2005_Comparison of a Robust and a Flatness Based Control for a Separated Shear Flow

8/17/2019 2005_Comparison of a Robust and a Flatness Based Control for a Separated Shear Flow

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COMPARISON OF A ROBUST AND A

FLATNESS BASED CONTROL FOR A

SEPARATED SHEAR FLOW

Ralf Becker and Rudibert King 1

Measurement and Control Group, Institute of Process and Plant Technology, Berlin University of Technology,

Hardenbergstr. 36a, 10623 Berlin, Germany

Abstract: Closed-loop flow control is gaining more and more interest in the lastfew years. Whereas most of the published results are based on simulation studies,this work explores the synthesis of black-box model based closed-loop controllersfor separated, wall-bounded shear flows in experiments. In this paper we comparea linear robust controller with a flatness based approach. To consider both theuncertainty of the model and the inherent time delay, the well-known flatness basedcontroller synthesis scheme had to be extended by a robust design approach and aprediction step. The actuated backward-facing step flow is chosen as a benchmarkconfiguration, in which the length of the separated flow region is to be controlled.This configuration can be seen as a simple representation of the situation in aburner or behind a flame holder. Copyright c2005 IFAC

Keywords: robust, flatness, time delay, flow control, backward-facing step,separated flow, reattachment length

1. INTRODUCTION

Separated flows show positive and negative effectsdepending on the application. A desired recircula-

tion region as a result of a separation is needed incombustion chambers to keep the fuel mixture inthe reaction zone for a complete combustion. Byinfluencing a recirculation zone, the residence timebehaviour in a general mixing problem is altered.In contrast to that, negative consequences couldbe efficiency loss and noise production in turbo-machines and process plants, or drag increase andlift reduction for aeroplanes.

When shaping of the geometry has reached anoptimum or when passive means, such as vortexgenerators, have positive and negative effects,

1 Corresponding author: e-mail: [email protected], web: http://mrt.tu-berlin.de

active devices in open- or closed-loop control canfurther improve the performance by suction andblowing or acoustic actuation. However, most of the work published so far is dedicated to open-

loop control.

Experimental validations of closed-loop flow con-trol are still rather rare. The present investigationexplores the closed-loop control of separated flowsby active means in wind tunnel experiments toprofit from the well-known advantages, such asdisturbance rejection and set-point tracking. As abenchmark problem, the flow around a backward-facing step is investigated here. For feedback flowcontrol three different approaches concerning thecontroller synthesis can be distinguished. First,strategies based on a numerical solution of the

governing Navier Stokes equations (NSE) are pro-posed. This is done by (Hinze, 2000), for instance,who also gives a review about this field. How-


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ever, those approaches are still not applicable inreal-time. In a second promising approach, low-dimensional models based on the NSE are de-rived, with the intention to describe the nonlinearphysics and to synthesise nonlinear controllers.The use of those Galerkin- and vortex modelsfor controller synthesis and their application to

experimental setups is still in a state of research,see e.g. (Baker and Christofides, 2002), (Noacket al., 2003) and (Pastoor et al., 2003). Firstexperimental results are presented in (Glauser et al., 2004) and (Siegel et al., 2004).

Third, (Allan et al., 2000) and (Becker et al.,2005) propose low-dimensional black-box modelsfor controller synthesis as a feasible alternativefor experimental configurations. In (Allan et al.,2000) tuning rules are used for the control of ageneric model of an airfoil. Robust controllers aresynthesised and compared in simulation studies

and wind tunnel experiments in (Becker et al.,2005). This approach will be used here as well. Asurvey of several successful applications of robustand adaptive controllers for separated shear flowsis given in (King et al., 2004).

This paper is organised as follows: A descriptionof the configuration and its dynamics is givenin section 2. First, a simply structured, linearcontroller is designed employing robust methods,see section 3. However, the performance is limitedby the uncertainty of the linear design model.Second, to overcome this limitation, a nonlinearmodel that resolves the dynamical phenomenamore precisely in combination with a flatnessbased control approach is proposed in section 4.The command tracking performance of the robustand the flatness based control are compared insection 5. Finally, the results are summarised.

