Identification-Based Closed-Loop Control Strategies for ... - IEEE Xplore

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Identification-Based Closed-Loop Control Strategies for a Cylinder Wake Flow Ercan Atam, Lionel Mathelin, and Laurent Cordier

Abstract— Four closed-loop control strategies are discussed to reduce the drag of a cylinder wake flow: Linear Quadratic Gaussian (LQG) control, gain-scheduled LQG (GS-LQG) control, gain-scheduled PI control, and multimodel predictive control (M-MPC). The control models are obtained in an input–output framework through ARMAX (for LQG control), multi-ARMAX (for GS-LQG control and M-MPC), and multi-ARX (for gainscheduled PI control). The use of system identification for the underlying flow control problem gets rid of the difficult task of developing accurate and robust reduced-order models for the Navier–Stokes (NS) equations. The control is introduced through sucking of fluid through the cylinder surface. The drag on the cylinder is reduced for all control methods. The robustness of all control strategies is tested against unmodeled disturbances and/or dynamics through a detailed simulation of an NS equationbased model by varying in time the Reynolds number around its nominal value 200. For the considered cylinder wake, the M-MPC approach is the best solution. The application of the presented closed-loop control algorithms for the cylinder drag control as a benchmark problem constitutes promising solutions for other related flow control problems in industries. Index Terms— ARX-ARMAX, closed-loop fluid flow control, cylinder flow drag minimization, gain-scheduling, LQG, multimodel predictive control (M-MPC), system identification.

I. I NTRODUCTION

F

LUID flow control problems exist in a multitude of applications ranging from everyday life to industrial processes. Examples include ground, maritime and airborne transportation, renewable energy, such as wind turbines or waterturbines, mitigation of combustion instabilities, and improvement of industrial processes in which fluid flows or heat transfer convection is present (see [1] for a recent review). Given the economic and societal challenges raised, closed-loop flow control can be a critical enabler to improve efficiency and performance and to save energy. In spite of many available methods and theoretical achievements in control theory, closed-loop control of systems governed by a large number of nonlinear ordinary differential equations (ODEs) in strong interaction remains difficult. For fluid flows of practical interest, the situation is even worse,

Manuscript received April 12, 2016; revised July 6, 2016; accepted July 24, 2016. Date of publication September 15, 2016; date of current version June 9, 2017. Manuscript received in final form August 22, 2016. Recommended by Associate Editor C. Prieur. E. Atam and L. Mathelin are with the Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, Centre National de la Recherche Scientifique, Rue John von Neumann, Campus Universitaire d’Orsay Bât. 508, F-91405 Orsay, France (e-mail: [email protected]; [email protected]). L. Cordier is with the Centre National de la Recherche Scientifique, École Nationale Supérieure de Mécanique et d’Aérotechnique, Institut PPRIME and the Institut Supérieur de l’Aéronautique et de l’Espace, Université de Poitiers, F-86962 Chasseneuil-du-Poitou, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2016.2604779

since they are governed by a system of nonlinear partial differential equations (PDEs) called the Navier–Stokes (NS) equations. These equations are well known for introducing strong nonlinearities, high dimensionality, and time-delays. Compared with other control fields, these features make fluid flow control a challenging area, especially since real applications are targeted. Flow control can be directly formulated as an optimal control problem. Within this framework, the NS equations are used as state equations and are discretized in the spatial variables, resulting in a very large state vector (from millions to billions of spatial nodes). In this case, an optimal controller can be designed using an adjoint-based optimization, which may require considerable computational efforts [2]. As a simplification, NS-model linearizations are sometimes derived around an operating point to design a computationally lowcost controller. In either case, the control is of open-loop type. Another strategy is to first obtain a low-order ODE model from the governing PDE. This approach, called model order reduction in the literature, was developed extensively in the past 15 years. In this context, the system is characterized through its internal description, which in addition to the input and the output describes the whole flow field. In [3], a reduced-order vortex model is exploited to understand the physical mechanisms and then design an efficient feedback control. More often, a control algorithm is directly coupled to the reduced-order model (ROM) of the system. Most popular ways of obtaining ROMs are the celebrated proper orthogonal decomposition (POD) method [4], the more recent dynamic mode decomposition [5], and optimal mode decomposition [6]. Although POD-based modeling and the subsequent control have proved to be success in some applications, for instance [7] and [8], they are generally limited due to the fact that an accurate POD-based modeling of a system described by an NS model requires a significant number of modes, typically above ten, and often lacks robustness with respect to variations in the operating conditions and perturbations. This results in a set of tens of nonlinear ODEs, which is difficult, if not impossible, to control in general. In the linear framework, one could use balanced POD [9] or rely on the eigensystem realization algorithm [10], which typically requires lower modes to capture the input–output dynamics. A modeling alternative is to describe the map between the actuator actions and the sensor measurements, i.e., to adopt an external description and to rely on system identification techniques. The main advantage of such an approach is that no physics-based modeling of the system is necessary. Identification is then ideally suited for the practical conditions of use, since modeling just requires an appropriate set of input–output data (called identification data). Although just

