On Commutation of Reduction and Control: Linear FeedbackControl of a von Karman Street

Imran Akhtar, Jeff Borggaard, Miroslav Stoyanov and Lizette ZietsmanInterdisciplinary Center for Applied Mathematics

Virginia Tech, Blacksburg, VA 24061, USA

Design of feedback controls for distributed parameter systems in fluid flows remains a formidable task forresearchers. One popular approach is “reduce-then-contol” which has been successfully implemented by firstdeveloping a reduced-order model and then applying the control theory. However, this approach has severaldrawbacks, such as ensuring the unknown feedback functional gains are well represented in the reduced-basis.Control of vortex shedding past a circular cylinder has become a canonical problem to test or validate a flowcontrol design. Control of the von Karman vortex street has many applications, e.g. when designing oceanplatforms–as the vortex shedding induces an oscillatory fatigue load on structural components. In this paper,we follow “control-then-reduce” approach to control von Karman vortex street (periodic shedding) in a chan-nel using cylinder rotation as the actuation. The approach is to linearize the Navier-Stokes equations aboutthe desired (unstable) steady-state flow and design the control for the regulator problem using distributed pa-rameter control theory. The Oseen equations are discretized using finite element methods and the resultingLQR control problem requires the solution to algebraic Riccati equations with very high rank. The feedbackgains are computed using model reduction in a “control-then-reduce” framework. Model reduction is usedto efficiently solve both Chandrasekhar and Lyapunov equations. The reduced Chandrasekhar equations areused to produce a stable initial guess for a Kleinman-Newton iteration. The high-rank Lyapunov equations as-sociated with Kleinman-Newton iterations are solved by applying a novel model reduction strategy. Numericalresults for a 2-D cylinder wake problem at a Reynolds number of 100 demonstrate that this approach workswhen perturbations from the steady-state solution are small enough. In this study, we map the functional gainscomputed for the flow past a cylinder in a channel onto an exterior flow past a cylinder to analyze the perfor-mance of the controller. We present numerical results for various flow conditions to test the robustness of thecontrol mechanism.

I. INTRODUCTION

Control of flow separation and bluff body wakes has its fundamental significance in flow physics and its practicalimportance in aerodynamic and hydrodynamic applications. Since Roshko1 measured the vortex shedding periodbehind a bluff body, many researchers have investigated this phenomenon experimentally and numerically for a widerange of Reynolds numbers. The flow behind a circular cylinder has become the canonical problem for studying suchexternal separated flows.2–7 Flow-induced vibrations are a direct consequence of the fluctuating hydrodynamic forces(lift and drag) on the cylinder. These fluctuations may have adverse effects on the performance and life of a structure.Control of the flow over a circular cylinder has received considerable attention because of its canonical nature andbeing a typical unstable flow. There are various control mechanisms, which have been employed to suppress vortexshedding on a circular cylinder. These methods include cylinder rotation, transverse motion, blowing/suction on thecylinder surface, and acoustic actuation.

The control of flows by cylinder rotation has a long history in fluid dynamics (cf. Prandtl8). The feasibility ofusing rotary oscillation to stabilize the cylinder wake has been shown in several experimental9–12 and numerical13–15

studies. This has motivated numerous feedback control studies, e.g.11, 16–19

Significant progress has been made in developing various control strategies; however, the control of fluid flowremains an active field of research. Due to the inherent nonlinearity in the Navier-Stokes equations and complexityof infinite-dimensional flow dynamics, design and application of a control system still remains a challenging task. Inorder to reduce the complexity of the governing equation, low-dimensional models are often developed. Several recentworks consider a “reduce-then-control” approach to the problem. For example, Lee et al.19 proposed a phenomeno-logical model based on a van der Pol oscillator to design a feedback control. A more direct approach to develop

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open-loop or closed-loop controllers for nonlinear equations is proper orthogonal decomposition (POD)-based modelreduction.17, 18, 20–34 The POD-Galerkin control methodologies are constructed typically in three steps: (a) computationof the POD eigenfunctions from the data ensemble of the flow field and (b) Galerkin projection of the Navier-Stokesequations onto a space spanned be a small number of POD eigenfunctions including a control mechanism, and (c)build a feedback control law for the reduced-order model, then apply this law to the full order system However, thisrequires care in selecting the reduced-basis since it is typically also used to represent the feedback control law (knownas a functional gain35).

