CONTROL OF SEPARATION POINT IN PERIODIC FLOWS INCLUDING DELAY EFFECTS

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1 IFAC Workshop on Adaptation and Learning in Control and Signal rocessing, and IFAC Workshop on eriodic Control Systems, Yokohama, Japan, August 3 September 1, 24 CONTROL OF SEARATION OINT IN ERIODIC FLOWS INCLUDING DELAY EFFECTS Tamas Insperger,1 Francois Lekien,2 Hayder Salman George Haller Gabor Stepan Budapest University of Technology and Economics, Department of Applied Mechanics, Budapest, 1521, Hungary Massachusetts Institute of Technology, Department of Mechanical Engineering, Cambridge, MA 2139, USA University of Cambridge, Centre for Mathematical Sciences, Cambridge CB3 WA, UK Abstract: We discuss the stability of a digital control algorithm for flow separation in a time-periodic fluid flow The analysis concentrates on the destabilizing effect of a delay existing in the control loop This delay is inevitable and corresponds to the time needed for the effect of the controller to propagate into the flow and reach the separation profile For the setting discussed in this paper, the delay is large relative to the time scales of the system We show that the controller remains robust to time delays if an appropriate gain is used Keywords: flow control, stability, time delay, sampling, controllers 1 INTRODUCTION Flow separation is the detachment of fluid particles from a solid surface (Gad and Bushnell, 1991, Greenblatt, 2) Inducing separation at selected positions enables the transport of materials, such as fuel, from separation points to designated flow domains, such as the flame region in a combustor To design such a transport mechanism, one first needs to control the location of the separation point itself (Wang et al, 23) The development of closed-loop separation controllers has unfortunately been hindered by a limited understanding of unsteady separation itself Sears and Telionis (1975) proposed that at un- 1 inspi@mmbmehu 2 lekien@mitedu steady separation points, both the shear stress and the velocity should simultaneously vanish in a frame moving with the separation velocity This criterion, however, requires the a priori knowledge of the separation velocity However, the separation velocity is usually unavailable if the separation point is unknown, which makes such a criterion difficult to use in practice By contrast, Haller (24) took a Lagrangian view on unsteady separation and derived a mathematically rigorous criterion for the separation location in twodimensional unsteady flows Building on these results, Haller et al (22) developed a control algorithm for flow separation in two-dimensional time-periodic Stokes flows They linearly superimposed a time-periodic shear flow and two wall-jet flows that modeled synthetic jet actuators blowing parallel to a no-slip boundary 451

2 They managed to induce unsteady separation at prescribed locations on test flows, but found that their control law became unstable when used with full Navier-Stokes simulations In this paper, we modify the closed-loop controller of Haller et al (22) to include inertia effects that are present in Navier-Stokes flows, but are absent in the Stokes approximation of these flows Specifically, we include a time delay in the sensor inputs to account for the time needed for the actuator to affect the velocity at the desired point of separation With a properly tuned proportional gain, we achieve stability for this modified separation control algorithm Our work fits into a more general theme of modern control theory that is concerned with the avoidance of instabilities in controllers affected by sampling delay (Craig, 1986, Stepan and Haller 1995) The phase space for such a delayed system is infinite-dimensional (Stepan, 1989) If, in addition, the governing equations of the system are time-periodic, then the stability of any steady state must be studied through infinite-dimensional Floquet theory (Insperger and Stepan, 2, 22a) A paradigm for such delayed time-periodic systems is the delayed Mathieu equation (Insperger and Stepan, 22b) 2 A MODEL FOR SEARATION OINT CONTROL The location of the flow separation point is controlled by the use of two wall-mounted jets (actuators), as shown in Figure 1 By computing the required strength of the wall jet blowing or suction, the change in the oncoming unsteady background flow is counteracted to hold the separation point at the desired location γ T T u y (γ,,t)dt =, (1) ( vyy (γ,,t) β ) dt =, (2) where (u(x, y, t),v(x, y, t)) is the velocity field and β is a parameter characterizing the strength of the separation (Haller, 24) We assume that, close to the wall, a linear superposition of 3 components is a good approximation of the wall shear (skin friction) Hence, u y (x,,t)=u ybg (x,,t)+u yac1 (x,,t) + u yac2 (x,,t), (3) v yy (x,,t)=v yy bg (x,,t)+v yy ac1 (x,,t) + v yy ac2 (x,,t), (4) where bg refers to background flow, ac1 andac2 refers to the two actuators This linear superposition of the flows is intended to be a design approximation We rely on the robustness of the controller to stabilize the flow, even with a large modeling error The actuator flows are decomposed into temporal and spatial modes u yacj (x, y, t) =q ac j (t)û yacj (x, y), (5) v yy ac j (x, y, t) =q ac j (t)ˆv yy ac j (x, y), (6) j =1, 2 As part of our control algorithm, we update the temporal components of the jets at each sampling time Changes in the jet flow do not immediately affect the separation point We model this effect with a time delay τ As a first approximation, we assume that τ is a multiple of the sampling period t = t n+1 t n Hence, τ = m t Ateachstep, the intensity of each jet, as seen at the separation point γ,is updated by a (delayed) quantity q ac j (t) =q ac j (t T )+ q n m ac j, j =1, 2, t [t n,t n+1 ], (7) where the control law is defined as qac1 n m = ˆv yy ac2 (γ,)ε 1 (t n m ) û yac2 (γ,)ε 2 (t n m ) [ˆv yy ac1 (γ,)û yac2 (γ,) û yac1 (γ,)ˆv yy ac2 (γ,)] t, (8) Fig 1 Schematic view of a surface subject to a background flow, where the position of the separation point γ is controlled by the use of two wall jets The unsteady separation criteria for T periodic incompressible flows at point (x, y) =(γ,) are qac2 n m = û yac1 (γ,)ε 2 (t n m ) ˆv yy ac1 (γ,)ε 1 (t n m ) [ˆv yy ac1 (γ,)û yac2 (γ,) û yac1 (γ,)ˆv yy ac2 (γ,)] t (9) Here, is the proportional gain and can be modified to achieve stability The objective of the control loop is to satisfy conditions (1) and (2) In general, the separation objective has not been achieved completely and ε 1 and ε 2 represent the 452