2. FLOW CONFIGURATION

2.1 General description

The backward-facing step flow has been estab-lished as a benchmark problem for separated flowsin fluid dynamics. A variety of information aboutthe flow process and actuation mechanisms isavailable, see for example the survey in (Beckeret al., 2005). A sketch of the flow field is givenin Fig. 1 where the oncoming flow detaches atthe edge of the step and reattaches downstream.Three different regimes exist in the wake: a sep-aration region, also called separation bubble, areattachment zone and a newly developing bound-ary layer. The separation bubble consists of the

recirculation zone and the shear layer above.

Here, the goal is to control the size of the separa-tion bubble given by the reattachment length xR.

Fig. 1. Flow field downstream of a backward-facing step

While negative skin-friction, i.e. reverse flow, oc-curs in the recirculation zone, downstream a newboundary layer develops causing positive skin-friction, i.e. forward flow. The reattachment po-sition xR is characterised by zero friction.

The shear layer above the recirculation zone isgoverned by the Kelvin-Helmholz instability phe-nomenon. Perturbations are amplified and leadto two-dimensional shear layer roll-up into span-

wise coherent vortical structures which grow andconvect downstream. Particularly the vortices areresponsible for entrainment of fluid out of thebubble. By stimulating their growth, the increasedoutflow leads to a significant reduction of thebubble size xR.

The vortex generated entrainment mechanism isenhanced by acoustic actuation of the detachingboundary layer at the edge of the step as shown inFig. 1. A slot-loudspeaker actuator is used for this.Therefore, the forcing frequency should be nearthe natural instability frequency of the system.As the forcing frequency is fixed to the optimalone, the amplitude of the harmonic loudspeakersignal finally affects the initial size of the growingvortices, and thus, the reattachment length xR. Incontrol notation the amplitude of the loudspeakersignal is taken as the control signal u(t) and thereattachment length as the plant output y(t) =xR(t).

2.2 Flow parameters

A Reynolds number of ReH = 4000 correspondingto U ∞ = 3.04 m/s giving the dimensionless free-stream velocity and a step height of H = 20mmare chosen. To compare the results with fluiddynamical investigations, all distances are madedimensionless by the step height H and all timesby the ratio H/U ∞ in the following. Details aboutthe experimental setup are described in (Becker et al., 2005).

An online measurement of the output signal xR(t)cannot be done in a straightforward manner. Inthe present investigation, wall pressure fluctua-

tions are measured by a microphone array, anda Kalman filter based scheme is used to estimatethe reattachment length xR(t). Details about thisapproach can be found in (Becker et al., 2003).


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0 50 100 150 200 250 300

5

6

7

t U∞ / H

x R /

H experimentblack−box model

00.05

u [ V ]

0 50 100 150 200 250 300

5

6

7

t U∞

/ H

x R /

H

00.05

u [ V ]

Fig. 2. Identification of black-box models fromstep responses (top: switching u(t) on; bot-tom: switching u(t) off).

2.3 Dynamical behaviour

In spite of the nonlinear and infinite dimensionalNavier Stokes equations, it is surprising that thedynamical behaviour of the reattachment lengthxR looks very simple in step experiments, seeFig. 2. It can be approximated by stable linearblack-box models GP (s) of first or second orderwith a time-delay, see again (Becker et al., 2005).Due to the nonlinear characteristics of the inves-tigated system, large ranges for the model pa-rameters have to be accepted for various step

heights of the input signal. To avoid a significantdetuning of the controller, it is desirable to re-duce the model uncertainty. As the steady-stategain K P = lims→0 GP (s) of all identified modelsshows a highly nonlinear dependence on the sizeof the control signal u, i.e. the forcing amplitude,the inverse f −1 of the stationary operating pointcharacteristics y = K P (u)u = f (u) (u = const.,t → ∞) can be used to compensate for thisnonlinearity.

