1 Introduction
The growing constraints on energy saving in the transportation industry have spurred greater interest in vehicle aerodynamics. Improvement of the aerodynamic performance of road vehicles can increase their efficiency, in particular at highway speeds and under gusty driving conditions. At highway speeds, overcoming aerodynamic drag represents the largest portion of the total power expenditure, over
\(65\%\) of the total power expense according to Hucho and Sovran (
1993) and McCallen (
2004). This explains why most aerodynamic developments on automobiles during the last decades have focused on drag reduction. Most existing approaches to minimize vehicle drag on commercial road vehicles are passive (Gilhaus and Renn
1986). They rely on aerodynamic shaping, such as well-rounded contours, rear slope angle and add-on devices (Cooper
2003; Krajnović
2014). Such passive approaches, however, are restricted by design and practical considerations and cannot be ‘turned off’ when not needed.
Recently, more research activities have focused on active flow control (AFC) solutions. Cattafesta III and Sheplak (
2011) give an extensive overview of possible actuation mechanisms, whereas Choi et al. (
2008) present the most common AFC approaches on bluff bodies. Of the many available approaches, a large subset is considered as an academic exercise, such as rotary (Dennis et al.
2000; Bergmann et al.
2005), streamwise (Cetiner and Rockwell
2001) and transverse (Carberry et al.
2003; Blackburn and Henderson
1999) oscillation of a bluff body. Of the practical and realistic AFC mechanisms, many were investigated on a generic bluff body and the Ahmed body (Ahmed et al.
1984), such as synthetic jets (Li et al.
2022), pulsed pneumatic actuation (Joseph et al.
2013; Rouméas et al.
2009; Pastoor et al.
2008), microjets with steady blowing (Aubrun et al.
2011), blowing and suction slots (Krajnović and Fernandes
2011) and Coanda blowing (Freund and Mungal
1994; Oswald et al.
2019; Herrmann et al.
2020). AFC on heavy truck vehicles were also researched using, for example, synthetic jet (El-Alti et al.
2010) and Coanda actuators (Englar
2001,
2004; Haffner et al.
2020a,
b).
Most of the aforementioned studies focused mainly on drag reduction as the aerodynamic property to optimize. However, drag reduction measures usually affect other aerodynamic characteristics, particularly side-wind stability. Such conflicts are even more likely to occur with rear-end treatment, which is where most AFC systems are usually installed.
The term crosswind stability describes the behavior of a vehicle under flow conditions with lateral velocity components. These flow conditions can be caused by atmospheric winds as well as other vehicles, for example during passing maneuvers, and are strongly dependent on road-side obstacles or the preceding vehicle. Side-wind stability is important for passenger comfort of all ground vehicles. It is however critical and a safety issue for vehicles with large projected side area, such as trucks and buses. Accidents linked to crosswind can happen as it is reported by several governmental agencies (Destatis
2010; Starnes
2006). Side-winds and side-gusts originate from different sources like weather, surrounding traffic or the topology of the terrain next to the road. Several numerical (Guilmineau et al.
2013; de Villiers et al.
2009) and experimental (Gilhaus and Renn
1986; Chadwick et al.
2000) studies were conducted to understand the dynamics of such flows. Both steady (Howell
1993; Baker
1991) and unsteady (Chadwick et al.
2000; Kitoh et al.
2009) crosswind investigations were researched. From these aforementioned studies and others, it was observed that a reduction of the lateral projected area in the back will reduce the rear side force and thereby the yaw moment. However, this relation is not valid for well-rounded rear-end configurations (C and D-pillars). Hence, unsteady effects are expected to play a major role for a heavy vehicle that has a boxy shape with many sharp edges, where the flow is dominated by a large time-averaged recirculation bubble behind the base.
Despite the numerous studies on side-wind effects and driving stabilities, very few have tackled this issue using AFC. To our knowledge, only Englar (
2001,
2004) and Pfeiffer (
2016); Pfeiffer and King (
2018) applied Coanda blowing on the four rear edges of a truck to reduce drag
and to control the yaw moment. Whereas Englar (
2001,
2004) only implemented open-loop AFC, Pfeiffer (
2016); Pfeiffer and King (
2018) successfully applied it as a closed-loop control. The latter authors were able to achieve a maximum drag reduction of 15% and complete authority over the yaw moment. However, an automatic and optimized reduction in actuation power for both drag minimization and yaw control has yet to be attempted.
The above-mentioned studies have one theme in common, which is efficiency. Greener, more efficient and safer vehicles are now important objectives for the car industry, whose emission reduction targets are restricted by the European union (European Parliament, Council of the European Union
2007). This requires that the power gained through active drag reductions be as large as possible in comparison with the expanded power of the blown jet. This is typically quantified through the net power savings,
$$\begin{aligned} \textrm{NPS} = \frac{ (D_0-D) V_{\infty } - \rho _{\textrm{jet}} V_{\textrm{jet}}^3 A_{\textrm{jet}}}{\frac{1}{2} \rho _{\infty } V_{\infty }^3 A_{\textrm{ref}}}\>, \end{aligned}$$
(1)
where
D is the drag force,
\(D_0\) is the reference drag without actuation, and
\(A_{\textrm{ref}}\) is a reference area. The subscripts ‘jet’ and
\(\infty\) denote blown jet and freestream properties, respectively.
