift, Ylift and input matrix U defined, we solve the the least-squares problems (12)to find the linear system matrices. It is very important for the lifting function g to have ameaningful dependence on ui, . . . , ui+nd

. This allows EDMD to approximate the dynamics

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of the extended state space and discern the effect of previous inputs in the evolution of thestate.

The linear predictor in this case would be

z+ = Az +Bu

ζ = Cz, (19)

Here ζ denotes the prediction of the “embedded” state ζ (note that when ζ is employedfor controller design, typically only the part of ζ corresponding to the most recent outputprediction is used).

4 Model predictive control

The methodology presented in the last section allows us to construct a model of theflow in the form of a linear dynamical system (8). In this work, we will apply MPC to thislinear model to control the original nonlinear flow, but other techniques from modern controltheory could be applied as well; see the survey [51] or the book [52] for an overview of MPC.In the context of MPC, we formulate the control objective as minimization of a cost functionover a finite-time horizon. The general strategy is to use the model in eq. (8) to predict thesystem evolution over the horizon, and use these predictions to compute the optimal inputsequence minimizing the given cost function along this horizon. Then we apply only the firstelement of the computed input sequence to the real system, thereby producing a new valueof the output, and repeat the whole process. This technique is sometimes called the recedinghorizon control. In the following, we describe the notation and some mathematical aspectsof this technique. The distinguishing feature when using the lifted linear predictor (8) isthat the resulting MPC problem is a convex quadratic program (QP) despite the originaldynamics being nonlinear. In addition, the complexity of solving the quadratic problemcan be shown to be independent of the size of the lift if the so-called dense form is used [1],thereby allowing for a rapid solution using highly efficient QP solvers tailored for linear MPCapplications (in our case qpOASES [53]).

Let N be the length of the prediction horizon, and uiN−1i=0 and yiNi=1 denote the

sequence of input and output values over that horizon. A very common choice of costfunctions is the convex quadratic form,

J(uiN−1

i=0 , yiNi=1

)= y>NQNyN + q>yN (20)

+N−1∑i=1

y>i Qiyi + u>i Riui + q>i yi + r>i ui

+ u>0 R0u0 + r>0 u0,

where Qi=0,...,N and Ri=0,...,N−1 are real symmetric positive-definite matrices. The above costfunction can be used to formulate many of the common control objectives including thetracking of a reference signal. For example, assume that we want to control the flow suchthat its output measurements follow an arbitrary time-dependent output sequence denoted

8

by yrefi. We can formulate this objective as minimization of the distance between yiand yrefi, and the corresponding cost function over the finite horizon would be

J1

(uiN−1

i=0 , yiNi=1

)=

N∑i=1

(yi − yref

i

)>Q(yi − yref

i

), (21)

=N∑i=1

y>i Qyi − 2(yrefi

)>Qyi +

(yrefi

)>Qyref

i .

where Q is the weight matrix that determines the relative importance of measurements iny. Note that the last term in the above equation is not dependent on the input or output,and therefore it does not affect the optimal solution. By dropping this term, and lettingqi = −Q>yref

i , we obtain

J1

(uiN−1

i=0 , yiNi=1

)=

N∑i=1

y>i Qyi + q>i yi, (22)

which is a special form of eq. (20). In the numerical examples presented in this paper, wewill use this type of cost function.

The MPC controller solves the following optimization problem at each time step of theclosed loop operation(

u?i N−1i=0 , y?i Ni=1

)= arg min J

(uiN−1

i=0 , yiNi=1

)s.t. zi+1 = Azi +Bui, i = 0, . . . , N

yi = Czi (23)

Eyi yi + Eu

i ui ≤ bi, i = 0, . . . , N − 1,

ENyN ≤ bN

z0 = g(ζc),

where ζc is the delay-embedded vector of measurements

ζc = [h(xk−nd+1), . . . ,h(xk), uk−nd, . . . , uk−1]> (24)

The matrices Exi=0,...,N−1, Eu

i=0,...,N−1 and EN define polyhedral state and input con-straints. This is a standard form of a convex quadratic programming problem which can beefficiently solved using many available QP solvers - in our case qpOASES [53]. The com-putational complexity can be further reduced by expressing the lifted state variables z interms of the control inputs u, thereby eliminating the dependence on the possible very largedimension of z; see [1] for details.

