Data-driven design of model-free control for reference model tracking based on an ultra-local model: Application to vehicle yaw rate control

Abstract

Lateral vehicle dynamics control is important for autonomous driving. This paper presents a data-driven design of model-referenced model-free control (DD-MR-MFC) based on an ultra-local model for vehicle yaw rate control. The characteristics of lateral vehicle dynamics systems depend on vehicle velocities and weights. For this system, fixed proportional–integral–derivative (PID) controllers cannot provide the desired control performance. Additionally, although model-based control can be applied to lateral vehicle dynamics, the modeling process is time-consuming. To efficiently design controllers that can realize the desired performance, we adopt a model-free approach. In this study, the control law of practical MR-MFC is derived by extending the traditional MFC based on an ultra-local model and using a data-driven design method. The MFC approach can be applied to nonlinear systems with few parameters, and the data-driven method provides optimized parameters from single-experiment time-series data without the need for repeated experiments and system model to be controlled. Additionally, the processing cost is considerably low because the controller parameter can be obtained using least-square methods. The effectiveness of the proposed method is verified using a multibody vehicle simulator. The yaw rate tracking performance is examined under different velocities and loads. Results showed that the root-mean-square error of the proposed method is approximately 1/100th of that when using a fixed PID controller optimized using a data-driven method.

Keywords

Data-driven method virtual reference feedback tuning (VRFT)model-free control model matching vehicle yaw rate control

Introduction

Vehicle yaw rate control is important for autonomous driving. The target yaw rate is calculated for reducing the tracking error between the actual and target paths of an automated driving vehicle. A yaw rate controller calculates the steering angle as a control input signal for reducing the tracking error between the yaw rate and its target value. The desired yaw rate response enables desired path planning^1–3 because path-tracking control is converted into yaw rate control.⁴ The popular conventional methods employ proportional–integral–derivative (PID) control.^5,6 PID control is popular in industry owing to its simplicity^7,8; however, when the vehicle velocity varies, a fixed PID controller cannot realize the desired yaw rate response⁹ because the lateral dynamics of the vehicle system become time-dependent.¹⁰ A variable vehicle velocity changes the cut-off frequency and the direct current (DC) gain of lateral dynamics. These time-variant properties of the vehicle deteriorate the control performance of normal PID control. We can adopt model-based approaches, such as model predictive control,^11,12 sliding mode control,¹³ and robust gain-scheduled control.¹⁴ However, controller performance is dependent on modeling accuracy. Additionally, it may be difficult to build accurate models for industrial systems.

In the past decade, model-free control (MFC) or data-driven control that does not need a plant model has garnered attention because this control can simplify the process of system identification and can be easily used. First, we describe MFC. Many types of MFC have been explored, including MFC based on ultra-local models,¹⁵ model-free adaptive control,¹⁶ controller parameter tuning for vibration control,¹⁷ PID tuning via simultaneous perturbation stochastic approximation,¹⁸ intelligent PID control for Type-1 diabetes,¹⁹ and model-free adaptive iterative learning control for robustness and event-triggered systems.^20,21 Among the numerous types of MFC, we focus on an MFC based on an ultra-local model¹⁵ because it can be applied to nonlinear systems despite its simple controller. This control law is called MFC or intelligent PID control.^15,22 MFC can achieve the desired control response even when the controlled objects exhibit nonlinearity and time-variant properties. Such MFC is efficient because it involves few tunable parameters. For industrial systems, it is crucial to maintain a small number of parameters; therefore, MFC has been widely studied.^1,23–25 However, efficient MFC performance cannot be achieved without optimizing the MFC parameters. Next, we describe data-driven control. Various types of data-driven control have been proposed.²⁶ In this study, we considered data-driven tuning. In particular, virtual reference feedback tuning (VRFT)^27–29 is well-known among the data-driven parameter tuning methods. VRFT considers the model-referenced (MR) problem and allows tuning the controller parameters using single-experiment time-series data without modeling the system. VRFT has been widely studied with regard to its applications in multi-input–multi-output systems,^30,31 linear parameter-varying systems,³² sparse gain–scheduled control,³³ model-free controllers,^34,35 and look-up table type controllers.³⁶ Additionally, VRFT has been used in industrial systems such as vehicle lateral control,⁹ bicycle control,³⁷ internal-combustion engines,³⁸ and clutch actuators.³⁹

In this paper, we propose a data-driven MR MFC (DD-MR-MFC) design method for vehicle yaw rate control. The proposed method combines the controller structure of the MR MFC (MR-MFC) and data-driven design method based on the VRFT framework. Practical MR-MFC is realized by extending the standard MFC to realize model matching for a vehicle system. We demonstrate that the traditional MFC is not suitable for the MR problem and derive MR-MFC that contains feedback and estimation terms. The estimation term of the traditional MFC adopts algebraic identification techniques that provide an accurate estimate but incur high calculation cost. Thus, as a practical solution to this issue, we adopt low-pass filters in the proposed method. When accurate estimation is possible, the feedback controller can be uniquely and analytically determined. In practice, an estimation error may arise because of the phase delay caused by the filter. Thus, the MR-MFC parameters should be tuned; accordingly, we adopt the VRFT framework for the data-driven method. The desired yaw rate control, which depends on vehicle velocity, can be achieved by introducing practical MR-MFC that can be designed based on single-experiment time-series data by introducing data-driven tuning.

