Funnel cruise control

Abstract

We consider the problem of vehicle following, where a safety distance to the leader vehicle is guaranteed at all times and a favourite velocity is reached as far as possible. We introduce the funnel cruise controller as a novel universal adaptive cruise control mechanism which is model-free and achieves the aforementioned control objectives. The controller consists of a velocity funnel controller, which directly regulates the velocity when the leader vehicle is far away, and a distance funnel controller, which regulates the distance to the leader vehicle when it is close so that the safety distance is never violated. We provide a rigorous proof for the feasibility of the overall controller design. The funnel cruise controller is illustrated by a simulation of three different scenarios which may occur in daily traffic.

Introduction

With traffic steadily increasing, simple cruise control (see e.g. Åström & Murray, 2008), which holds the velocity on a constant level, becomes less useful. A controller which additionally allows a vehicle to follow the vehicle in front of it while continually adjusting speed to maintain a safe distance is a suitable alternative. Various methods which achieve this are available in the literature, see e.g. the survey (Xiao & Gao, 2010) on adaptive cruise control systems. A common method is the use of proportional–integral–derivative (PID) controllers, see e.g. Åström and Murray, 2008, Ioannou and Chien, 1993, Ioannou et al., 1993 and Yanakiev and Kanellakopoulos (1998), which however are not able to guarantee any safety.

Another popular method is model predictive control (MPC), where the control action is defined by repeated solution of a finite-horizon optimal control problem. A two-mode MPC controller is developed in Bageshwar, Garrard, and Rajamani (2004), where the controller switches between velocity and distance control. The MPC method introduced in Li, Li, Rajamani, and Wang (2011) incorporates the fuel consumption and driver desired response in the cost function of the optimal control problem. In Magdici and Althoff (2017) a method which guarantees both safety and comfort is developed. It consists of a nominal controller, which is based on MPC, and an emergency controller which takes over when MPC does not provide a safe solution.

Control methods based on control barrier functions which penalize the violation of given constraints have been developed in Ames, Grizzle, and Tabuada (2014) and Mehra, et al. (2015). While safety constraints are automatically guaranteed by this approach, it may be hard to find a suitable control barrier function. Furthermore, these methods are implemented as open-loop control inputs using quadratic programs, which require knowledge of the model parameters and hence are not robust in general. Another recent method is correct-by-construction adaptive cruise control (Nilsson, et al., 2016), which is also able to guarantee safety. However, the computations are based on a so called finite abstraction of the system (which is already expensive) and changes of the system parameters require a complete re-computation of the finite abstraction.

Drawbacks of the aforementioned approaches are that either safety cannot be guaranteed (as in Ioannou and Chien, 1993, Ioannou et al., 1993, Yanakiev and Kanellakopoulos, 1998) or the model must be known exactly (as in Ames et al., 2014, Bageshwar et al., 2004, Li et al., 2011, Magdici and Althoff, 2017, Mehra, et al., 2015, Nilsson, et al., 2016). However, the requirements on driver assistance systems are increasing steadily. It is expected that in the near future autonomous vehicles will completely take over all driving duties. Therefore, a cruise control mechanism is desired which achieves both: under any circumstances (in particular, in emergency situations) the prescribed safety distance to the preceding vehicle is guaranteed and at the same time the parameters of the model, such as aerodynamic drag or rolling friction, need not be known exactly, i.e., the control mechanism is model-free. The latter property also guarantees that the controller is inherently robust, in particular with respect to uncertainties, modelling errors or external disturbances. Another requirement on the controller is that it should be simple in its design and of low complexity, and that it only requires the measurement of the velocity and the distance to the leading vehicle. We stress that in a lot of other approaches as e.g. Bageshwar et al., 2004, Ioannou and Chien, 1993, Li et al., 2011 and Magdici and Althoff (2017) the position, velocity and/or acceleration of the leading vehicle must be known at each time.