Once the optimal input sequence u?i N−1i=0 is computed, we apply its first element u?0 to the

system to obtain a new output measurement which updates the current state ζc; the wholeprocess is then repeated in a receding horizon fashion. Algorithm 1 summarizes the closed-loop control operation, and the entire algorithm for implementation of the Koopman-MPCis illustrated in Figure 1 .

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Algorithm 1 Koopman MPC – closed-loop operation

Initialization: h(x−nd), . . . ,h(x−1), u−nd

, . . . , u−1

1: for k = 0, 1, . . . do2: Measure h(xk).3: Set ζc = [h(xk−nd+1), . . . ,h(xk), uk−nd

, . . . , uk−1]>.4: Set z0 := g(ζc)5: Solve (23) to get an optimal solution (u?i )

Ni=1

6: Apply u?1 to the nonlinear system

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Figure 1: Schematic representation of Koopman-MPC framework for identification and closed-loop control of nonlinear flows.

5 Numerical Examples 2

5.1 Burgers equation

As the first example, we consider the Burgers equation with periodic boundary condition,

∂v

∂t+ v

∂v

∂z= ν

∂2v

∂z2+ f(z, t), z ∈ [0, 1], t ∈ [0,∞). (25)

v(0, t) = v(1, t) (26)

where ν is the kinematic viscosity. Note that we have used z to denote the spatial coordinatesin the flow examples, hoping that it will not be confused with the Koopman-linear state in

2The MATLAB implementation of the examples is available at https://github.com/arbabiha/

KoopmanMPC_for_flowcontrol.

10

(8). Similar to [32], we assume the forcing f(z, t) is given by

f(z, t) = u1(t)f1(z) + u2(t)f2(z), (27)

= u1(t)e−(

15(z−0.25))2

+ u2(t)e−(

15(z−0.75))2

(28)

with the control input u = (u1, u2) ∈ R2.The control objective is to follow the reference state

vref(z, 0 ≤ t < 2) =1

2,

vref(z, 1 ≤ t < 4) = 1,

vref(z, 4 ≤ t < 6) =1

2, (29)

starting from the initial condition,

v(z, 0) = ae−(

5(z−0.5))2

+ (1− a) sin(4πz). (30)

with a ∈ [0, 1] chosen randomly, and with the input signals constrained as |u1|, |u2| < 0.1.To construct the Koopman-linear system, we have used 50 two-second long trajectories

with ν = 0.01. Each trajectory starts from a random initial condition as in (30). The inputcontrol at each time instant is randomly drawn from the uniform distribution on (u1, u2) ∈[−0.1, 0.1]2. The Burgers equation upwind finite-difference scheme for advection and centraldifference for diffusion term, with 4th-order Runge-Kutta time stepping performed on 150spatial grid points with time steps of 0.01 second.

In case of full-state measurements, we use the vector of values of v at the computationalgrid points, the kinetic energy of v, and the constant observable (ψ(v) = 1). The costfunction to be minimized is the kinetic energy (L2-norm) of the state tracking error,

e(t) =

∫ 1

0

|v(z, t)− vref(z, t)|2dz. (31)

For the sparse measurement scenario, we assume that we only have access to the vectorof sparse measurements vs = h(v) which consists of values of v at 10 random grid points. Weform the Koopman-linear state vector by including delay embedding of vs with embeddingdimension nd = 5, the kinetic energy of instantaneous measurements ‖vs‖2, and the constantobservable. That is

ζc = [vs(ti−4), . . . , vs(ti), u(ti−4), . . . , u(ti−1)]>, (32)

g(ζc) = [ζ>c , ‖vs(ti)‖2, 1]> ∈ R60. (33)

We define the tracking error as

e(t) =1

m‖vs(t)− vsref(t)‖2, (34)

11

and the predicted objective function used within the MPC is then

J =

∫ T

0

e(t) dt,

where the prediction horizon is set to T = 0.1. After spatio-temporal discretization, thisobjective readily translates to the form (20) with N = 10.

