Funnel cruise control with input saturation

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Funnel Cruise Control with Input Saturation

Master Thesis

Applied Mathematics Chair of Hybrid Systems

Author:

Steven F. Horstink

Supervisor:

Dr. Felix L. Schwenninger

Graduation committee:

Prof. dr. H.J. Zwart Dr. ir. D. Dresscher

September, 2020

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Preface

It has been a tumultuous time to write a thesis, considering the outbreak of a virus that resulted in a pandemic.

However, it has granted me the opportunity to write the most part of this thesis at my parents’ home, for which I am grateful. I was able to write this thesis in the calm and quiet backyard of Lichtenvoorde, while my mother forbade me to go out to do groceries out of fear that I would contract the virus. Being a student at the University of Twente and living on its campus is a wonderful experience, but also one without rest. Working at my parents’

home has been very beneficial for my thesis, hence I would like to express my gratitude to you, my beloved parents, for allowing me to use all my time to study and for the social interaction when I was not studying.

During the half year in which I did my research, I had regular meetings with Felix via Skype per my request, which lasted nearly two hours every time. He was quick to reply to e-mails and no question was too much. This kind of supervision has been exactly what I needed, for which I thank you, Felix.

I would also like to take this opportunity to thank you, Anaïs, for always listening to me rambling on about my research, encouraging me and even helping me put it together. Thank you, Dieuwertje, Maaike and Pascal, for the (WhatsApp) fun we had in times of depletion of social interaction.

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Abstract

Funnel Cruise Control was developed as an adaptive cruise control with a combination of distance and speed control, based on the concept of funnel control. It is able to guarantee the safety of the driver by keeping a minimal distance from the preceding vehicle, tracks a desired speed without violating the former condition and can be applied to any model that describes the dynamics of a vehicle, as long as the model satisfies some structural properties. While the performance of Funnel Cruise Control is promising, the presence of saturation bounds on the input threatens the desirable properties of the Funnel Cruise Control. In this thesis, a condition is stated in the form of certain threshold functions that serve as bounds on the saturation bounds. If the condition is satisfied, then the cruise control objectives are achieved. A second approach is considered in which the distance funnel is widened while the Funnel Cruise Control is in use, which aids in case of sudden braking of the preceding vehicle. Additionally, an adjustment to the Funnel Cruise Control is proposed such that design options are available, e.g. faster control performance.

Keywords: cruise control, autonomous driving, adaptive control, funnel control, input saturation, input constraint

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List of Theorems

I Existence of solution . . . 9

II Funnel Control . . . 10

III Funnel Cruise Control . . . 17

IV Funnel Cruise Control with Input Saturation . . . 41

V Funnel Cruise Control with Input Saturation and Restricted Relative Speed . . . 46

List of Definitions

1 Operator class

T

. . . 6

2 System class

S

. . . 6

3 Reference class

R

. . . 7

4 Funnel functions class

Φ

. . . 7

5 Performance funnel

F

_ϕ. . . 7

6 Gain functions class

A

. . . 8

7 Carathéodory function . . . 8

8 C-solution . . . 8

9 Maximal solution . . . 9

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Nomenclature

The list below describes the symbols that will be used frequently throughout this thesis.

[−]

dimensionless parameter;

Ò

≥0

B [0, ∞)

^;

Ò

>0

B (0, ∞)

^;

kx k B p

hx, x i,

the Euclidean norm for

x ∈ Ò

^N^;

B

_ε

(x ) B y ∈ Ò

^N

kx − y k < ε

;

C I → Ò

^N

set of continuous functions

I → Ò

^N

, I ⊂ Ò

an interval;

AC

_loc

I → Ò

^N

set of locally absolutely continuous fuctions

I → Ò

^N

, I ⊂ Ò

an interval;

L

^∞

I → Ò

^N

space of measurable essentially bounded functions

I → Ò

^N with the following norm;

kx k

∞

B

^ess-sup

t ∈I

kx (t )k

;

L

^∞_loc

I → Ò

^N

space of measurable, locally essentially bounded functions

I → Ò

^N^;

W

^1,∞

I → Ò

^N

space of bounded functions

x ∈ AC

_loc

I → Ò

^N

with derivative

x ∈ L Û

^∞

I → Ò

^N

and the following norm;

kx k

_1,∞

B kx k

∞

+ k Ûx k

∞;

x |

J restriction of

x

:

I → Ò

^N ^to

J ⊂ I.

Vectors and vector-valued functions are inbold.

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Section 1 Introduction

As the number of vehicles on the road increases, the probability of traffic accidents increases as well. Considering that most of these accidents are due to human error, we may improve road safety by taking away some of the control from the driver and placing it in the hands of a system designed to safely guide a vehicle through traffic.

While conventional cruise control (CCC) was created to relieve the driver from maintaining a constant speed, it does not aid in reducing the risk of accidents. Adaptive cruise control (ACC), however, is an improvement of CCC with regards to safety: it is able to detect a preceding vehicle and adjust vehicle speed to maintain a specified distance from the preceding vehicle. There exist many kinds of ACC systems, based on different methods, requiring different information about the vehicle or its environment and having different properties.

A recently developed ACC system, called Funnel Cruise Control (FCC), claims to guarantee the preservation of a minimal safe distance in all situations while requiring no information about the vehicle or its environment. As contained in its name, it is an application of funnel control.

Funnel control is a type of control law that tracks a desired reference signal such that the error between the output and reference signal evolves within a prespecified funnel, see Figure 1.1. It may be applied to non-linear high-gain MIMO systems of relative degree one (i.e. the first derivative of the system state can be controlled by the input), although extensions exist to higher relative degree. It has applications in for example temperature, current and speed and position control. One of its desirable features is that the control law is model-free and may be parametrized arbitrarily. In general, however, asymptotic tracking is not achieved, although it is not a requirement in ACC systems.

Figure 1.1: Illustration of funnel control applied to a SISO system.

While the computed input by the control law remains bounded, its minimum and maximum values depend on the system, its parameters and the specific reference signal and cannot be computed in advance. In the context of cruise control, it may well be that the preceding vehicle has a larger braking performance than the follower vehicle. If so, then the required braking performance computed by the FCC may exceed the maximum braking performance available and a car crash is no longer inconceivable. This phenomenon is called input saturation and poses a threat to the safety guarantees provided by FCC. The type of input saturation that we consider is

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Section 1 Introduction

defined by the saturation function in (1.1), of which a simple example is given in Figure 1.2.

uu

u(t ) B

 

 



 





u, u(t ) ∈ (−∞, u], u(t ), u(t ) ∈ (u, u ), u, u(t ) ∈ [u, ∞).

(1.1)

In this thesis, we respond to the question of how to avoid or circumvent input saturation in Funnel Cruise Control.

Two approaches are considered, the first of which can be interpreted as a derivation of minimal vehicle performance conditions such that driver safety can be guaranteed in normal driving situations. The second approach is a method to increase the chances of maintaining a safe distance in case of the existence of input saturation bounds and thus deals with a typical situation that compromises driver safety.

0 0.2 0.4 0.6 0.8 1

Time [s]

-1 -0.5 0 0.5 1

(a)u(t ).

0 0.2 0.4 0.6 0.8 1

Time [s]

-1 -0.5 0 0.5 1

(b) ^0.8₋_0.8u(t ) .