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ATAM et al.: IDENTIFICATION-BASED CLOSED-LOOP CONTROL STRATEGIES FOR A CYLINDER WAKE FLOW

a few, there have recently been some applications of flow control using system identification methods during the control model development phase [11]–[14]. For instance, in [13], a linear transfer function of the system is derived and used to synthesize a linear controller. Among the popular black-box model structures are ARMAX/ARX [15] and state space (using subspace identification) [16]. ARMAX/ARX-based identification is easy, and it has been proved that its model structure is a good representative of many systems of practical interest [11], [14]. In contrast, state-space identification has the advantages of offering more flexibility (through internal states) to represent the system dynamics. Moreover, subspace identification gives the estimates of the covariance matrices of the system noise. The estimation of noise covariances is important, because, during the control phase, state estimation is required, and the noise covariance matrices are typically used as parameters in a state estimation algorithm (for example, Kalman filter), which indirectly affects the performance of the controller based on the identified model and the estimated states. The main advantage of the ARMAX/ARX models over state-space models of the same order is that an ARMAX/ARX model requires fewer parameters to be estimated, and this, in turn, may imply a better robustness of the control model against parametric uncertainties. In this brief, we minimize the wake-induced drag of a cylinder in a laminar uniform flow at Reynolds numbers varying around Re = 200. The cylinder wake flow is a prototype of bluff body (see [17] for a review on the control of bluff body flows), as such, has become a classic benchmark, and has been considered by many authors. Among others, [18] explained the gain window phenomenon observed in experimental and numerical studies at a Reynolds number of 60. In [7], a POD-based ROM coupled with a trust-region optimization approach has been used to derive an optimal control policy, while in [19], a robust control strategy has been proposed relying on both an ROM, robust with respect to uncertainties in the operating conditions, and on a robust formulation of the optimal control problem. These two approaches rely on an ROM of the physical system at hand. Two important limitations can be identified. First, the ROM is derived to approximate the state of the system as well as possible, but may not be accurate when used to evaluate the cost function involved in the control problem. For example, the POD-based ROM is accurate in representing the flow field in the energy norm sense but may not be accurate for approximating the drag of the cylinder. Indeed, the drag is mainly depending on the flow state in a localized region around the cylinder for which the POD-ROM is not necessarily sufficiently accurate. Second, to derive the ROM, knowledge of the full system is required, for instance, the velocity field. This typically restricts the applicability of such approaches to configurations where sophisticated visualization techniques can be employed (e.g., particle-image velocimetry) or where detailed numerical simulations are available. In this brief, neither model nor detailed observation of the system is assumed. Sensors are spatially restricted to lie on the cylinder surface and the flow field cannot be