In the present work we propose a novel control design which satisfies the above requirements. Our control design is based on the funnel controller which was developed in Ilchmann, Ryan, and Sangwin (2002), see also the survey (Ilchmann & Ryan, 2008) and the recent paper (Berger, Lê, & Reis, 2018). The funnel controller is a low-complexity model-free output-error feedback of high-gain type. The funnel controller is an adaptive controller since the gain is adapted to the actual needed value by a time-varying (non-dynamic) adaptation scheme. It has been successfully applied e.g. in temperature control of chemical reactor models (Ilchmann & Trenn, 2004), control of industrial servo-systems (Hackl, 2017) and underactuated multibody systems (Berger, Otto, Reis, & Seifried, 2019a), speed control of wind turbine systems (Hackl, 2014, Hackl, 2015b, Hackl, 2017), current control for synchronous machines (Hackl, 2015a, Hackl, 2017), DC-link power flow control (Senfelds & Paugurs, 2014), voltage and current control of electrical circuits (Berger & Reis, 2014), oxygenation control during artificial ventilation therapy (Pomprapa, Alfocea, Göbel, Misgeld, & Leonhardt, 2014) and control of peak inspiratory pressure (Pomprapa, Weyer, Leonhardt, Walter, & Misgeld, 2015).

In our design we will distinguish two different cases. If the preceding vehicle is far away, i.e., the distance to it is larger than the safety distance plus some constant, then a velocity funnel controller will be active which simply regulates the velocity of the vehicle to the desired pre-defined velocity. If the preceding vehicle is close, then a distance funnel controller will be active which regulates the distance to the preceding vehicle to stay within a predefined performance funnel in front of the safety distance. The combination of these two controllers results in a funnel cruise controller which guarantees safety at all times. We like to stress that the distance funnel controller does not directly regulate the position of the vehicle, but a certain weighting between position and velocity; hence, a relative degree one controller suffices.

A conference proceedings version of the present paper has been published in Berger and Rauert (2018). In the present journal version we consider the effect of additional disturbances in the model and have added a full proof of the main result in Theorem 2.1. Furthermore, we explain the controller design in more detail and discuss the presence of control constraints.

Also recall that $\wedge$ is the operator of the logical conjunction.

In the present work we consider the framework of one vehicle following another, see Fig. 1.

By $x_l$ we denote the position of the leader vehicle, while $x$ and $v$ denote the position and velocity of the follower vehicle. The change in momentum of the latter is given by the difference of the force $F$ generated by the contact of the wheels with the road and the forces due to gravity $F_g$ (including the changing slope of the road), the aerodynamic drag $F_a$ and the rolling friction $F_r$. Detailed modelling of these forces can be very complicated since all the individual components of the vehicle have to be taken into account. Therefore, we use the following simple models which are taken from Åström and Murray (2008, Sec. 3.1):

$$F_g:{\mathbb{R}}_{\ge 0}\to \mathbb{R},t\mapsto mgsin\theta \left(t\right),$$$$F_a:{\mathbb{R}}_{\ge 0}\times \mathbb{R}\to \mathbb{R},(t,v)\mapsto \frac{1}{2}\rho \left(t\right)C_{d}A{v}^2,$$$$F_r:\mathbb{R}\to \mathbb{R},v\mapsto mg{C}_{r}sgn\left(v\right),$$

where $m$ (in kg) denotes the mass of the (following) vehicle, $g=\text{9.81}\text{m/s2}$ is the acceleration of gravity, $\theta \left(t\right)\in \left[-\frac{\pi }{2}\text{rad},\frac{\pi }{2}\text{rad}\right]$ and $\rho \left(t\right)$ (in kg/m³) denote the slope of the road and the (bounded) density of air at time $t$, resp., $C_d$ denotes the (dimensionless) shape-dependent aerodynamic drag coefficient and $C_r$ the (dimensionless) coefficient of rolling friction, and $A$ (in m²) is the frontal area of the vehicle.

