A data-driven Koopman model predictivecontrol framework for nonlinear flows

Hassan Arbabi, Milan Korda and Igor Mezic ∗

April 14, 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 flow control with examples isavailable at https://github.com/arbabiha/KoopmanMPC_for_flowcontrol.

1

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 costly tocontrol using well-developed strategies of modern theory. The recent advances in numeri-cal computation has led to partial success with active control of flows using models basedon first principles (i.e. Navier-Stokes equation for incompressible flows) [2, 4–6]; however,these methods suffer from two major shortcomings: first, the obtained models are still high-dimensional and computationally costly for implementation, and second, they often rely onlocal linearization around equilibria or a trajectory of the flow which makes the model validonly 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 constructionof (autonomous) state space models via Galerkin projection of the Navier-Stokes equationsonto the modes obtained by proper orthogonal decomposition (POD) of data [7–9], or iden-tification of linear input-output systems using balanced POD [10, 11]. There are also a fewapplications of system identification methods to construct linear input-output models purelyfrom data, including the eigensystem realization algorithm [12,13], as well as subspace iden-tification and autoregressive models [14, 15]. The application of the above techniques to avariety of problems has shown great promise for low-dimensional modeling and control ofcomplex 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 second,the models obtained are linear (but possibly high-dimensional) due to the fact that the Koop-man operator is a linear operator irrespective of whether the dynamical system is linear ornot. The linearity of the Koopman models is especially advantageous since it makes themamenable to the plethora of mature control strategies developed for linear systems. The con-troller design procedure follows the Koopman model predictive control framework of [1]. Inthe first step of our approach, we build a finite-dimensional approximation of the controlledKoopman operator from the data, using a variation of the extended dynamic mode decom-position algorithm (EDMD) [18], with a particular choice of observables assuring linearityof the resulting approximation. The ideal data would include measurements on a numberof system trajectories with various input sequences. In the second step, we apply the modelpredictive control (MPC) to these linear models to obtain the desired objectives. The distin-guishing feature is that a linear MPC, solving a convex quadratic programming problem, isutilized, thereby enabling a rapid solution of the underlying optimization problem, which isnecessary for real-time deployment. This methodology can be also applied to problems withsparse measurements, i.e., problems with a limited number of instantaneous measurements

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(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 works is given in sec-tion 1.1. Section 2 gives a review of the Koopman operator theory for dynamical systemswith input. In section 3, we describe the EDMD algorithm for construction of the Koopman-linear model from measurement data. In section 3.1, we discuss using delay-embedding toconstruct and control Koopman-linear models from sparse measurements. An overview ofthe model predictive control framework is given in section 4. In section 5, we present twonumerical examples: the Burgers’ system on a periodic domain and the 2D lid-driven cavityflow. We formulate the control problem for these cases using various objectives and demon-strate our approach for both full-state and sparse measurements. We discuss the results andthe outlook in 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 usingdata. 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 optimization of the switching instants between the input values. This methodologywas successfully used for tracking reference output signals in Burgers equation and incom-pressible flow past a cylinder. The work in [33] proposed to remove the restriction of theinput to a finite set, by interpolating between the Koopman-linear systems for each inputvalue which led to an improvement of the control performance. In [1] (which this work isbased on), a more general extension of the Koopman operator theory to systems with inputwas presented, and used to construct linear predictors especially suitable for model predic-tive control, demonstrating the effectiveness of the approach (among other examples) on thecontrol of the Korteweg-de Vries PDE. The results in [1] showed superiority of the controlledKoopman-linear predictors constructed from data to models obtained by local linearization

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and Carleman’s representation both for prediction and for feedback control. Let us alsomention the earlier work in [34] that utilized KMD to construct the normal forms, for dy-namics of the flow past an oscillating cylinder; the input forcing appeared as a bilinear termin the normal forms for this flow. See also [35] for the application to pulse-based control ofmonotone systems, as well as [36] to system identification and [37] to 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. However, the convergence of DMD algorithms in the context ofKoopman operator approximation was just recently shown in [38,39]. For systems with sparsemeasurements, the EDMD is modified to include the delay embeddings of instantaneousmeasurements. The delay-embedding technique is classical in system identification literature(see, e.g., [40] for a comprehensive reference) and control as well as in linear and nonlineartime-series analysis (e.g., [41]). In the field of dynamical systems, the classical referenceis the work of Takens [42] on geometric reconstruction of nonlinear attractors. The workin [43] suggested the combination of this technique with the DMD algorithm for identificationof nonlinear systems and its role in approximation of the Koopman operator was studiedin [38,44]; the use for control, in the Koopman operator context, 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 our 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 thestate 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)

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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 controlled systems

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. The extended statespace is denoted by S = M × `(U). Now, given an observable g : S → R we can define thenon-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)

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)