The effectiveness of the proposed method (DD-MR-MFC) is verified using multibody vehicle (truck) simulators to consider various vehicle velocities and vehicle weights. Additionally, the performance of the proposed method is compared with the performances of other controllers, such as the standard PID tuned by VRFT and MR-MFC tuned by an analytical scheme. Simulation results showed that the proposed method provides the best tracking performance for the yaw rate and that its parameters can be designed using the proposed data-driven method. The contributions in this study can be summarized as follows:

MR-MFC is introduced to achieve the desired tracking performance. The analytical tuning scheme is derived by assuming ideal estimation.

The proposed DD-MR-MFC method is built on the VRFT framework without assuming ideal estimation. In this method, the design parameters of practical MR-MFC are directly optimized from data without a system model.

DD-MR-MFC method is applied to lateral vehicle dynamics control using multibody vehicle (truck) simulators for examining its performance under various vehicle velocities and weights. Results show that the proposed method provides the desired control performance.

The remainder of this paper is structured as follows. Section “Problem formulation” presents the problem formulation. Section “Controller structure” describes the traditional MFC and proposed practical MF-MFC. Section “Data-driven design for MR-MFC” explains the data-driven tuning method for practical MR-MFC. The numerical simulation results are presented in Section “Simulation verification.” Finally, Section “Conclusion” concludes this paper.

[Notation] To enhance readability, we denote the output time signal y(t) of G(s) relative to the input time signal u(t) as y = G(s)u and the time-derivative signal $\overset{\cdot}{y}$ as $\overset{\cdot}{y} = sy$ . Throughout this paper, we omit the notation “s” from G(s) if the context clarifies that G(s) is a rational function with respect to s.

Problem formulation

We explored MR yaw rate control. Figure 1 provides an overview of the system configuration for yaw rate control. Figure 1(a) shows a system of a front-wheel steering vehicle. In the figure, $V \in R$ represents the vehicle velocity; $u \in R$ represents the steering angle; and $y \in R$ represents the yaw rate. The front wheels are controlled by inputting the steering angles to the left and right. Figure 1(b) shows the MR yaw rate control system. The front-wheel steering angle $u \in R$ is computed based on the errors between the target yaw rate $r \in R$ and actual yaw rate $y \in R$ . The target yaw rate is obtained from an autonomous driving system. $T_{d}$ represents a reference model (a user-defined transfer function), and $y_{d} \in R$ is the desired yaw rate (a reference model output). We aim to match the actual and desired yaw rates.

Figure 1.

Control system configuration: (a) front-wheel-steering vehicle and (b) model-referenced (MR) control.

Furthermore, we briefly describe the characteristics of lateral vehicle dynamics because a model-free approach is adopted. The physical model of the general nonlinear two-track vehicle dynamics is described in Appendix 1. Lateral vehicle dynamics depend on vehicle velocity, as shown in Figure 2. This figure shows the frequency response of a vehicle model.¹ At higher vehicle velocities, the DC gain is higher and the cut-off frequency is lower, while at lower velocities, the DC gain is lower and the cut-off frequency is higher. Therefore, lateral vehicle dynamics have parameter-varying characteristics for a given vehicle velocity. Additionally, the weight of a truck changes according to the loads it carries, that is, the vehicle dynamics characteristics are variable. To address these problems, we adopt a model-free approach; in this approach, we do not use the model of the system to be controlled. We aim to construct the yaw rate MFC and introduce the data-driven design method for the proposed MFC from single-experiment data to realize model matching as follows:

J_{MR} = \frac{1}{N} \sum_{t = 1}^{N} (y (t) - T_{d} r (t)),

(1)

where $T_{d}$ represents a user-defined transfer function.

Figure 2.

Frequency response characteristics of the vehicle dynamics.

Controller structure

In this section, we describe the traditional MFC based on an ultra-local model and introduce practical MR-MFC.

