Control, Observation and Energy Regulation of Wake Flow Instabilities Gilead Tadmor Electrical & Comp. Eng. Northeastern University Boston, MA 02115, USA. tadmor@ece.neu.edu Bernd R. Noack and Andreas Dillmann Hermann-F¨ ottinger-Institute of Fluid Mechanics Techn. Univ. Berlin Straße des 17 Juni 135 D-10623 Berlin, Germany Johannes Gerhard, Mark Pastoor and Rudibert King Measurement & Control Gr. Institute for Process and Plant Technology Techn. Univ. Berlin Hardenbergstraße 36a D-10623 Berlin, Germany Marek Morzy´ nski Inst. Combust. Engines & Basics of Machine Design Pozna´ n Univ. Techn. ul. Piotrowo 3, PL 60-965 Pozna´ n, Poland Abstract— A three-dimensional Galerkin model is used in feedback design to regulate the perturbation kinetic energy in the flow around a cylinder. The objective may vary from stabilization in order to reduce drag to mixing enhancement. The Landau model [1] includes an oscillatory state pair and a shift mode, exchanging energy with the mean flow. Given the model’s simplicity, it is essential to maintain closed-loop dynamics close to the system’s natural dynamic range which is represented by an invariant manifold and a natural frequency, adding a design constraint addressed in this note. I. THE PROBLEM The complexity of computational fluid dynamics (CFD) models is a major hindrance to implementable feedback control [2]. This note presents a benchmark design based on a very low order Galerkin model and highlights control design aspects peculiar to the use of such models. The regulation of laminar vortex shedding behind a cylinder has been adopted as our benchmark control problem from the fluid flow community [3]- [6]. The system is considered at the reference Reynolds number of 100 which is far above the laminar shedding regime’s critical value of 47 [7] but far below the transitional range in which three-dimensional instabilities characterize the flow [8]. This paper is focused on the control and system theoretic aspects. Model development issues are discussed in [9], [10] and we shall therefore be content here with a brief review. Reduced order Galerkin models (GM) are widely used in fluid dynamics [11]. Here, the GM utilizes an orthonormal 1 Galerkin approximation of the attractor u(x,t)= us(x)+ 3  i=1 ai (t) ui (x), (1) where us(x) is the unstable steady Navier-Stokes solution and the Karhunen-Loe` ve modes ui (x), i =1, 2, capture some 96% of the perturbation energy of the von K´ arm´ an oscillatory instability. Yet, a model based on these two modes alone cannot capture any of the system dynamics. That is the purpose of the shift mode u3(x) ∝ us(x) − u0(x) which is an orthonormalized mean-field correction between the natural mean flow u0(x) and the steady solution. This shift mode resolves the energy exchange between the base flow and the oscillatory perturbation. Figure 1 depicts these four modes in terms of stream-lines. Actuation is effected with a local volume force, f (x), in the near-wake, thus mimicking, for instance, a Lorentz force in a magneto-hydrodynamical flow. Both 1 Signal norms (with the subscript “2”), inner products and orthogonality is understood in the appropriate (spatial or temporal) L 2 sense. Euclidean norms (with no subscript) and inner products are used for Euclidean vectors. To simplify notations, the base flows u 0 and us are not normalized in (1) and elsewhere a single and a two degrees of freedom actuation, are considered. In the former case, the orientation of the force is fixed: f (x)= α1u1(x)+α2u2(x), with fixed αi . This ansatz ignores the residual field which is left out of the GM model. In the latter case, two identically structured actuators will stimulate mutually orthogonal fields, fi (x), i =1, 2. A single or multiple point velocity field sensor (such as hot-wire anemometers), captured by the variable s, are postulated, located sufficiently far downstream and displaced from the axis of symmetry near the maximum of fluctuation energy. As will be seen below, velocity field transients are dominated by a single harmonic, which is therefore easily identifiable. Under slow variations in a3(t) (relative to the dominant period), residual effects of both the constant us and the slowly varying a3u3, can be removed by band-pass filtering. It is thus assumed that, excluding noise, the sensor reading s is a linear combination of a1 and a2. Figure 2 depicts the system with a single actuator and a single sensor. Under this description, the Galerkin system is given by ˙ a = A(a)a + Bǫ, s = Ca (2) where a := [a1,a2,a3] T , ǫ is the actuation command, and A(a) :=   µ −ω −βa1 ω µ −βa2 δa1 δa2 −ρ   (3) The coupling terms βai and δai and the growth parameter µ> 0 are typically small relative to the natural oscillation frequency ω and dissipation parameter ρ> 0. A single actuator is represented by a scalar ǫ and B = b[cos(θ), sin(θ), 0] T . Associated with a double actuator, ǫ := [ǫ1,ǫ2] T , and B = b   cos(θ) − sin(θ) sin(θ) cos(θ) 0 0   reflects the assumed orthogonal symmetry of the two volume force fields. The sensor coefficient for a single sensor is C =[c1,c2, 0]. With multiple sensors, the matrix C will comprise several rows of a similar form. The control task set forth is the design of feedback regulation of the perturbation kinetic energy, represented by K(t)=0.5‖a(t)‖ 2 , about a set reference, K*. Stabilization of the reference K* =0 is most commonly discussed, motivated by such objectives as reducing drag or mechanical vibrations. Enhancement of K beyond its open loop value may be motivated by a mixing objective. An energy regulation objective, respects the basic characteristics of the system’s unactuated, oscillatory dynamics — as opposed to other setpoint or orbit tracking tasks. Indeed, an intrinsic limitation Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 WeM10-4 0-7803-7924-1/03/$17.00 ©2003 IEEE 2334