Poiseuille Flow Controller Design via the Method of Inequalities - - PowerPoint PPT Presentation

▶
poiseuille flow controller design via the method of
SMART_READER_LITE
LIVE PREVIEW

Poiseuille Flow Controller Design via the Method of Inequalities - - PowerPoint PPT Presentation

Oct 30, 2022 238 likes •455 views

Poiseuille Flow Controller Design via the Method of Inequalities James F. Whidborne 1 John McKernan 1 George Papadakis 2 1. Department of Aerospace Sciences, Cranfield University 2. Division of Engineering, Kings College London UKACC Control

slide-1
SLIDE 1

Poiseuille Flow Controller Design via the Method of Inequalities

James F. Whidborne1 John McKernan1 George Papadakis2

  • 1. Department of Aerospace Sciences, Cranfield University
  • 2. Division of Engineering, King’s College London

UKACC Control 2008 – p. 1/20

slide-2
SLIDE 2

Introduction — feedback flow control

  • Problem of stabilizing fluid flows by feedback is a topic of much interest
  • Fluid flow dynamics are often highly non-normal — that is their eigenvectors are closely

aligned — and this non-normality is one factor that makes fluid systems hard to control

  • Traditionally, fluid dynamicists have assessed the stability of systems using Lyapunov’s

first method — hence differences between measured and predicted flow stability

  • Plane Poiseuille or channel flow is the unidirectional flow between two infinite

parallel planes — laminar and stable for low Reynolds numbers — but at high Reynolds numbers the flow becomes unstable resulting in turbulence

  • Experiments show that the flow undergoes transition to turbulence for Reynolds

number as low as 1000

  • However, eigenvalue predictions show the flow to be stable at Reynolds numbers

below approximately 5772

  • Non-normal nature of the dynamics makes the flow very sensitive — an initial

perturbation will grow to very large values before decaying — this can drive the system into regions where the non-linearities are significant and trigger turbulence

  • Hence system dynamics can thus be considered as conditionally linear

UKACC Control 2008 – p. 2/20

slide-3
SLIDE 3

Introduction — transient energy growth

Consider the asymptotically stable linear time-invariant system described by the initial value problem ˙ x = Ax, x(0) = x0, Transient energy (or energy of perturbations) is a measure of the size of the perturbations of the state following a unit initial perturbation: E(t) := max

  • x(t)2 : x0 = 1
  • Transient energy has clear physical meaning and is a fundamental notion in the study of

turbulence and transition Consequently, the maximum transient energy growth following some energy-bounded initial state perturbation is often used as a performance measure for fluid flow systems where maximum transient energy growth is defined as

  • E := max {E(t) : t ≥ 0}

In practice we require appropriate weights on the states E(t) = max

Wx(0)=1 Wx(t)2

UKACC Control 2008 – p. 3/20

slide-4
SLIDE 4

Transient energy growth in Poiseuille flow

From an initial state x(0) with unit kinetic energy density E(0) = xT (0)Wx(0) = 1, large transient growth in the kinetic energy density E(t) of the state occurs before an eventual exponential decay of energy at the rate of the least-stable constituent eigenmode.

500 1000 1500 2000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

t E(t)

UKACC Control 2008 – p. 4/20

slide-5
SLIDE 5

Feedback control of plane channel flow

Objective is to maintain laminar flow by measuring the shear at the wall and using the controller to actively modify the boundary conditions by blowing/suction at the walls

(Controller) Plane Poiseuille Flow Flow Disturbance Lower Wall Upper Wall Actuation Sensing Spanwise, z Wall−normal, y Streamwise, x

UKACC Control 2008 – p. 5/20

slide-6
SLIDE 6

Feedback control of plane channel flow

  • Incompressible fluid flow is described by the Navier-Stokes and the continuity equations

˙

  • U +
  • U · ∇
  • U = − 1

ρ ∇P + µ ρ ∇2 U ∇ · U = 0

  • U is velocity, P is pressure, ρ is density, µ is viscosity
  • Laminar flow has a parabolic stream-wise velocity profile

Ub = ((1 − y2)Ucl, 0, 0), Pb with no slip occurring at the bounding parallel planes

  • It undergoes transition to turbulence when small disturbances

u = (u, v, w), p about the steady base profile, grow spatially and temporally to form a self-sustaining turbulent flow