Traditional MFC based on the ultra-local model

The controlled object is a nonlinear single-input single-output system that can be expressed as follows:

\begin{matrix} y^{(m)} (t) = f_{p} (y^{(m - 1)} (t), \dots, \overset{\cdot}{y} (t), u^{(l)} (t), \dots, \overset{\cdot}{u} (t), u (t)), \end{matrix}

(2)

where $f_{p} ()$ represents an unknown nonlinear function; $u \in R$ represents the control input; $y \in R$ represents the output; and $l$ and $m$ represent unknown orders of the input and output, respectively. Furthermore, $y^{(m)} (t)$ represents the $m$ -th derivative of $y$ with respect to $t$ . The ultra-local model proposed in MFC^15,22 is expressed as follows:

\begin{matrix} y^{(n)} (t) = α u (t) + F (t), \end{matrix}

(3)

where $n \geq 1$ represents the order; $α \in R$ represents a design parameter; and $F \in R$ represents the unmodeled dynamics and disturbance. Assuming $n = 1$ as in previous studies,^1,24,25 (3) becomes

\begin{matrix} \overset{\cdot}{y} (t) = α u (t) + F (t) . \end{matrix}

(4)

Figure 3 shows the traditional MFC based on an ultra-local model; in the figure, r represents the set point and P represents the system to be controlled. Based on the ultra-local model, the MFC law is given as

\begin{matrix} u (t) = α^{- 1} (- \hat{F} (t) + \overset{\cdot}{r} (t) + u_{fb} (t)) \end{matrix}

(5)

with

\begin{matrix} \hat{F} (t) = \overset{\cdot}{y} (t) - α u (t - δ), \end{matrix}

(6)

where $\hat{F} (t)$ represents the estimated value of $F$ and $δ$ represents an extremely short time interval. In addition, $r$ represents a differentiable signal generated by passing the original target value through a low-pass filter. Further, $u_{fb} (t)$ represents the feedback control input generated by the feedback controller. In this study, the feedback controller follows the PI control law:

u_{fb} (t) = K_{i} \int^{} e (t) dt - K_{p} y (t)

(7)

with

\begin{matrix} e (t) = r (t) - y (t) . \end{matrix}

(8)

Remark 1. From the block diagram shown in Figure 3 and the MFC law (5), we can interpret that the terms $- α^{- 1} \hat{F} (t)$ , $α^{- 1} \overset{\cdot}{r} (t),$ and $α^{- 1} u_{fb} (t)$ represent the disturbance observer, feedforward control, and feedback control, respectively. In other words, MFC can be considered a disturbance observer–based controller, which is an extended form of the classical PID controller.⁴⁰

Remark 2. MFC based on the ultra-local model is a considerably useful approach because it can be applied to nonlinear systems or linear parameter-varying (LPV) systems such as vehicle systems despite its simple controller. Hence, we apply MFC to vehicle yaw rate control.

Figure 3.

Model-free control (MFC) with an ultra-local model.

Practical MR-MFC

In this section, we introduce MR-MFC. Based on (4) and (5), the traditional MFC yields the following relationship:

\begin{array}{l} y = \frac{K_{i}}{s^{2} + K_{p} s + K_{i}} r + \frac{s}{s^{2} + K_{p} s + K_{i}} F_{d l t} \\ + \frac{s^{2}}{s^{2} + K_{p} s + K_{i}} r, \end{array}

(9)

where

\begin{matrix} F_{dlt} = F - \hat{F} . \end{matrix}

(10)

In (9), the first term represents feedback control; the second term represents the estimation error for $F$ ; and the third term represents feedforward control. When the time constant $δ$ is extremely small, that is, $F_{dlt} ≅ 0$ , equation (9) can be rewritten as

y ≅ \frac{K_{i}}{s^{2} + K_{p} s + K_{i}} r + \frac{s^{2}}{s^{2} + K_{p} s + K_{i}} r .

(11)

As shown in the problem formulation, we explore the model-matching problem as well as prior data-driven control studies. For model matching, the feedforward term is not necessary. For instance, when the reference model is given as

T_{d} = \frac{ω^{2}}{s^{2} + 2 τ ω s + ω^{2}},

(12)

the PI gain can be uniquely and analytically determined as

\begin{matrix} K_{p} = 2 τ ω and K_{i} = ω^{2}, \end{matrix}

(13)

where $τ$ and $ω$ represent the damping coefficient and natural angular frequency, respectively. Notably, the form of the reference model in (12) is the complementary sensitivity function, which is generally used for reference-tracking model-matching problems. The analytical solution (13) does not depend on the target plants so long as the assumption $F_{dlt} ≅ 0$ holds. The closed-loop transfer function is expressed as

G_{ry} ≅ \frac{K_{i}}{s^{2} + K_{p} s + K_{i}} .