Figure 1.2: Example function

u(t ) =

^sin

2πt

with saturation bounds

± 0.8

^.

We now outline the structure of this thesis. In Section 2 it is shown how FCC relates to existing ACC methods, the extent of funnel control in terms of its development and applications and the research that has been done regarding funnel control with input saturation. The concept of funnel control itself is discussed in detail in Section 3, in which we also state the definitions and theorems that will be used throughout this thesis. A full description of FCC is given in Section 4, complete with a theorem that shows the extent of the application of FCC to a system that describes the dynamics of a vehicle in motion. We finish that section with simulations showing the FCC in action and a discussion on why input saturation should be considered. The first approach mentioned above is discussed in Section 5, also finalizing with a theorem and simulations. The second approach is discussed in Section 6, which ends with simulations. While the first approach can be stated in the form of a theorem, the second approach is rather experimental, i.e. no conclusions may be drawn, but the method seems promising. The thesis closes with the conclusions in Section 7 and a discussion and the recommendations for further research in Section 8.

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Section 2 Overview of literature

It appears imperative that control of a vehicle should be shifted from the driver to an automatic system. A human error analysis by [30] revealed that 64% of crashes were related to driver errors; the major shortcomings are skill- based errors, decision errors and violations. It is estimated to be between 50% and 90% by [44] . Transportation accidents are one of the major causes of death in the world according to [12]. While CCC, which simply maintains a prescribed speed, might not aid specifically in reducing the number of crashes, ACC is specifically created to additionally maintain a safe distance from the preceding vehicle. It is able to detect the preceding vehicle using sensors such as radar, lidar and camera, the data of which may be used to adapt the speed accordingly. Future predictions show that most of the major cars companies will launch cars with built-in safety systems such as ACC, according to [12]. The next step towards an autonomous vehicle is providing communication between the vehicle and other vehicles (V2V) or the infrastructure (I2V) [51], but that is beyond the scope of this thesis.

Although detailed cruise control strategies are generally not made public by their developers [27], ACC systems are still developed using various methods. For example, [16] used a reinforcement learning approach which shows promising results. However, for the control to learn, an as close to real-life model of the behaviour of vehicles was required, which makes it difficult to implement. The use of barrier functions for the safety conditions and Lyapunov functions for the control objectives has been explored by [5]. The control then mediates between the two types of control functions to guarantee safety. However, the functions appear to be model-specific and hence the model information needs to be known. An overview of advantages and disadvantages of using proportional–integral–derivative (PID) control, model predictive control (MPC) and fuzzy logic control (FLC) as an ACC system is provided by [27]. It shows that PID control systems are easy to implement and tune, MPC systems can easily achieve optimal control and FLC systems can easily deal with the complexities of the cruise control problem. On the other hand, PID control and FLC cannot guarantee safety and MPC strongly depends on the model.

Earlier this year, [11] proposed an ACC based on funnel control, a control method developed by [33]. Since funnel control is able to ensure that the tracking error between output signal and reference signal evolves within a prespecified boundary, called the funnel boundary, it seems to be a good proposal. While funnel control pro- poses a time-varying control law, it is not a type of adaptive control, to which Karl Åström gave the pragmatic definition that an adaptive controller is “a controller with adjustable parameters and a mechanism for adjusting the parameters” [8]. Adaptive control should not be confused with ACC, which is a cruise control system with the ability to detect a preceding vehicle and adjust the speed of the follower vehicle appropriately. For a history on adaptive control the reader is referred to [22, 31].

Funnel control has many applications in robotics [23] and mechatronics [24], more specifically in temperature control [38], speed and position control [14, 19, 20, 21, 25, 37], as well as medical applications [46, 47] and the boundary controlled heat equation [49]. The latter is of particular interest, because it shows the feasibility of funnel control applied to systems described by partial differential equations. Such systems are explored in detail in [10]. The advantages of funnel control is that the control is adaptive, the input remains bounded and the control is inherently robust because it requires no system or parameter identification, i.e. it is model-free.

On the other hand, it has non-zero prescribed accuracy, which in practice implies that the tracking error remains near the funnel boundary. This may be resolved with the use of an internal model [19], a PI control [37], or may even be circumvented with a simple adjustment to the control law [40].

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Section 2 Overview of literature

Although originally funnel control was developed for non-linear high-gain systems of relative degree one, it has since its creation in 2002 been extended to systems of higher relative degree [9, 34, 35]. For a system plant which accepts only a binary input, for example

u ∈ {− 1, 1}

, a bang-bang control based on funnel control was developed [41] and extended to systems of higher relative degree [42]. Gain scaling, as an extension of the control proposed in [33], was developed in [26, 36] for faster control performance. It can also be used to specify a minimal gain permanently, or fix an arbitrary initial control

u

_v

( 0)

^[22].

The first result regarding funnel control with input saturation was published by [38]. This result has been gener- alized for linear and non-linear systems [28, 29] for more general forms of the saturation (1.1) that we consider.

Input saturation in a relative degree two system was treated in [25] as an extension of [29]. Saturation com- pensation was developed in [18] in which the funnel boundary is adjusted towards a boundary with a smaller convergence rate.

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Section 3 Funnel control

Consider a system

S = p, f ,T

described by a non-linear,

M

-input

u

,

M

-output

y

, differential equation of the form

( y (t ) Û = f p(t ), (T y )(t ), u(t ), y

_[−h,0]

= y

⁰

∈ C [−h, 0] → Ò

^M

,

^(3.1)

in which

p

may be considered a disturbance term and

T

is an operator on

y

. The initial condition may be passed as a continuous function

y

⁰defined on the interval

[−h, 0]

^,

h ∈ Ò

≥0. Even though such a system is restricted to a relative degree of one, it is able to describe many types of systems due to the non-linearity property of the function

f

. Later on, in Definitions 1 and 2, we give a full description of the operator class

T

and system class

S

to which the operator

T

and the system

S

are assumed to belong, respectively. If the objective is to track a reference signal

r ∈ R

, where

R

is a certain class of reference signals, then to such systems of class

S

a control of the form

u (t ) = −α ϕ(t ) ke(t )k e(t )

^(3.2)

can be applied, where

e B y − r

denotes the difference between output and reference signal, also called the tracking error,

ϕ ∈ Φ

denotes the inverse of a prespecified funnel boundary (specified by the user), where

Φ

is a certain class of functions, and

α

:

[ 0, 1) → Ò

≥0 is continuous, strictly increasing and unbounded.

Specifically, we require that

α (s ) → ∞

as

s → 1

⁻, which can be achieved using a relatively simple function such as

α (s ) =

_1−s¹ , for example. This way, the gain grows as the norm of the tracking error approaches the funnel boundary (provided that

ϕ( 0) ke(0)k < 1

), to the extent that contact between the error and the funnel boundary is precluded and the error evolves within the funnel as depicted in Figure 1.1. This entire process is illustrated in Figure 3.1, in which an output disturbance term

n

may be added which is tolerated by the proposed control (3.2), provided that

n

is of the same class as the reference signal

r

.

The control (3.2) was first introduced by [33] and, as described in Section 2, has seen many applications and has been extended to a broader class of systems. The fact that the error evolves within the funnel is a result of [33,

Figure 3.1: Feedback control.