  • Non-dimensionalizing the perturbation equations gives

˙

  • u +
  • Ub · ∇
  • u + (

u · ∇)

  • u +

Ub

  • = −∇p + 1

R ∇2 u ∇ · u = 0 where R := ρUclh/µ is the Reynolds number

UKACC Control 2008 – p. 6/20

slide-7
SLIDE 7

Linearized model

  • For control by wall transpiration, no-slip wall boundary conditions at y = ±h are

replaced by prescribed wall transpiration velocities, (u(±h) = 0, v(±h) = 0, w(±h) = 0)

  • Variations in span-wise and stream-wise directions are assumed to be periodic

ℜei(αx+βz), and flow disturbances grow in time, but not in space

  • Boundary control at wave numbers α and β respectively can be represented as linear

state-space system in the standard form x = Ax + Bu where linearized Navier-Stokes equations are evaluated at N locations in the wall-normal direction y

  • State variables x are wall-normal ve-

locity ˜ v and vorticity, ˜ η := ∂u/∂z − ∂w/∂x, perturbation Chebyshev coeffi- cients, plus the upper and lower wall v velocities

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 ←ΓDN

0 (8)

←ΓDN

1 (4.6)

←ΓDN

last(2.1)

←ΓD

0(2)

←ΓD

1(1.5)

←ΓD

last(2)

←fu

lmi(1)

←fl

lmi(1)

y

UKACC Control 2008 – p. 7/20

slide-8
SLIDE 8

Feedback control of plane channel flow

  • The outputs are measurements of wall shear stress and pressure
  • Control inputs u are the rates of change of transpiration velocity on the upper and lower

walls

  • The test case considered here is α = 0, β = 2.044, R = 5000
  • The model is discretized in the wall-normal direction with N = 20
  • This

test case is linearly stable but has the largest linear transient en- ergy growth

  • ver

all unit initial con- ditions, time and wave-number, and represents the very earliest stages

  • f

the transition to turbulence

500 1000 1500 2000 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

t E(t)

UKACC Control 2008 – p. 8/20

slide-9
SLIDE 9

Control of maximum transient energy growth

  • Can be minimized using a Q-paramterization — very high order controller
  • Upper bound can be minimized using LMI approach — quite conservative — state

feedback required We propose using the Method of Inequalities to

  • 1. design low order controllers
  • 2. design H∞-optimal controllers (in a mixed optimization approach)

that reduce the maximum transient energy growth

UKACC Control 2008 – p. 9/20

slide-10
SLIDE 10

Control of maximum transient energy growth

Now consider the linear time-invariant plant ˙ x(t) = Ax(t) + Bu(t), x(0) = x0, y(t) = Cx(t) with feedback controller ˙ xk(t) = Akxk(t) + Bky(t), xk(0) = xk0, u(t) = Ckxk(t) + Dky(t) The closed loop system is given by ˙ xc(t) = Acxc(t), xc(0) = xc0 where Ac :=

  • A + BDkC

BCk BkC Ak

  • ,

xc(t) :=

  • x(t)

xk(t)

  • UKACC Control 2008 – p. 10/20
slide-11
SLIDE 11

Control of maximum transient energy growth

The maximum transient energy growth of the plant is ˆ E = max

  • x(t)2 : x0 = 1, xk0 = 0, t ≥ 0
  • and can be evaluated by |

| |Φc(t)| | |2 where Φc(t) := [In 0nk]eAct[In 0nk]T and | | |·| | | denotes the spectral norm In order to limit the amount of effort generated by the controller in a closed loop system, the maximum control “transient energy growth” is defined as ˆ U := max

  • u(t)2 : x0 = 1, xk0 = 0, t ≥ 0
  • UKACC Control 2008 – p. 11/20
slide-12
SLIDE 12

The Method of Inequalities (MOI)

MOI is a computer-aided multi-objective approach where aim is to find p which satisfies φi(p) ≤ εi for i = 1 . . . n εi are largest tolerable values of φi (design goals) design parameter p is real vector (p1, p2, . . . , pq) For control systems, φi(p) may be functionals of system step response, e.g. rise time,

  • vershoot, integral absolute error, or functionals of the frequency response, eg bandwidth,

phase margin May include both time and frequency domain performance Solution obtained by numerical search algorithms