In order to describe the nonlinear behaviour of thereal process, a family Π of linear black-box models

GP (s) ∈ Π is identified from representative stepexperiments, see Fig. 3. The spread of the cut-off frequencies is caused by the different dynamics of the experiments with positive, i.e. switching on,and negative, i.e. switching off, steps u(t). How-ever, the slower switching off steps and the fasterswitching on steps are clustered, respectively.

3. ROBUST CONTROL

Due to the uncertainty of the identified models,

robust controllers are required to maintain closed-loop stability for all operating points. A linearnominal model Gn(s) with a multiplicative un-certainty description

10−3

10−2

10−1

−40

−20

0

20

40

M a g n i t u d e [ d B ]

10−3

10−2

10−1

−400

−200

0

P h a s e [ d e g ]

f H / U∞

Fig. 3. Frequency responses of the model familywith compensation of the nonlinear gain.

lM (ω) = maxGP ∈Π

|(GP ( jω) − Gn( jω))/Gn( jω)|

for the neglected or non-modelled dynamics is de-

rived from the model family Π, and these are thenused for robust controller synthesis. By searchinga nominal model Gn(s) with the smallest uncer-tainty radius, it turnes out that a simple first-order transfer function Gn(s) =

K 1+sT

e−T 0s can beused, see as well (Becker et al., 2005). A Padé-approximation of second order is used for thefollowing controller synthesis.

To find a trade-off between the performance, givenby the closed-loop sensitivity function, the restric-tion of the magnitude of the plant input signals,and the robustness, the mixed sensitivity problem

is solved by H∞-minimisation to shape the closed-loop transfer functions.

The obtained H∞-controller is tested in wind tun-nel experiments. Comparing the open- and closed-loop tracking responses in the case of switchingu(t) on (see first step response after tU ∞/H = 90in Fig. 4), the closed-loop response shows a slowerbehaviour than the open-loop response. This canbe explained with the wide spread of the cut-off frequencies of GP (s) ∈ Π, see Fig. 3. Due tothe requirement of robust stability, the crossoverfrequency of lM (ω) is an upper bound of the active

control frequency domain, i.e. the closed-loop gainhas to drop below 1 for higher frequencies. How-ever, in this case the crossover frequency of lM (ω)is near the lower cut-off frequencies of the clus-tered models GP (s) that describe the switching off processes. Hence, the limited validity of the linearnominal model Gn(s), given by lM (ω), limits theachievable closed-loop performance to the case of the slowest process behaviour, i.e. the switchingoff dynamics. Thus, the closed-loop transient timeof the first switching on step in Fig. 4 is as slowas in the switching off case. The transient open-

and closed-loop velocities are the same after u(t)is switched off. The different transient time afterthe second step in Fig. 4 is a result of the differentstarting level due to noise.


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0 100 200 300 400

5

6

7

8

x R /

H

r(t)y(t) open−loop (step response)y(t) H

∞

−control

0 100 200 300 4000

0.5

t U∞

/ H

u [ V ]

Fig. 4. Comparison between the dynamics of theopen-loop and the closed-loop (linear H∞-controller) systems with step responses.

Although the linear H∞-control shows a slowbehaviour, it is superior to open-loop control in

the presence of disturbances. The feature of dis-turbance rejection is demonstrated in the experi-ments of (Becker et al., 2005).

4. FLATNESS BASED CONTROL

Since the performance limitation of the robustcontroller is imposed by the uncertainty of thelinear design model, an extended model that de-scribes the real behaviour more precisely, in com-bination with a flatness based control, are pro-posed next. The idea is to exploit the correlation

between the black-box model parameters and theflow state, and to improve the control based onthis more precise description.