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3.1 Definitions Section 3 Funnel control

Thm. 7], restated as Theorem II later in this section. This section serves as a general introduction into the concept generally referred to as “funnel control” and as the first step towards understanding the concept of Funnel Cruise Control.

Although the differential equation 3.1 encompasses many types of systems, it has two major drawbacks. The first is that the system is restricted to a relative degree of one. The second is that the proposed control (3.2) depends directly on the error

e(t )

. Therefore, it is generally not possible to have a non-zero input with a zero error. In case

u (t ) = 0

implies that

e( Û t ) , 0

, it is inevitable to have

e(t ) Û = e(t ) = 0

simultaneously and so we may expect to have a non-zero error most of the time. As mentioned in Section 2, there are ways to resolve this if asymptotic tracking is required. Otherwise, gain scaling may be employed which amplifies the gain such that a smaller relative error

ϕ(t ) ke(t )k

can achieve the same gain as a greater relative error in case no gain scaling is applied. In fact, we will utilize the latter method for the Funnel Cruise Control and we demonstrate it in Section 4.4.

The definitions and theorems related to funnel control are required later on in this thesis and presented next.

Section 3.1 also treats the definitions of the operator class

T

and system class

S

, which are not used in the thesis but are included to state Theorems I and II, given in Section 3.2, in full. Since we are dealing with measurable functions, Appendix B provides a brief overview on measurability.

3.1. Definitions

The following definition relates to the operator

T

as introduced in the general funnel control system description (3.1). It can be described informally as causal, bounded-input, bounded-output (BIBO) and locally Lipschitz continuous. The definition is taken from [33].

Definition 1. Operator class

T

.

An operator

T

is said to be of class

T

if for some

h ∈ Ò

≥0and

M , Q ∈ Î

, the following hold:

(1)

T

:

C [−h, ∞) → Ò

^M

→ L

^∞_loc

Ò

≥0

→ Ò

^Q

;

(2) for every

δ > 0

, there exists

∆ > 0

such that for all

x ∈ C [−h, ∞) → Ò

^M

, sup

t ∈[−h,∞)

kx (t )k ≤ δ =⇒ T x (t ) ≤ ∆

for almost all

t ∈ Ò

≥0;

(3) for all

t ∈ Ò

>0, the following hold:

(1) for all

x, ξ ∈ C [−h, ∞) → Ò

^M

,

x (·) ≡ ξ(·)

on

[−h, t ] =⇒ T x (s) = T ξ(s)

for almost all

s ∈ [ 0, t ]

^; (2) for all

ζ ∈ C [−h, t ] → Ò

^M

, there exist

τ, δ, c > 0

such that, for all

x, ζ ∈ C [−h, ∞) → Ò

^M

with

x |

_{[−h,t ]}

= ζ = ξ|

[−h,t ]and

x (s ), ξ(s ) ∈ B

_δ

(ζ(t ))

for all

s ∈ [t , t + τ]

^, ess-sup

s ∈[t ,t+τ]

k(T x )(s ) − (T ξ)(s )k ≤ c

sup

s ∈[t ,t+τ]

kx (s ) − ξ(s )k .

A system of system class

S

is described as a triple

p, f ,T

, where

p

may be considered a disturbance term,

f

is a non-linear function that takes

p

,

T y

and

u

as its input and

T

is an operator of class

T

. The definition is taken from [33].

Definition 2. System class

S

.

Consider the controlled non-linear functional differential equation (3.1) where

h ∈ Ò

^≥0quantifies the “memory” of the system and, for some

P , Q ∈ Î

^,

(1)

p ∈ L

^∞

Ò → Ò

^P

;

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Section 3 Funnel control 3.1 Definitions

(2)

f ∈ C Ò

^P

× Ò

^Q

× Ò

^M

→ Ò

^M

;

(3) for every non-empty compact set

C ⊂ Ò

^P

× Ò

^Qand sequence

(u

_n

) ⊂ Ò

^M

\{ 0}

, the following property (akin to radial unboundedness or weak coercivity) holds:

ku

n

k → ∞

as

n → ∞ =⇒

^min

(v,w )∈C

hu

n

, f (v , w , u

_n

)i

ku

n

k → ∞

as

n → ∞

; (4)

T

:

C [−h, ∞) → Ò

^M

→ L

^∞_loc

Ò

^≥0

→ Ò

^Q

is of class

T

.

Then

S

is the class of nonlinear

M

-input

u

,

M

-output

y

systems

(p, f ,T )

of the form (3.1).

The objective of funnel control is to track a reference signal

r ∈ R

such that the tracking error

e

between plant output

y

and reference signal

r

evolves within a funnel

F

_ϕ specified by a function

ϕ ∈ Φ

. Definitions 3 to 5 specify these terms precisely.

Definition 3. Reference class

R

.

R

is the class of reference signals

r ∈ W

^1,∞

Ò

≥0

→ Ò

^M

. The important property of such signals is that

kr k

_1,∞

= kr k

∞

+ k Ûrk

∞

< ∞

, i.e. both the signal as well as its derivative are bounded.

Definition 4. Funnel functions class

Φ

.

Φ

is the class of funnel functions

ϕ

with the properties (1)

ϕ ∈ W

^1,∞

Ò

^≥0

→ Ò

^≥0

; (2)

ϕ(t ) > 0

^{for all}

t > 0

^; (3) lim inf_{t →∞}

ϕ(t ) > 0

^.

Definition 5. Performance funnel

F

_ϕ.

The performance funnel, of which an example is shown in Figure 1.1, denotes the time-varying set in which the tracking error

e

is supposed to evolve and is defined as

F

_ϕ :

Ò

≥0

→ Ò

^M

, t 7→ e ∈ Ò

^M

ϕ(t ) ke k < 1 .

The boundary of the performance funnel is denoted as

∂ F

_ϕ

(t ) = ϕ(t )

⁻¹

C ψ(t ).

Throughout this thesis, we frequently switch between

ϕ

and

ψ

. While

ψ

has a more intuitive interpretation as the funnel boundary,

ϕ

is used to describe the relative error

ϕ(t ) ke(t )k

. Note that specifically

ϕ( 0) = 0

is allowed, such that the initial condition

ϕ( 0) ke(0)k < 1

is trivially satisfied. In Appendix A, three types of functions are presented that are admissible as funnel boundaries

ψ

, based on the desired initial value

ψ( 0) = ψ

0, a desired limit

ψ

∞such that

ψ(t ) → ψ

∞as

t → ∞

and a desired Lipschitz constant

Λ

^.

Remark 1.