Our approach is to linearize the Navier-Stokes equations about the steady-state solution and develop a feedbackcontrol law in the form of a functional gain that prescribes the cylinder angular velocity to stabilize the (otherwiseunstable) steady-state solution. This approach is motivated by the Hartman-Grobman theorem and will stabilize manynonlinear systems.36 Instead of resorting to the “reduce-then-control” methodology that is typically applied to theseproblems, we use a “control-then-reduce” methodology. We first discretize the linear Oseen equations and seek toapproximate the solution to the full-order Riccati equations (the action of the solution matrix on the control inputvector).

Following Banks and Ito,37 we use the Chandrasekhar equations38–43 (on a related problem) to compute a stableinitial guess and use a Kleinman-Newton algorithm to solve the Riccati equations.44–47 The size of the discretized flowproblems prohibit direct solution of either the Chandrasekhar equations or the Lyapunov equations that result from theKleinman-Newton iterations. Thus, our approach is to apply model reduction methods directly towards the solution ofthese equations (or of the functional gain in the Riccati equation case). This leverages our recent work on algorithmsfor Chandrasekhar equations48 and high rank Lyapunov equations.49, 50

In Section II, we provide a short survey of our approaches for Chandrasekhar and Lyapunov equations. Followingthat survey, we introduce the von Karman vortex street stabilization problem, including two objective functions thatwill be used to pose the linear quadratic regulator problem. Numerical results are provided for the flow past a circularcylinder in a channel in Section III. In Section IV, we test the “robustness” of the control by mapping the functionalgains for the interior flow onto an external flow past a cylinder. We vary the initial conditions and also perturb thefreestream to analyze the control effectiveness.

II. CONTROL-THEN-REDUCE

The control problem of interest has the following form. Find a control u(�) that solves

minu

Z 10

�xT

1 (t)Qx1(t) + uT (t)Ru(t)dt (1)

with Q = QT � 0, R = RT > 0, subject to"E11 00 0

#"_x1(t)_x2(t)

#=

"A11 DT

D 0

#"x1(t)x2(t)

#+

"B1

0

#u(t);

where E11 has full rank. This matches the form of the discretized Oseen equations where x1 and x2 are the velocityand pressure fluctuation terms, respectively. The matrixE11 is the velocity mass matrix,A11 contains the diffusive and(non-normal) convective terms,D is the discretized divergence operator withDT the discretized gradient operator. Theterm B1u(t) represents the influence of the boundary control term. Note that this is a differential algebraic equation(DAE, also known in the control literature as a descriptor system, e.g.51). Our control input does not affect the massin the system, thus it does not feature in our discretized system above (B2 = 0). The discussion below is worked outfor DAEs with the above structure in,50 but to simplify the discussion here, we will restrict our attention to the purelydifferential case, E = E11 = I .

Note that the cylinder rotation that stabilizes the Oseen equations is described by

u(t) = �Kx1(t) (2)

where K = R�1BT � and � is the symmetric positive definite solution to Algebraic Riccati Equation (ARE)

AT � + �A��BR�1BT � +Q = 0:

This is a large ARE where matrices A, Q, and thus � are high rank. However, the number of control inputs m istypically small (m = 1 in our case). This structure makes the problem of finding K in (2) tractible. While � 2 Rn�n

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with n � 1, K 2 Rm�n. Thus, we can feasibly seek KT , the action of � on the vector (BR�1). We will furtherassume that while matrices A and Q may be high or full rank, products of the form Av, AT v and Qv for vectors v canbe computed efficiently. This is frequently the case for sparse matrices associated with the discretization of PDEs.

Large AREs are typically solved by Kleinman-Newton algorithms. These algorithms produce a sequence of largeLyapunov equations that need to be solved. However, the Kleinman-Newton iteration also requires a stable initialguess. In other words, either a �0 or a K0 must be provided such that

A�BR�1BT �0 < 0 or A�BK0 < 0;

respectively. We compute the stable initial guess by solving the Chandrasekhar equations for a low-rank Q, cf.37

Nearly all algorithms for large Lyapunov equations assume that Q is low-rank, see e.g.52–57 Our Q and thus �will be high-rank, but we will exploit the fact that B and thus K is low-rank to obtain an approximation by modelreduction methods.