Since the discontinuous nature of the rolling friction causes some problems in the theoretical treatment and in the vehicle following framework the velocities are typically positive, we approximate the $sgn$ function by the smooth error function

$$erf\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_0^z{e}^{-t^2}\mathrm{d}t,z\in \mathbb{R},$$

using the property that ${lim}_{\alpha \to \infty }erf\left(\alpha z\right)=sgn\left(z\right)$ for all $z\in \mathbb{R}$. Therefore, we will use the following model for the rolling friction:

$$F_r:\mathbb{R}\to \mathbb{R},v\mapsto mg{C}_{r}erf\left(\alpha v\right)$$

for sufficiently large parameter $\alpha >0$. For more sophisticated friction models we refer to Armstrong-Hélouvry, Dupont, and Canudas-de Wit (1994) and Leine and Nijmeijer (2004).

The force $F$ which is generated by the engine of the vehicle is usually given as torque curve (depending on the engine speed) times a signal which controls the throttle position, see Åström and Murray (2008, Sec. 3.1). Since the latter can be calculated from any given force $F$ and velocity $v$ (taking the current gear into account), here we assume that we can directly control the force $F$, i.e., the control signal is $u\left(t\right)=F\left(t\right)$. The equations of motion for the vehicle are then given by

$$\dot{x}\left(t\right)=v\left(t\right),$$$$m\dot{v}\left(t\right)=u\left(t\right)-F_g\left(t\right)-F_a\left(t,v\left(t\right)\right)-F_r\left(v\left(t\right)\right)+\delta \left(t\right),$$

with the initial conditions

$$x\left(0\right)=x^0\in \mathbb{R},v\left(0\right)=v^0\in \mathbb{R},$$

where $\delta \in {ℒ}^{\infty }({\mathbb{R}}_{\ge 0}\to \mathbb{R})$ is a bounded disturbance which captures modelling errors, uncertainties and noises, which may be caused by unexpected potholes in the road for instance.

Roughly speaking, the control objective is to design a control input $u\left(t\right)$ such that $v\left(t\right)$ is as close to a given favourite speed $v_{\mathrm{ref}}\left(t\right)$ as possible, while at the same time a safety distance to the leading vehicle is guaranteed, i.e., $x_l\left(t\right)-x\left(t\right)\ge x_{\mathrm{safe}}\left(t\right)$. The safety distance $x_{\mathrm{safe}}\left(t\right)$ should prevent collision with the leading vehicle and is typically a function of the vehicle velocity, but could also be a constant or a function of other variables. In the literature different concepts are used, see e.g. Hong, Park, Yoo, and Hwang (2016) and Santhanakrishnan and Rajamani (2003) and the references therein. A common model for the safety distance that we also use in the present paper is

$$x_{\mathrm{safe}}\left(t\right)={\lambda }_{1}v\left(t\right)+{\lambda }_2$$

with positive constants ${\lambda }_1$ (in s) and ${\lambda }_2$ (in m). The parameter ${\lambda }_1$ models the time gap between the leader and follower vehicle and ${\lambda }_2$ is the minimal distance when the velocity is zero. If for instance ${\lambda }_1=\text{0.5}\text{s}$, then it would take the following vehicle $\text{0.5}\text{s}$ to arrive at the leading vehicle’s present position.

We assume that the distance $x_l\left(t\right)-x\left(t\right)$ to the leader vehicle as well as the velocity $v\left(t\right)$ can be measured, i.e., they are available for the controller design. Apart from that, the controller design should be model-free, i.e., knowledge of the parameters $m,\theta \left(t\right),\rho \left(t\right),C_d,C_r$, and $A$ as well as of the initial values $x^0,v^0$ and the disturbance $\delta \left(t\right)$ is not required. This makes the controller robust to modelling errors, uncertainties, noise and disturbances. Summarizing, the objective is to design a (nonlinear and time-varying) control law of the form

$$u\left(t\right)=F\left(t,v\left(t\right),x_l\left(t\right)-x\left(t\right)\right)$$

such that, when applied to a system (2), in the closed-loop system we have that for all $t\ge 0$

• $x_l\left(t\right)-x\left(t\right)\ge x_{\mathrm{safe}}\left(t\right)$,

• $|v\left(t\right)-v_{\mathrm{ref}}\left(t\right)|$ is as small as possible such that (O1) is not violated.