An increase in the net power savings can be achieved through periodic blowing. Such observations were initially noted in studies investigating Coanda blowing on airfoils, where large aerodynamic gains were observed with the introduction of periodic actuation (Oyler and Palmer
1972; Jones et al.
2002; Semaan et al.
2021). Periodic Coanda actuation on ground vehicles (generic or otherwise) is less common. Barros et al. (
2016) and Haffner et al. (
2020b,
2021) have shown reduction on a 3D Ahmed body, where drag reduction of up to 20% is reported. No measurement of the net power savings was provided, which renders transferability of the results difficult.
An additional increase in the net power savings can be achieved through closed-loop flow control, which has been rapidly advanced over the past decades with the development of control theory, simulation methods and experimental techniques (Rowley and Williams
2006; Choi et al.
2008). Several control strategies can be found in the literature. A very simple strategy is a model-free approach with direct sensor feedback (Rapoport et al.
2003), or with sensor-based feedback, such as PID control (Roussopoulos
1993). This method is simple to implement and requires only parameter tuning (Killingsworth and Krstc
2006). The resulting control has been successfully applied to single-input single-output opposition and phasor control but comes with no guaranteed performance or optimality property. Furthermore, the large number of tunable parameters makes this approach impractical for multiple actuators, multiple sensors and complex dynamics. A well-investigated model-based control design for multiple sensors and actuators directly uses the input/output signals (black box model) under strong linearity assumptions (Rowley and Williams
2006; Pfeiffer
2016). For complex dynamics with unknown nonlinearities, this control strategy has severe challenges. In contrast, optimal control (Scott Collis et al.
2002; Gunzburger and Lee
2000) is based on a high-fidelity nonlinear Navier–Stokes discretization and minimizes a cost functional. The computational cost, however, excludes any real-time implementation in an experiment. In addition, sensor-based flow estimation may give rise to further technical difficulties.
Control-oriented reduced-order models (ROM) resolving just the key nonlinear actuation mechanisms are a compromise between the online-capable black-box models geared toward linear dynamics and the accuracy of the computationally expensive nonlinear Navier–Stokes-based approaches. These ROMs are typically obtained by performing a Galerkin projection of the Navier–Stokes equations onto the proper orthogonal decomposition (POD) basis (Graham et al.
1999; Noack et al.
2003; Weller et al.
2009; Semaan et al.
2016; Semeraro et al.
2011). Further simplifications of such models can be accomplished, such as the generalized mean-field model by Luchtenburg et al. (
2009). Due to truncation and other errors, such models require calibration, which is accomplished by either the addition of an eddy-viscosity term or by direct identification of the model coefficients. While reduced-order Galerkin models can be pivotal in understanding the actuation mechanisms, they tend to have an intrinsically narrow dynamic bandwidth which limits the use for control design.
Furthermore, many complex nonlinear control tasks, like feedback turbulence control and robot missions (Wahde
2008), are beyond the capabilities of a control-oriented dynamical model. Data-driven methods of machine learning (Bishop
2006) have become increasingly powerful for estimation, prediction and control of many systems. In particular, a control logic may be learned directly from the plant. A well-known example in fluid mechanics is feedback control for skin friction reduction in a wall turbulence simulation using a learning neural network for the control law (Lee et al.
1997). Another approach relies on cluster-based feedback control, which has been successfully applied to control a stalled flow over an airfoil (Nair et al.
2019). The first demonstration of automated learning of a nonlinear closed-loop law taming experimental turbulence was presented by Duriez et al. (
2017) and was shown to outperform all tested state-of-the-art control strategies. Moreover, late progress in reinforcement learning managed to learn feedback control laws (Rabault et al.
2019; Fan et al.
2020).
Recent successes of machine learning control (MLC) via genetic programming control (GPC) comprise drag reduction of the Ahmed body with and without yaw angle (Li et al.
2019,
2018), mitigation of vortex-induced vibration of a cylinder (Ren et al.
2019,
2020), individual pitch control of a floating off-shore wind turbine (Kane
2020), mixing layer control (Parezanović et al.
2016; Li et al.
2020), separation control of a turbulent boundary layer (Debien et al.
2016), recirculation zone reduction behind a backward facing step (Gautier et al.
2015), jet mixing enhancement (Wu et al.
2018; Zhou et al.
2020) and control synthesis for the spatial stabilization of a robot (Diveev and Mendez Florez
2021). GPC has been successful in multiple experiments, where the learned solutions outperformed existing control strategies by exploiting unpredictable nonlinearities. Lately, MLC learning speed was greatly increased thanks to the introduction of intermediate gradient-based steps to exploit local gradient information (Cornejo Maceda et al.