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observed. This situation is realistic and likely to be common in practice. Therefore, the physical system is modeled through an ARMAX model or a set of ARMAX/ARX models, in an input–output framework. Next, a series of different closed-loop control methods are considered. As a first control method, an Linear Quadratic Gaussian (LQG) controller with an integral action is designed using an ARMAX model structure. The reason for choosing LQG is its robustness property [20]. The main variables affecting the system dynamics and cylinder wake drag are the Reynolds number and the control input level. As a result, a gain-scheduled LQG (GS-LQG) control based on multi-ARMAX models is used with the Reynolds number as the scheduling parameter. The other two control methods considered are GS-PI where the control input is a scheduling variable and multimodel predictive control (MMPC), where each MPC controller is valid within a specific input range. The performances of all control methods are compared. All controllers were tested on the nominal NS model at Re = 200, and their robustness is also verified through extensive simulations with a time-varying Reynolds number around the nominal value. In all control methods, the target is drag minimization, which is controlled by sucking some fluid through the cylinder surface. The objective of the control is of tracking type. The brief is structured as follows. In Section II, the cylinder wake flow and its induced drag are defined. ARMAX/ ARX-based modeling of drag is carried out in Section III. The four control strategies, tested to control the cylinder wake drag, namely LQG, GS-LQG, GS-PI, and M-MPC, are described in Section IV, and their performances are compared in Section V. In Section VI, we discuss briefly the robustness issue. Finally, we give the main findings of this brief in Section VII. II. P ROBLEM D EFINITION The 2-D incompressible flow around a circular cylinder of diameter D submitted to a uniform incident flow with constant velocity U∞ is considered at a nominal Reynolds number Re = U∞ D/ν = 200, with ν the kinematic viscosity of the fluid. The value of the Reynolds number is above the critical value Rec  49 of the onset of vortex shedding, so that the uncontrolled flow wake exhibits a periodic vortexshedding pattern. The aim of this brief is to design closed-loop controllers in order to reduce the cylinder total drag coefficient down to around 1 (unit reference tracking). This objective is formulated under a cost function form. The control is achieved by uniformly aspirating some fluid through the porous cylinder surface, hence introducing a nonzero normal velocity at its surface. The suction intensity at each time t, hereafter denoted u(t), is assumed to be in the range u = [−0.25, 0]. To remain consistent with real-life problems, the control intensity is restrained from getting too large. Furthermore, the sensors are only available at the cylinder surface cyl . For the subsequent robustness test of the control strategies, the Reynolds number is assumed to be a stochastic process uniformly distributed in the range Re = [150, 260]. For Re > 270, the closedloop system was not stable for the tested control methods. The control problem is shown in Fig. 1.

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Fig. 1. Sketch of the control problem. The goal is to determine a deterministic control law u(t) (spatially uniform suction at the cylinder boundary) to reduce the cylinder drag for inflows with uncertain Reynolds numbers. Measurements are only available on the cylinder surface cyl .

The flow is simulated by solving the NS equations in their dimensionless ψ - ω form (stream function—vorticity) [21]. For any {u, Re} ∈ u × Re , we solve ⎧ ⎨ ∂ω + v · ∇ω = 1 ∇ 2 ω, v = ∇ ∧ (ψ e ) z ∂t Re (1) ⎩ 2 ωe z = ∇ ∧ v ∇ ψ = −ω, together with the boundary conditions on the cylinder surface v · τ = 0, v · n = u

(2)

with v the fluid velocity vector and e z the direction normal to the 2-D {x, y} plane of the flow. Vectors τ and n are tangential and normal to the cylinder surface, respectively. The drag coefficient d depends on the vorticity field and can be written  ω  (3) τ − pn · (U∞ e x )d d= cyl Re with p the fluid pressure and e x the longitudinal unit vector. The O-shaped computational domain is 30D in diameter around the cylinder, comprising 180 cells both in the radial and azimuthal directions, leading to a 180 × 180-cell grid. The governing equations are solved using a second-order centered finite-difference scheme for the linear terms, while the nonlinear terms (convection) are discretized with a fourthorder upwinded scheme. The time stepping is carried out through the second-order Euler scheme with a time step of 10−2 time units, leading to about 1000 time steps per vortexshedding period. Both the mesh and the time discretization have been carefully checked to be refined enough to get accurate results. Further details on the numerical simulation can be found in [21]. The generation of snapshots covers about ten vortex-shedding periods. III. ARMAX/ARX M ODELING OF D RAG In this section, we will obtain a single ARMAX model to be used for LQG, two sets of multi-ARMAX models to be used for GS-LQG and M-MPC, and multi-ARX models to be used for GS-PI. An ARMAX system with input u, output y, and noise e is represented by the polynomial model A(q −1 )y(k) = B(q −1 )u(k − n k ) + C(q −1 )e(k) where A(q −1 ) = 1 + α1 q −1 + α2 q −2 + · · · + αna q −na B(q −1 ) = β0 + β1 q −1 + β2 q −2 + · · · + βnb q −nb

