# Robust control of drag and lateral dynamic response for road vehicles exposed to cross-wind gusts

## Abstract

A robust closed-loop active flow control strategy for road vehicles under unsteady cross-wind conditions is presented. It is designed based on black-box models identified from experimental data for a 3D bluff body equipped with Coanda actuators along the rear edges. The controller adjusts the blowing rates of the actuators individually, achieving a drag reduction of about \(15\%\) while simultaneously improving cross-wind sensitivity. Hereby, the lateral vehicle dynamics and driver behavior are taken into account and replicated in the wind tunnel via a novel model support system. The effectiveness of the control strategy is demonstrated via cross-wind gust experiments.

## 1 Introduction

Most wind-tunnel investigations of road vehicles are conducted under low-turbulent free-stream conditions, mainly with the objective to reduce aerodynamic drag and thus increase fuel efficiency. However, there has been a growing interest in how unsteady flow phenomena affect vehicle aerodynamics. These effects can be classified into three different groups: time-varying external flow conditions, for example, due to side-wind gusts; unsteady effects created by the lateral or vertical vehicle motion; and self-excited flow characteristics such as wake instabilities. Sims-Williams (2011) gives an overview over the related test methods and results obtained by different researchers. Accordingly, cross-wind gusts with scales of 2–20 vehicle lengths are the most critical, because the observed flow phenomena can differ significantly from quasi-steady conditions. Furthermore, lateral vehicle dynamics and driver behavior are sensitive to disturbances in this frequency range. Wagner (2004) and Krantz (2011) point out the importance of lateral vehicle dynamics and driver behavior when assessing a vehicle’s cross-wind sensitivity and they present corresponding models.

Approaches to improve the cross-wind sensitivity are mainly limited to passive means such as shape optimization (Krajnović and Basara 2009) or adding airflow break-away edges (Krantz 2011). Although active flow control (AFC) methods for bluff bodies are gaining more and more attention, they are still mostly investigated under steady, low-turbulent flow conditions with the objective of minimizing aerodynamic drag. Here, most researchers investigate the Ahmed body with actuation on the rear slant (Brunn et al. 2007; Muminović et al. 2008; Aubrun et al. 2011; Joseph et al. 2012; Park et al. 2013; Gilliéron and Kourta 2013), whereas there are only a few reports about the successful application of AFC to square-back configurations. To this end, Littlewood and Passmore (2012) use steady blowing through a single slot at the upper rear edge of a simplified car model for aerodynamic drag reduction, whereas Englar (2001, 2004) applies pneumatic actuation, exploiting the Coanda effect along all four rear edges of a generic truck-trailer configuration. Englar also suggests asymmetric blowing for generating a yaw moment to counter the effects of cross-wind.

Most of the research on AFC for bluff bodies is carried out with open-loop actuation that is operated at constant amplitudes or frequencies. However, AFC, as opposed to passive means, offers the potential to adapt the actuation to changing flow conditions using feedback. In this paper, we present a closed-loop AFC strategy for road vehicles exposed to cross-wind gusts. The feedback controller is designed based on black-box models identified from input–output measurements. It adjusts the blowing rates at the rear edges of a square-back 3D bluff body individually. The controller reduces aerodynamic drag and controls a weighted sum of side-force and yaw-moment coefficients to reduce cross-wind sensitivity. Thus, the lateral vehicle dynamics and driver behavior are taken into account. The resulting vehicle motion is calculated in real time based on appropriate dynamic models and is executed online by a novel experimental setup. The effectiveness of the controller in reducing the drag coefficient while improving lateral stability is demonstrated via cross-wind gust experiments.

## 2 Experimental setup

### 2.1 Cross-wind tunnel

The experiments are conducted in a special cross-wind tunnel, the concept of which is based on a similar tunnel used by Dominy and Ryan (1999) at Durham University. Figure 1 shows a CAD representation of the cross-wind facility at Technische Universität Berlin. A blowing open-jet wind tunnel with a nozzle exit width and height of \(0.7 \, \hbox {m} \times 0.5 \, \hbox {m}\) is used to generate an axial velocity of up to 20 m/s with a turbulence level of less than \(3 \%\) in the empty test section. A raised splitter plate \(50 \, \hbox {mm}\) above the test-section floor reduces the wind-tunnel boundary layer. The vehicle model is installed at the intersection of the two wind-tunnel axes, at a distance of 0.64 times the vehicle length between the beginning of the raised floor and the front of the body. A second blowing wind tunnel with two outlets, with dimensions of \(1.93 \, \hbox {m} \times 0.5 \, \hbox {m}\) each, is installed at an angle of \(20^\circ\) with respect to the axial jet. When the axial wind-tunnel is set to its maximum velocity of 20 m/s, cross-wind angles of up to \(15^\circ\) can be achieved by running the cross-wind fan at maximum speed. During the cross-wind gust experiments, the transient velocity and cross-wind angle are monitored by a Cobra-shaped 5-hole probe (Aeroprobe, PC5-TIP-2-5-C240-152-025, absolute accuracy better than \(0.4^\circ\)) that is installed at the ceiling of the test section above the model front.