The results of the controlled simulation for both scenarios are depicted in fig. 2. Inboth cases, the state successfully tracks the reference signal; however, the controller built viasparse measurements is slightly delayed compared to the case of full-state measurement whichcan be attributed to the construction of the Koopman-linear state using delay-embedding.We note that the tracking error during the transient phases is caused by input saturation asdocumented by the plot of the control input signal.

Robustness with respect to the parameter ν. One question that arises in the contextof low-dimensional modeling is wether the models constructed at some parameter valuewould be robust enough for prediction at other values. In order to test the robustness ofKoopman-linear model in case of Burgers, we use the model constructed above using sparsemeasurements to control the Burgers system at various values of ν ∈ [10−4, 0.1]. The resultsin fig. 3 indicate that Koopman-linear model constructed at the parameter regime ν = 0.01is remarkably effective over a wide parameter range, and the control performance is veryrobust. As expected, however, the input signal and tracking error in the diffusion-dominatedregime (large ν) is less fluctuating, as the diffusion helps the controller to stabilize the statearound the spatially-uniform reference state in (29).

5.2 2D lid-driven cavity flow

In the second example, we consider an incompressible viscous flow in a square cavitywhich is driven by motion of the top lid. The dynamics of the cavity flow is governed by theNavier-Stokes equation, which, in terms of the stream function variable reads

∂

∂t∇2ψ +

∂ψ

∂z2

∂

∂z1

∇2ψ − ∂ψ

∂z1

∂

∂z2

∇2ψ =1

Re∇4ψ, (z1, z2) ∈ [−1, 1]2, t ∈ [0,∞), (35)

ψ

∣∣∣∣z1=±1

= ψ

∣∣∣∣z2=−1

= 0 and∂ψ

∂z2

∣∣∣∣z2=1

= f(z1, t), (36)

where ψ(z1, z2, t) is the stream function, Re is the Reynolds number, and f1(z1, t) is thevelocity of the top lid which acts as the forcing on the system. We assume that we cancontrol the top lid velocity,

f(z1, t) = (1 + u(t))(1− z21)2, (37)

with the control input u ∈ R.The autonomous cavity flow with Re ≤ 10000 converges to a steady velocity profile (i.e.

fixed point in the state space) which consists of a large central vortex with downstreamcorner eddies. At around Re = 10500, a Hopf bifurcation makes the fixed point unstable

12

Figure 2: Koopman-MPC for control of Burgers system. Input signals, tracking error and stateevolution in for closed-loop simulation using Koopman-linear models constructed by full-state measurements(150 observables) and sparse measurements (10 observables).

and the solutions up to Re = 15000 converge to a limit cycle. In this regime, the boundaryof the central vortex oscillates due to the periodic shedding of vortices from the downstreamcorners. At higher Reynolds, the flow dynamics grows more complicated and ultimatelybecomes chaotic at high Reynolds. More details on the dynamics and the numerical schemeused to solve (35) can be found in [54].

We consider two control problems for the lid-driven cavity flow:

13

Figure 3: Robustness of Koopman-MPC with respect to parameter ν. The tracking error andstate evolution for controlling the flow at various values of ν using a controller constructed at ν = 0.01 .

Problem 1) We aim to stabilize the limit cycling flow at Re = 13000 around the fixed pointsolution at Re = 10000. This problem has the trivial solution u0 = −3/13, sincethe effective Re is proportional to the top velocity and u = u0 would set back theflow to the fixed point at Re = 10000. To avoid the trivial solution, we use theinput constraints −2/13 < u < 2/13.

Problem 2) Using the same input constraints, we try to stabilize the limit cycling flow at Re =13000 around the unstable fixed point solution at the same Re. This problem isspecially challenging since the linearization around this fixed point has eigenvalueswith positive real part that are not controllable. This implies that the nonlinearsystem is not stabilizable and there is no linear or nonlinear (regular) feedbackcontrol that could achieve the full stabilization [55].