The final control design will consist of two different funnel controllers for appropriate relative degree one systems. While, in view of the control objective, the system (2) cannot be rewritten as a relative degree one system, this is possible when velocity and distance control are considered separately. This separate consideration may serve as a motivation and therefore we briefly recall the concept of funnel control here.

The first version of the funnel controller was developed in Ilchmann et al. (2002) and this version will be sufficient for our purposes. We consider nonlinear relative degree one systems governed by functional differential equations of the form

$$\dot{y}\left(t\right)=f\left(d\left(t\right),\left(Ty\right)\left(t\right)\right)+\gamma u\left(t\right),$$$$y\left(0\right)=y^0\in \mathbb{R},$$

where $\gamma >0$ is the high-frequency gain and

• $d\in {ℒ}^{\infty }({\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^p)$, $p\in \mathbb{N}$, is a disturbance;

• $f\in \mathcal{C}({\mathbb{R}}^p\times {\mathbb{R}}^q\to \mathbb{R})$, $q\in \mathbb{N}$;

• $T:\mathcal{C}({\mathbb{R}}_{\ge 0}\to \mathbb{R})\to {ℒ}_{loc}^{\infty }({\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^q)$ is an operator with the following properties:

  • $T$ maps bounded trajectories to bounded trajectories, i.e, for all $c_1>0$, there exists $c_2>0$ such that for all $\zeta \in \mathcal{C}({\mathbb{R}}_{\ge 0}\to \mathbb{R})\cap {ℒ}^{\infty }({\mathbb{R}}_{\ge 0}\to \mathbb{R})$ we have $T\left(\zeta \right)\in {ℒ}^{\infty }({\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^q)$ and

    $${\Vert \zeta \Vert }_{\infty }\le c_1⟹{\Vert T\left(\zeta \right)\Vert }_{\infty }\le c_2.$$

  • $T$ is causal, i.e., for all $t\ge 0$ and all $\zeta ,\xi \in \mathcal{C}({\mathbb{R}}_{\ge 0}\to \mathbb{R})$:

    $${\left.\zeta \right|}_{[0,t]}={\left.\xi \right|}_{[0,t]}⟹{\left.T\left(\zeta \right)\right|}_{[0,t]}\stackrel{\mathrm{a.e.}}{=}{\left.T\left(\xi \right)\right|}_{[0,t]},$$

    where “a.e.” stands for “almost everywhere”.

  • $T$ is locally Lipschitz continuous in the following sense: for all $t\ge 0$ and all $\xi \in \mathcal{C}([0,t]\to \mathbb{R})$ there exist $\tau ,\delta ,c>0$ such that, for all ${\zeta }_1,{\zeta }_2\in \mathcal{C}({\mathbb{R}}_{\ge 0}\to \mathbb{R})$ with ${\left.{\zeta }_i\right|}_{[0,t]}=\xi$ and $|{\zeta }_i\left(s\right)-\xi \left(s\right)|<\delta$ for all $s\in [t,t+\tau ]$ and $i=1,2$, we have

    $${\Vert {\left.\left(T\left({\zeta }_1\right)-T\left({\zeta }_2\right)\right)\right|}_{[t,t+\tau ]}\Vert }_{\infty }\le c{\Vert {\left.\left({\zeta }_1-{\zeta }_2\right)\right|}_{[t,t+\tau ]}\Vert }_{\infty }.$$

The functions $u,y:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ are called input and output of the system (6), resp. The operator $T$ is typically the solution operator corresponding to a (partial) differential equation which describes the internal dynamics of the system. More general classes involving nonlinear equations, higher relative degree and unbounded operators are discussed e.g. in Berger et al., 2018, Berger et al., 2019b, Berger et al., 2020, Ilchmann and Ryan, 2009 and Ilchmann, Ryan, and Townsend (2007).