C(q −1 ) = 1 + γ1 q −1 + γ2 q −2 + · · · + γnc q −nc

(4)

Fig. 2. Spectral analysis of drag for sampling period determination (u = 0).

with q −1 denoting the backward time shift operator. The variable n k ≥ 1 ∈ N is the delay in the input. As mentioned in Section I, ARMAX models have successfully modeled many complex systems thanks to their ability in representing colored noise. It is modeled in (4) by the term C(q −1 )e(k), meaning that the overall effect of process/measurement noise on the output is modeled by filtering of the white noise process e(k) through the linear filter C(q −1 ). An ARX model structure is a special case of ARMAX with C(q −1 ) ≡ 1. Here, the output is selected to be the drag of the cylinder, the input is the normal velocity of the fluid at the cylinder surface (suction), and the typical process noise is due to perturbations in the inflow velocity, or equivalently, the Reynolds number. After selecting the model structure, the next issue is the selection of the identification input to excite the system. Here, we excite the NS model given in (1) with a set of inputs to get the corresponding drag values. Defining an identification data set first requires the determination of the sampling period (ts ) of the system. The determination of ts is crucial: a too large sampling period violates the wellknown Nyquist sampling criterion, while a too small sampling period may be short for control input calculations and/or signal processing computations. The strategy we follow to determine the sampling period is that we consider the unforced response of the system, and start to sample this response from a high sampling rate of 100 Hz to a small sampling rate of 1 Hz. The spectral analysis of the drag values for the unforced response at these sampling frequencies is given in Fig. 2, where d100 , d10 , and d1 denote drag values sampled at 100, 10, and 1 Hz, respectively. We see that the maximum frequency associated with a significant level of energy included in the uncontrolled drag is 0.67 Hz. As a result, the sampling frequency must be at least 2 × 0.67 = 1.34 Hz, meaning that the maximum sampling period should be 1/1.34  0.75 s, which is hereafter the retained sampling period ts . Another issue is the selection of the model order for ARMAX/ARX. In [11], the ARMAX/ARX model orders were selected based on some correlation analysis or some physical arguments. In this brief, we start with a low-order model and then compare the multistep ahead prediction performance of an identified model with the drag values in the identification and

ATAM et al.: IDENTIFICATION-BASED CLOSED-LOOP CONTROL STRATEGIES FOR A CYLINDER WAKE FLOW

TABLE I

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state-space form as

I DENTIFICATION M ODELS U SED IN D IFFERENT C ONTROL M ETHODS AND T HEIR E IGHT-S TEP A HEAD P REDICTION P ERFORMANCES

x(k + 1) = Ax(k) + Bu(k) + Ge(k) y(k) = C x(k) + β0 u(k) + γ0 e(k) where

⎡ ⎡

0

⎢ ⎢ A=⎢ ⎣ In−1 ⎡

validation data. If the performance of the identified model is acceptable, it then can be decided that the model captures well the main dynamics and can, hence, be used for the controller design. The final issue is the selection of the excitation input, which depends on the assumed model order in some sense. An ARMAX model involves n p = n a + n b + n c + 1 and an ARX model n p = n a + n b + 1 unknown model parameters. The excitation input signal should include at least n p /2