4.1 Black-box model for the flatness based controller synthesis

The underlying philosophy here is that a moreprecise description can be obtained by taking thestate depending time constants for the switchingon and off processes into consideration. To do so,

the flow process is empirically described by thefollowing time delay-free system

[T + sign(ẏ∗(t))∆T ] ẏ∗(t) + y∗(t) = u(t) (1)

and the output time delay

y(t) = y∗(t − T 0) . (2)

y∗(t) is an artificial system output without timedelay. The time constant of the system is chosendepending on the sign of ẏ∗(t), i.e. the directionof motion of y∗(t). Thus, the faster responses,

after switching u(t) on, are described by the timeconstant T − ∆T , and the slower responses, afterswitching u(t) off, by T + ∆T . This accountsfor a hysteresis-type effect known for such flow

systems. Due to the compensation f −1, see above,the steady-state gain of Eq. (1) is one.

The model (1,2) approximates the process byswitching between the two first order modelsGon(s) and Goff (s) according to

Gon/off (s) = Y (s)U (s)

= Y ∗

(s)U (s)

e−sT 0

= 1

1 + (T ∓ ∆T )s

G∗on/off (s)

e−sT 0 . (3)

In fact, the time constants of the system responsesafter non-stepwise changes in u(t) are somewherebetween the values T ± ∆T . Both, the simplefirst order models and the switching between thetwo time constants T ± ∆T are simplifications of the real behaviour. On the one hand, overshoots

and oscillations cannot be described by thosePT1-models. On the other hand, the switchingcriterium, sign(ẏ∗(t)), is a simplification, too, andits evaluation is uncertain due to measurementnoise. Concluding these facts, robustness in thesense of the uncertain time constant and of anincorrect switching between T ± ∆T is desirablefor the control to follow.

4.2 Control

Roughly speaking, a system is differentially flat

if it is possible to find an output vector y =y(x,u, u̇, . . . ,u(α)) with dim y = dim u, such thatall states x and all inputs u are expressed bythose outputs and a finite number of its deriva-tives according to x = x(y, ẏ, . . . ,y(β)) and u =u(y, ẏ, . . . ,y(β+1)). Since the behaviour of the flatsystem can be parameterised by the flat output y,it is possible to plan trajectories in output space.The basic approach of a two degrees-of-freedomdesign is to separate the nonlinear controller syn-thesis problem into the design of a feedforwardtracking control, followed by a feedback control

for the stabilisation around the desired trajectoryin the presence of uncertainties and disturbances.The feedback control is designed such that thetracking error dynamics is linear.

Both the input u(t) and the internal state y∗(t)of the system (1,2) can be parameterised by y(t),i.e.

u(t) = [T + sign(ẏ(t + T 0))∆T ] ẏ(t + T 0) +

+y(t + T 0) (4)

and

y∗(t) = y(t + T 0) . (5)

Hence, the physical output y(t) contains the flat-ness property. Due to the inherent time delay T 0,


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u(t) and y∗(t) are parameterised by the futureoutput y(t + T 0). This extension of the conceptof flatness to time delay systems is proposed by(Mounier and Rudolph, 1998).

With the control law

u(t) = [T + sign(ẏ(t + T 0)

ẏ∗(t)

)∆T ] v(t + T 0)

v∗(t)

+

+ y(t + T 0)

y∗(t)

(6)

v(t + T 0)

v∗(t)

= ẏd(t + T 0)

ẏ∗d(t)

−

−q (y(t + T 0)

y∗(t)

− yd(t + T 0)

y∗d(t)

) (7)

the closed loop reads

[T + sign(ẏ∗

(t))∆T ] ẏ∗

(t) + y∗

(t) =[T + sign(ẏ∗(t))](ẏ∗d(t) − q (y

∗(t) − y∗d(t))) + y∗(t)

The superscript ∗ denotes the future variablesat t + T 0 that are predicted at time instant t.Introducing the future error e(t + T 0) = e

∗(t) =y∗(t) − y∗d(t) it follows from the above equationthat the error equation is linear in the nominalcase, according to ė∗(t) + qe∗(t) = 0. The con-troller design variable q has to be chosen positive,such that e∗(t) decreases to zero, thereby keepingthe system (1,2) around the desired trajectoryy∗d(t) = yd(t + T 0) in the presence of disturbances

and model uncertainties.