### 2.2 Bluff body

*D*, side force

*S*, and yaw moment

*N*are defined for the coordinate system located at the center of the wheelbase, as shown in Fig. 2. The 5-hole probe measures dynamic and static pressures

*q*and \(p_{s}\), respectively, free-stream velocity \(\mathrm{u}_\infty\), and cross-wind angle \(\beta _w\). The Reynolds number \(Re_{L} = \mathrm{u}_\infty L / \nu\) is defined for model length

*L*and kinematic viscosity \(\nu\). Trip tapes are installed 15 mm behind the front of the body. Their efficiency to promote transition location from laminar to turbulent boundary layer was verified by hot-wire measurements. The nondimensional coefficients for the aerodynamic forces and moments are given by the following:

*i*, and \(\overline{p}_{s}\) refers to the mean, nominal static pressure. Four pneumatic actuators along the vehicle’s rear edges are used for active flow control. They run separately and their velocities can be adjusted individually to apply an asymmetric actuation during cross-wind gusts. To this end, the supply pressures \(p_{a,r}\) and \(p_{a,l}\) to the right and left actuators are controlled separately by a pair of Piezo pressure regulators (Hoerbiger, tecno basic PRE-U). The upper and lower actuator slots have the pressure supply \(p_{a,ul}\). It is adjusted by a third Piezo pressure regulator. Figure 2 shows a close-up view of the actuator geometry. The pressurized air is accelerated through a duct and exits the actuator through a slot with a width of \(w_{\text {S}} = 0.3 \, \hbox {mm}\). The actuator jet stays attached to an adjacent rounded surface due to the Coanda effect and is thus redirected towards the base of the bluff body. The momentum coefficient characterizing the actuation amplitude is defined by the following:

*j*with cross-sectional surface \(A_{a,j}\). At maximum wind-tunnel velocity, the actuators can achieve an overall momentum coefficient \(c_\upmu \le 0.063\). A digital signal processor (dSpace, DS 1005 PPC) running at a sampling frequency of 1 kHz is used for data acquisition, control of the actuators, and real-time simulation of the models for lateral vehicle dynamics and driver behavior.

### 2.3 Lateral vehicle dynamics

#### 2.3.1 Single-track model

*m*and moment of inertia \(J_{z}\) is assumed to be restricted to the horizontal plane, with a center of gravity located at road level at a distance of \(L_{f}\) and \(L_{r}\) from the front and rear axles. The tire properties and steering stiffness are represented by linearized front and rear cornering stiffness coefficients \(C_{\alpha f}\) and \(C_{\alpha r}\). In the classic single-track model, the front wheel angle \(\delta _{\text {f}}\) is the only input variable. Here, we substitute it by the steering angle \(\delta = i_s \delta _{f}\), with steering gear transmission ratio \(i_{s}\). As described by Sackmann and Trächtler (2003), the model is augmented by additional input variables \(F_{y}\) and \(M_{z}\), accounting for the side force and yaw moment acting on the vehicle’s center of gravity due to cross wind. Assuming a constant velocity \(\text {v}\) and small values of the yaw angle \(\psi\), side-slip angle \(\beta\), and steering angle \(\delta\), linearizing the equations for translational and rotational balance of the vehicle yields

*s*and the transfer functions \(G_{\text {a}_l \delta }(s)\) and \(\varvec{G}_{\text {a}_l d} (s)\) for the vehicle’s lateral acceleration response to steering input \(\delta\) and to disturbance inputs \(\underline{d}\). Note that the disturbances side force \(F_{y}\) and yaw moment \(M_{z}\) are defined here with respect to the center of gravity as is common in driving dynamics. For the real-time simulation of the lateral vehicle dynamics, they are calculated from the measured aerodynamic side force

