Drag reduction of a car model by linear genetic programming control

Abstract

We investigate open- and closed-loop active control for aerodynamic drag reduction of a car model. Turbulent flow around a blunt-edged Ahmed body is examined at $Re_{H}\approx 3\times 10^{5}$ based on body height. The actuation is performed with pulsed jets at all trailing edges (multiple inputs) combined with a Coanda deflection surface. The flow is monitored with 16 pressure sensors distributed at the rear side (multiple outputs). We apply a recently developed model-free control strategy building on genetic programming in Dracopoulos and Kent (Neural Comput Appl 6:214–228, 1997) and Gautier et al. (J Fluid Mech 770:424–441, 2015). The optimized control laws comprise periodic forcing, multi-frequency forcing and sensor-based feedback including also time-history information feedback and combinations thereof. Key enabler is linear genetic programming (LGP) as powerful regression technique for optimizing the multiple-input multiple-output control laws. The proposed LGP control can select the best open- or closed-loop control in an unsupervised manner. Approximately 33% base pressure recovery associated with 22% drag reduction is achieved in all considered classes of control laws. Intriguingly, the feedback actuation emulates periodic high-frequency forcing. In addition, the control identified automatically the only sensor which listens to high-frequency flow components with good signal to noise ratio. Our control strategy is, in principle, applicable to all multiple actuators and sensors experiments.

Appendices

Appendix A: Linear genetic programming

A control law maps $N_s$ sensor signals into $N_b$ actuation commands. For simplicity, we assume a single-input plant, i.e. $N_b=1$. We assume this control law can be represented by a given maximum number of instructions. These instructions change the content of $N_\mathrm{r}$ registers, $r_1, \ldots , r_{N_\mathrm{r}}$. The registers may be variables or constants. As concrete example, we assume that the first $N_s$ registers are initialized with the sensor signals, the next $N_b=1$ register represents the actuation command, initially zero, and the next registers contain $N_\mathrm{c}$ constants. These constants are the same for all considered control laws in one optimization.

An instruction includes an operation on one or two registers and assigns the result of the operation to a destination register, e.g. the instruction $r_1:=r_2+r_3$ includes two operands, the register $r_2$ and $r_3$, and assigns the result to $r_1$. One instruction with two operands can be coded as an array of four integers referring to the two operands, the operator and the destination register, respectively. Note that for the instruction with one operand only an array of three integers is required. However, to maintain a unified representation, a fourth integer is equally assigned but ignored. Consequently, the set of $N_i$ instructions can be coded as a matrix $\mathcal {M}$ with dimension $N_{i}\times 4$. An example with $N_{i}=5$ is presented in Fig. 17. Constant registers are write-protected. This means that the constants cannot be destination registers and their values are initialized at the beginning of a run from a user-defined range. One or more variable registers are defined as output register(s). The remaining variable registers are referred to as input registers. For the decoding, the input registers are initialized by the sensor values and the output register(s) by zero. The destination registers are updated after each instruction. After executing all the instructions, the final expression of the output register yields the control law K. This matrix representation can interpret the instructions efficiently by casting the integer values.

There is only a finite number of control laws for a given number of registers $N_\mathrm{r}$, of operations $N_\mathrm{o}$ and of constants $N_\mathrm{c}$:

$$\begin{aligned} \left[ N_\mathrm{r} \times N_\mathrm{r} \times N_\mathrm{o} \times (N_\mathrm{r}-N_\mathrm{c}) \right] ^{N_i} . \end{aligned}$$

This number is, however, astronomical, even accounting for different matrices leading to the same control law. Already the simple matrix of Fig. 17 has over $1.9 \times 10^{14}$ different realizations. Despite the discrete nature of possible control laws, almost any reasonably smooth control law can be approximated by such a set of instructions with suitable number of instructions. Evidently, a combinatorial search of control laws and testing in an experiment is not an option. In contrast, evolutionary algorithms as genetic programming are a near optimum choice. In fact, formulating a function from a set of instructions is the constitutive element of LGP (Brameier and Banzhaf 2007), which is a variant of genetic programming. The term linear in LGP refers to the linear sequence of instructions, and not to superposition principle like in differential equations. The method in itself can provide highly non-linear functions as exemplified in Fig. 17.