distinct frequencies [22] for the unknowns to be identified correctly. The chosen excitation signal is a multisine signal (30 sines) varying in the interval [−0.25, 0] for the single ARMAX model to be used in LQG and for the multiARMAX models to be used in GS-LQG, and in the ranges [−0.25, −0.2], [−0.2, −0.1], and [−0.1, 0] for the multiARMAX/ARX models to be used in M-MPC and GS-PI. Five ARMAX models will be identified for GS-LQG, three ARMAX models for M-MPC, and three ARX models for GS-PI. Each of these ARMAX/ARX models will be identified using inputs in the above-specified input ranges. For GS-LQG, five ARMAX models corresponding to Re grid values from the set {160, 180, 200, 220, 240} will be used. Except ARMAX models for GS-LQG, all models are identified at the nominal Re = 200. Control input is considered as a gain-scheduling variable in GS-PI and as a controller switching variable in M-MPC. The multistep ahead prediction performance of all identified models will be assessed by the index Predm defined as Predm = 1 − ((y − yˆm )/(y − mean(y)), where yˆm is the m-step ahead response from an identified model. The used identified models and their Predm index are summarized in Table I for a validation data set and m = 8. Both identification and validation data sets consist of 800 data points. The ARMAX model in (4) can be written as y(k) = − +

na 

αi y(k − i ) +

i=1 nc 

nb 

−α1

−αn γ0 .. .

⎤

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −αnb +1 β0 ⎥ ⎢ ⎥ B =⎢ ⎥ ⎢ βnb − αnb β0 ⎥ ⎢ ⎥ .. ⎣ ⎦ . β1 − α1 β0

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −αnc +1 γ0 ⎥ ⎢ ⎥, C = [0, . . . , 0, 1]T . G=⎢ ⎥ ⎢ γnc − αnc γ0 ⎥ ⎢ ⎥ .. ⎣ ⎦ . γ1 − α1 γ0 The state-space representation for the ARX models is obtained by setting γi = 0, i = 1, 2, . . . , n c . At this point, one may wonder: 1) why we need to transform the ARMAX/ARX model into an equivalent state-space model and 2) since the ARMAX/ARX model has an equivalent state-space model, then why do not we directly identify a state-space model? The answer to the first question is that we will use LQG and MPCbased control approaches, which require a state-space form. As to the second point, as noticed, the equivalent state-space form for a given ARMAX/ARX model has a special structure, which involves many zeros, and hence, the ARMAX/ARX model identification is simpler. If a direct state-space model structure was identified, it would be very unlikely to obtain the state-space model given in (6), which has many zeros in A, C matrices. Moreover, the identified state-space model will involve many parameters to be identified, which, in turn, may tend to make the identified model less robust against parametric uncertainty when compared with an ARMAX/ARX model (or its equivalent state-space form). IV. D RAG C ONTROL In this section, four different control methods for controlling cylinder wake drag will be presented along with their performance. A. LQG-Based Drag Control Consider the state-space model (6) with β0 = 0

βi u(k − i − n k )

x(k + 1) = Ax(k) + Bu(k) + Ge(k) y(k) = C x(k) + γ0 e(k) + ξ(k)

i=0

γi e(k − i )

⎤ −αn −αn−1 ⎥ ⎥ .. ⎥, . ⎦

−αn β0 .. .

(6a) (6b)

(7a) (7b)

(5)

i=0

with γ0 ≡ 1. Assuming that n a ≥ n b , n a ≥ n c , and n = n a be the size of the state vector x, (5) can be represented in the

where now y denotes the noisy output with measurement noise ξ . The objective is the reduction of the cylinder drag to an acceptable level, which means tracking of a constant reference value dref equal to 1 in our application. We want

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integral action in the tracking, and hence, consider that the tracking error behaves as ˙ = dref − y = dref − C x − γ0 e − ξ

(8)

which can be discretized in time as (k + 1) = (k) + ts (dref (k) − C x(k) − γ0 e(k) − ξ(k)). (9) The augmented system, including as a state, becomes        x B x(k + 1) A 0 + u = 0 (k + 1) −ts C 1 ⎤ ⎡   e G 0 0 ⎣ ξ ⎦ (10) + −ts γ0 −ts ts dref

Fig. 3.