*S*and yaw moment

*N*by

#### 2.3.2 Driver model

#### 2.3.3 Implementation and scaling to wind-tunnel conditions

Coefficients of single-track model and virtual driver

Vehicle | Driver |
---|---|

\(L = 5.6 \, \hbox {m}\) | \(\tau = 0.2 \, \hbox {s}\) |

\(L_{wb} = 3.2 \, \hbox {m}\) | \(T_{S} = 0.2 \, \hbox {s}\) |

\(L_{f} = 1.71 \, \hbox {m}\) | \(T_{I} = 0.7 \, \hbox {s}\) |

\(L_{r} = 1.49 \, \hbox {m}\) | \(T_{P} = 0.7244 \, \hbox {s}\) |

\(d_{CG} = -0.107 \, \hbox {m}\) | \(\phi _{r} = 35^\circ\) |

\(m = 3000 \, \hbox {kg}\) | \(\omega _{c} = 0.35 \cdot 2 \pi \, \hbox {rad/s}\) |

\(J_{z} = 7300 \, \hbox {kg}\,{\hbox {m}^2}\) | \(V_M = 0.4017 \,^\circ \!/\hbox {m}\) |

\(C_{\alpha f} = 157500 \, \hbox {N/m}\) | |

\(C_{\alpha r} = 302500 \, \hbox {N/m}\) | |

\(i_{s} = 17.5\) |

## 3 Flow characteristics

### 3.1 Unsteady cross-wind gust response

The flow around the bluff body at zero yaw separates at the rear edges and forms a highly turbulent, three-dimensional wake. This leads to a low time-averaged base pressure coefficient \(\overline{c}_{p,b} \approx -0.12\) and a high mean drag coefficient \(\overline{c}_{D} \approx 0.43\).

The transient responses of force, moment, and pressure coefficients are shown in Fig. 5 for a gust with a maximum cross-wind angle of \(\beta _{w} \approx 11^\circ\). The depicted time series are phase-averaged over ten identical experiments. The gust begins at \(t^* = t \text {u}_\infty / L \approx 0\) with an increase in normalized total pressure fluctuation \(p'_t/\overline{p}_t\). Here, the total pressure measured by the five-hole probe is decomposed by \(p_t(t) = \overline{p}_t + p^{\prime }_t(t)\) into a steady mean component \(\overline{p}_t\) and an unsteady component \(p^{\prime }_t(t)\). The cross-wind angle starts to increase at \(t^* \approx 3\), approximately when the total pressure reaches its maximum. The response of the drag coefficient is characterized by a significant delay with respect to cross-wind angle. Furthermore, its transient evolution correlates very well with the base pressure coefficient, indicating that the increase in drag is mostly caused by a modification of the wake during the gust. By contrast, the cross-wind gust response of side-force and yaw-moment coefficients is much faster. Their increase is mostly caused by pressure changes at the front of the bluff body’s sides that are immediately affected by the cross-wind angle \(\beta _{w}\), see \(c_{p_{11}}\) and \(c_{p_{17}}\). In comparison, the response of the pressure readings \(c_{p_4}\) and \(c_{p_{24}}\) at the rear sides is much smaller and occurs delayed by approximately 1 convective time unit. This causes a small delay in the build-up of the side-force coefficient during the gust and an overshoot of the yaw-moment coefficient with respect to the cross-wind angle.

In an application to a real vehicle, the force and moment coefficients would not be available as measurement variables. However, the flow characteristics indicate that the main effects can be estimated from surface-pressure measurements. To this end, surrogate variables \(\hat{c}_{D}\), \(\hat{c}_{S}\), and \(\hat{c}_{N}\) are calculated from a weighted sum of the pressure coefficients \(c_{p,b}\), \(c_{p_4}\), \(c_{p_{11}}\), \(c_{p_{17}}\), and \(c_{p_{24}}\), located on the bluff body’s base and sides, respectively. Their weighting parameters were determined by linear regression from a series of steady-state measurements for several Reynolds numbers, cross-wind angles, and actuation amplitudes (Pfeiffer 2016). As can be seen from Fig. 5, the surrogate variables (plotted in red) match the transient evolution of the force and moment coefficients (plotted in blue) very well. They are thus well suited as measurement variables for the closed-loop controller.

### 3.2 Actuated flow

*j*, respectively. Overcoming the drag force \(D_0\) at driving speed \(\mathrm {u}_\infty\) requires the power

## 4 Model identification

The feedback controller for the bluff body is designed based on dynamic black-box models identified from experimental data. This approach has been used successfully in many previous applications of closed-loop AFC (see, e.g., Pastoor et al. 2008; Pfeiffer and King 2012). Here, we identify separate models for the actuator dynamics and the actuated flow dynamics to describe their respective effect on the overall plant dynamics.

