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, King’s College London UKACC Control 2008 – p. 1/20
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
Introduction — transient energy growth Consider the asymptotically stable linear time-invariant system described by the initial value problem x = Ax , ˙ x (0) = x 0 , Transient energy (or energy of perturbations) is a measure of the size of the perturbations of the state following a unit initial perturbation: � � � x ( t ) � 2 : � x 0 � = 1 E ( t ) := max 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 � Wx (0) � =1 � Wx ( t ) � 2 E ( t ) = max UKACC Control 2008 – p. 3/20
Transient energy growth in Poiseuille flow From an initial state x (0) with unit kinetic energy density E (0) = x T (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. 5000 4500 4000 3500 3000 E ( t ) 2500 2000 1500 1000 500 0 0 500 1000 1500 2000 t UKACC Control 2008 – p. 4/20
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 Upper Wall Wall−normal, y Streamwise, x Plane Poiseuille Flow Disturbance Spanwise, z Flow Lower Wall Actuation (Controller) Sensing UKACC Control 2008 – p. 5/20
Feedback control of plane channel flow • Incompressible fluid flow is described by the Navier -Stokes and the continuity equations � � U = − 1 ρ ∇ P + µ ˙ ρ ∇ 2 � � � � U + U · ∇ U ∇ · � U = 0 � U is velocity, P is pressure, ρ is density, µ is viscosity • Laminar flow has a parabolic stream-wise velocity profile � U b = ((1 − y 2 ) U cl , 0 , 0) , P b 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 � � � � = −∇ p + 1 ˙ � u + � R ∇ 2 � u + � U b · ∇ u + ( � � u · ∇ ) � U b u ∇ · � u = 0 where R := ρU cl h/µ is the Reynolds number UKACC Control 2008 – p. 6/20
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 ℜ e i ( 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 1 ← f l lmi (1) 0.8 0.6 lmi (1) ← f u • State variables x are wall-normal ve- 0.4 ←Γ D last (2) locity ˜ v and vorticity, ˜ η := ∂u/∂z − 0.2 ←Γ D 1 (1.5) ∂w/∂x , perturbation Chebyshev coeffi- 0 y ←Γ D cients, plus the upper and lower wall v 0 (2) −0.2 velocities ←Γ DN last (2.1) −0.4 ←Γ DN 1 (4.6) −0.6 ←Γ DN 0 (8) −0.8 −1 0 0 0 0 0 0 0 0 UKACC Control 2008 – p. 7/20
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 5000 • This test case is 4500 linearly stable but 4000 has the largest 3500 linear transient en- ergy growth over 3000 all unit initial con- E ( t ) 2500 ditions, time and wave-number, and 2000 represents the very 1500 earliest stages of the transition to 1000 turbulence 500 0 0 500 1000 1500 2000 t UKACC Control 2008 – p. 8/20
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
Control of maximum transient energy growth Now consider the linear time -invariant plant ˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x 0 , y ( t ) = Cx ( t ) with feedback controller x k ( t ) = A k x k ( t ) + B k y ( t ) , ˙ x k (0) = x k 0 , u ( t ) = C k x k ( t ) + D k y ( t ) The closed loop system is given by x c ( t ) = A c x c ( t ) , ˙ x c (0) = x c 0 where � � � � A + BD k C x ( t ) BC k A c := , x c ( t ) := x k ( t ) B k C A k UKACC Control 2008 – p. 10/20
Control of maximum transient energy growth The maximum transient energy growth of the plant is � � � x ( t ) � 2 : � x 0 � = 1 , x k 0 = 0 , t ≥ 0 ˆ E = max | 2 where and can be evaluated by | | | Φ c ( t ) | | Φ c ( t ) := [ I n 0 n k ] e A c t [ I n 0 n k ] 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 � � � u ( t ) � 2 : � x 0 � = 1 , x k 0 = 0 , t ≥ 0 ˆ U := max UKACC Control 2008 – p. 11/20
The Method of Inequalities (MOI) MOI is a computer -aided multi-objective approach where aim is to find p which satisfies φ i ( p ) ≤ ε i for i = 1 . . . n ε i are largest tolerable values of φ i (design goals) design parameter p is real vector ( p 1 , p 2 , . . . , p q ) For control systems, φ i ( p ) may be functionals of system step response, e.g. rise time, overshoot, 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
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 α 0 ( p ) ≤ ǫ α ˆ E ( p ) ≤ ǫ ˆ E ˆ U ( p ) ≤ ǫ ˆ U E are prescribed tolerable values of α 0 , ˆ E , and ˆ where ǫ α , ǫ ˆ E , and ǫ ˆ U respectively and α 0 = max {ℜ ( λ i ( A c )) } i UKACC Control 2008 – p. 13/20
Proportional controller The design goals are set at ǫ α = − 1 × 10 − 5 ǫ ˆ E = 1000 ǫ ˆ 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 � � 0 . 7474 0 . 8655 0 . 3259 − 0 . 7862 K = 0 . 7855 0 . 7855 0 . 2061 0 . 7757 was obtained with a performance α 0 = − 1 . 7750 × 10 − 3 ˆ E = 2781 . 4 ˆ U = 9 . 7519 UKACC Control 2008 – p. 14/20
Proportional controller 3000 2500 2000 E ( t ) 1500 1000 500 0 0 500 1000 1500 t Transient Energy UKACC Control 2008 – p. 15/20
Recommend
More recommend