(14)

Thus, model-matching is realized without the feedforward term. In addition, the feedforward term derived from the ultra-local model may negatively affect model matching. Thus, MR-MFC comprises a feedback controller and an estimator as shown in Figure 4. The desired closed-loop response can be realized under the control law given as

\begin{matrix} u (t) = α^{- 1} (- \hat{F} (t) + u_{fb} (t)) . \end{matrix}

(15)

In practice, noise reduction is necessary to estimate F for an actual system. Algebraic identification techniques for estimating F have been proposed earlier¹⁵:

\hat{F} (t) = - \frac{6}{L^{3}} \int_{t - L}^{t} ((L - 2 σ) y (σ) + α σ (L - σ) u (σ)) d σ,

(16)

where L depends on the sampling period and noise level. This approach realizes accurate estimation. However, it may be difficult to implement the mass-product controller because the order of the estimator increases for short sampling periods. Hence, we use low-pass filters that are more suitable for implementation. The estimator is expressed as

\begin{matrix} \hat{F} (t) = W_{F} (W_{y} \overset{\cdot}{y} (t) - α W_{u} u (t - δ)), \end{matrix}

(17)

where $W_{F}$ , $W_{y}$ , and $W_{u}$ represent the low-pass filters of the estimator, system output, and control input, respectively. We use the following low-pass filters:

W_{F} = \frac{1}{τ_{f} s + 1},

(18)

W_{y} = \frac{1}{τ_{y} s + 1},

(19)

W_{u} = \frac{1}{τ_{u} s + 1},

(20)

where $τ_{f}$ , $τ_{y}$ , and $τ_{u}$ represent time constants of the low-pass filters. From (17), the control law can be obtained as

\begin{matrix} u (t) = α^{- 1} (- W_{F} (W_{y} \overset{\cdot}{y} (t) - α W_{u} u (t - δ)) + u_{fb} (t)) . \end{matrix}

(21)

In general, the PI gain determined using (13) cannot provide the desired tracking performance because the analytical solution (13) is based on the assumptions that $F_{dlt} ≅ 0$ without a filter and that noise is absent. However, practical MR-MFC involves low-pass filters for noise and a phase lag is generated, resulting in an error in the estimated $F$ . To address this problem, we adopt a data-driven design for MR-MFC, as described in the next section.

Figure 4.

Model-referenced MFC (MR-MFC) with an ultra-local model.

The difference between this study and previous studies on MFC tuning is described here. In a previous study,⁴¹ a data-driven design based on fictitious reference feedback tuning was proposed for extended MFC; however, a linear system was targeted and $F_{dlt} ≅ 0$ was assumed. In such cases, we can obtain optimized parameters from the analytical scheme (13) without using a data-driven approach. A study on MR-MFC⁴² did not consider the closed-loop stability analysis, practical MR-MFC, or verification of application to vehicle lateral control for trucks. In another study,¹ the PI gain was uniquely determined from (13) by adopting the Kalman filter as an estimator to achieve the highest possible estimation accuracy. This approach realized good performance by tuning the design parameters of the Kalman filter; however, it required repeated testing, imposing a burden on the designer. In this study, the PI gain of MR-MFC is directly tuned using single-experiment data. Hence, it is not necessary to perform repeated experiments and modeling of the system to be controlled.

Stability analysis

Let us now consider the stability analysis of MR-MFC. Practical MR-MFC includes the error estimation $F_{dlt}$ . Thus, from (9), the transfer characteristics of the MR-MFC are given as

\begin{matrix} y = G_{1} r + G_{2} F_{dlt} \end{matrix}

(22)

with

G_{1} = \frac{K_{i}}{s^{2} + K_{p} s + K_{i}},

(23)

G_{2} = \frac{s}{s^{2} + K_{p} s + K_{i}} .

(24)

Let us assume bounded error estimation, that is, $| | F_{dlt} | |_{\infty} < \infty$ . If the real parts of the poles of $G_{1}$ and $G_{2}$ have negative values, then the closed-loop system is stable.

Data-driven design for MR-MFC

In this section, we first describe the standard VRFT, which involves data-driven tuning, and then describe the data-driven method for MR-MFC.

Standard VRFT

VRFT²⁸ is a direct controller parameter-tuning method for a linear time-invariant (LTI) feedback controller with a prespecified structure. Tuning is achieved using input/output data in an open-loop test without a system model. Furthermore, VRFT is an MR data-driven control that automatically tunes the control parameter $ρ$ so that the closed-loop transfer characteristic from the target value (setpoint) $r$ to the output $y$ matches the reference model $T_{d}$ determined by the designer. The VRFT procedure is briefly described in this section. Figure 5 depicts the concept of VRFT. The plant input and output data $D = {u (t), y (t) | t = 1, \dots, N}$ are acquired from a test. The virtual reference signal $r_{v}$ is expressed as

\begin{matrix} r_{v} (t) = T_{d}^{- 1} y (t) . \end{matrix}

(25)

Using $r_{v} (t)$ , the VRFT objective function $J_{VR}$ is expressed as

J_{VR} (ρ) = \frac{1}{N} \sum_{t = 1}^{N} {(u (t) - u_{v} (t))}^{2}

(26)

with

\begin{matrix} u_{v} (t) = C (ρ) (r_{v} (t) - y (t)), \end{matrix}

(27)

where $C$ represents the LTI feedback controller and $ρ$ represents the controller parameters. By introducing prefilter $L_{F}$ , the VRFT cost function (26) can be brought close to the MR cost function (1) as follows:

J_{VR} (ρ) = \frac{1}{N} \sum_{t = 1}^{N} {(u_{L} (t) - C (ρ) e_{L} (t))}^{2}

(28)

with

\begin{matrix} u_{L} (t) = L_{F} u (t), e_{L} (t) = L_{F} (r_{v} (t) - y (t)) . \end{matrix}

(29)

Figure 5.