UKACC Control 2008 – p. 12/20

slide-13
SLIDE 13

The flow control problem

Poiseuille flow control problem can be formulated as

Problem 1. Find a p ∈ P and hence a K(p) such that

α0(p) ≤ ǫα ˆ E(p) ≤ ǫˆ

E

ˆ U(p) ≤ ǫˆ

U

where ǫα, ǫˆ

E , and ǫˆ E are prescribed tolerable values of α0, ˆ

E, and ˆ U respectively and α0 = max

i

{ℜ(λi(Ac))}

UKACC Control 2008 – p. 13/20

slide-14
SLIDE 14

Proportional controller

The design goals are set at ǫα = −1 × 10−5 ǫˆ

E = 1000

ǫˆ

U = 10

Several low-order structures were tried, but no controller was found that satisfied problem. After a small number of iterations, the proportional controller K =

  • 0.7474

0.8655 0.3259 −0.7862 0.7855 0.7855 0.2061 0.7757

  • was obtained with a performance

α0 = −1.7750 × 10−3 ˆ E = 2781.4 ˆ U = 9.7519

UKACC Control 2008 – p. 14/20

slide-15
SLIDE 15

Proportional controller

500 1000 1500 500 1000 1500 2000 2500 3000

t E(t)

Transient Energy

UKACC Control 2008 – p. 15/20

slide-16
SLIDE 16

P+D controller

The P+D controller structure K(s) = Kp + Kd

  • s

s + a

  • was tried and after some iteration a controller with a = 14.9058 and

Kp =

  • 0.6151

0.4184 1.0055 0.4233 1.4066 0.0563 −0.0266 0.3148

  • Kd =
  • 26.2916

42.0612 4.3891 4.8970 13.1916 92.8310 4.1459 13.3138

  • was obtained that gave a performance

α0 = −1.7806 × 10−3 ˆ E = 2737.4 ˆ U = 8.913 This is only a small improvement on the proportional controller The transient energy E(t) is similar to the proportional controller

UKACC Control 2008 – p. 16/20

slide-17
SLIDE 17

Mixed optimization

The MoI can be combined with Mc- Farlane and Glover’s loop-shaping design procedure (LSDP) in a mixed

  • ptimization approach by using the

parameters of the weighting func- tions as the design parameters W1(s) Ks(s) G(s) W2(s)

✲ ✛ ✲ ✛

Problem 2. For the system above find a p ∈ P and hence a (W1, W2)(p) and K such that

γ0(p) ≤ ǫγ ˆ E(p) ≤ ǫˆ

E

ˆ U(p) ≤ ǫˆ

U

where (W1, W2)(p) is a pair of fixed order weighting functions with real parameters p = (p1, p2, . . . , pq) and

γ0 = inf

K

  • W −1

1

K W2

  • (I − GK)−1

W −1

2

GW1

  • ∞

UKACC Control 2008 – p. 17/20

slide-18
SLIDE 18

Mixed optimization

The design goals are ǫγ = 5 ǫˆ

E = 1000

ǫˆ

U = 10

A simple proportional diagonal structure W1 = diag([p1, p2]) was found to be best. The post-plant weighting was set to be the identity W2 = I. The resulting controller performance was fairly insensitive to the values of p, thus W1 = I was chosen. The design goal, ǫˆ

E was

not satisfied, and interestingly, the best value of ˆ E was marginally greater than that for the proportional controller. The resulting performance was γ0 = 1.818 ˆ E = 2884.1 ˆ U = 8.876

UKACC Control 2008 – p. 18/20

slide-19
SLIDE 19

Mixed optimization

500 1000 1500 500 1000 1500 2000 2500 3000

t E(t)

Transient Energy

UKACC Control 2008 – p. 19/20

slide-20
SLIDE 20

Conclusions

  • Design of both low order controllers and H∞-optimal controllers using the MoI has

been performed

  • LQR state feedback controllers gave a value of ˆ

E = 883 — considerably lower than the best value obtained in this study of ˆ E = 2737

  • Optimal H∞ was unable to improve on the P+D controller — indicates little advantage

in dynamic controllers and that the output feedback problem as posed here does not have any solutions

  • With an increase in computational power, the convex optimization methods could be

applied to this problem and determine what is the minimal ˆ E and whether a solution exists

  • The problem of model order reduction for the maximum transient energy growth

problem would aid the design of controllers

UKACC Control 2008 – p. 20/20