In the non-nominal case with a time-varying timeconstant T P (t) of the real system, stability of theerror equation for T − ∆T ≤ T P (t) ≤ T + ∆T can be shown. The error e∗(t) tends to zero whenẏ∗d(t) → 0 which is given in all practical problemsfor the backward facing step.

4.3 Implementation

To implement the above control law (6, 7), the

internal state y∗(t) = y(t+T 0) has to be estimatedby a prediction. Therefore, the well-known Smith-predictor structure is used, see Fig. 5. A timedelay-free model G∗(s) is simulated parallel tothe process to predict y∗(t). Depending on thesign of dŷ∗(t)/dt, G∗(s) is chosen according toG∗(s) = G∗on(s) (switching on process) or G

∗(s) =G∗off (s) (switching off process), respectively. Inthe Smith-predictor approach the nominal modelG∗(s)e−sT 0 is simulated as well parallel to theprocess. In the nominal case with d = 0, y(t)and the output of G∗(s)e−sT 0 cancel such that

the prediction y∗

(t) = ŷ∗

(t) of G∗

(s) is used asfeedback, and the control system behaves as if thetime delay is arranged after the control structureaccording to y(t) = y∗(t − T 0). One price payed

Fig. 5. Extension of the flatness based control (6,7) to predict the internal state y∗(t) = y(t +T 0) online and in real-time.

using the Smith-predictor is that the feedbacksignal consists of the prediction of the futureoutput y∗(t) = y(t + T 0) superimposed by actualdisturbances d(t) and signal components resultingfrom model mismatch, i.e. of a corrected signalŷ∗(t). This results in a performance limitationof the feedback control for disturbance rejection.An alternative prediction approach is proposed by

(Mounier and Rudolph, 1998).

4.4 Robust design

In order to analyse the closed-loop stability notwith model (3), but with a better descriptionusing the identified model family GP ∈ Π, theloop transfer functions

LP ( jω) =Ŷ ∗( jω)

E ( jω)

= qT nGP ( jω) + G∗( jω)(1 − e−jωT 0)

1 − GP ( jω) − G∗( jω)(1 − e−jωT 0) (8)

are evaluated by the Nyquist criterium. T n = T ±∆T denotes the nominal time constant. The goalis to guarantee both robust stability in the senseof uncertainties of the model and of incorrectswitching between G∗(s) = G∗on(s) and G

∗(s) =G∗off (s), and between T n = T − ∆T and T n =T + ∆T . The controller gain q is chosen suchthat LP ( jω) in the ’worst case’ does not encirclethe critical point −1. To do so, all possible loop

transfer functions LP ( jω), i.e. all combinationsbetween G∗(s) and T n for the two nominal cases,respectively, and the models GP (s) ∈ Π of thefamily are considered. In order to obtain a highperformance, q is designed as high as possible.

Since the feedback loop only stabilises the track-ing of the reference y∗d(t), its performance limi-tation does not limit the feedforward commandtracking performance. The ’worst case’ crossoverfrequency of LP ( jω) is a measure for the distur-bance rejection performance, and this is approx-imatively as twice as high in comparison to the

robust H∞-control. This is due to both the differ-ent control structures and the more conservativeunstructured uncertainty description applied inthe H∞-design.


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0 20 40 60 80 100

5

6

7

t U∞

/ H

x R /

H

H∞-control: flatness based control:r(t) yd(t) = y

∗

d(t− T 0)

y(t) y(t)

Fig. 6. Comparison between the dynamics of thelinear H∞-controller and the flatness basedcontroller with closed-loop step responses.