Fig. 17

a An example of matrix $\mathcal {M}$ comprising five instructions ($N_i=5$). The matrix is displayed in the centre of the figure. The five instructions are shown on the right side of the matrix. Let $\mathcal {R}=\{r_1, r_2, r_3, r_4, r_5, r_6\}$ denote the set of registers, indexed by the integer numbers $\{1,\ldots ,6\}$. The first four registers are variables, i.e. they can be assigned a new value. The last two registers are constants and therefore write-protected. The operand(s) of instructions are coded in the first two columns of the matrix. They can assume any value from $\{1,\ldots ,6\}$. The operator set $\mathcal {O}=\{+, -, \times , \div , \exp \}$ is indexed by an integer number $\{1,\ldots ,5\}$ and coded in the third column of the matrix. The last column encodes the destination registers, which can be one of the variables from $\{r_1,\ldots ,r_4\}$. b Interpretation of the matrix $\mathcal {M}$. In this example, we have three input registers $\{r_1, r_2, r_3\}$ and one output register $r_4$. Input registers are initialized by the sensors and output register by zero. Step 1 shows the updated registers after implementing the first instruction. Based on this result, we implement the second instruction and obtain the updated result in step 2, etc. The final expressions are obtained after implementing all five instructions. The expression of output register $r_4$ is the targeted function K

The genetic operations (elitism, crossover, mutation and replication) are directly applied to the matrices as depicted in Fig. 18.

Fig. 18

An example showing the realisation of genetic operations on the individuals for a fixed number of instructions

Appendix B: Feedback control using Morlet filters

In this section, we describe the use of Morlet wavelet filter (MF) to extract frequencies of interest in the sensor signals. In time domain, the Morlet wavelet $\psi$ is a cosine function modulated by a Gaussian envelope. It is then defined for a frequency $f_{c}$ as:

$$\begin{aligned} \psi (t)=\dfrac{1}{\sqrt{2\pi }\sigma }\exp \Bigg(-\frac{t^2}{2\sigma ^2}\Bigg)\cos (2\pi f_{c}t). \end{aligned}$$

(15)

In frequency domain, MF is a band-pass filter which attenuates the undesired frequencies outside the range $[f_{c}-\lambda /2, f_{c}+\lambda /2]$, where $\lambda$ represents the bandwidth which is governed by the parameter $\sigma$. In our applications, only the fourth sensor $s_4$ identified for the optimal SIMO control (see Sect. 6.1) is chosen as the output of the plant, resulting in SISO (Single-Input Single-Output) system (see Sect. 6.2). To avoid the confusion, we denote the fourth sensor $s_4$ as s and its fluctuation $s^\prime _4$ as $s^\prime$. The sensor ${\varvec{s}}$ in the feedback control law $b=K({\varvec{s}})$ is defined as ${\varvec{s}}=[\hat{s}_1,\ldots ,\hat{s}_5, \overline{s},s^\prime ]$, where

$$\begin{aligned} \hat{s}_i(t)= & {} \int _{0}^{\tau _P}\psi _i(\tilde{t})s^\prime (t+\tilde{t}-\tau _P)\text {d}\tilde{t},\quad i=\{1,\ldots ,5\}\nonumber \\ \overline{s}(t)= & {} \frac{1}{\tau _P}\int _{t-\tau _P}^{t}s(t)\text {d}t\\ s^\prime (t)= & {} s-\overline{s}(t).\nonumber \end{aligned}$$

(16)

$\psi _i$ represents the ith Morlet wavelet and $\overline{s}$ is the moving average of the signal over a period of $\tau _P=0.1$ s. For $i=\{1,\ldots ,5\}$, we set $f_{c_i}=\{100, 200, 250, 320, 400\} Hz$. The corresponding Strouhal numbers are $St_{H_{c_i}}=f_{c_i} H/U_\infty =\{2, 4, 5, 6.5, 8\}$. Figure 19 represents the five wavelets in the time and frequency domains.

Fig. 19

Morlet wavelets in time domain (left) and frequency domain (right)

One may notice that the centre frequencies in the frequency domain are slightly different compared to the values of $f_{c_i}$. This is related to the frequency resolution of the MF which is determined by the wavelet length $\tau _P$ considered in (16). In the present study, the wavelet includes 200 points for a time window of $\tau _P=0.1$ s within the frequency $f_{RT}=2$ kHz. This leads to a frequency resolution of about $\Delta f=10$ Hz ($\Delta St_{H}=0.2$). The spectra can then be shifted within $\Delta St_{H}=0.2$ with respect to the set ones.