3 error between the left-hand side and the righthand side of the separation conditions (1) and (2) Their precise definition is [ tn 2 ε 1 (t n )= q ac j (t)û yacj (γ,) ε 2 (t n )= t n T tn t n T + u ybg (γ,,t) ] [ 2 q ac j (t)ˆv yy ac j (γ,) + v yy bg (γ,,t) β 3 STABILITY ANALYSIS ] dt, (1) dt (11) Using equation (7) and considering that the background flow is T periodic, equations (1) and (11) yield ε 1 (t n+1 ) ε 1 (t n ) tn+1 2 = t n [ q n m ac1 ε 2 (t n+1 ) ε 2 (t n ) tn+1 2 = t n [ q n m ac1 q n m ac j ûyac1(γ,) + q n m q n m ac j û yacj (γ,) dt ac2 ûyac2(γ,) ] t, (12) ˆv yy ac j (γ,) dt ˆv yy ac1(γ,) + qac2 n m ˆv yy ac2(γ,) ] t (13) Equations (12) and (13), together with the control law given by equations (8) and (9) lead to the following recursive formula for the error dynamics ε 1 (t n+1 )=ε 1 (t n )+a 1 ε 1 (t n m )+a 2 ε 2 (t n m ), (14) ε 2 (t n+1 )=ε 2 (t n )+b 1 ε 1 (t n m )+b 2 ε 2 (t n m ), (15) where a 1 = 1, a 2 =,b 1 =andb 2 = 1 in this case We define ε 1 (t n ) ε 2 (t n ) ε 1 (t n 1 ) ε 2 (t n 1 ) ε 1 (t n 2 ) ɛ(t n )= ε 2 (t n 2 ) ε 1 (t n m+1 ) ε 2 (t n m+1 ) ε 1 (t n m ) ε 2 (t n m ) representing the errors on the controlled quantities as a vector of R 2m+2 Equations (14) and (15) define the evolution of the error as the action of an operator A m acting on the (2m + 2)-dimensional space ɛ(t n+1 )=A m ɛ(t n ), (16) where 1 a 1 a 2 1 b 1 b A m = The stability of the controlled system depends on the eigenvalues of the matrix A m in equation (16) If the eigenvalues are inside the unit circle of the complex plane, then the controller is asymptotically stable (Lakshmikantham and Trigiante, 1988) The robustness of the controller increases when the modulus of the critical eigenvalue decreases If the time delay is neglected (ie, m = ) and the gain is set to = 1, the matrix in equation (16) reads A = ( ) 1+a1 a 2 = b 1 1+b 2 ( ) (17) Both eigenvalues vanish for this set of parameters This so-called dead beat control does not work in real systems due to the effect of unavoidable delays in the control loop If we consider the case m = 1 (and = 1, again), the corresponding coefficient matrix is 1a 1 a A 1 = 1b 1 b 2 1 = (18) All the eigenvalues of this matrix are of unit norm As a result, the system with m = 1 is at the border of stability With increasing delay m, the size of the resulting coefficient matrix increases, and the modulus of the critical characteristic multiplier grows The critical characteristic multipliers corresponding to different values of m are shown in Table 1 and in Fig 2 For m>1and = 1, the modulus of the characteristic multiplier is larger than 1, that is, the system is always unstable However, by decreasing the gain, one can recover the stability of the system Fig 3 shows the stability chart of the controlled system in the plane (m, ) The stability bound- 453