In the remainder of this section, we will provide a short discussion of the reduce-then-control approach as well asan example where this approach fails to produce an accurate solution. We then summarize our methods for solvingChandrasekhar48 and high-rank Lyapunov equations.49, 50

A. A Limitation of Reduce-Then-Control

A popular approach for solving distributed parameter LQR control problems is the reduce-then-control approach.Namely, a pair of reduced bases V;W 2 Rn�r are selected which approximate the state variables well and producethe best input-output characteristics of the high-order system.Thus, given the system

E _x(t) = Ax(t) +Bu(t) (3)y(t) = Cx(t); (4)

we represent the state x(t) 2 spanfV g, x(t) � V xr(t), then define the reduced-order system�WTEV

�_xr(t) = Arxr(t) +Bru(t)yr(t) = Crxr(t)

where Ar = WTAV , Br = WTB, Cr = CV , and Qr = V TQV (often Qr = CTr Cr). The LQR problem is

easily solved for this reduced-order system and u(t) = �Krxr(t) is used as the controller for the full-order systemu(t) = �KrV V

Tx(t). The challenge is to compute a suitable pair of reduced spaces.

B. Chandrasekhar Equations

The Chandrasekhar equations were introduced in the control literature in the 1970’s to solve Riccati equations (cf.38–40).In the 1980’s, they were used as a methodology for solving the infinite horizon control problem when the system in-volves a distributed parameter system (cf.41–43). The Chandrasekhar equations have the form

� _K(t) =R�1BTL(t)LT (t); K(0) = 0 2 IRm�n (5)

� _L(t) = (A�BK(t))TL(t); L(0) = CT 2 IRn�p: (6)

The solution to the control problem (1)–(4) is then given by

K = limt!�1

K(t):

This approach replaces the need to find the (dense) n�n solution to the ARE by the integration of (m+p)n equationsbackwards in time towards a steady state solution. A significant advantage of this approach is that A does not need tobe stable for the Chandrasekhar equations to converge. All that is required is that (A;B) be a controllable pair.

While the reduced storage costs of the Chandrasekhar equations make some large problems tractible that may notbe otherwise solvable, the slow convergence towards a steady state solution magnify the computational costs associatedwith the integration. The methodology we proposed in48 uses an efficient numerical integration of the system (5)–(6)given in37 until K(t) begins to converge over [T; 0] (T < 0). At this point, we perform model reduction on (6) andintroduce a reduced problem to continue integration backwards from t = T . We observed that when A � BK(t) is

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stable, the high frequency components of L are quickly damped and effective model reduction on (6) can be carriedout. We find an empirical basis for L using the proper orthogonal decomposition (POD)58 (introduced in the controlliterature under names such as principal component analysis59 and reduced-basis methods,27, 28, 60 among others) on aninput collection consisting of stored values of the trajectory L.

Finally, we approximate the solution L in the orthogonal basis generated by columns of V ,

L(t) = V c(t) where c(t) 2 IRr:

Thus, (5)–(6) have the form

� _K(t) = R�1BTV c(t)cT (t)V T (7)

� _c(t) = V T (A�BK(t))TV c(t) (8)

from final conditions K(T ) and c(T ) = V TL(T ). These equations could be integrated directly since the equationthat needs to be handled implicitly, (8), is low-dimensional. It may also be advantageous to introduce a new variable� = KV in (7), although introducing a new level of approximation.

The derivation of the Chandrasekhar equations introduces the factorization

� _�(t) = L(t)LT (t)

into the differential Riccati equation. Thus,

� = limt!�1

�(t) =Z 0

�1L(t)LT (t) dt

using the final condition �(0) = 0, and

� =Z T

�1L(t)LT (t) dt| {z }

�res

+Z 0

T

L(t)LT (t) dt: (9)

The last term above is integrated using the Chandrasekhar equation, while we approximate the first integral using alow-dimensional basis for L. The expression (9) above shows a clear connection between finding a good basis for Land a good basis for �res.