Concept of virtual reference feedback tuning (VRFT).

Data-driven tuning of the proposed controller

Herein, we introduce DD-MR-MFC based on VRFT. Based on practical MR-MFC, the control input is given as

\begin{matrix} u (t) = - Λ W_{F} W_{y} sy (t) + Λ [\begin{matrix} K_{p} K_{i} \end{matrix}] [\begin{matrix} - y (t) \\ \frac{1}{s} e (t) \end{matrix}], \end{matrix}

(30)

with

\begin{matrix} Λ = α^{- 1} {(1 - W_{F} W_{u} e^{- s δ})}^{- 1} . \end{matrix}

(31)

By replacing $u$ with a virtual input $u_{v}$ , we can express the VRFT cost function for MR-MFC as

J (w) = \frac{1}{N} \sum_{t = 1}^{N} {(d (t) - w^{T} ξ (t))}^{2},

(32)

with

\begin{matrix} w = {[\begin{matrix} K_{p} & K_{i} \end{matrix}]}^{T}, \end{matrix}

(33)

\begin{matrix} d (t) = L_{F} (u (t) + Λ W_{F} W_{y} sy (t)), \end{matrix}

(34)

and

\begin{matrix} ξ (t) = L_{F} Λ [\begin{matrix} - y_{0} (t) \\ \frac{1}{s} (T_{d}^{- 1} - I) y (t) \end{matrix}] . \end{matrix}

(35)

Because the cost function is convex, the optimized parameters can be obtained using the least-squares method:

\begin{matrix} w^{*} = {(Z^{T} Z)}^{- 1} Z^{T} D \end{matrix}

(36)

with

\begin{matrix} Z = {[\begin{matrix} ξ (1) & ξ (2) & \dots & ξ (N) \end{matrix}]}^{T} \end{matrix}

(37)

and

\begin{matrix} D = {[\begin{matrix} d (1) & d (2) & \dots & d (N) \end{matrix}]}^{T} . \end{matrix}

(38)

Algorithm

The algorithm for the direct data-driven tuning method of MR-MFC is as follows:

[Step1] Obtain the input/output data via a test.

[Step2] Set the reference model and prefilter $L_{F} = T_{d}$ .^9,31

[Step3]After calculating $Z$ and $D$ as shown in (37) and (38) from the data obtained in Step 1 and setting the reference model in Step 2, use (36) to obtain the weight coefficients (i.e. MFC design parameters) that minimize the cost function.

In implementing the controller, we use the controller (21) with the tuned parameters.

Remark 3. The VRFT approach does not guarantee closed-loop stability. However, it is an attractive approach when the system is complex or models are unavailable because of high development costs (in terms of the engineering time and required hardware).²⁵

Simulation verification

We verified the effectiveness of the proposed method using a high-accuracy multibody vehicle simulator. We compared the performances of the proposed method, initial PI controller, popular data-driven approach using the PI controller tuned by VRFT, MR-MFC tuned by (13), and MR-MFC tuned by the proposed method. Additionally, the effectiveness of the controller performance for different vehicle weights was examined.

Simulation environment

We used a high-accuracy vehicle simulator that combines MATLAB/Simulink and TruckMaker (IPG Inc.).⁴³ TruckMaker is exclusively used for testing commercial vehicles. As the vehicle models in TruckMaker are designed based on various experimental data and mechanism analyses, they can reproduce nonlinear vehicle characteristics with high accuracy. In the simulation, we built a vehicle model and a running course using TruckMaker and created a controller using MATLAB/Simulink. We set the sampling period of the controller to 0.01 s and discretized the controller. Figure 6 shows the multibody vehicle in TruckMaker. This vehicle model implemented in TruckMaker was modeled as a multidegree-of-freedom multibody dynamics system to reliably simulate the vehicle motion characteristics of the truck ISUZU ELF (Table 1).⁴⁴

Figure 6.

Vehicle model in TruckMaker.

Table 1.

Vehicle parameters.