With regards to Definition 4, it appears that different authors use different notations to describe the set

Φ

. Specifically, [11] assumes a different definition compared to the definition assumed in [33], i.e. it additionally requires that

ψ |

_[ε,∞)

∈ W

^1,∞

[ε, ∞) → Ò

^≥0

for all

ε > 0

. As to not confuse the reader, we may show that the two definitions are equal: assume that

ϕ ∈ Φ

. From property (1) in Definition 4 we obtain that

kϕ k

_∞

< ∞

and

k Û ϕ k

_∞

< ∞

and that

ϕ

is Lipschitz continuous. Let us denote

Λ

ϕas the Lipschitz constant of

ϕ

. From properties (2) and (3) we get that

ψ |

_[ε,∞)

∈ L

^∞

[ε, ∞) → Ò

>0

for all

ε > 0

, from which we may assume that there exists a constant

K B

^sup

t₀,t₁≥ε

|ψ(t

₀

)ψ(t

₁

)| < ∞

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3.1 Definitions Section 3 Funnel control

for all

ε > 0

^{. Since}¹_x

−

_y¹

=

^{y −x}_{x y} we have that for almost all

t

₀

, t

₁

≥ ε

^,

t

₀

, t

1,

Λ

ϕ

≥ |ϕ(t

₁

) − ϕ(t

₀

)|

|t

₁

− t

₀

| =

1 ψ(t₁)

−

_ψ(t¹

0)

|t

₁

− t

₀

| = 1 K

|ψ(t

₁

) − ψ(t

₀

)|

|t

₁

− t

₀

| ,

hence

ψ |

_[ε,∞)is Lipschitz continuous with Lipschitz constant

Λ

ψ

≤ K Λ

ϕ and so

ψ |

_[ε,∞)

∈ W

^1,∞

[ε, ∞) → Ò

≥0

for all

ε > 0

^.

Finally, the proposed control in (3.2) utilizes a gain function

α

with

ϕ(t ) ke(t )k

, called the relative error, as the argument. Such gain functions belong to the class

A

.

Definition 6. Gain functions class

A

.

A

is the class of gain functions

α

with the properties (1)

α ∈ C [ 0, 1) → Ò

>0

;

(2) for all

s

₀

, s

₁

∈ [ 0, 1)

^with

s

₀

< s

₁:

α (s

₀

) < α(s

₁

)

; (3) lim_{s →1}⁻

α (s ) = ∞

^.

Next, consider the initial-value problem

( x (t ) Û = F t, x(t ), T x (t ),

x |

_[−h,0]

= x

⁰

∈ C [−h, 0] → Ò

^N

,

^(3.3)

where

D ⊂ Ò

^N^,

T

:

C [−h, ∞) → Ò

^N

→ L

^∞_loc

Ò

≥0

→ Ò

^Q

and

F

:

[−h, ∞) × D × Ò

^K

→ Ò

^N^{. The} existence of a solution to the differential equation (3.3) may be inferred from [33, Thm. 5], restated as Theorem I later in this section. With the input

u

as in (3.2), the system (3.1) may be written in the form of (3.3) such that a solution to (3.1) can be proven to exist. The equality between the two types of systems may be found in the proof of [33, Thm. 7] and is not shown in this thesis. This theorem is restated as Theorem II later in this section and proves the claims made in the introduction of this section. To fully understand the terms used in the theorems previously mentioned, we require three more definitions. Definitions 7 and 8 are taken from [32].

Definition 7. Carathéodory function

A function

F

:

[−h, ∞) × Ò

^N

→ Ò

^M is a Carathéodory function if (1)

F (t , ·)

is continuous for almost all

t ∈ Ò

^≥0;

(2)

F (·, w )

is measurable for each fixed

w ∈ Ò

^N^;

(3) for each compact set

C ⊂ Ò

^N there exists

κ ∈ L

¹_loc

[−h, ∞) → Ò

^≥0

such that

kF (t , w )k ≤ κ(t )

for almost all

t ∈ [−h, ∞)

and all

w ∈ C

.

It is generally required for the existence of a solution to a differential equation that the governing function, such as the function

F

in (3.3), satisfies the conditions posed in Definition 7. The resulting solution to the differential equation is then referred to as a solution in the sense of Carathéodory.

Definition 8. C-solution

The pair

x, I

is said to be a solution of the initial value problem (3.3) on the interval

I = [−h, ω)

with

ω ∈ ( 0, ∞]

in the sense of Carathéodory (abbreviated; a C-solution) if

x |

_[−h,0]

= x

⁰^,

x |

[0,ω)

∈ AC

loc

[ 0, ω) → Ò

^N

and it satisfies the differential equation (3.3) almost everywhere on

[ 0, ω)

^.

Finally, a solution that exists on a certain interval can sometimes be extended to a greater interval of existence, of which the greatest such interval corresponds to the maximal solution.

(19)

Section 3 Funnel control 3.2 Theorems

Definition 9. Maximal solution

Let

(x, I)

denote a solution

x

of (3.3) on the interval

I

. Then

(x

^e

, I

^e

)

is called an extension of

(x, I)

if it is also a solution,

I ⊆ I

^eand

x

^e

|

_I

= x

. Furthermore,

(x, I)

is called a maximal solution if and only if for any extension

(x

^e

, I

^e

)

we have

I

^e

⊆ I

.

3.2. Theorems

As previously mentioned, we seek to find a solution (and its properties) of the system described by (3.1) in which the input

u

is given by (3.2). The resulting system may be written in the form of (3.3), for which the theorem below provides the required information.

Theorem I. Existence of solution

Let

D ⊂ Ò

^N be non-empty and open, let

T

be a class

T

operator and

x

⁰

∈ C [−h, 0] → Ò

^N

with

x

⁰

( 0) ∈ D

^{. Assume}

F

:

[−h, ∞) × D × Ò

^K

→ Ò

^N is a Carathéodory function.

Then there exists a C-solution

x ∈ C [−h, ω] → Ò

^N

of the initial-value problem (3.3) with

ω ∈ ( 0, ∞]

and

x |

[0,ω)

⊂ D

. Every solution can be extended to a maximal solution. Furthermore, if

F

is locally essentially bounded and

x

:

[−h, ω) → Ò

^N^,

x [ 0, ω) ⊂ D

, is a maximal solution with

ω < ∞

, then, for every compact set

C ⊂ D

, there exists

t

⁰

∈ [ 0, ω)

^{such that}

x (t

⁰

) < C

^.

The proof of this theorem can be found in [33]. This theorem will also be used later in this thesis for a different type of system.

Remark 2.

Theorem I states that if

x

:

[−h, ω) → Ò

^N^,

x [ 0, ω) ⊂ D

, is a maximal solution, then either

ω = ∞

or, for every compact set

C ⊂ D

, there exists

t

⁰

∈ [ 0, ω)

^{such that}

x (t

⁰

) < C

. Another way to formulate this is by stating that the closure of the graph of the solution is not a compact subset of

D

. For example, the proof of [11, Thm.

2.1] states that the closure of the graph of the solution

(x , v )

is not a compact subset of

D

and refers to [55, §10, Thm. XX], which states that every solution can be extended to the right of the boundary of

D

. In [55, §6, Thm.

VII] this statement is defined as follows: if

G

is the closure of the graph of the solution, then

G

is not a compact subset of

D

. They then proceed to provide the equivalent formulation that at least one of the following three cases must occur:

(1)

ω = ∞

, i.e. the solution exists for all

t ∈ Ò

≥0;

(2)

ω < ∞

and the limit of the solution goes to infinity as

t → ω

⁻;

(3)

ω < ∞

and the solution comes arbitrarily close to the boundary of

D

as

t → ω

⁻.

In case the boundary of

D

lies at infinity the second case can occur, which is referred to as a finite blow-up time solution. A formulation in the form of these three cases might give a better understanding compared to the original formulation. This formulation will also be used in the proof of Theorem III, which will be presented in the next section.

The next theorem is the main result of [33] and lies at the foundation of funnel control.