However, as in,37 we will consider the associated Riccati equation for the “tail” of the integration. If we write

K = K(T ) +Kres; (10)

then Kres = R�1BT �res and �res is the solution to

(A�BK(T ))T �res + �res (A�BK(T ))��resBR

�1BT �res + L(T )L(T )T = 0:

However, instead of using a Kleinman-Newton iteration on this Riccati equation, we project this onto V . Define P asthe solution to

~ATP + P ~A� P ~BR�1 ~BTP + c(T )c(T )T = 0: (11)

where ~A = V T (A�BK(T ))V and ~B = V TB. Since r is small, the solution of (11) for P is easy to obtain. Ourapproach is summarized below.

Algorithm II.1 (Long-Time Integrator for (5)–(6)) Given A, B and C.

1. Integrate (5)–(6) until time T < 0, store K(T ).

2. Build a reduced-basis V for L(t), t < T .

3. Solve (11) for P , set ~Kres = R�1BTV PV T .

4. Use (10) to obtain K.

Steps 3) and 4) above are mathematically equivalent to integrating (7)–(8) from t : T ! �1. However, the smallsize of the Riccati problem (11) removes the need to integrate the reduced systems (7)–(8). Borggaard and Stoyanov48

successfully implemented this algorithm in approximating K for the 2D advection-diffusion-reaction problem.

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C. High Rank Lyapunov Solver

We turn our attention to the algebraic Riccati equation

AT � + �A��BBT � +Q = 0:

where the state weighting matrix Q has high rank. A Kleinman-Newton approach will be used to find the feedbackgains. Assume that an initial guess K0 is given (or computed as above) such that (A�BK0) is stable. Iterates of thecorrecting Kleinman-Newton algorithm45, 61 solve

ATn�1�n + �nAn�1 +KT

n�1Kn�1 +Q = 0; (12)

where An � A � BKn, Qn � KTnKn + Q, and the next iterate can be computed as Kn = BT �n. Note that under

suitable conditions,44 Kn ! K. Thus, solving Lyapunov equations (12) with high rank matrices An and Qn is acomputational challenge that must be overcome to apply our control approach. We emphasize that B is assumed to below rank and KT

n = �nB is computed by finding the product of the solution �n with a few (or one) vectors b.Note that �n is symmetric positive definite and is generally dense. We will make use of the identity

�n =Z 1

0

eATn tQne

Ant dt:

Let z(t) = eAntb, then_z(t) = Anz(t); z(0) = b;

and �nb =R1

0eAT

n tQnz(t) dt.Define

�(t) = �nz(t) =Z 1

0

eATn sQne

Ansz(t) ds;

thus we are looking for �(0) = �nz(0) = �nb = KTn . Note that �(t) =

R1teAT

n (s�t)Qnz(s)ds, so � solves

� _�(t) = ATn�(t) +Qnz(t): (13)

We introduce a similar approach as before (cf.49, 50), integrate z(�) in time until t = T , when good model reductioncan be performed on the system. Thus, we find V such that z(t) � V zr(t) where zr solves _zr = Arzr from T . Wethen seek an approximation of �(T ) such that we can integrate the adjoint equation (13) back to t = 0. The finalcondition for � can be approximated as follows

�(T ) =Z 1

0

eATn sQnz(s+ T )ds

�Z 1

0

eATn sQnV zr(s+ T )ds

=Z 1

0

eATn sQnV e

Arszr(T )ds

= Szr(T )

where S is the solution to the mixed order Sylvester equation

ATnS + SAr +QnV = 0:

This can be solved with a straightforward modification of a Smith iteration.50, 62, 63 We summarize the algorithm below.

Algorithm II.2 (High Rank Lyapunov Solver for (12)) Given An and Qn.

1. Integrate z(t) until T and save the history.

2. Obtain a reduced-basis V for z(t), t > T by POD.

3. Form the reduced order system _zr(t) = Arzr(t).

4. Solve the mixed high and low order Sylvester equation

ATnS + SAr +QnV = 0:

5. Integrate � _�(t) = ATn�(t) +Qnz(t) over [0; T ] using �(T ) = Szr(T ) and values of z(�) stored in step 1).

6. �(0) � �nb = KTn .

