Review Article

Adaptive and Model-Based Control Theory Applied to Convectively Unstable Flows

[+] Author and Article Information
Nicolò Fabbiane, Shervin Bagheri, Dan S. Henningson

Department of Mechanical Engineering,
Linnè FLOW Centre,
Royal Institute of Technology (KTH),
Stockholm S-10044, Sweden

Onofrio Semeraro

Laboratoire d'Hydrodynamique (LadHyX),
CNRS-Ecole Polytechnique,
Palaiseau 91128, France

1http://www.mech.kth.se/~nicolo/ (Nicolò Fabbiane)

Manuscript received December 20, 2013; final manuscript received April 11, 2014; published online June 17, 2014. Assoc. Editor: James J. Riley.

Appl. Mech. Rev 66(6), 060801 (Jun 17, 2014) (20 pages) Paper No: AMR-13-1104; doi: 10.1115/1.4027483 History: Received December 20, 2013; Revised April 11, 2014

Research on active control for the delay of laminar–turbulent transition in boundary layers has made a significant progress in the last two decades, but the employed strategies have been many and dispersed. Using one framework, we review model-based techniques, such as linear-quadratic regulators, and model-free adaptive methods, such as least-mean square filters. The former are supported by an elegant and powerful theoretical basis, whereas the latter may provide a more practical approach in the presence of complex disturbance environments that are difficult to model. We compare the methods with a particular focus on efficiency, practicability and robustness to uncertainties. Each step is exemplified on the one-dimensional linearized Kuramoto–Sivashinsky equation, which shows many similarities with the initial linear stages of the transition process of the flow over a flat plate. Also, the source code for the examples is provided.

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Fig. 1

Scheme of a Blasius boundary-layer flow developing over a flat plate. A disturbance modeled by d grows exponentially while convected downstream. The actuator u is used to attenuate the disturbance before it triggers transition to turbulence; the actuation signal is computed based on the measurements provided by the sensor y. The output z, located downstream of the actuator, estimates the efficiency of the control action.

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Fig. 2

Response to a small, localized initial condition in a Blasius boundary-layer flow. A Tollmien–Schlichting wave-packet emerges and grows exponentially while propagating downstream. Contours of the streamwise component of the velocity are shown as a function of the streamwise direction (x) and time (t). The location along the normal-direction y is chosen in the vicinity of the wall.

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Fig. 3

The real frequency ωr and its imaginary part ωi are shown as a function of the spatial frequency α, in (a) and (b), respectively. The relation among the spatial and temporal frequencies is given by the dispersion relation (8). Positive values of ωi characterize unstable waves (gray region).

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Fig. 4

Response to a small, localized initial condition in a 1D KS flow (6) with ℜ = 0.25,P = 0.05, and V = 0.4. The contours are shown as a function of the streamwise direction (x) and the time (t). The initial condition triggers a growing and traveling wave-packet, similar to the 2D boundary-layer flow shown in Fig. 2. [script00.m].

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Fig. 5

Spatial support of the inputs and outputs along the streamwise direction. All the elements are modeled as a Gaussian function (14), with σd = σu = σy = σz = 4.

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Fig. 6

Top frame (a) shows the spatiotemporal response to white noise d(t). (b) The velocity contours are shown as a function of the streamwise direction (x) and time (t). The signals y(t) and z(t) are shown for two different realizations (black and gray lines) in (c) and (d), respectively. Dashed lines indicate the standard deviation of the signals. [script01.m].

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Fig. 8

Controllability (Gc,u) and observability (Go,y) Gramians, normalized by their trace; the absolute values are reported in logarithmic scale as a function of the streamwise direction (x). Due to the symmetry, only the upper/lower triangular part of each Gramian is shown. [script03.m].

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Fig. 7

Time discrete impulse response (○) between the input u to the output z; due to the presence of strong time delays in the system, a lag of t ≈ 550 is observed. The relevant part of the kernel is reconstructed via a FIR filter (◻). [script02.m].

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Fig. 9

Schematic figure showing the 5 transfer functions defining the closed-loop system (35). The transfer functions Pyd,Pzd describe the input/output behavior between the disturbance d and the outputs y and z, respectively; Pyu and Pzu relate the actuator u to the two outputs y and z, respectively, while Kuy is the compensator transfer-function. Because of the convectively unstable nature of the flow, Pyu is negligible for the chosen sensor/actuator locations; thus it does not allow any feedback.

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Fig. 10

The disturbance generated by the impulse response of the system at the actuator location u in (a) is shown as a function of the streamwise direction (x) and time (t). The wave-packet is detected only by the output z (c); due to the convective nature of the flow, the sensor placed upstream of the actuator can not detect the propagating disturbance, and the resulting signal is practically null (b). [script02.m].