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3.2 Theorems Section 3 Funnel control

Theorem II. Funnel Control

Let

α ∈ A

and let

ϕ ∈ Φ

. For every system

(p, f ,T ) ∈ S

, every reference signal

r ∈ R

, and all initial data

y

⁰

∈ C [−h, 0] → Ò

^M

with

ϕ( 0) y

⁰

( 0) − r (0) < 1

, the control (3.2) applied to system (3.1) yields a closed-loop initial-value problem which has a solution defined on

[−h, ∞)

. Furthermore,

y

and

u

are continuous and bounded and there exists

ε ∈ ( 0, 1)

^{such that}

ϕ(t ) ke(t )k ≤ 1 − ε

^{for all}

t ∈ Ò

≥0.

The proof of this theorem can be found in [33]. The latter statement on the existence of

ε ∈ ( 0, 1)

implies that the tracking error is uniformly bounded away from the funnel boundary by a minimal distance

ε

, see Figure 3.2. There- fore, the solution does not come arbitrarily close to the boundary of the set

D B y ∈ Ò

^M

ϕ(t ) k y − r (t )k <

1

^{. Since}

ϕ(t ) k y (t ) − r (t )k < 1

is guaranteed to hold for

t ∈ [ 0, ω)

^and

r

is essentially bounded, we have also that the solution

y

is bounded. This is how it can be shown with the help of Remark 2 that

ω = ∞

^.

Figure 3.2: Illustration showing how the error is uniformly bounded away from the funnel boundary.

(21)

Section 4 Funnel Cruise Control

In this section, we provide the full framework regarding the objectives and the construction of the Funnel Cruise Control. As mentioned in Section 2, it was developed earlier this year by [11]. It is based on the funnel control (3.2), although it does not suffice to say that it is simply an application of funnel control, which will be explained in Section 4.1. Furthermore, as an improvement on the proposed control by [11], we will incorporate more design options into the control law by including gain scaling as mentioned in Section 3.

Section 4.1 provides all specifications of the Funnel Cruise Control, including the control objectives. Although Funnel Cruise Control is model-free, we show feasibility of the control by applying it to a specific model derived in Section 4.2. Subsequently, we show in Section 4.3 that the tracking objectives of Section 4.1 are guaranteed to be achieved for any set of appropriately chosen funnel, gain and gain scaling functions and reference signals.

To demonstrate how the application of Funnel Cruise Control unfolds in practice, we show simulations of the control in Section 4.4 applied to the system introduced in Section 4.2. Subsequently, we show in Section 4.5 that input saturation is a safety concern and hence the validity of the objective of this thesis.

4.1. Control design

As introduced in Section 1, the objective of adaptive cruise control is twofold:

(1) maintain a minimal safe distance from the preceding vehicle at all times;

(2) regulate the speed of the vehicle such that a prescribed speed is maintained without violating the first objective.

As mentioned in Section 2, we may assume that a vehicle is able to measure its velocity and the distance between itself and the preceding vehicle. This information can be used to construct a control law that satisfies the two objectives above. Because it is imperative that the first objective is satisfied at all times, the second objective should be relaxed when the preceding vehicle is within critical range. Furthermore, the first objective is trivially satisfied when the preceding vehicle is not within a critical range. In other words, the first objective may then be neglected, such that the vehicle is only concerned with maintaining the prescribed speed. Such a control law cannot be constructed by simply applying funnel control as described in Section 3, since funnel control is concerned with keeping an error within a prescribed funnel at all times. By the aforementioned reasoning, it is impossible to create a funnel such that both the distance and speed are kept within a certain range. In fact, we desire a control that is able to let the distance or speed error escape the funnel at some point without collapsing the system. Hence, the control proposed by [11] consists of two controls, a speed and distance control, cooperating in a certain fashion such that the two objectives are achieved. The two controls are designed using funnel control.

In this section, we first introduce the two types of control in Sections 4.1.1 and 4.1.2 as variants on the control (3.2).

The combination of the two types of control will be presented in Section 4.1.3. Let

x

and

v

denote the position and speed of the follower vehicle, respectively.

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4.1 Control design Section 4 Funnel Cruise Control

4.1.1. Speed control

The speed control is concerned with tracking a reference speed

v

ref

∈ W

^1,∞

Ò

≥0

→ Ò [

^m

/

s

]

, such that the speed error

e

_v

[

m

/

s

]

, defined as

e

_v

(t ) B v (t ) − v

ref

(t ),

(4.1)

evolves within the funnel

F

_ϕ_v, see Definition 5, specified by a function

ϕ

_v

∈ Φ

such that

ψ

_v

B ϕ

v⁻¹. We now introduce the control law that describes the speed control. Choose

α

_v

∈ A

and

κ

_v

∈ Φ

. Then the speed control is defined as

u

_v

(t ) B −κ

v

(t )α

v

ϕ

_v

(t ) |e

v

(t )| e

_v

(t ).

(4.2) The gain function

α

_v serves the same purpose as the gain function used in (3.2). The gain scaling function

κ

_v may be utilized to make the computed control input more aggressive. Since we require

κ

_v to be continuous and positive, it suffices to choose

κ

_v

∈ Φ

. This definition of the speed control is a generalization regarding the distance control proposed in [11], which was originally defined as (4.2) with

κ

_v

(t ) = 1

^and

α

_v

(s ) =

_1−s¹₂^{. In} Section 4.4, we show how gain scaling may influence the resulting trajectory of the speed error.

4.1.2. Distance control

As discussed earlier, the distance control can be used while the vehicle is within a certain distance range of the preceding vehicle. This distance range is specified by the interval

F

_ϕ_d, where

ϕ

_d

∈ Φ

, translated by a minimal safety distance

d

safe

[

m

]

behind the preceding vehicle. The safety distance

d

safemay be specified by any bounded function. However, the preference is given to constant time gap over a constant clearance distance [51]. Hence, we employ the safety distance

d

safe

(t ) B λ

1max

0, v(t ) + λ

2 (4.3)

with

λ

₁

, λ

₂

∈ Ò

^≥0, such that there exists always a minimal distance

λ

₂

[

m

]

between the two vehicles and an additional time gap of

λ

₁

[

s

]

seconds. The resulting distance distance range is then described by the interval

d

_safe

(t ), d

safe

(t ) + 2ψ

d

(t )

, such that the objective of the distance control is to track a reference distance

d

_ref

[

m

]

, defined as

d

ref

(t ) B d

^safe

(t ) + ψ

d

(t ),

(4.4)

which lies in the center of the distance range. This principle is illustrated in Figure 4.1. Although the control objective is to simply keep a minimal distance, tracking the reference distance

d

_refin this case implies that a preferred distance of

d

_refis kept from the preceding vehicle, with a minimal distance of

d

_safe, such that the control objectives regarding safety are achieved. To ensure that the reference distance is well defined, we impose the condition that

ϕ( 0) , 0

, which is otherwise generally allowed in funnel control. The distance error is defined as the error between the position

x [

m

]

of the follower vehicle (the vehicle we control) and a reference position

x

_ref

[

m

]

, which specifies the position of the center of the distance range behind the preceding vehicle,

x

ref

(t ) B x

`

(t ) − d

safe

(t ) − ψ

_d

(t ),

(4.5)

Figure 4.1: Illustration of distance funnel control (Image courtesy of [4]).