The construction of the Koopman-linear system is similar to the Burgers system; we haveused 300 two-second long trajectories of the system with control inputs that are randomlydrawn from [−3/13, 3/13]. The initial condition for each trajectory is a random convexcombination of the stable fixed point at Re = 10000, a point on the limit cycle and the

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unstable fixed point at Re = 13000. The unstable fixed point is computed via the methodproposed in [56]. For the full-state observation, we use the values of the stream functionon the 50 × 50 computational grid, the kinetic energy and the constant observable. Forthe case of sparse measurement, we use the values of stream functions at k = 2, 5, 50, 100random points inside the flow domain, the l2 vector norm of observed stream function values,and the constant observable. According to section 3.1, the dimension of the state space forthe Koopman-linear system built from sparse measurements will be n = 16, 31, 256, 506respectively, which is considerably smaller than the Koopman-linear system with full-stateobservation (n = 2502).

Let ψref denote the stream function at the target fixed point. In the case of the full-statemeasurements, we define the tracking error to be the kinetic energy of the flow distance fromthe reference state, i.e.,

e(t) = ek(t) :=

∫Ω

|v(t)− vref |2dz1dz2, (38)

where v = (∂ψ/∂z2,−∂ψ/∂z1) is the velocity field, and Ω is the flow domain.In the case of sparse measurements, let ψs be the vector of stream function measurements.

Then the tracking error will be the l2-norm of distance from the reference measurements,that is,

e(t) = ‖ψs(t)− ψs,ref‖2. (39)

The objective function of the MPC is then given by

J =

∫ T

0

e(t) dt,

where the prediction horizon is set to T = 0.2. After spatio-temporal discretization, thisobjective function readily translates to to the form (20) with N = 20.

Figure 4 shows the kinetic energy of the state discrepancy (ek defined in (38)) in applyingthe Koopman-MPC to problem 1. All the closed-loop simulations start from the same initialcondition on the limit cycle. The Koopman-MPC, except for k = 1, 2, is successful inconsiderably reducing the flow distance from the desired state over finite time. The controlinputs and the flow evolution for some values of k and full-state observation is shown infig. 5. In the case of full-state observation, the input signal is mostly saturated at the lowerbound which results in a lower effective Re for the flow, and hence getting closer to thefixed point at Re = 10000. However, the controller occasionally uses bursts to speed upthe stabilization. The effect of these intermittent bursts on the control can be deduced bycomparison with the control input with k = 5 which is saturated at the lower bound at alltimes.

Figure 4 also suggests that the control performance of the Koopman-linear systems gen-erally scales with the number of measured observables, i.e., larger number of observablesresults in better control performance. This indicates that there is a reasonable tradeoff be-tween the sparsity of measurements and the control performance. Moreover, the full-stateobservation offers less than 10 percent improvement over k = 50 in the terminal discrepancy,which indicates that the cavity flow dynamics is approximately low-dimensional and it can

15

be effectively captured using low-dimensional models from data. We have observed that thechoice of measurement location in the flow domain may significantly affect the control per-formance for very small k, e.g. k = 1, 2, 5, and results reported in the figures only representthe typical behavior of controllers built on sparse measurements.

Figure 4: Control Performance for stabilization around the steady solution at Re = 10000.Normalized kinetic energy of flow distance for cavity flow controllers built by Koopman-MPC with variousnumber of measurements (k), as well as LQR based on local linearization of Navier-Stokes.

A standard technique for flow stabilization is to use linearized Navier-Stokes with linearcontrol strategies. Figure 4 shows the performance of such technique (labeled boundedNS-LQR) in achieving the control objective of the first problem. In this method, the Navier-Stokes equation is linearized around the reference fixed point, and an optimal state feedbackgain is computed that would minimize the cost function in (38) over an infinite-time horizonfor the linearized system (see Appendix for detail). At each time step, the computed optimalinput is bounded by the constraints identical to the MPC setting and then applied to thenonlinear system. This method results in an input signal which is saturated at the lowerbound and therefore its performance is identical to the case of Koopman-MPC with k = 5measurements. This method is successful in substantially reducing the tracking error, butunlike the Koopman-MPC framework, it is not capable of exploiting the nonlinearities farfrom the fixed point to speed up the stabilization. Moreover, this method is model-basedand its performance is likely to degrade when uncertainties in estimating fluid properties, orinput modeling errors are present.

