Closed-loop fluid flow control with a reduced-order model gain-scheduling approach

Closed-loop fluid flow control with a reduced-order model gain-scheduling approach L. Mathelin 1 M. Abbas-Turki 2 L. Pastur 1,3 H. Abou-Kandil 2 1 LIMSI - CNRS (Orsay) 2 SATIE, Ecole Normale Supérieure (Cachan) 3 Université Paris-Sud 11 (Orsay) GDR Contrôle Poitiers Dec. 06-08, 2010

Control of fluid flows Context Some issues... more and more complex flows complex models (> 10 7 spatial DOFs, > 10 5 temporal DOFs): 3-D unsteady and turbulent, high Reynolds number flows, coupled physics (e.g., fluid / structure / thermal). strict specifications calls for efficient control algorithms and not just crude ones, real-time control... and some remedies?... One hence needs a light and robust model a reduced-order model which retains the essential dynamics and features for a whole range of operating conditions, an accurate and stable way to time-integrate it, a control objective formulated in the state space, a linear, yet accurate, framework for control.

Drag control Drag control (F D ) of a cylinder in laminar cross-flow: a gentle test bench for the methodology. Non-linear problem can not be directly put under the standard LTI form and the matrices involved remain dependent on the state vector X (X 1... X N ) T and control µ. can not take advantage of LTI tools such as Riccati-based feedback to design the control command. To circumvent this difficulty, trajectory tracking. The reference trajectory is determined using a classical open-loop approach through an optimal control technique X (t) and µ (t), [off-line] The deviation of the system from this optimal trajectory is controlled. [on-line]

Configuration Simulation parameters: The reduced model has 9 state space variables: X R 9. The deviation from optimal trajectory is due to uncertainty in the Reynolds number (Re) which is modeled by a Gaussian stochastic process of exponential temporal correlation of 2 sec (non-dimensional time unit). Its mean is 200 and its variance is 3.8. The control is blowing through the surface of the cylinder.

Phase-portraits of the ROM and the detailed model

Formulation The evolution of the vector state X is approximately described by a first-order non-linear model of the form: Ẋ = F (X, µ). Deviations from the optimal trajectory meant to remain small linear framework. Denoting δx X X and δµ µ µ, the state vector deviation dynamics follows: δx = X F (X, µ) δx + µf (X, µ) δµ. pre-compute the Jacobians for a given number of operating points along the trajectory.

Drag control cont d Operating points (X(t m), µ(t m)) are not known a priori approximation: which formally rewrites δx X F (X (t m), µ (t m)) δx + µf (X (t m), µ (t m)) δµ, δx A (X (t m), µ (t m)) δx + B (X (t m), µ (t m)) δµ, = A tm δx + B tm δµ. Matrices A tm and B tm are known this problem is now described by a LTV model obtained by interpolating the LTI models: δx = A tm δx + B tm δµ with m = 0, 1,..., M, where M is the number of operating points.

Controllability of the reduced system The controllability Gramian is defined as W C (t 0 ) + φ(s) B(t 0 + s) B (t 0 + s) φ (s) ds, t=0 W C (t 0 ), = n max φ n B n Bn φ n t, n=1 with φ the transition matrix: φ(s) = A(t 0 + s) φ(s), with φ(0) = I.

Controllability of the reduced system

Synthesis of the controller First, we introduce the performance output z, which quantifies the distance of the actual trajectory to the optimal one, defined as: with z = C 1m δx + D 12m δµ, C 1m = 1 2 X F D (X (t m), µ (t m)) (I + S X F D ), D 12m = 1 2 µf D(X (t m), µ (t m)) (I + S µf D ). Only state variables and control components that tend to increase z, at first order, are considered. The matrices S X F D and S µf D elements are given by the sign of X F D (X (t m), µ (t m)) and µf D (X (t m), µ (t m)) elements, respectively.

Synthesis of the controller cont d Second, synthesizing LTI controllers for a certain number of operating points and interpolating them to get a Linear Parameter Varying (LPV) controller. LTI controller design: For each operating point, LTI controller design aims at finding a suitable criterion to reach the performance objective of reducing z. LPV controller design: LPV controller design aims at finding a suitable set of LTI controllers, such that there interpolation keeps the stability of the closed-loop system without deteriorating the performance objective.

LTI controller design Exogenous input w. The problem is shown in this standard form: G : z y w δµ A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 where matrices A = X F (X ; µ ) and B 2 = µf (X ; µ ). Lower Linear Fractional Transformation (LLFT). It is still to determine the matrices B 1, C 1, C 2, D 11, D 12, D 21 and D 22.

LTI controller design To simplify the determination of the matrices of standard form: we assume that w acts directly on z and not on δx B 1 = 0, we consider full observability C 2 = I and D 22 = 0. Since w is not a random vector, a H -criterion seems more appropriate. Therefore, the problem of synthesis of the controller K is to minimize the H -norm of the LLFT: J(G; K ) = min K F L (G; K ). We use a direct interpolation of the state-space matrices of the LTI controllers to determine the matrices of the K t. This approach can be applied only if the state-space data of the controller are continuously time-varying.

Gain scheduling Set of linearization Continuity between controllers is a compromise between the following propositions: Single LTI controller stabilizing G t no stability problem but poor performance that can induce the non validity of the linear models. A large number of operating points may lead to an unstable interpolation, due to the discontinuity of the controllers. Solution: selection criterion to give the compromise between the number of controllers (stability) and the drag attenuation (performance). For the LTI controller K ti : The main criterion is the stabilization of the system for all operating points t m. The other criterion is performance which constrains the H -norm of the transfer F L (G tm ; K ti ) to remain below a threshold τ. Set of operating points: { } S i (τ) = (G tm, K ti ) F : J(G tm, K ti ) J(G tmi, K ti ) (1 + τ), where F is the set of stable state space variables induced by the closed-loop controller K ti.

Gain scheduling Stability Criterion: For two controllers K ta and K tb stabilizing the LTV system on the time interval [t a, t b ], if the closed-loop systems admit the same Lyapunov matrix P solution of: A clta P + PA T cl ta < 0 and A cltb P + PA T cl tb < 0, t [t a, t b ], with A clta and A cltb the corresponding closed-loop dynamic matrices ( ) AT + B A clta = 2t D Kt B 2t C Kt, B Kt C 2 A Kt then the linear interpolation of controllers K ta and K tb stabilizes exponentially the LTV system on the interval [t a, t b ]. τ is chosen such that the controllers K ti and K ti+1 verify the stability criterion for all the operating points included in S i (τ) K t = (1 α(t)) K ti + α(t)k ti+1, with α = t t i t i+1 t i.

Configuration Synthesized controllers: LTI controller designed from the initial operating point and stabilizing the system on the whole domain of simulation, LPV controller given by the interpolation of 20 LTI controllers selected among 249 (τ = 0.05). Note: The interpolation of 249 controllers gives an unstable system.

Results LTI controller z δµ

Results LPV controller z

As a conclusion... For controlling a fluid flow in a robust framework: Low dimensional subspace to allow for efficient control methods to remain applicable. Open-loop control to derive an optimal trajectory in the state space [off-line], closed-loop control to prevent the system from departing too much from it despite uncertainties and exogenous perturbations. [on-line] Interpolation of LTI controllers results in a closed-loop LPV robust control. H control achieves better overall performance than H 2 owing to its robustness w.r.t. controllers interpolation artefacts.