(23)

Section 4 Funnel Cruise Control 4.1 Control design

where

x

_`

∈ W

^1,∞

Ò

≥0

→ Ò [

^m

]

denotes the position of the preceding vehicle. Thus, we define the distance error

e

_d

[

m

]

as

e

_d

(t ) B x (t ) − x

ref

(t ) = x(t ) − x

`

(t ) + d

safe

(t ) + ψ

d

(t ).

(4.6) In this form, the distance control has the appearance of a position control, which contradicts the term “distance control”. This choice was made to ensure that the distance control has the same form as the speed control, introduced in the preceding section, in the sense that a positive distance error will imply that a negative input should be applied. The proof of the Funnel Cruise Control theorem, Theorem III, will then be easier to follow.

Moreover, the distance error is directly related to the position error as the negation of the position error, i.e. if

d [

m

]

denotes the distance to the preceding vehicle,

d (t ) B x

`

(t ) − x (t ),

^(4.7)

then

d (t ) − d

ref

(t ) = x

`

(t ) − x (t ) − d

safe

(t ) − ψ

d

(t )

⁽^4.6)

= −e

d

(t ).

In that sense, the choice for the appearance of a position control is rather a choice of aesthetics, while we refer to it as the distance control. We now introduce the control law that describes the distance control. Choose

α ∈ A

and

κ

_d

∈ Φ

. Then the distance control is defined as

u

_d

(t ) B −κ

d

(t )α

d

ϕ

_d

(t ) |e

d

(t )| e

_d

(t ).

(4.8) Again, the gain function

α

_d serves the same purpose as the gain function used in (3.2) and the gain scaling function

κ

_d may be utilized to make the computed control input more aggressive. This definition of the distance control is a generalization regarding the distance control proposed in [11], which was originally defined as (4.2) with

κ

_v

(t ) = 1

^and

α

_v

(s ) =

_1−s¹2.

Remark 3.

In the above, we specified that the position of the preceding vehicle,

x

_`, is assumed to be bounded. By doing so, we deviate from the assumptions made in [11], in which

x

_` is allowed to be unbounded. While theoreti- cally it makes sense not to impose boundedness on

x

_`, which would imply that the distance

d

may grow to be unbounded, it makes no difference practically and it simplifies the proof of Theorem III, specifically Step 2.

4.1.3. Final control design

We now present the final control design as a combination of the two controls as introduced in [11]. Although the speed and distance controls have been defined, it is not yet clear how to make the two controls interact while

e

_v

(t ) ∈ F

_ϕ_v

(t )

and

e

_d

(t ) ∈ F

_ϕ_d

(t )

hold simultaneously. In order to explain, let the triple

t , x (t ), v (t )

denote the trajectory of the position and speed of the vehicle at time

t ∈ Ò

^≥0. For any time

t

, we define a set

D

relatively open¹in

Ò

^≥0

× Ò

²in which the triple

t , x (t ), v (t )

is allowed to evolve. The set

D

is constructed in such a way that the control objectives stated in the introduction of Section 4.1 are automatically satisfied as long as

t , x (t ), v (t )

∈ D

. The set

D

should be defined in such a way that a desired reference speed is followed unless the follower vehicle is too close to the preceding vehicle, in which case a desired reference distance is followed. The set

D

that achieves this objective is the non-convex set

D B D

v

∪ D

_d

∪ D

_v^d

,

(4.9)

where

D

_v

B

n (t , x , v ) ∈ Ò

^≥0

× Ò

²

v − v

ref

(t ) ∈ F

_ϕ_v

(t ), x − x

ref

(t ) ≤ −ψ

_d

(t )o , D

_d

B

n (t , x , v ) ∈ Ò

^≥0

× Ò

²

v − v

ref

(t ) ≤ −ψ

_v

(t ), x − x

ref

(t ) ∈ F

_ϕ_d

(t )o , D

_v^d

B

n (t , x , v ) ∈ Ò

^≥0

× Ò

²

v − v

ref

(t ) ∈ F

_ϕ_v

(t ), x − x

ref

(t ) ∈ F

_ϕ_d

(t )o .

(4.10)

1There exists an open setC ⊂Ò≥0×Ò²^{such that}D= C ∩ Ò≥0×Ò²^.

(24)

4.1 Control design Section 4 Funnel Cruise Control

These sets are visualized in Figure 4.2. If the vehicle is located behind the distance range, then

x (t ) − x

ref

(t ) ≤

−ψ

_d

(t )

and so

t , x (t ), v (t ) ∈ D

_v. Because the vehicle is located behind the distance range, clearly the speed control should dominate. Therefore, we impose that the final control design,

u(t )

, is such that

u(t ) = u

v

(t )

if

t , x (t ), v (t ) ∈ D

_v. Similarly, if the vehicle is located within the distance range, but is not yet within range of the reference speed, then the vehicle should simply be concerned with maintaining the reference distance until the reference speed is within range. This is the case when

v (t ) − v

_ref

(t ) ≤ −ψ

v

(t )

holds, which is analogous to

t , x (t ), v (t ) ∈ D

_d. Therefore, we impose that

u(t ) = u

d

(t )

if

t , x (t ), v (t ) ∈ D

_d. Finally, if the vehicle is located within the distance range and also within range of the reference speed, then

t , x (t ), v (t )

∈ D

_v^d. Since it should be allowed that the preceding vehicle accelerates and the follower vehicle needs not concern himself anymore with keeping a safety distance, it should be possible for the follower vehicle to transfer from

D

_v^d into

D

_v, or from

D

_d into

D

_v^d. Similarly, the preceding vehicle could slow down and the follower vehicle might transfer from

D

_v^d into

D

_d, or from

D

_v into

D

_v^d. This means that it should be possible for the vehicle to move freely from one of the sets in (4.10) into another (although direct transfers from

D

_v to

D

_d and vice-versa are prohibited). However, we cannot let happen that

v (t ) − v

ref

(t ) ≥ ψ

v

(t )

, because this would simply mean that the vehicle is moving too fast for no reason. Similarly, we cannot let happen that

x (t ) − x

ref

(t ) ≥ ψ

d

(t )

, because this would imply that a prescribed safety distance has been violated. To avoid that either the speed of the vehicle is increased while the distance between the vehicles is within the distance range, or that the speed of the vehicle is increased while the reference speed is already achieved, we specify that the minimum of the computed values of the two controls is applied as the input while

t , x (t ), v (t ) ∈ D

_v^d. At last, we present the final control design,

u(t ) B

 

 



 





u

_v

(t ), t , x (t ), v (t ) ∈ D

_v

, u

_d

(t ), t , x (t ), v (t ) ∈ D

_d

,

min

u

_d

(t ), u

_v

(t ) , t , x (t ), v (t ) ∈ D

_v^d

,

(4.11)

which is the governing input control used throughout this thesis.

Figure 4.2: Illustration of the set in which the position and speed of the vehicle is allowed to evolve.

Remark 4.

Note that the control (4.11) is fully dependent on the trajectory of the position

x

and speed

v

, while the funnel functions

ϕ

_v and

ϕ

_d, the gain functions

α

_v and

α

_d, the gain scaling functions

κ

_v and

κ

_d and the reference signals

v

refand

x

_`are determined beforehand. In that sense, it might have been more appropriate to write

u

as a function of

t

,

x

and

v

. However, to avoid lengthy notation, and because

x

and

v

depend on

t

as well, it is more convenient to write

u

as a function of

t

only.