The performance of the Koopman-MPC framework for stabilization around the fixedpoint at Re = 13000 (problem 2) is shown in fig. 6. Recall that the target fixed point is notstabilizable and no feedback solutions exist that could asymptotically bring the state to thefixed point. Nevertheless, the controllers based on Koopman-linear systems are capable ofsubstantially reducing the tracking error (e.g. down to 40 % with k = 100). The behaviorof controllers is similar to the previous problem, i.e., they tend to decrease the effectiveRe and use occasional bursts to accelerate the stabilization. An interesting observation isthat the controllers built on delay-embedding of measurements perform better than the one

16

Figure 5: Closed-loop control of cavity flow evolution with Koopman-MPC. Discrepancy in thevorticity of controlled state, and input signal, for full-state and sparse measurements. The measurementlocations are marked via crosses in the leftmost column. The performance of bounded NS-LQR is identicalto the Koopman-MPC with k = 5.

17

with full-state measurements. This shows the effectiveness of using nonlinear observablessuch as delay-embedded measurements to predict the nonlinear evolution. Note that for thisproblem, the linearized system around the fixed point is not stabilizable and there is no clearstart point for designing feedback control based on linearization techniques commonly usedin flow control.

Figure 6: Control Performance for stabilization around the steady solution at Re = 13000:Normalized kinetic energy of state discrepancy and the input signals for cavity flow controllers built byKoopman-MPC with various number of measurements (k). The steady solution is not stabilizable and thereis no optimal feedback solution for LQR.

Computation time Table 1 summarizes the average computational time3 required to eval-uate the control input at each time step of the closed-loop operation. We report separatelythe computation time tembed required to build the state of the Koopman linear system byembedding the available measurements and the time tMPC required to solve the optimizationproblem (23) in the dense form4 using the active set qpOASES solver [53]. As evident fromthe table, combination of the Koopman linear representation and convex quadratic program-ming of the MPC framework leads to computation of the control input in a fraction of amillisecond. Note that in both examples the bulk of the computation time is spent on em-bedding the sparse measurements to build the Koopman linear state; this step requires datamanipulation carried out purely in MATLAB and could be sped up by a tailored implemen-tation (e.g., in C). Of course, in a real-world implementation of this framework on nonlinearflows, other factors including the time to record and process the physical measurementsshould also be considered.

3The computations were carried out in MATLAB running on a 3.40 GHz Intel Xeon CPU and 64 GBRAM.

4The conversion of the optimization problem (23) to the dense form consist in solving for the statevariables (z1, . . . , zN ) in terms of the control inputs (u0, . . . , uN−1) and the initial state z0 using the linearrecursion z+ = Az + Bu; the result of this straightforward linear algebra excercise can be found in theappendix of [1].

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Table 1: Computation time for Koopman MPC of cavity flow and Burgers equations and NS-LQRcontrol for cavity flow. The symbol “—” signifies a negligible embedding time in the case of fullstate measurement.

# of measurements k embedding dimension n tembed [sec] tMPC [sec]

Burgers 10 62 4.71 · 10−5 2.28 · 10−7

150 152 — 2.95 · 10−7

Cavity 1 11 1.51 · 10−4 8.96 · 10−6

2 16 1.54 · 10−4 5.05 · 10−5

5 31 1.52 · 10−4 4.19 · 10−5

50 256 1.55 · 10−5 3.07 · 10−5

100 506 1.66 · 10−4 7.59 · 10−5

2500 2502 — 4.44 · 10−6

NS-LQR 2500 2501 — tLQR = 6.55 · 10−5

6 Conclusion and outlook

In this work, we discussed the application of the Koopman-linear MPC framework, firstproposed in [1], for data-driven control of nonlinear flows. The key idea is to approximate theKoopman operator from data to obtain finite-dimensional linear systems that approximatethe nonlinear global evolution of the system and use these systems as the predictor in themodel predictive control framework. The combination of Koopman-linear representation ofthe dynamics and MPC leads to a convex quadratic programming problem that is solvedat each time step; this is accomplished using highly efficient and tailored solvers for linearMPC. Moreover, the proposed framework is based solely on data and therefore robust touncertainties and errors in available models of the nonlinear system. In the problems consid-ered in this work, the Koopman MPC framework showed superior performance compared tofeedback strategies based on local linearization and with sub-millisecond computation time.