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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 is not required to construct the Koopman-linear model. Bycollecting data on the dynamical system, 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, i.e., thereare fewer observables than sample points), it is more efficient to compute the matrices bysolving 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 lifted state z = g(x). Theprediction 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., considering several consecutive output measurements) is a classicaltechnique ubiquitous in system identification literature (e.g., [40]) but also in the theory ofdynamical systems for a geometric reconstruction of nonlinear attractors [42]. It has alsobeen utilized in the context of Koopman framework in [20, 38, 50]. The key idea behinddelay embedding is that it provides samplings of extra observables to realize the Koopmanoperator. To be more precise, if we have a sequence of measurements on a single observableh at the nd time instants ti, ti+1, . . . , ti+nd−1, we can think of them as sampling of the nd

observables [h, Kh, . . . , Knd−1h] at the single time instant ti. Here, we describe how we canincorporate delay-embedding into identification and control of Koopman-linear models. Theonly requirement for identification is that we should have access to at least nd + 1 sequentialtime 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 eq. (9), but now assume that they containa string of sequential samples with length nd + 1, i.e., for some j, we have

x+i = T (xi, ui) = xi+1, 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 as before, we use the same process as aboveto compute the linear system matrices. It is very important for the lifting function g to havea meaningful dependence on ui, . . . , ui+nd

. This allows EDMD to approximate the dynamics

7

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

Now we can construct a linear predictor

z+ = Az +Bu

ζ = Cz, (19)

with the matrices A, B, C obtained by solving the least-squares problems (12) with Xreplaced by X. Here ζ denotes the prediction of the “embedded” state ζ (note that whenζ is employed for controller design, typically only the part of ζ corresponding to the mostrecent output prediction 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,thereby allowing for a rapid solution using highly efficient QP solvers tailored for linear MPCapplications (in our case qpOASES [53]); see [1] for details.

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 follows 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 mainly 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. The closed-loop operation is summa-rized in Algorithm 1. The entire algorithm, i.e., the predictor construction (identfication)and closed-loop operation, is summarized in fig. 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?0 to the nonlinear system

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Figure 1: Summary of the predictor design procedure (identification) and closed-loop control.

5 Numerical Examples

5.1 Burgers equation 2

As the first example, we consider the Burgers equation which is a one-dimensional PDEwith advection-diffusion dynamics 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 which determines the relative strength of the diffusion toadvection. Note that we have used z to denote the spatial coordinates in the flow examples,hoping that it will not be confused with the Koopman-linear state in eq. (8). We use the

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

KoopmanMPC_for_flowcontrol.

10

control setup considered in [32] and 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 (u1, u2) ∈ R2. To construct the Koopman-linear system, we have used50 two-second long trajectories with ν = 0.01. The initial conditions are convex combinationsof Gaussian and Sinusoidal profiles,

v(z, 0) = a1e−(

5(z−0.5))2

+ (1− a1) sin(4πz). (29)

where a1 ∈ [0, 1] is randomly chosen for each trajectory. The input control at each timeinstant is randomly drawn from the uniform distribution on (u1, u2) ∈ [−0.1, 0.1]2. Thecomputational scheme for the system evolution consists of upwind finite-difference schemefor advection and central difference for diffusion term, with 4th-order Runge-Kutta timestepping performed on 100 spatial grid points with time steps of 0.01 second.

The objective of the control for the system would be 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, (30)

given a random initial condition described in eq. (29) with the input signals constrained as|u1|, |u2| < 0.1.

Full-state measurements: As Koopman observables, we use the vector of values ofv at the computational grid points, the kinetic energy of v, and the constant observable(ψ(v) = 1). The cost function to be minimized is the kinetic energy (L2-norm) of the statetracking error,

e(t) =

∫ 1

0

|v(z, t)− vref(z, t)|2dz. (31)

Sparse measurements: We assume that we only have access to the vector of sparsemeasurements ys which consists of values of v at m = 10 random grid points. We form theKoopman-linear state vector by including delay embedding of ys with nd = 5 delay times,the kinetic energy of instantaneous output ‖ys‖2 and the constant observable (the dimensionof the Koopman-linear state space is 5× 10 + 4× 2 + 1 + 1 = 60). The tracking error, to beminimized is

e(t) =1

m‖ys(t)− ysref(t)‖2. (32)

The predicted objective function used within the MPC is then

J =

∫ T

0

e(t) dt,

11

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, we observe that withsparse measurements the controller is slightly delayed compared to the case of full-statemeasurement which can be attributed to the construction of the Koopman-linear systemstate using delay-embedding. We note that the tracking error during the transient phase iscaused by input saturation as documented by the plot of the control input signal.

Figure 2: Control of Burgers’ system (eq. (25)) using the Koopman-linear models constructed based onfull-state measurements (150 observables) and sparse measurements (10 observables).

12

Robustness with respect to the parameter ν regime: One question that arises inthe context of low-dimensional modeling is wether the models constructed at some parametervalue would be robust enough for prediction at other values. In order to test the robustnessof Koopman-linear model in case of Burgers, we apply the model constructed above usingsparse measurements to various values of ν ∈ [10−4, 0.1]. The results in fig. 3 indicates thatKoopman-linear model constructed at the parameter regime ν = 0.01 shows a remarkablerobustness in a wide parameter range. As expected, the input signal and tracking error in thediffusion-dominated regime (large ν) is less fluctuating, as the diffusion helps the controllerto stabilize the state around the spatially-uniform reference state in (30).