(25)

Section 4 Funnel Cruise Control 4.2 Dynamics of a vehicle in motion

4.2. Dynamics of a vehicle in motion

Although the Funnel Cruise Control (4.11) is model-free and can be applied to any system that satisfies certain structural properties, we derive here a model of the dynamics of a vehicle in motion to be used to prove that application of the Funnel Cruise Control to such a system achieves the control objectives set in Section 4.1.

Additionally, we use this model for the simulations in Sections 4.4, 5.5, 6.5 and 8.1.

A simple model of the longitudinal dynamics of a vehicle in motion is given in [7]. More details are provided by [17, 48], which contain additional models, e.g. the tyre slip and the driveline dynamics. However, to demonstrate the FCC the simple model by [7] will suffice.

By Newton’s second law we have that the sum of the forces acting on a vehicle equals the mass of the vehicle multiplied by its acceleration. The forces acting on the vehicle are primarily the force

F [

N

]

generated by the car, determined by the input

u(t )

, the force

F

_g

[

N

]

due to gravity, the aerodynamic drag

F

_a

[

N

]

and the rolling friction

F

_r

[

N

]

. We assume that we can directly control the force generated by the engine of the car, such that

u(t ) = F (t ) [

^N

]

. The equations of motion for the car are given by

( x (t ) Û = v(t ),

m Û v (t ) = u(t ) − F

g

x (t ) − F

_a

t , v (t ) − F

_r

v (t ) + δ(t ),

^(4.12) with the initial conditions

x ( 0) = x

⁰

∈ Ò, v(0) = v

⁰

∈ Ò.

^(4.13)

The forces are defined as

F

_g :

Ò → Ò, x 7→ mg

sin

θ(x ) ,

F

_a :

Ò

≥0

× Ò → Ò, (t , v ) 7→

sgn

(v ) 1

2 ρ(t )C

_d

Av

²

,

(4.14)

F

_r :

Ò → Ò, v 7→

sgn

(v ) mg C

r

.

The speed of the wind may be added to the vehicle speed in the function for the aerodynamic drag force

F

_a. However, in this thesis we neglect the speed of the wind for simplicity. The parameters used in the equations above are listed below.

m

the mass of the vehicle

[

kg

]

;

g

the gravity of Earth

[

m

/

s²

]

;

θ ∈ W

^1,∞

Ò → h

− π 2 , π

2 i ,

the slope of the road depending on the position of the vehicle

[

rad

]

;

ρ ∈ W

^1,∞

Ò

≥0

→ Ò

≥0

,

the density of the air

[

m

/

s²

]

;

C

_d the shape-dependent aerodynamic drag coefficient

[−]

;

A

the frontal area of the vehicle

[

m²

]

;

C

_r the rolling friction coefficient

[−]

;

δ

a bounded disturbance term

[

N

].

With the rolling friction

F

_r

(v )

as defined in (4.14), there exists a discontinuity at

v = 0

due to the sign function sgn

(v )

. Since discontinuities are not supported in the context of funnel control (consider Theorem II, in which

f

is required to be a Carathéodory function), we approximate the sign function by the smooth error function erf

(αv )

for sufficiently large

α > 0

, see Figure 4.3. Therefore, we redefine the rolling friction as

F

_r :

Ò → Ò v 7→

erf

(αv ) mg C

r

,

(4.15)

(26)

4.2 Dynamics of a vehicle in motion Section 4 Funnel Cruise Control

with

α = 100

. Although the function that describes the aerodynamic drag also contains the sign function, it is not discontinuous because of the sign function and therefore does not require this approximation. Furthermore, we can simplify the equations of motion (4.12) by defining the function

f

:

Ò

≥0

× Ò

²

→ Ò, (t , x , v ) 7→ 1 m

F

_g

(x ) + F

a

(t , v ) + F

r

(v ) − δ(t )

(4.16)

with

F

_g and

F

_aas in (4.14) and

F

_ras in (4.15), such that we may write (4.12) as

( x (t ) Û = v(t ),

v (t ) Û =

_m¹

u(t ) − f t , x (t ), v (t ) .

^(4.17) We have now completed defining our model and (4.17) will be used as the main model of the vehicle dynamics in this thesis. An example of vehicle parameters is presented in Table C.1 in Appendix C.

-1 -0.5 0 0.5 1

-1.5 -1 -0.5 0 0.5 1 1.5

Figure 4.3: The smooth error function erf

(αv )

for

α = 100

^.

It should be noted that two small adaptations were made on the forces acting on the vehicle in comparison to the model which was taken from [7]. Firstly, to ensure that the aerodynamic drag acts on the vehicle in the direction opposite to the direction of motion, we include the sign function in the function that describes the aerodynamic drag. This does not endanger continuity of the function, since¹₂

ρ(t )C

_d

Av

²is already continuous for all

v ∈ Ò

. Secondly, we let the slope of the road depend on the (longitudinal) location of the vehicle, rather than on time. This is to impose that the vehicle will encounter the same road slope if it encounters the same location more than once. This might happen when the vehicle drives backwards after having moved forwards, for example. Ultimately, the slope of road will again (implicitly) depend on time.

Although the equations of motion given by (4.17) provide a very basic model for the dynamics of a vehicle, Funnel control is, as mentioned in Section 3, a model-free controller. It can therefore be applied to any such model that describes the motion of a vehicle, on the condition that the function that describes the forces acting on the vehicle can be described as Carathéodory functions, see Definition 7. What this means precisely will be made clear from Step 1 of the proof of Theorem III, stated in the next section.

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Section 4 Funnel Cruise Control 4.3 A theorem on Funnel Cruise Control

4.3. A theorem on Funnel Cruise Control

In Section 4.1 we defined the Funnel Cruise Control which is designed to satisfy the two control objectives specified in the introduction of Section 4.1. The next theorem, based largely on [11, Theorem 2.1], finalizes the idea of Funnel Cruise Control by showing the implications of applying the control (4.11) to the system (4.17). Keep in mind that, although the control (4.11) is model-free, Theorem III is not model-free because it is proven explicitly for the model presented in Section 4.2. However, as previously mentioned, the Funnel Cruise Control may be applied to any such system that satisfies certain structural properties, which will become clear in the proof of Theorem III.

We emphasize that the Funnel Cruise Control itself is not restricted to the model described by the system (4.17).

Theorem III. Funnel Cruise Control

Consider the system (4.17) with initial conditions (4.13) and assume that the position of the preceding vehicle

x

_`is in

W

^1,∞

Ò

^≥0

→ Ò

^.

Choose a desired speed

v

ref

∈ W

^1,∞

Ò

≥0

→ Ò

^with

v

ref

(t ) ∈ Ò

≥0for all

t ∈ Ò

≥0, safety distance parameters

λ

₁

> 0

^,

λ

₂

∈ Ò

≥0, functions

κ

_v

, ϕ

_v

, κ

_d

, ϕ

_d

∈ Φ

, such that

ϕ

_d

( 0) , 0

^and

0, x

⁰

, v

⁰

∈ D

, with

D

as in (4.9), and functions

α

_v

, α

_d

∈ A

.