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III. FLOW CONTROL PROBLEM

We now return to the main motivation for this paper: feedback control of Navier-Stokes equations. We adopt themethodology “control-then-reduce”, explained in Section II, to develop an optimal control law. In other words, wewill compute the gains by employing reduction techniques in solving the Riccati equation.

Consider the Navier-Stokes equations given by

@

@t~v = �r � � (~v)� ~v � r~v �rp; (14)

0 = r � ~v: (15)

over the domain = f(�; �) 2 (�5; 5)� (�5; 15)g n

�(�; �) : �2 + �2 � 0:5

shown in Figure 1. The fluid enters the domain on the left and leaves out the right. A quadratic velocity profile isspecified on the left of the domain and zero stress on the right boundary. No-slip homogeneous boundary conditions onthe top and bottom walls. The Reynolds number based on cylinder diameter is 100 (� = 0:01), well above the criticalvalue for vortex shedding. We seek to stabilize the steady-state solution using Dirichlet boundary control tangent to thecylinder surface. The control can be physically implemented by spinning the cylinder and the control input u(t) 2 Ris the tangential velocity on the cylinder. For details of the implementation, see.50

u(t)

Figure 1. Computational mesh for rotating cylinder flow.

We discretize (14)-(15) on the mesh given in Fig. 1. This results in a non-linear differential algebraic system of27; 085 equations. The steady-state solution is computed by finding the root of the right hand side of (14)-(15) and isplotted in Fig. 2. At this value of �, the steady-state is unstable. If we simulate the flow from a zero initial condition weobserve that the flow appears to approach the equilibrium, see Fig. 3(a), however, it eventually converges to a periodicorbit corresponding to counter rotating vortices that are alternately shed from the cylinder as seen in Fig. 3(b). Ourobjective is to stabilize the steady-state equilibrium and thus steer the flow to the profiles in Fig. 2. This should beachieved by specifying the tangential velocity of the fluid at the cylinder wall.

We linearize Navier-Stokes around the equilibrium and then discretize the variational form of the problem. Weuse the Taylor-Hood finite element pair, with quadratic elements for the velocity profile and linear elements for thepressure. The resulting discretization of the Oseen equations has the form of a linear DAE. The structure matches thatof the equations considered in Section II:"

E11 00 0

#"_x1(t)_x2(t)

#=

"A11 DT

D 0

#"x1(t)x2(t)

#+

"B1

0

#u(t);

where E11 2 Rn1�n1 is symmetric positive definite, A11 2 Rn1�n1 , B1 2 Rn1�m, D 2 Rn2�n1 and vector functionsx1(�), x2(�) and u(�) have lengths n1, n2 and m respectively. In the above DAE, x1(t) is the vector with nodal valuesof the deviation of the velocity from Fig. 2 and x2(t) contains the nodal values of the pressure fluctuation.

Our objective is to specify u(�) such that kx1(t)k22 ! 0 as t!1. However, we also need to consider an auxiliarygoal that leads to a low rank control output to apply our control design approach outlined in Section II. Thus, we alsoconsider the minimization of �Z

q � ~vd�2

; (16)

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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Figure 2. Unstable Steady-State Flow.

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

(a) t = 40

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

(b) t = 100

Figure 3. Streamwise velocity component.

where the weight q is defined as

q(�; �) =

((1; 1); (�; �) 2 [1; 15]� [�5; 5](0; 0); otherwise.

(17)

If we introduce finite elements to discretize (16), we obtain a discrete output y(t) = Cx1(t).The control problems we consider seek to minimize

J(u(�)) =12

Z 10

hx1(t); Qx1(t)i+ hu(t); Ru(t)i dt; (18)

subject to the linear DAE found from discretizing the Oseen equations. Note that we will consider the rank one stateweighting matrix Q = CTC when solving the Chandrasekhar equations for a stabilizing gain or the finite elementmass matrix Q = E11 for our desired (high-rank) control objective. We set the control weight R = 1.