The funnel controller for systems (6) is of the form

$$u\left(t\right)=-k\left(t\right)e\left(t\right),e\left(t\right)=y\left(t\right)-y_{\mathrm{ref}}\left(t\right),$$$$k\left(t\right)=\frac{1}{1-\phi {\left(t\right)}^{2}e{\left(t\right)}^2},$$

where $y_{\mathrm{ref}}\in {\mathcal{W}}^{1,\infty }({\mathbb{R}}_{\ge 0}\to \mathbb{R})$ is the reference signal, and guarantees that the tracking error $e\left(t\right)$ evolves within a prescribed performance funnel

$${ℱ}_{\phi }≔\left\{(t,e)\in {\mathbb{R}}_{\ge 0}\times \mathbb{R}\left|\phi \left(t\right)\left|e\right|<1\right.\right\},$$

which is determined by a function $\phi$ belonging to

$$\Phi ≔\left\{\phi \in {\mathcal{W}}^{1,\infty }({\mathbb{R}}_{\ge 0}\to \mathbb{R})\left|\begin{array}{c}\phi \left(s\right)>0\text{ for all }s>0\text{ and}\hfill \\ \text{for all }\epsilon >0:\hfill \\ {\left.(1∕\phi )\right|}_{[\epsilon ,\infty )}\in {\mathcal{W}}^{1,\infty }([\epsilon ,\infty )\to \mathbb{R})\hfill \end{array}\right.\right\}.$$

The funnel boundary is given by the reciprocal of $\phi$, see Fig. 2. The case $\phi \left(0\right)=0$ is explicitly allowed, meaning that no restriction is put on the initial value since $\phi \left(0\right)\left|e\left(0\right)\right|<1$; the funnel boundary $1∕\phi$ has a pole at $t=0$ in this case.

An important property is that by boundedness of $\phi$ there exists $\lambda >0$ such that $1∕\phi \left(t\right)\ge \lambda$ for all $t>0$. The funnel boundary is not necessarily monotonically decreasing and widening the funnel over some later time interval might be beneficial, e.g., when periodic disturbances are present. For typical choices of funnel boundaries see e.g. Ilchmann (2013, Sec. 3.2).

In Ilchmann et al. (2002), the existence of global solutions of the closed-loop system (6), (7) is investigated. To this end, $y:[0,\omega )\to \mathbb{R}$, $\omega \in (0,\infty ]$, is called a solution of (6), (7), if $y\left(0\right)=y^0$ and $y$ is weakly differentiable and satisfies the differential equation in (6) with $u$ as in (7) for almost all $t\in [0,\omega )$; $y$ is called maximal, if it has no right extension that is also a solution. Note that uniqueness of solutions of (6), (7) is not guaranteed in general.

The following result is proved in Ilchmann et al. (2002).

Theorem 1.1

Consider a system (6) with initial value $y^0\in \mathbb{R}$, a reference signal $y_{\mathrm{ref}}\in {\mathcal{W}}^{1,\infty }({\mathbb{R}}_{\ge 0}\to \mathbb{R})$ and a funnel function $\phi \in \Phi$ such that $\phi \left(0\right)|y^0\left(0\right)-y_{\mathrm{ref}}\left(0\right)|<1$. Then the controller (7) applied to (6) yields a closed-loop system which has a solution, and every maximal solution $y:[0,\omega )\to \mathbb{R}$ is global (i.e., $\omega =\infty$), all involved signals $y,k$ and $u$ are bounded, and the tracking error evolves uniformly within the performance funnel in the sense

$$\exists \epsilon >0\forall t>0:\left|e\left(t\right)\right|\le \phi {\left(t\right)}^{-1}-\epsilon .$$

In Section 2 we present a novel funnel cruise controller which satisfies the control objectives as stated in Section 1.3. The controller is basically the conjunction of a velocity funnel controller and a distance funnel controller, both formulated for appropriate relative degree one systems. Those controllers are presented separately before the final controller design is stated and feasibility is proved. In Section 3 the performance of the controller is illustrated for some typical model parameters and scenarios from daily traffic. Some conclusions are given in Section 4.