4 Table 1 Critical characteristic multipliers, moduli of the multipliers and number c of unstable characteristic multipliers for different values of m and =1 m µ 1 c 1 5 ± i ± i ± i ± i ± i ± i ± i ± i m Fig 2 Moduli of critical characteristic multipliers for different values of m and = =7 >1 unstable =8 =9 = m Fig 3 Stability chart for gain ( ) and delay parameter (m) aries correspond to µ = 1 As shown on Fig 3, the system can be stabilized, even for large delays, with properly chosen small gains For example, if m = 2, Fig 3 shows that the controller is stable only if the proportional gain is < < 618 Two numerical simulations are presented with realistic flow parameters for gains =6 and =65 in Figs 4 and 5, respectively Note that for = 6, the control was stable and the desired separation profile was achieved, while for the case =65 (ie, above the limit read in Fig 3), the controller was unstable 3 The simulation of the controlled flow is based on the semi-analytic approximation of the background and the jet flows, which we obtained by solving randtl s boundary layer equation with appropriate boundary conditions (Schlichting and Gersten, 1999) 3 Other figures and movies of simulations can be seen at 4 CONCLUSION The control of separation points in periodic flows with wall jet actuators includes large time delays related to the time scale of the flow dynamics We have shown how the time delay destabilizes such system A proportional controller was tuned based on stability charts to obtain stable control The results were also verified by numerical simulations For future work, we plan to use a numerical solver for Navier-Stokes equations and the proposed geometry, and to test the controller on an actual flow that is completely independent of the linear superposition used in the control algorithm In this case, the period of the flow and the linearized components will be determined empirically (from skin friction sensors) and the approximate flow that we use to design the controller is simply a rough approximation ACKNOWLEDGMENT This research was supported in part by the Zoltán Magyary ostdoctoral Fellowship of Foundation for Hungarian Higher Education and Research, by the Hungarian National Science Foundation under grant no OTKA F47318 and OTKA T43368, by the AFOSR Grant F and NSF Grant DMS REFERENCES Craig, J J, 1986, Introduction to Robot Mechanics and Control, Addison-Willey, Reading Gad-el-Hak, M and D M Bushnell, 1991, Separation Control: Review, Journal of Fluids Engineering, 113, pp 5-3 Greenblatt, D and I J Wygnanski, 2, The Control of Flow Separation by eriodic Excitation, rogress in Aerospace Sciences, 36, pp Haller, G, Y Wang, H Salman, J S Hesthaven, Banaszuk, A, 22, Control of Lagrangian Coherent Structures, IUTAM Symposium on Unsteady Separated Flows, Toulouse, France Haller, G, 24, Exact Theory of Unsteady Separation for Two-Dimensional Flows Journal of Fluid Mechanics, in press Insperger, T, Stepan, G, 2, Remote control of periodic robot motion, roceedings of 13th CISM-IFToMM Symposium on Theory and ractice of Robots and Manipulators, Zakopane, pp Insperger T, Stepan, G, 22a, Semi-Discretization Method for Delayed Systems, International Journal for Numerical Methods in Engineering, 55(5), pp

5 Fig 4 Simulations with m =2, =6 stable flow control Fig 5 Simulations with m =2, =65 unstable flow control Insperger, T, Stepan, G, 22b, Stability chart for the delayed Mathieu equation, roceedings of The Royal Society, Mathematical, hysical and Engineering Sciences, 458(224), pp Lakshmikantham, V, Trigiante, D: Theory of Difference Equations, Numerical Methods and Applications, Academic ress, London, 1988 Schlichting, H, Gersten, K, 1999, Boundary- Layer Theory, 8th Revised and Enlarged Edition, Springer-Verlag, Berlin Sears, W R, Telionis, D, 1975, Boundary- Layer Separation in Unsteady Flows SIAM Journal of Applied Mathematics, 28, pp 215 Stepan, G, 1989, Retarded Dynamical Systems, Longman, Harlow Stepan, G, Haller, G, 1995, Quasiperiodic oscillations in robot dynamics, Nonlinear Dynamics, 8, pp Wang, Y, G Haller, A Banaszuk and G Tadmor, 23, Closed-Loop Lagrangian Separation Control in a Bluff Body Shear Flow, hysics of Fluids, 15(8), pp

Boundary-Layer Theory

Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22