A. Chandrasekhar DAE Applied to Navier-Stokes

The control problem given by the linear DAE system and (18) presents different challenges from the advection-diffusion-reaction equation considered earlier. A number of modifications to Algorithm II.1 are needed to handle theDAE, most notably the replacement of the ODE (6) by a DAE for L. Details of these modifications are provided in.50

Our algorithm is applied to the problem with a rank one state weighting matrix. Since the term (A � BK(t))in the Chandrasekhar equations is unstable for initial times, the integration needed to continue until T = �60. Thesystem (A � BK(�60)) is stable. Further integration from T = �60 to T = �150 only affected the value of

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K(T ) by � 3%. The additional POD correction for the tail of the system only affected the 5th significant digit.The resulting streamwise and normal components of the functional gain, denoted by Kc, are shown in Fig. 4, where

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

(a) Streamwise component (Kcx)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

(b) Normal component (Kcy)

Figure 4. Velocity Fluctuation Gain from Chandrasekhar Equations.

subscripts indicate the streamwise and normal components and superscripts indicates the method by which the gainsare computed, i.e. Chandrasekhar (c).

B. Lyapunov DAE Applied to Navier-Stokes

The Kleinman-Newton iteration is used to compute the linear feedback gain from the stabilizing initial guess Kc

computed in Section A. Note that the gain Kc stabilizes the linear DAE, and even though it corresponds to a differentweight CTC, it will suffice as an initial guess for a Kleinman-Newton iteration for an LQR problem with weight E11.

At each step, the Kleinman-Newton iteration requires the solution to a Lyapunov equation. We use AlgorithmII.2 with modifications described in50 to handle the DAE case. For the first iteration, we were required to integratethe equation for z(t) in the Lyapunov solver to T = 20 before a suitable reduction could be computed. The methodconverged in 7 iterations with kKl

7 �Kl6k(L2)2 < 10�10. The solution is plotted in Fig. 5 and shows the streamwise

and normal component of the functional gain, where subscripts indicate the streamwise and normal components and

-25 -20 -15 -10 -5 0 5 10 15 20 25

(a) Streamwise component (Klx)

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

(b) Normal component (Kly)

Figure 5. Velocity Fluctuation Gain from High-Rank ARE.

superscript l indicates the Lyapunov method to compute gains.The gain corresponding to the full rank weight is notably different from the low rank weight. The gain in Fig. 5

is larger in norm and has a different structure, most notably around the walls of the cylinder. Note that, in general,minimizing y(t) in (16) does not guarantee that the velocity fluctuation x1(t) is reduced.

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C. Local and Global Stabilization for the Control

We test the algorithms by numerically implementing the control within the dynamical system. The control input iscomputed from the control law (2) using the gains obtained from “control-then-reduce” approach.

In both cases of low- and high rank, the linear gains stabilizes the system in a (small) neighborhood of the equi-librium of the nonlinear Navier-Stokes system. When we simulate the flow starting from zero initial conditions, andusing the feedback control, we observe that the solution converges to the equilibrium. It also converges for a set ofinitial conditions that lie close to the equilibrium. It is important to note that the steady-state or equilibrium point towhich the flow is stabilized is in fact lies in the unstable manifold of the full Navier-Stokes solution and correspond tounstable equilibrium point. Figure 6 shows the snapshots of the streamwise velocity component in which the flow isinitialized with zero velocity in the flow field and it reaches the desired output.

(a) (b)

(c) (d)

Figure 6. Snapshots of streamwise velocity components with control actuation and zero flow field as the initial condition.

However, if the flow field is initialized with von Karman vortex street, the linear gains are unable to stabilize theflow to the the steady-state. In this case, the solution converges to a different limit cycle with a flow pattern closer tothe steady state pattern as shown in Figure 7. Thus, we find that the output is sensitive to initial conditions and thesolution may lie in a different manifold. The question as to whether it is possible to achieve global stabilization forthis control mechanism remains open.

IV. “ROBUSTNESS” OF GAINS

In this section, we apply the control law developed for the flow past a cylinder in a channel to the flow past acylinder in open flows. In other words, we test the “robustness” of the computed functional gains on an exterior flowpast a circular cylinder. The distribution of the functional gain suggests that the most of the activity lies in the wake ofthe cylinder. Thus, we map the gains, obtained for the high-rank case, computed for the flow past a circular cylinderin a channel (Kl) to the external flow past the cylinder as

Kext =

(Kl; (�; �) 2 [�0:2; 14]� [�4; 4]0; elsewhere.

(19)

In addition, we use the time-average flow field of the exterior flow over a shedding cycle, shown in Fig. 8, to computethe velocity fluctuations.