5. COMMAND TRACKING PERFORMANCERESULTS OF BOTH CONTROLS

A comparison of the command tracking responsesof both controls is displayed in Fig. 6. Both,the inherent time delay T 0 and the uncertaintylM (ω) of the linear design model Gn(s) limit theperformance of the robust control, resulting in atransport delay of T 0U ∞/H = 8 convective timeunits after the stepwise decrease of r(t) and thefollowing slow transient behaviour. However, theflatness based control can compensate for the de-lay by an appropriate choice of the future refer-ence y∗d(t) = yd(t + T 0). Since the flatness basedcontrol splits command tracking and disturbancerejection, the feedforward control is not limited.Consequently, even faster references y∗d(t) could

be followed. Only the disturbance rejection per-formance is limited by the time delay and theuncertainty of the design model G∗(s).

6. CONCLUSIONS

A fast and cheap controller synthesis for theseparated flow around a backward facing step canbe performed employing robust methods such asH∞-controller design. However, the inherent timedelay and the uncertainty of the linear design

model imposes a limitation on the achievableperformance, i.e. the performance is limited to thecase of the slowest process behaviour.

A faster command tracking performance can beobtained by a nonlinear design model that resolvesthe different state depending dynamics of the flowprocess in combination with a flatness based con-trol. Although the nonlinear design model con-tains significant uncertainties, an arbitrary fastcommand tracking performance can be achieveddue to the splitting between open-loop commandtracking and closed-loop disturbance rejection.

Two extensions had to be added to the classicalflatness based control design scheme. First, asthe control signal is computed from the futureoutput, the prediction approach of the well-known

Smith predictor was used for the online computa-tion in real-time. Second, the feedback control fordisturbance rejection was designed to guaranteerobust stability. One price paid is that in contrastto the H∞-controller, a lot of experience for thedetermination of the nonlinear black-box designmodel and its uncertainty description is needed.

REFERENCES

Allan, B.G., J.-N. Juang, D.L. Raney, A. Seifert,L.G. Pack and D.E. Brown (2000). Closed-loop separation control using oscillatoryflow excitation. Technical Report NASA/CR-2000-210324, ICASE Report No. 2000-32 .

Baker, J. and P.D. Christofides (2002). Drag re-duction in transitional and linearized chan-nel flow using distributed control. Inter. J.Contr. 75, 1213–1218.

Becker, R., M. Garwon and R. King (2003). De-velopment of model-based sensors and theiruse for closed-loop control of separated shearflows. In: Proc. of the European Control Con- ference ECC 2003 . University of Cambridge,UK.

Becker, R., M. Garwon, C. Gutknecht, G. Bärwolff and R. King (2005). Robust control of sep-arated shear flows in simulation and exper-iment. Journal of Process Control, accepted for publication .

Glauser, M.N., H. Higuchi, J. Ausseur and

J. Pinier (2004). Feedback control of sepa-rated flows. AIAA-Paper 2004-2521.

Hinze, M. (2000). Optimal and instantaneous con-trol of the instationary Navier-Stokes equa-tions. Habilitation thesis. Technische Univer-sität Berlin. Berlin.

King, R., R. Becker, M. Garwon and L. Hen-ning (2004). Robust and adaptive closed-loopcontrol of separated shear flows. AIAA-Paper 2004-2519 .

Mounier, H. and J. Rudolph (1998). Flatnessbased control of nonlinear delay systems: achemical reactor example. Int. J. Control 71, 871–890.

Noack, B.R., K. Afanasiev, M. Morzynski, G. Tad-mor and F. Thiele (2003). A hierarchy of low-dimensional models of the transient andpost-transient cylinder wake. J. Fluid Mech.497, 335–363.

Pastoor, M., R. King, B.R. Noack and A. Dill-mann (2003). Model-basedcoherent-structure control of turbulent shearflows using low-dimensional vortex models.AIAA-Paper 2003-4261.

Siegel, S., K. Cohen and T. McLaughlin (2004).

Experimental variable gain feedback con-trol of a circular cylinder wake. AIAA-Paper 2004-2611.

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2005_Comparison of a Robust and a Flatness Based Control for a Separated Shear Flow