Then the control (4.11) applied to the system (4.17) with initial conditions (4.13) yields a closed-loop system which has a solution

(x , v )

^:

Ò

≥0

→ Ò

²^{such that}

t , x (t ), v (t ) ∈ D

for all

t ∈ Ò

≥0. Furthermore, all involved signals

x , v

and

u

are continuous and bounded and the error is uniformly bounded away from the boundary of

D

, i.e. there exists

ε > 0

such that for all

t > 0

^{we have}

e

_v

(t ) ≤ ϕ

_v

(t )

⁻¹

− ε, e

_d

(t ) ≤ ϕ

d

(t )

⁻¹

− ε,

ε ≤

max

0, ϕ

v

(t )

⁻¹

+ e

v

(t ) +

^max

0, ϕ

d

(t )

⁻¹

+ e

d

(t ) .

Although essentially we could have simply referred to the proof of [11, Theorem 2.1] rather than proving Theorem III further below, it seemed better fitting to include the entirety of the proof for completeness, in addition to the fact that [11, Theorem 2.1] was proven only for

α

_v

(s ) = α

d

(s ) =

_1−s¹2 and

κ

_v

(t ) = κ

d

(t ) = 1

. In view of Remark 3, proving that the system states remain bounded has become shorter. Finally, although it is stated in [11] that Funnel Cruise Control results in a continuous input, it is not explicitly proven there. In the proof that follows below, continuousness of the the input is proven in Step 6.

Furthermore, note that we do not consider the absolute value of the tracking errors in the first two inequalities.

Despite the confusing notation of the third inequality, it has a straightforward interpretation. This formulation of the boundedness of the error implicitly implies that

 

 





 



|e

v

(t )| ≤ ϕ

v

(t )

⁻¹

− ε ∧ e

_d

(t ) ≤ −ϕ

d

(t )

⁻¹

, t , x (t ), v (t ) ∈ D

_v

,

|e

d

(t )| ≤ ϕ

d

(t )

⁻¹

− ε ∧ e

_v

(t ) ≤ −ϕ

v

(t )

⁻¹

, t , x (t ), v (t ) ∈ D

_d

,

|e

v

(t )| ≤ ϕ

v

(t )

⁻¹

− ε ∧ |e

d

(t )| ≤ ϕ

d

(t )

⁻¹

− ε, t , x (t ), v (t ) ∈ D

_v^d

,

with

D

_v

, D

_d and

D

_v^d as defined in (4.10). This is derived in Step 3 of the proof, accompanied by the clarifying illustration of Figure 4.4. It is a consequence of Remark 2, similar to the comment below Theorem II, accompanied by Figure 3.2.

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4.3 A theorem on Funnel Cruise Control Section 4 Funnel Cruise Control

Proof of Theorem III.

The proof is split into six steps.

Step 1

We show that there exists a maximal solution of (4.17) with initial conditions (4.13) and input

u

as in (4.11).

By assumption we have

0, x

⁰

, v

⁰

∈ D

, with

D

as defined in (4.9). Define

F

:

Ò

^≥0

× Ò

²

→ Ò

²

, (t , x , v ) 7→ v

1

m

u(t ) − f (t , x , v )

! ,

with

u

as in (4.11) and

f

as in (4.16). Note that

F

is well defined almost everywhere in

Ò

^≥0

× Ò

², but here we only consider the domain

D

. Then the closed-loop initial-value problem of the control (4.11) applied to the system (4.17) with initial conditions (4.13) is equivalent to

Û x (t ) v (t ) Û

!

= F t, x(t ), v(t ), x ( 0) v ( 0)

!

= x

⁰

v

⁰

!

.

(4.18)

In order to apply Theorem I to (4.18), we need that

F

is a Carathéodory function, see Definition 7, which we will show now.

The functions

F

_g

, F

_aand

F

_r are continuous as functions of

t , x

and

v

and

δ

is measurable as a function of

t

. Since a continuous function of measurable functions is again measurable, see Lemma 1 in Appendix B, we have that

f

t

and continuous as a function of

x

and

v

. Let

C ⊂ Ò

³ be a compact set such that

C ⊂ D

. Then there exists

v ∈ Ò

^≥0such that

|v | ≤ v

for all

(t , x , v ) ∈ C

, from which follows that

|F

_a

(t , v )| ≤

¹₂

ρ(t )C

_d

Av

²,

|F

_r

(v )| ≤ mg C

_r and

F

_g

(x )

≤ mg

for all

(t , x , v ) ∈ C

. Recall that

ρ ∈ W

^1,∞

Ò

^≥0

→ Ò

^≥0

. If we define

κ

₁

(t ) B 1

2m ρ(t )C

_d

Av

²

+ g (C

r

+ 1) + kδ k

∞

,

then

f t , x , v

≤ κ

₁

(t ) ∈ L

¹_loc

[ 0, ∞) → Ò

≥0

for almost all

t ∈ Ò

≥0, by which

f

is a Carathéodory function.

By the same reasoning, we may prove that the input

u

is a Carathéodory function. Keep in mind that

u

depends on

x

and

v

as well, although we write

u(t )

to avoid lengthy notation, see Remark 4. First, we consider

u

_v for

(t , x, v ) ∈ D

_v

∪ D

_v^d, see (4.10). Since

v

ref

∈ W

^1,∞

Ò

^≥0

→ Ò

, we have that

v − v

ref

(t )

t

v

. Since

ϕ

_v

∈ Φ

,

ϕ

_v

(t ) |v − v

ref

(t )| ∈ [ 0, 1)

^for

(t , x, v ) ∈ D

_v

∪D

_v^d and

α

_v

(s )

is continuous for

s ∈ [ 0, 1)

, we have that

α

_v

ϕ

_v

(t ) |v − v

ref

(t )|

is also measurable as a function of

t

v

. Finally, since

κ

_v

∈ Φ

, we also have that

−κ

v

(t )α

v

ϕ

_v

(t ) |v − v

ref

(t )| v − v

_ref

(t )

t

v

. Let

C ⊂ Ò

³be a compact set such that

C ⊂ D

_v

∪ D

_v^d. Then, by definition of

D

_v

∪ D

_v^d, there exists

v ∈ Ò

≥0such that for some

c > 0

^{we have}

|v − v

ref

(t )| ≤ v − v

ref

(t ) ≤ ϕ

v

(t )

⁻¹

− c

for all

t , x , v ∈ C

. Therefore, for all

(t , x , v ) ∈ C

, we have that

κ

_v

(t )α

v

ϕ

_v

(t ) |v − v

ref

(t )| v − v

_ref

(t )

≤ κ

v

(t )α

v

1 − ϕ

v

(t )c 1 − ϕ

v

(t )c

C κ

2

(t ) ∈ L

¹_loc

,

by which

u

_vis a Carathéodory function for

(t , x, v ) ∈ D

v

∪ D

_v^d.

Analogously, we may prove that

u

_d is a Carathéodory function for

t , x , v

∈ D

_d

∪ D

_v^d, provided that

x

_`

− x − λ

₁

v − λ

₂

− ψ

d is measurable as a function of

t

x

and

v

. This is true since

x

_`

∈ W

^1,∞

Ò

≥0

→ Ò

^and

ϕ

_d

∈ Φ

(recall that

ψ

_d

= ϕ

⁻_d¹^).

Finally, since the minimum of two continuous functions is again continuous, we have that the control (4.11) is a Carathéodory function for all

(t , x, v ) ∈ D

. Together with

f

being a Carathéodory function, we now have that also

F

is a Carathéodory function.

Funnel cruise control with input saturation