Figure 3: Robustness of Koopman-linear system constructed at ν = 0.01 for controlling the flow at variousvalues of ν.

5.2 2D lid-driven cavity flow

In the second example, we consider the flow of an incompressible viscous flow in a squarecavity which is driven by motion of the top lid. The dynamics of the cavity flow is governed

13

by the Navier-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,∞), (33)

ψ

∣∣∣∣z1=±1

= ψ

∣∣∣∣z2=−1

= 0 and∂ψ

∂z2

∣∣∣∣z2=1

= f(z1, t), (34)

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, (35)

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 unstableand the solutions up to Re = 15000 converge to a limit cycle. More details on the dynamicsand the numerical scheme used to solve eq. (33) can be found in [54].

As the objective of control, we aim to stabilize the flow at Re = 13000 (the autonomousdynamics is limit cycling) around the fixed point solution at Re = 10000. We also bound theinput signal such that −2/13 < u < 2/13 to avoid the trivial solution given by u0 = −3/13(note that the effective Re is proportional to the top velocity and this choice of u0 setsback the flow into the fixed point at Re = 10000). The construction of the Koopman-linearsystem is similar to the Burgers system; we have used 250 two-second long trajectories of thesystem with control inputs that are randomly drawn from [−3/13, 3/13]. For the full-stateobservation, we use the values of the stream function on the 50 × 50 computational grid,the kinetic energy and the constant observable. For the case of sparse measurement, we usethe values of stream functions at k = 2, 5, 50, 100 random points inside the flow domain, thel2 vector norm of observed stream function values and the constant observable. Accordingto section 3.1, the dimension of the state space for the Koopman-linear system built fromsparse measurements will be n = 16, 31, 256, 506 respectively which is considerably smallerthan the Koopman-linear system with full-state observation (n = 2502).

Let ψref denote the steady solution at Re = 10000. In the case of the full-state obser-vations, we define the tracking error to be the kinetic energy of the flow distance from thereference state, i.e.,

e(t) = E(ψ(t)− ψref) =

∫Ω

|v(t)− vref |2dz1dz2, (36)

where v = (∂ψ/∂z2,−∂ψ/∂z1) is the velocity field. In case of sparse measurements given bythe vector of observation in ψs, the cost function for the MPC will be the l2-norm of distancefrom the state, that is,

e(t) = ‖ψs(t)− ψs,ref‖2. (37)

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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 in controlled simulationsstarting from an initial condition on the limit cycle. All the Koopman-linear systems, exceptk = 2, are successful in considerably reducing the flow distance from the desired state overfinite time. The control inputs and the flow evolution for some values of k and full-stateobservation is shown in fig. 5. In the case of full-state observation, the input signal generallytends to its lower bound which results in a lower effective Re for the flow, and hence gettingcloser to the fixed point at Re = 10000; however, the controller also uses intermittentbursts to speed up the stabilization. The effect of 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 systemsgenerally scales with the number of measured observables, i.e., larger number of observablesresults in better control performance. However, this improvement in the performance startsto saturate at a relatively low k, e.g., the full-state observation offers less than 10 percentimprovement over k = 50 in the final discrepancy. This indicates that the flow dynamics canbe approximated accurately with low-dimensional Koopman models and there is a reasonabletradeoff between the number of observables and control performance. We have observed thatthe effect of observable selection significantly affects the performance of the controller for thecase of k < 5, and results reported in the figures represent the typical behavior of controllersbuilt on sparse measurements.

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 strahgtforward linear algebra excercise can be found in theappendix of [1].

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Figure 4: Normalized kinetic energy of flow distance for cavity flow controllers built by full state and sparsemeasurements. k is the number of measurements used to build the Koopman-linear system.

Table 1: Computation time for Koopman MPC of cavity flow and Burgers equations. The symbol“—” signifies a negligible embedding time in the case of full state 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

6 Conclusion and outlook

In this work, we discussed the application of the Koopman model predictive controlframework, first proposed in [1], for data-driven control of nonlinear flows. The key idea isthat the Koopman operator acts linearly on the (infinite-dimensional) space of observableson a dynamical system. Using the EDMD algorithm [18], we computed finite-dimensionalapproximations of the Koopman operator in the form of an input-output linear system solelyfrom time-series data. The obtained Koopman-linear system was then used within a linearMPC to achieve the flow control objectives. The application of this framework to the caseof Burgers equation and lid-driven cavity flow showed considerable promise for application

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Figure 5: Discrepancy in the vorticity of controlled state, and input signal, for full-state and sparsemeasurements. k is the number of stream function measurements used to build the Koopman-linear system.The measurement locations are marked via crosses in the left column.

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to more complex flows.Future work should focus on optimizing the data collection process to obtain more ac-

curate and efficient Koopman linear models. This requires addressing two problems: first,finding efficient methods for sampling the extended state space of the nonlinear system, andsecond, identifying observables that provide the best finite-dimensional approximation of theKoopman operator in the space of observables.

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, Pro-gram managers Dr. Matthew Munson and Dr. Samuel Stanton. The research of M. Kordawas partly supported by the Swiss National Science Foundation under grant P2ELP2165166.

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