LQG control system structure for cylinder wake drag control.

which can be, more compactly, expressed as x e (k + 1) = Ae x e (k) + Be u(k) + G e w(k) where x e = [x ]T , w = [e ξ dref ]T , and      A 0 B G Ge = Be = Ae = −ts C 1 0 −ts γ0

0 −ts

(11)  0 . ts

Using (7), the state vector x is estimated by a Kalman filter x(k ˆ + 1) = A x(k) ˆ + Bu(k) + L(y(k) − C x(k)) ˆ

(12)

where the noise covariance data Re = E(ee T ), Rξ = E(ξ ξ T ), Reξ = E(eξ T ), and with E(·) the expectation operator, together with system matrices in (7), are used to determine L through an algebraic Riccati equation. The optimal control problem with the integral action then becomes the determination of the control law   xˆ  −K cont xˆe u = [K xˆ K ] to minimize the cost function +∞

J (u) =

 1  T xˆe (k)Q e xˆe (k) + u T (k)Q u u(k) 2

(13)

k=0

0  0 and Q u 0 are the weight 0 Q matrices for the state xˆe and the control input u. Note that a larger Q compared with Q xˆ and Q u means a larger integral action. The solution of this optimization problem is given as where Q e =

Q

xˆ

T K cont = Q −1 u Be S

(14)

where S is the positive semidefinite solution to the following discrete-time algebraic Riccati equation:  −1  T   Be S A e + Q e . S = AeT S Ae − AeT S Be Q u + BeT S Be (15) The controlled system with an integral action and a Kalman filter for state estimation is shown in Fig. 3.

Fig. 4. Nominal system Re = 200) responses for the uncontrolled case (duc , u = 0) and controlled case (d, u) with the LQG controller (Q u = 0.96). The targeted drag is dref = 1.

1) Control With Nominal Re = 200: The ARMAX model is identified for the nominal operating condition Re = 200. During the LQG control design phase, we assume Q e = In xe , where n xe is the number of extended states x e , and we vary Q u . It was observed that a set of LQG controllers could be designed by taking the control weight Q u in the range [0.03, 30]. The choice Q u = 0.96 was optimal, and the performance of the corresponding LQG controller on the NS model with Re = 200 is shown in Fig. 4 in terms of drag coefficient d and command u, where the uncontrolled drag variation (duc ) is also shown. 2) Control With Time-Varying Reynolds Number: Although the performance of the LQG controller on the nominal NS-model was very satisfactory, it is important that the controller performs well in variable conditions around the operating condition. Indeed, a robust controller is crucial and highly desired in practice. We test the robustness of the LQG controller in a scenario where the Reynolds number Re significantly varies over time around the nominal value Re = 200. The Reynolds number is here taken to be as realistic as possible, and is modeled as a random process, mimicking an actual situation where perturbations make the effective Reynolds number to vary. The random process is considered with a Gaussian correlation kernel in time, and the variance is about 30 % of the mean value. The variation of Re, u, and d for the controlled system is shown in Fig. 5 for a much longer time period of 1200 time units. The Reynolds

ATAM et al.: IDENTIFICATION-BASED CLOSED-LOOP CONTROL STRATEGIES FOR A CYLINDER WAKE FLOW

Fig. 7.

Fig. 5. Reynolds number variation, controlled drag, and the corresponding control input for LQG control. The targeted drag is dref = 1.

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GS-PI implementation scheme.

of 23 LQG controllers. Such an effort may not justify its use even if it gives better results than LQG. 3) It is well known that GS control approaches may have poor performance and instability problems [23]. Given that the nonlinearity of the cylinder flow at Re = 200 is already high, it seems that interpolation-based GS-LQG is not an appropriate control method. As a result, we leave this topic as a future study where linear parametervarying (LPV) identification combined with LPV control will be considered, which is a generalization of interpolation-based GS approaches toward a theoretical direction to guarantee stability and performance level. C. Gain-Scheduled PI-Based Drag Control

Fig. 6. Drag coefficient and the corresponding input using GS-LQG for time-varying Reynolds number case. The targeted drag is dref = 1.

