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An efficient control approach: The need for an upstream sensor
A central issue in flow control is the estimation of the present and future flow states from sensor observations. Indeed, the efficiency of the controller strongly relies on this estimation process. This is feasible if a state-space description of the system is available. If only an input-output description is available, the state estimation problem amounts to the prediction of the present and future outputs. In what follows, the particular case of convection-dominated flows is considered. It is shown that an appropriate estimation imposes restrictions on sensor placements. Interestingly, classical control tools such as the observability concept may produce very misleading results. To better understand the intrinsic properties of convection-dominated flows, a shift operator A is introduced as an idealized model. By definition, this operator transforms a state vector Xk = (x1, x2, x3, x4, x5, x6)T , at discrete time k, into a shifted vector Xk+1 = (0, x1, x2, x3, x4, x5)T , at discrete time k + 1. Such an operator may for instance be encountered in flow simulations when a transport equation is discretized with a unit Courant-Friedrichs-Lewy (CFL) coefficient. In this section, it is used to model flows where convection is the dominating feature. In addition, a white-noise perturbation wk excites the system at its most upstream location. The governing equations may then be written in state-space form as Xk+1 = AXk + Bwwk,
A realistic procedure: Subspace Identification
Most of the classical control design techniques resort to a model to describe the fluid sys-tem under consideration. This model may be obtained from numerical simulations, even though they are idealizations of reality. For instance, set-up imperfections and particular actuator specifics are typically difficult to take into account in simulations. Hence, due to possible differences between model and reality, the control may be suboptimal, inef-ficient or, in the most critical cases, even unstable. To remedy this problem, the model may be designed based directly on experimental measurements. This procedure, referred to as system identification, is presented in figure 2.4 for the case of subspace techniques. A first stage consists of exciting the system from its input (u) and of simultaneously recording the output (y). Note that u and y may be vectors if the system has multiple inputs or multiple outputs. In a second stage, the resulting input/output signals are used to fit a model in innovation form qk+1 = Aqk + Buk + Lek, (2.5) yk = Cqk + ek, (2.6)
where qk is the state vector at time k, A the state matrix, B the input matrix, C the output matrix and L the Kalman gain. In addition, ek is a Gaussian white noise with second-order moment (covariance) E(ekeHp ) = Rδpk. Finally, the last procedural stage is the design of a controller based on the previously identified model. This may be achieved by using, e.g., a Linear Quadratic Control framework or any other control design approach. Hence, the realistic aspect of the technique presented in this article rests on the use of system identification. This methodology has the strong advantage of being directly applicable to real experiments. In convection-dominated flows, however, some precautions have to be taken to ensure robustness of the results.
A robust strategy: The feed-forward approach
As illustrated in section 2.1.1 using the concept of a visibility length, information in convection-dominated flows essentially travels downstream. Hence, the actuator excitation u has no significant effect on the upstream sensor ys presented in figure 2.3. Applying the system identification procedure directly, as described in the previous section, i.e. with ys treated as an output, would therefore either fail or be very non-robust. For this reason, the feed-forward approach consists of treating the upstream sensor ys as an input in the identification and control procedure. In other words, the spy sensor ys provides information about the incoming perturbations generated by w. Since the input w is unknown, the feed-forward strategy consists of replacing w by ys in both the identification and control designs.
More precisely, a system is identified with inputs ys and u and with output y. This identified model can directly be used for the estimation since all inputs are assumed to be known (w has been replaced by ys). The controller may then be obtained from the solution of an algebraic Riccati equation as for instance in the Linear Quadratic Regu-lator (LQR) framework. Following this formulation, the control was applied for varying spy sensor locations and the resulting efficiency (reduction factor of the objective sen-sor amplitude due to the control) is displayed in figure 2.5. Fifty realizations of system identification and control were performed for each spy sensor placement, and the average is represented by a continuous black line. The standard deviation is also indicated in the form of error bars. In this figure, it is observed that the control efficiency drops precipitously as the spy sensor is placed downstream of the actuator, which is in line with the previous results of section 2.1.1. In addition, the feed-forward identification and control may be compared with optimal, but unrealistic, results obtained from an LQG control (red plus symbols) and an LQR implementation (green horizontal line). When the sensor is placed sufficiently upstream, the realistic technique presented in this article gives a control efficiency of the same order as optimal controllers that assume either full knowledge of the system (LQG) or even full knowledge of the state (LQR). Finally, we note that the standard deviation of the efficiency, indicated by the error bars, is rather small when compared to the total efficiency, indicated by the continuous blue line. A robust implementation of this approach is therefore ensured. To proceed to more realistic configurations, the technique is now applied numerically to the flow over a two-dimensional backward-facing step at Re = 350. Perturbations are generated by three independent noise sources upstream of the step. They are convected by the flow and are amplified as they travel through the strong shear region downstream of the step (ex-hibiting Kelvin-Helmholtz instabilities). This phenomenon, visualized by the averaged perturbation norm, is illustrated in figure 2.6a. In order to reduce this amplification an actuator is placed just upstream of the step. Two spy sensors, upstream of this actuator, describe the incoming perturbations. They are located, respectively, on the upper wall and on the lower wall. Finally, the control is intended to minimize the fluctuations of the signal measured downstream of the step by an objective sensor. To this end, the previously described feed-forward identification and control procedure is applied and the resulting averaged perturbation norm is shown in figure 2.6b. Perturbations downstream of the step have been reduced by approximately one order of magnitude.
Paper 2: Data-based model-predictive control design for convection-dominated flows
In this paper, an alternative feed-forward identification and control formulation is pre-sented, which is more amenable to a practical implementation in experiments. Rather than resorting to a state-space representation of the input-output relationship, the im-pulse response is first identified and an appropriate control strategy is then applied. This technique relies on a least-squares minimization procedure in order to determine the impulse response of the system. It has the advantage of manipulating quantities that have clear physical meanings such as perturbation convective speeds and characteristic frequencies. The intermediate validation of the results is therefore easier, which a key advantage regarding a physical implementation of the method. In the sequel, a Finite Impulse Response (FIR) model is adopted. This formulation is typically used in the general class of Model Predictive Controllers (MPC).
Table of contents :
1 Introduction
1.1 Active Flow Control
1.2 Flow Instabilities in Convection-Dominated Flows
1.3 Model Design
1.4 Controller Design
1.5 Outline
2 Summary of the articles
2.1 Paper 1: Control of amplifier flows using subspace identification techniques
2.1.1 An efficient control approach: The need for an upstream sensor .
2.1.2 A realistic procedure: Subspace Identification
2.1.3 A robust strategy: The feed-forward approach
2.2 Paper 2: Data-based model-predictive control design for convection-dominated flows
2.2.1 Choice of model structure and identification
2.2.2 Control Design
2.2.3 Results
2.3 Paper 3: Experimental control of natural perturbations in channel flow
2.3.1 Experimental set-up
2.3.2 Controller design and results
3 Conclusions and Outlook
4 Control of amplifier flows using subspace identification techniques
5 Data-based model-predictive control design for convection dominated flows
6 Experimental control of natural perturbations in a channel flow