### 4.1 Actuator dynamics

### 4.2 Actuated flow dynamics

### 4.3 Overall plant model for control design

*f*. An upper bound for the uncertainty is chosen here as a scalar transfer function

## 5 Control design

### 5.1 Robust \(H_\infty\) feedback controller

### 5.2 Reference filter

*k*is chosen equal to the steady-state derivative of \(y_2\) with respect to \(\beta _{w}\) of the natural flow. As can be seen from the lower plot in Fig. 11, this results in a similar steady-state behavior of \(y_2\) for natural and controlled flow for version 2 of the reference value calculation. Therefore, a more efficient, approximately symmetric actuation can now be applied under constant side-wind conditions, allowing for a lower setpoint for the drag coefficient.

## 6 Results

The developed closed-loop AFC strategy for the 3D bluff body was tested in wind-tunnel experiments with cross-wind gusts and real-time replication of the lateral vehicle dynamics and driver behavior. The control objective here is to achieve maximum net power savings by providing an efficient drag reduction while reducing lateral deviation and yaw rate during the gust.

Experimental results are shown in Fig. 12 for the controlled flow relative to the natural flow for gusts with a maximum cross-wind angle \(\beta _{w} \approx 10^\circ\). As can be seen from Fig. 12a, c, the controller successfully regulates the outputs \(\underline{y}\), so that they follow their setpoint trajectories \(\underline{r}\). This ensures a significant drag reduction relative to the natural, uncontrolled flow. Only at the beginning of the gust is a brief deviation of \(y_1\) from its setpoint \(r_1\) visible. The second output variable \(y_2\) represents the weighted sum of side-force and yaw-moment coefficients, and approximates their combined effect on the lateral vehicle response. Controlling \(y_2\) to its reference trajectory \(r_2\) results in an increase in side force during the cross-wind transient (Fig. 12e). By contrast, the yaw moment, shown in Fig. 12g, is first reduced to negative values to compensate for the effect of the side-force increase on the lateral vehicle response.

When the cross-wind angle \(\beta _{w}\) approaches a steady value, both the side-force and yaw-moment coefficients reach values similar to those from the natural flow. Thus, a larger control input is only needed during fast changes of \(\beta _{w}\), as in 12i, and the controller suppresses disturbances acting on \(y_2\) only in a frequency range where the driver is not able to react. A significant reduction of the peaks in lateral deviation \(\text {y}_l\), lateral acceleration \(\mathrm {a}_l\), and yaw rate \(\dot{\psi }\) is achieved by closed-loop AFC, as can be seen in Fig. 12b, d, f, respectively. This increases driving safety and comfort for the driver, who has to compensate for the effect of steady cross-wind only by slowly turning the steering wheel.

## 7 Conclusion

Our experimental results for a 3D bluff body exposed to cross-wind gusts show how closed-loop active flow control can be used to improve the steady and transient aerodynamic characteristics of road vehicles with respect to drag reduction and side-wind sensitivity. The nonlinear, parameter-dependent characteristics of the actuated flow require a continuous adjustment of the individual blowing velocities of the Coanda actuators located along the rear edges to ensure efficient drag reduction under time-varying flow conditions. This is achieved by applying feedback control, which offers the additional benefit of favorably shaping the frequency response of the aerodynamic force and moment coefficients. Here, we also take the lateral vehicle dynamics and driver behavior into account. This is investigated with a novel model support system that is capable of replicating the lateral vehicle motion in the wind-tunnel based on a real-time simulation of a single-track model and a virtual driver.

The design of the proposed closed-loop AFC strategy is carried out in two steps. First, a multivariable feedback controller for drag, side-force, and yaw-moment coefficients is designed by mixed-sensitivity \(H_\infty\) loopshaping, such that it provides high tracking performance and fast disturbance suppression. Here, we achieve a nominal bandwidth of 5.6 Hz for the drag coefficient and 6.4 Hz for side-force and yaw-moment coefficients. The controller guarantees robust stability for a set of black-box MIMO models identified from experimental data under all relevant flow conditions. In a second step, the control algorithm is augmented by a look-up table for the most efficient setpoint of the drag coefficient and by a dynamic reference filter for side-force and yaw-moment coefficients. This allows the transient aerodynamic characteristics to be adjusted, such that lateral deviation, yaw rate, and steering effort are significantly reduced during cross-wind gusts. Our AFC strategy provides a drag reduction of about \(15 \%\), which corresponds to about \(8 \%\) net power savings when taking the actuation effort into account. These results demonstrate the advantages of closed-loop AFC for road vehicles over simple open-loop actuation or conventional passive means of aerodynamic optimization that do not offer the possibility of adapting to changing flow conditions.

## Notes

### Acknowledgements

This work was funded by the German Research Foundation (DFG) in the context of the research projects Ki 679/9-1 and Ki 679/9-2 “Regelung instationärer Strömungen um stumpfe Körper unter Berücksichtigung der Fahrzeugquerdynamik”

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