An important direction for the future work would be to optimize the data collection pro-cess to obtain more accurate and efficient Koopman linear models. This requires addressingtwo problems: first, finding efficient methods for sampling the extended state space of thenonlinear system; in this work we used random initial condition and random input sequencesin the domain of interests to generate data for the EDMD algorithm. The second problem isidentifying observables that provide the best finite-dimensional approximation of the Koop-man operator in the space of observables. Using machine learning techniques (e.g, [57, 58])combined with sampling approaches (e.g., [59]) within the Koopman-MPC framework couldautomatize the choice of observables as well as improve control performance.

Acknoweldgements

The authors would like to thank Dr. Sebastian Peitz for a constructive exchange of ideason the subject. This research was supported in part by the ARO-MURI grant W911NF-17-1-0306, with program managers Dr. Matthew Munson and Dr. Samuel Stanton. The

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research of M. Korda was partly supported by the Swiss National Science Foundation undergrant P2ELP2165166.

Appendix: Model-based optimal control of cavity flow

In this section, we describe the design of LQR controller for lid-driven cavity flow basedon linearization around the steady solutions. Consider the Navier-Stokes equation in (35)written as

∂

∂tEψ = f(ψ, u) (40)

and let ψ0 be the fixed-point solution corresponding to the input u0, i.e., f(ψ0, u0) = 0. Thelinearized Navier-Stokes equations around this equilibrium is given by

∂

∂tEψ = Aψ (41)

with

E : = ∇2(·),

A :=1

Re∇4(·)− ∂

∂z2

(·) ∂

∂z1

∇2ψ0 +∂

∂z1

(·) ∂

∂z2

∇2ψ0 −∂ψ0

∂z2

∂

∂z1

∇2(·) +∂ψ0

∂z1

∂

∂z2

∇2(·) .

and ψ is stream function in the linearized equations. Similar to (37), the control input tothe system is the amplitude of the top velocity lid, which results in the following boundaryconditions,

ψ

∣∣∣∣∂Ω

= 0,∂ψ

∂n

∣∣∣∣z1=±1 or z2=−1

= 0, and∂ψ

∂z2

∣∣∣∣z2=+1

= u(t)(1− z21)2, (42)

with u as the deviation from the base input u0.In order to transform the boundary control problem into the standard linear time-

invariant (LTI) format, we introduce the extension function

H(z1, z2) =1

4(1− z2

1)2(1 + z2)2(z2 − 1), (43)

and use the change of variables

η(z1, z2) = ψ(z1, z2)−H(z1, z2)u(t), (44)

The linear system in the new variable reads

∂

∂tEη = Aη + AHu− EHdu

dt, (45)

with homogeneous boundary condition,

η

∣∣∣∣∂Ω

=∂η

∂n

∣∣∣∣∂Ω

= 0. (46)

20

The above is in fact an LTI descriptor system which can be written as

∂

∂t

[E 00 1

] [ηu

]=

[A AH0 0

] [ηu

]+

[−EH

1

]du

dt.

or in a more compact form,

Ex = Ax + B ˙u, (47)

where x = [η, u]> is the embedded state. We are interested in finding the optimal inputdu/dt (and u) for the above system that minimizes the cost function,

J(x, u) =

∫ ∞0

[ ∫Ω

|v|2dA+ α1u2 + α2( ˙u)2

]dt (48)

where v = (∂ψ/∂z2,−∂ψ/∂z1) is the velocity field of the linearized system.We have used the Chebyshev collocation scheme [60] to spatially discretize the linear

system in (47) and the cost function in (48) . We have chosen α1 = α2 = 10−6 to minimallypenalize the input and avoid infinitely large solutions. If the linear system is stabilizable (forexample in the case of steady solution at Re = 10000), solving the continuous-time algebraicRiccati equation (ARE) (see e.g. [61] for descriptor formulation of ARE), would give theoptimal feedback gain k = [kη ku]. The LQR optimal input u is then computed by timestepping the following ordinary differential equation

˙u = −kuu− kηη = −kuu− kη(ψ −Hu),

and the input u = u0 + u is applied to the nonlinear system. If the fixed point is not linearlystabilizable, such as the steady solution at Re = 13000, then ARE does not have a solutionand there is no stabilizing input.

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