2021), reducing the number of needed evaluations from
\({\mathcal {O}}\)(1000) to
\({\mathcal {O}}\)(100). The acceleration of the optimization process enables new testing opportunities such as learning under unsteady (operating) conditions to improve the control robustness. We refer to an unsteady condition as one that varies with time. This is particularly relevant, because in all aforementioned studies using MLC, the control laws were learned under steady experimental conditions. The benefits of learning under unsteady conditions include improved robustness and automatable identification of an optimal control law. This comes with the price of increased testing time as transient conditions are needed during training. Tang et al. (
2020) bypassed such constraint in numerical simulations with parallelization of the operating conditions for reinforcement learning. The derived control policy was able to efficiently reduce the drag for the trained conditions even for the out-of-design operating conditions. Experimentally, Asai et al. (
2019) identified a feedback control law for stall suppression on an airfoil model in unsteady conditions. The learning was carried out for both fixed and varying angles of attack to determine the actuation command for a discrete (on/off) actuator. However, both learning strategies identified similar controls independent of the operating conditions.
In this study, we leverage the gradient-enriched machine learning control (gMLC) algorithm to identify optimal control laws under unsteady conditions. We report on the first use of machine-learned multiple-input multiple-output control under changing operating conditions. In Sect.
2, we review the gMLC algorithm and illustrate its capabilities on a three-dimensional generic truck model equipped with five pneumatic actuators. Details of the experimental setup and the acquisition techniques are provided in Sect.
3. We present the results of the optimally actuated vehicle under constant and time-varying yaw angle in Sect.
4. Also provided is a comparison between the dynamically identified gMLC performance and that of a linearly interpolated controller from three gMLC control laws at steady conditions. A summary and conclusions are presented in Sect.
5.
5 Summary and conclusions
In this study, we successfully apply gradient-enriched machine learning control (gMLC) to identify an optimized control law under unsteady operating conditions. This marks a departure from previous MLC implementations on steady operating conditions. We report on the first use of machine-learned multiple-input multiple-output control under changing operating conditions. The approach is applied on a coupled oscillator system and a generic truck experiment with five pneumatic actuators. The results demonstrate the method’s capabilities in identifying the most efficient forcing for control under dynamically changing conditions.
The first gMLC implementation under unsteady conditions is performed on a modified coupled oscillator that includes a time-varying coupling parameter. The objective is to stabilize an unstable damped oscillator \((a_1,a_2)\) by exploiting the nonlinear coupling to two linearly stable oscillators \((a_3,a_4)\) and \((a_5,a_6)\) through a forcing term b that acts on both the second and third oscillators. The coupling is achieved through a parameter \(c \in [0;1]\) that defines the coupling level between oscillators. The coupling is exploited to generate unsteady conditions by sinusoidally varying c during the learning process. The results demonstrate the gMLC capability to dynamically adapt to the varying conditions by switching its forcing between the second and the third oscillator depending on the coupling. Moreover, the forcing is performed at the corresponding oscillator frequency without prior explicit inclusion of this information.
The second gMLC deployment is an experimental implementation on a generic truck model with multiple actuators and sensors under unsteady yawing conditions. Actuation is performed by five pneumatic actuators; four are Coanda actuators located at the trailing edges of the body, whereas the fifth is a vertical slot at the model base center, which can be described as a pneumatic splitter plate. The instantaneous feedback signals are provided by the yaw angle encoder, 18 fast pressure sensors distributed over the model and enriched features thereof. The control objective aims for efficient drag reductions at low yaw angles and side force reductions at high yaw angles. Again, gMLC automatically identifies a control law that minimizes this cost function and satisfies the aerodynamic objectives. Moreover, the resultant controller is independent of the explicit yaw angle and only relies on the surface pressure sensor signals and their features. This opens the possibility for commercial application as the wind direction is generally not known.
The control strategy highlights the coupling between the drag and the side-force on this model; side force minimization using active flow control inevitably reduces drag. This results from the control strategy similarly affecting the wake, shrinking the re-circulation region and the fluctuation level.
For both applications, the directly identified optimized control law under unsteady conditions is compared to a parametric control strategy that interpolates between individual gMLC optimized control laws trained at three steady conditions. The gMLC-based parametric control is marginally better than the directly identified one, albeit at approximately three times the training time. Moreover, the interpolated control for the generic truck experiment requires the explicit yaw angle input which is typically not available in a real-life application. We note that problems with a more complex dynamic response would require more than three conditions to interpolate the controller, thus rendering the gMLC-based parametric control strategy even more expensive. In summary, training a control law directly under unsteady conditions remains clearly favorable when accounting for the training cost despite the slight superiority of gMLC-based parametric control.
The current gMLC success opens a new avenue of automated control strategies for flow problems under unsteady conditions. Possible applications range from pitching airfoils to gust encounters, which the authors are currently pursuing.