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Fig. 11

Control gain K computed using the LQR technique for wz = 1 and wu = 1, (see Sec. 3.1.1). [script04.m]

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Fig. 12

MPC strategy: the controller is computed over a finite time-horizon Tc, based on the predicted time-horizon Tp. Once the solution is available, the control signal is applied on a shorter time windows Ta. In the successive step, the time-window slides forward in time and the optimization is performed again, starting from a new initial condition at t = Ta. The procedure is iterated while proceeding forward in time.

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Fig. 13

Control design in presence of constraints: the gray regions indicate the limits imposed to the amplitude of the control signal u(t). The control u(t) is designed following two different strategies: LQR with a saturation function (–) and constrained MPC (–), see Sec. 3.2.2. The LQR solution (– –) is introduced as reference. The performances of the controllers are shown in terms of rms-velocity reduction in Fig. 14.

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Fig. 14

Control of the KS equation. The rms velocity as a function of the x direction is analyzed; the uncontrolled configuration (–) is compared to three different control strategies already considered in Fig. 13 (same legend).

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Fig. 15.

In (a) the LQR solution (Sec. 3.1.1) is compared to the MPC gains computed for two different times of optimization Tp without constraints, see Sec. 3.2.3. The optimization times are compared to the impulse response Pzu(t) (b). Note that for longer time Tp, covering the main dynamics of the impulse response Pzu(t), the MPC and LQR solutions are equivalent.

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Fig. 16

Kalman estimation gain L computed for Rd = 1 and Rn = 0.1, (see Sec. 4.1.1). [script06.m].

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Fig. 17

Spatiotemporal evolution of the response of the system to a disturbance d(t) (a), compared to the estimated full-order state, using a Kalman filter (b); the contours are shown as a function of the streamwise direction (x) and time (t). The error-norm between the original state and the estimated state is shown in (c). The vertical blue, dashed line indicates when the estimator is turned on. [script06.m].

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Fig. 18

In (a) the evolution of E˜zy(i) is calculated by an adaptive LMS filter and shown as a function of the discrete time (iΔt). The estimation starts at t = 4000, as indicated by a blue dashed line (–). As the iteration progresses, the error-norm constantly reduces (b). [script07.m].

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Fig. 19

Impulse responses (yz) of the estimator as a function of the discrete time. Red circles (○) correspond to the FIR time-discrete Kalman-filter-based kernel E˜zy(i) and the blue squares (◻) to the one identified by the LMS algorithm. [script07.m].

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Fig. 20

Block diagram of the closed-loop system. The compensator, consisting of a controller coupled to an estimator, computes the control signal u(t) given the measurement y(t). The minimization of the measurement z(t) is the target parameter of the controller. Note that in a feedforward controller, the output z can be used to add robustness to the compensator (for instance, in adaptive filters, Sec. 5.4).

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Fig. 21

Spatiotemporal response in presence of a white noise input d(t) for the closed-loop system (a) and the compensator (b); the disturbance is shown as a function of the streamwise direction (x) and time (t). The measurement y(t), feeding the compensator, is shown in (c). At t = 4000 (– –), the compensator starts its action and after a short lag the actuator is fed with the computed control signal u(t). The perturbation is canceled, as shown in the contours reported in (a) and the output z(t) minimized (t  > 5000). [script08.m].

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Fig. 24

Robustness to uncertainties of the system: FXLMS control gain K˜uy(i) (◻) is shifted along the time-discrete coordinate if compared to the static LQG gain (○) to compensate for the unmodeled shift in actuator position. [script10.m].

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Fig. 22

The rms velocity as a function of the streamwise location x is shown for the uncontrolled case (–), the LQG (–), the LQR (– –) and the opposition controller P − τ (–). [script08.m, script09.m].

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Fig. 23

Robustness to uncertainties of the system: the actuator is displaced of 5 length units from its nominal position. The performance of the adaptive filter FXLMS (– – and ·-) is compared to the LQR (– –), LQG (–) and P-τ (–) compensators; as a reference, the uncontrolled case is shown (–). The rms-velocity is shown as a function of the streamwise direction (x). The adaptive filter performs reasonably well in the presence of unmodeled dynamics; the performances are enhanced by the use of a online identified P˜zu (– –). The performances of the LQG (–) and P-τ (–) compensators are significantly reduced (compare with Fig. 22). [script10.m].

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Fig. 25

Two strategies are possible to compute a reduced-order compensator, reduce-then-design and design-then-reduce. In general, the two paths do not lead at the same results.

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Fig. 26

Control configuration for a 3D flow developing over a flat plate. A possible configuration consists of localized sensors and actuators placed along the spanwise direction.




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