number changes in the range [150, 260], around 30% of its nominal value. We find that the robustness performance of the LQG controller is very satisfactory. B. Gain-Scheduled LQG-Based Drag Control In this section, we investigate the possibility of obtaining better control results using GS-LQG. For each ARMAX model corresponding to a Re value in the set {160, 180, 200, 220, 240}, an LQG controller was designed. Linear interpolation is used between system matrices with the current Re value. The performance of the GS-LQG controller is assessed by using the same Reynolds number variation as in Fig. 5. Fig. 6 shows that its performance is inferior to LQG. The possible reasons may be as follows. 1) The Reynolds number variation is not sufficiently small, which is a key factor for interpolation-based GS control methods [23]. 2) The system matrices of controllers are not sufficiently close to each other. This means that a very dense resolution is required in the Re grid set (for example, having a grid set where Re increment is 1–5). However, this will require a more demanding control design: for example, an increment of 5 for Re ∈ [150, 260] will require identification of 23 ARMAX models and design

First, the identified models ARX j , j = {1, 2, 3}, are used to design three PI controllers PI j . The proportional and integral gains of the controllers, K p1 = 0.12, K p2 = 0.15, K p3 = 0.42, K i1 = 0.008, K i2 = 0.04, and K i3 = 0.12, are obtained using the MATLAB control system toolbox for PID tuning. Afterward, considering the control input u as the scheduling variable, the proportional and integral gains of the gain-scheduled PI controller are given through the following double-sigmoid functions: K p2 − K p1    K p = K p1 + 1 + exp s1 u − u Max 1 K p3 − K p2    (16a) + 1 + exp s2 u − u Max 2 K i2 − K i1    K i = K i1 + 1 + exp s1 u − u Max 1 K i3 − K i2    (16b) + 1 + exp s2 u − u Max 2 = −0.1, u Max = −0.2, and s1 = s2 = 600 where u Max 1 2 (the sharpness variables for the double-sigmoid functions). The implementation schematic of the GS-PI controller is shown in Fig. 7. The performance of the GS-PI controller on the NS system is shown in Fig. 8 with the same Reynolds number variation as in Fig. 5. The performance of the GS-PI controller is satisfactory and close to the performance of the LQG controller. D. Multi-MPC-Based Drag Control Let the state-space models corresponding to identified ARMAX models be given as x i (k + 1) = Ai x i (k) + Bi u i (k) + G i e(k) yi (k) = Ci x(k), i = 1, 2, 3.

(17a) (17b)

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Fig. 9. Drag and the corresponding input using M-MPC for time-varying Reynolds number case. The targeted drag is dref = 1. Fig. 8. Controlled drag and the corresponding input using GS-PI for the time-varying Reynolds number case. The targeted drag is dref = 1.

TABLE II C OMPARISON OF C ONTROLLERS ’ P ERFORMANCE IN T ERMS OF RMSE (%) n 2 1/2 FOR dref = 1. RMSE = ((1/n) i=1 (y(i) − dref ) )

The MPC problems are formulated as min

Np  

yˆi (k + j ) − dref (k + j )

2

s.t.

(18a)

j =1

xˆi (k + 1) = (Ai − G i Ci )xˆi (k) + Bi u i (k) + G i y(k) (18b) xˆi (k + ) = A −1 xˆi (k + 1) + i

−2 

− j −2

Ai

Bi u i

j =0

× (k + j + 1), = 2, 3, . . . , N p

(18c)

yˆi (k + ) = Ci x(k ˆ + )x(k + ), = 1, 2, . . . , N p (18d) u i ≤ u i ≤ u¯ i

(18e)

where k is the current time, N p = 8 the prediction horizon, y the measured output, and yˆi the predicted output by the i th ARMAX model. Equations (18b)–(18d) give the prediction responses for models (17) up to N p -step ahead [24]. We chose u 1 = −0.125 (instead of −0.1), u 2 = −0.225 (instead of −0.2), and u¯ 2 = −0.075 (instead of −0.1) to allow some overlap between controllers in order to minimize the number of controller switchings for better transient performance. Let si (k) : u i ≤ u i (k) ≤ u¯ i and define εi (k)  | yˆi (k + 1) − dref (k + 1)|. Then, the switching rule between M-MPCs was set as follows: find index “i ” such that si is “true” and εi (k) is minimum. Then, u(k) = u i (k). The performance of M-MPC is shown in Fig. 9. The results are very satisfactory. We also observe that during system control, the first MPC is not used, since the control input never entered the range [u 1 , u¯ 1 ] = [−0.125, 0].