Name		Value
Loading platform	Length	4480 mm
	Width	1775 mm
	Height	2150 mm
Vehicle body	Length	6380 mm
	Width	1915 mm
	Height	3135 mm
	Height of loading platform	925 mm
Weight	Body	2920 kg
Weight	Load	350 kg

Initial run results

This section describes the initial results. The desired transfer function is defined as follows:

T_{d} (s) = \frac{1}{{(0.25 s + 1)}^{2}} .

(39)

This reference model has poles with a negative real part, which helps avoid overshoot. The order of the reference model is determined using (12). The time constant is determined based on previous studies.⁴⁵ In practice, the reference model parameters can be set based on prior system information. The fixed PI controller

C_{ini} = K_{p} + \frac{K_{i}}{s}

(40)

is used in a closed-loop test for measuring the initial data. $K_{p}$ and $K_{i}$ are set as 0.5 and 0.8, respectively. Figure 7 shows the initial simulation results with the initial PI controller, where the yaw rate (output), steering angle (input), and vehicle velocity are plotted. We acquire the input/output signals at vehicle velocities of 5 and 60 km/h. Figure 8 shows the time-series data with the initial PI controller at different velocities, where the yaw rate (output) and steering angle (input) are plotted. In the simulations, we set the target yaw rate as ramp waves, where the broken green line (Figure 8) indicates the desired transfer function response. The errors between the desired and actual yaw rate response are extremely large because the PI gain of the initial control system was not tuned.

Figure 7.

Time-series data for the initial proportional–integral (PI) controller.

Figure 8.

Time-series data for the initial proportional–integral (PI) controller at different velocities.

Tuning results

We examined the performances of the PI controller tuned via VRFT (DD-PI), MR-MFC with (13), and MR-MFC tuned using the proposed method (DD-MR-MFC). Table 2 summarizes the tuned parameters. Based on the results of the stability analysis (Section “Stability analysis”), we confirmed that for MR-MFC and DD-MR-MFC, the real part of the poles has negative values. We compared the yaw rate responses by varying the vehicle velocity in the range of 5–60 km/h.

Table 2.

Controller parameters.

	Initial PI controller	PI controller (DD-PI)	Comparison (MR-MFC)	Proposed (DD-MR-MFC)
$τ_{y}$	–	–	0.5	0.5
$τ_{u}$	–	–	0.05	0.05
$τ_{F}$	–	–	0.1	0.1
$α$	–	–	100	100
K_p	0.5	0.0331	8	127
K_i	0.8	1.69	16	253

First, we confirmed the results of the tuned standard PI controller. Figure 9 shows the numerical simulation results for the fixed-gain PI control tuned via VRFT, which is a commonly used data-driven method; this figure shows the plotted yaw rate (output) and steering angle (input) values. The broken green line in the figure indicates the desired transfer function response. The actual yaw rate is close to the desired responses only at a vehicle velocity of 30 km/h. At other vehicle velocities, the desired and actual yaw rates are different. This implies that the standard PI controller, which is an LTI controller, cannot achieve high tracking performance for the lateral vehicle dynamics that depend on vehicle velocity (see section “Problem formulation”). Thus, the result indicates that a controller structure that is applicable to nonlinear (or LPV) systems is necessary for realizing the desired vehicle lateral dynamics control.

Figure 9.

Time-series data with the fixed PI control tuned by VRFT (DD-PI).

Next, we consider the MR-MFC tuned using equation (13). Figure 10 shows the time-series data for this control, where the yaw rate (output) and steering angle (input) values are plotted. This figure shows that the actual yaw rates do not follow the desired responses for different vehicle velocities. As can be seen, the tracking error at low velocities is larger than that at high velocities. This is because the dynamic characteristics undergo greater changes at low velocities than at high velocities. In the analytical tuning scheme (13), we assume that the low-pass filter lag is not considered for estimating F. However, in practice, a low-pass filter is essential for dealing with noise, and the MR-MFC is equipped with such a filter. Thus, the controller with the parameters tuned by (13) provides a low tracking performance. This implies that it is essential to obtain optimal parameters in addition to selecting an appropriate structure of MR-MFC, which has the potential for application to nonlinear (or LPV) systems.

Figure 10.

Time-series data with MR-MFC parameters tuned by (13).

Finally, we tested the proposed method. Figure 11 shows the time-series data applied to the proposed method, that is, DD-MR-MFC, in which the MR-MFC is tuned using a data-driven method. The yaw rate (output) and steering angle (input) values are plotted in the figure. The desired and the actual yaw rate signals almost overlap. The proposed method provides a high tracking performance for vehicle velocities of 5–60 km/h. Table 3 lists the root-mean-square errors (RMSEs) of all methods. The data in the table show that the proposed method provides the best performance. The RMSE for DD-MR-MFC was approximately 1/100th of that for DD-PI and approximately 1/400th of that for the MR-MFC tuned using (13). Thus, the effectiveness of the MR-MFC structure and data-driven tuning method was established. Furthermore, the performance of MR-MFC tuned by (13) was lower than that of DD-PI control in the velocity range of 5–20 km/h. This is because the estimation delay increases in low-velocity ranges where a large change in the steering angle is required. In DD-MR-MFC, the controller parameters are optimized to compensate for the estimation error. A detailed discussion is provided in the next section.