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(a) (b)

(c) (d)

Figure 7. Snapshots of streamwise velocity components with control actuation and von Karman vortex street as the initial condition.

We then compute the control input as

u = �Z

Kextx (vx � �vx) +Kext

y (vy � �vy)d (20)

where Kextx and Kext

y are the mapped functional gains in the streamwise and normal directions.

(a) Streamwise component (�vx) (b) Normal component (�vy)

Figure 8. Time average of the exterior flow field ~v = (vx; vy).

We validate the effectiveness of the control by initializing the flow field from two different initial conditions: zeroflow field ~v(x; 0) = 0 and potential flow field ~v(x; 0) = V pot. The control input computed for the two cases areplotted in Fig. 9. In case of zero initial condition, the control input starts out from zero and gradually reached a steady-state behavior. In other words, the cylinder starts rotating gradually and achieves a steady rotation about a zero mean.However, in case of the potential flow, the cylinder starts oscillating vigorously for a short period of time and thenreaches the same steady state control as in the case of zero initial condition. High magnitude transient response is dueto the absence of separation bubble in the wake of potential flow solution. As the simulation starts, the control rotatesthe cylinder with large magnitudes to speed-up the viscous separation leading to create the wake and then graduallysettling to a periodic control input. It is interesting to note that the output of both initial conditions does not convergeto the desired steady-state solution, rather it settles to the limit cycle solution obtained for the initial condition of vonKarman vortex street for the internal flow.

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Time

Control

10 20 30 40 50

-1

0

1

Potential flow

Zero IC

Figure 9. Rotational velocity of the control input.

To further investigate the effectiveness of the gains computed on the exterior flow, we alter the angle of incidenceof the freestream. We consider two cases of angle of incidence: AOI = 5� and 10�, and compare it to the resultsobtained for the zero AOI . We initiate the flow field with ~v(x; 0) = V pot. We compute the responses of the controlmechanism and compare the steady-state control input in Fig. 10 along with the schematic of AOI in the freestream.In all the three cases compared, the flow field settles to the same limit cycle solution obtained in the previous cases.However, there is a constant shift or a bias in the control input when the freestream has an angle. The shift or biasis more for AOI = 10� than that for AOI = 5� although peak-to-peak magnitude and phase of the control inputsignal remains almost constant. The results obtained are also intuitive and can be explained by the negative shift in thesignal which corresponds to more clockwise rotation of the cylinder. The shift in the cylinder rotation in the clockwisedirection tries to streamline the flow towards the desired output. The results indicate that the control input, although

Time

Control

180 185 190 195 200

-1

0

1

0o5o10o

Figure 10. Rotational velocity of the control input.

far from perfect, tends to include the effect of perturbations in the inflow variations in the freestream. It also showssome degree of “robustness” in the control mechanism and its capability to perform beyond the parametric space forwhich it was originally developed.

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V. CONCLUSIONS AND FUTURE WORKS

We present a feasible “control-then-reduce” approach to computing feedback control laws for complex fluid flowsystems. We considered the flow past a circular cylinder and made an attempt to control it through cylinder rotations.We believe that this problem is particularly challenging in that rotation required to relieve the negative pressure bubblein the cylinder wake responsible for the vortex shedding actually causes a negative pressure bubble on the oppositeside. Thus, it is very challenging to design a control for this specific actuation. We intend to benchmark this approachagainst other common wake stabilization strategies.

This study is in fact a step forward in implementing the control algorithm developed to solve Riccati equationand compute the gains. The response of the control mechanism in the environments for which it was not designedoriginally highlights some degree of robustness in the control. There are certainly a number of remaining questionsthat will be answered in future works.

In future, we will include the lift-coefficient within our cost function and will design the control to reduce thefluctuation forces on the cylinder. Finally, we intend to extend this study to three dimensional flows. The fact that thiscontrol mechanism is independent of the flow direction makes it an attractive active control strategy if it can be provento be effective in realistic three dimensional flows.

VI. ACKNOWLEDGMENTS

The authors gratefully acknowledge partial support from the Air Force Office of Scientific Research under grantsFA9550-05-1-0449, FA9550-07-1-0273, and FA9550-08-1-0136.

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