is moderate. GS-LQG has the poorest performance among all for the time-varying Reynolds number case, for which possible reasons were given in Section IV-B. Remark: The following attempts failed to produce a controller, which works on the NS system under the time-varying Reynolds number scenario. 1) A nonlinear model of the system was obtained through POD and used in the LPV control scheme. It was difficult to obtain an accurate state-space model for the drag, and the applied LPV controller on the NS cylinder system was not stable. 2) An NARX model of the system was identified and used in nonlinear MPC, but this did not work: possibly due to a combination of poor model identification, the lack of stability, and local optima for nonlinear MPC. 3) H2, H∞ , and LQG without an integral action control using different linear identification models failed. 4) Control models obtained from state-space, output error, and Box-Jenkins black-box polynomial models did not work for the tried control techniques. ARX and ARMAX models were accurate probably due to a lower number of parameters to identify.

V. C OMPARING C ONTROLLERS

VI. F RAMEWORK FOR A PPROXIMATE ROBUSTNESS A NALYSIS

The comparison of the performance of all controllers is made in terms of root-mean-square error (RMSE) values. The results are shown in Table II for a long simulation period of 1200 time units. In the time-varying Reynolds number case, the evolution of Re (as a stochastic process uniformly distributed in the range [150, 260]) is the same as in Fig. 5. It can be observed that the best control strategy is M-MPC. The performance gap between the GS-PI and LQG

A framework for robustness analysis of the system under fluctuating Reynolds number and model uncertainty is important for this kind of system. The plant model is a nonlinear partial differential equation (NS equations). Hence, an exact robustness analysis of the closed-loop system is not possible, since a closed-loop expression in the state-space form for the combined controller and plant model cannot be obtained. However, an approximate robustness analysis can be performed

ATAM et al.: IDENTIFICATION-BASED CLOSED-LOOP CONTROL STRATEGIES FOR A CYLINDER WAKE FLOW

based on an LPV model of the system with some uncertainty level and the design of a robust LPV controller. Such a model is given as x(k + 1) = A(ρ(k),  A )x(k) + B(ρ(k),  B )u(k) + G(ρ(k), G )e(k) y(k) = C(ρ(k), C )x(k)

(19a) (19b)

where ρ = Re is the online measurable time-varying Reynolds number, and  X is the uncertainty in the matrix X. The model in (19) can be obtained by interpolation of a set of linear models derived for a grid of Reynolds number values and assuming some level of uncertainty in the obtained system matrices. A method for robust linear parameter-varying controller synthesis and robustness analysis of the type of systems in (19) is presented in [25]. VII. C ONCLUSION Closed-loop fluid flow control is a crucial and challenging area in the flow control community. The difficulty is associated with both the complexity of the NS model and with the limitations of the available closed-loop control techniques for PDE systems. As a result, development of simplified but accurate control model approaches constitutes a very important first step toward obtaining a solution to the problem. Depending on the form of the control model, the selection of a proper closed-loop control strategy constitutes the second step. The success of the second step, the controller design, is highly dependent on the accuracy of control model. In this brief, different identification-based control models and four closed-loop control strategies were considered for cylinder wake drag control problem. The control objective was to reduce the total drag on the cylinder by 40% through reference tracking. The first method designed an LQG controller with integral action, the second method a GS-LQG controller with Reynolds number as the scheduling variable, the third method a GS-PI controller where the control input was the scheduling variable, and the last one an M-MPC controller. All controllers were tested on the cylinder flow system for two cases: with a nominal Re = 200 and with a time-varying Re ∈ [150, 260] to test the robustness of the controllers. Among them, M-MPC was the best solution, and GS-LQG had the poorest results for the time-varying Reynolds number scenario. This benchmark study has illustrated the possibility and power of identification-based modeling and design of different control approaches for handling complex fluid flow control systems. Future extensions of the present work will be on: 1) identification of LPV polynomial models and the design of the corresponding LPV controllers and 2) recursive polynomial identification and the design of the corresponding adaptive controllers, so that the controllers remain effective for a wider range of Reynolds numbers, including the turbulent regimes.

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