Figure 11.

Time-series data with the proposed method (DD-MR-MFC). The yaw rate of the desired and proposed methods almost overlap.

Table 3.

Root-mean-square errors (RMSEs) of all the examined methods.

Velocity (km/h)	RMSE (×10⁻⁶)
Velocity (km/h)	Initial PI	Fixed PI (DD-PI)	Comparison (MR-MFC)	Proposed (DD-MR-MFC)
5	109	20.7	197	0.170
10	26.0	0.384	68.7	0.0217
20	4.67	4.91	19.1	0.0619
30	4.32	10.0	8.49	0.0943
40	5.98	13.3	4.88	0.113
50	7.47	15.3	3.34	0.123
60	8.62	16.5	2.57	0.130
Average	23.7	11.6	43.4	0.102

We confirm the effectiveness of the proposed method under varying vehicle velocity and various setpoints. Figure 12 shows the time-series data for changing the vehicle velocity using the proposed method; the yaw rate (output), steering angle (input), and vehicle velocity are plotted. The actual yaw rate follows the desired signal. In addition, we confirmed that the proposed method provides robust performance under different vehicle loads. This consideration is particularly important for trucks, which are vehicles that tend to carry widely varying loads. Notably, we used the controller designed for a load of 350 kg (Table 2). Table 4 lists the RMSEs for the same target signal and velocity as those presented in Figure 12 and for a load of 3000 kg in addition to 350 kg. The data presented in Table 4 show that the proposed method provides the best tracking performance for different loads. These results show that the proposed DD-MR-MFC is effective for the lateral dynamics control of trucks despite the simple controller. Such performance cannot be easily achieved using the model-based approach because it requires an accurate vehicle model, including real-time weight estimation. Additionally, the model-based controller tends to be complex. Thus, the effectiveness of the proposed method was validated.

Figure 12.

Time-series data for the case of changing vehicle velocity with DD-MR-MFC. The yaw rate of the desired and response lines almost overlap.

Table 4.

RMSEs of the tuning results obtained under different loads.

Load (kg)	RMSE (×10⁻⁶)
Load (kg)	Initial PI	Fixed PI (DD-PI)	Comparison (MR-MFC)	Proposed (DD-MR-MFC)
350	228	370	320	6.19
3000	426	350	201	5.83

Discussion

We applied the proposed DD-MR-MFC method to vehicle lateral dynamics control. The effectiveness of the proposed method was verified by comparing its performance with the performances of other methods, namely, DD-PI and the MR-MFC tuned using the analytical tuning scheme. The compared methods did not provide good tracking performance with respect to yaw rate. Because DD-PI used the LTI controller, it could not be applied to nonlinear lateral dynamics although PI gain is optimized. In case of MR-MFC, the analytical scheme (13), which requires the assumption of ideal estimation, did not provide the optimal parameters because of the estimation errors of the low-pass filters. Conversely, the proposed method realized high tracking performance for variable vehicle velocities and weights. The proposed method provided the controller parameters that compensate for the effect of the estimation error. The simulation results showed that MR-MFC optimized using a data-driven method is effective for vehicle lateral dynamics control. The proposed data-driven tuning method does not need the model to be controlled and provides the optimal control parameters from single-experiment time-series data. Moreover, this method incurs considerably low optimization cost because it uses the least-squares method. Additionally, the proposed practical MR-MFC has a very simple structure, and it is easy to implement the mass-produced controller.

Furthermore, we discuss the parameters listed in Table 2. The difference between the PI gains of DD-PI and MR-MFC obtained using (13) is explained. When the analytical solution (13) is used, we assume that the error of estimation F is zero ( $F_{dlt} ≅ 0$ ). Based on (4) and (15), the transfer function from $u_{fb}$ to $y$ is expressed as

y (t) = \frac{1}{s} u_{fb} .

(41)

That is, the feedback component of MR-MFC treats (41) as the target plant. Notably, so long as the assumption of $F_{dlt} ≅ 0$ holds, the analytical solution does not depend on the actual plants. In contrast, the standard PI controller (DD-PI) targets the vehicle lateral dynamics presented in (A1), and the optimized parameters change with the characteristics of the targeted systems. Although the gain of DD-PI is less than that of MR-MFC obtained using (13) in the simulation results, this gain may become larger if the target system is changed. Briefly, the gain of MR-MFC is plant-independent and unique, while that of DD-PI is plant-dependent. Therefore, the aforementioned methods provide different gains. Further, we explain why the PI gain of DD-MR-MFC was larger than that of MR-MFC obtained using (13). Unlike the analytical tuning scheme for MR-MFC, DD-MR-MFC does not assume $F_{dlt} ≅ 0$ because the estimator includes low-pass filters and estimation errors may occur. Therefore, the PI gain of the proposed method was larger than that of MR-MFC with (13) to compensate for the effect of the estimation error.

Conclusion

In this study, we propose data-driven MR-MFC, termed DD-MR-MFC, based on an ultra-local model for vehicle yaw rate control. DD-MR-MFC combines the controller structure of the MR-MFC and the VRFT-based data-driven design methods. The effectiveness of the proposed method was verified using a vehicle simulator. Moreover, its performance was compared with those of other methods, such as the standard PI control tuned via VRFT and the MR-MFC tuned using an analytical tuning scheme based on the closed-loop characteristics of MR-MFC. The current methods did not achieve the desired yaw rate control; conversely, the proposed method realized the desired yaw rate response using single-experiment data without modeling the system to be controlled. In future research, we aim to verify the validity of the proposed method using an actual automated vehicle.

Footnotes

Appendix 1

The general two-track vehicle model with front wheel steering is described, as shown in Figure A1.^46–48 According to Newton’s second law, the vehicle dynamics can be expressed as

(A1)

\begin{matrix} {\begin{matrix} m ({\overset{\cdot}{v}}_{x} - v_{y} γ) = F_{xfl} + F_{xfr} + F_{xrl} + F_{xrr} \\ m ({\overset{\cdot}{v}}_{y} + v_{x} γ) = F_{yfl} + F_{yfr} + F_{yrl} + F_{yrr} \\ I_{z} \overset{\cdot}{γ} = l_{f} (F_{yfl} + F_{yfr}) - l_{r} (F_{yrl} + F_{yrr}) + w \end{matrix} \end{matrix}

with $w = {(F_{x f r}^{t} - F_{x f l}^{t}) \cos (δ) + (F_{y f l}^{t} - F_{y f r}^{t}) \sin (δ)} \frac{d}{2} + (F_{xrr} - F_{xrl}) \frac{d}{2}$ . Herein, $m$ and $I_{z}$ represent the vehicle mass and moment of inertia around the yaw axis, respectively; $v_{x}$ , $v_{y}$ , and $γ$ represent the longitudinal velocity, lateral velocity, and yaw rate of the center of mass (CM), respectively; $l_{f}$ and $l_{r}$ represent the distances from the CM to the front and rear axles, respectively; $d$ represents the track width; $F_{xfj}$ and $F_{xrj}$ represent the longitudinal axis tire forces; and $F_{yfj}$ and $F_{yrj} (j \in {l, r})$ represent lateral axis tire forces of the vehicle. $F_{yfj}$ and $F_{yrj}$ are given as follows:

(A2)

\begin{matrix} {\begin{matrix} F_{xfj} = F_{xfj}^{t} \cos (δ) - F_{yfj}^{t} \sin (δ) \\ F_{xrj} = F_{xrj}^{t} \\ F_{yfj} = F_{yfj}^{t} \sin (δ) + F_{yfj}^{t} \cos (δ) \\ F_{yrj} = F_{yfj}^{t} \end{matrix}, \end{matrix}

where $δ$ represents the steering angle; and $F_{xij}^{t}$ and $F_{yij}^{t} (i \in {f, r}, j \in {l, r})$ represent longitudinal and lateral axis tire forces, respectively. $F_{xij}^{t}$ and $F_{yij}^{t}$ can be written as follows:

(A3)

\begin{matrix} {\begin{matrix} F_{xij}^{t} = f_{x} (s_{ij}, α_{ij}, F_{zij}, v_{xij}) \\ F_{yij}^{t} = f_{y} (s_{ij}, α_{ij}, F_{zij}, v_{xij}) \end{matrix}, \end{matrix}

where $s_{ij}$ , $α_{ij}$ , and $F_{zij} (i \in {f, r}, j \in {l, r})$ represent the longitudinal slip ratio, side slip angle, and vertical force of the tires, respectively; and $f_{x}$ and $f_{y}$ represent nonlinear functions. In addition, the rotational dynamics of four wheels are given as

(A4)

J_{wi} ω_{ij} = T_{dij} - T_{bij} - F_{xij}^{t} R_{ij}

with $T_{dfj} = 0 (j \in {l, r})$ , where $J_{wi} (i \in {f, r})$ represents the moment of inertia of the wheel; $ω_{ij}$ represents the rotational velocity of the wheel; $T_{dij}$ and $T_{bij} (i \in {f, r}, j \in {l, r})$ represent the wheel drive and brake torque, respectively; and $R_{ij} (i \in {f, r}, j \in {l, r})$ represents the radius of the wheel.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Shuichi Yahagi

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