Structural Intelligent Platooning by a Systematic LQR Algorithm

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Structural Intelligent Platooning by a Systematic LQR Algorithm

GUSTAV HAMMAR & VADIM OVTCHINNIKOV

Master’s Degree Project

Stockholm, Sverige 2010

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Abstract

Oil price has rapidly increased in the past centuries and made the fuel consumption a hot topic in the transport business. This was also empha- sized by a study made by the heavy duty vehicle manufacturer, Scania CV AB, who concluded that one third of their operational cost could be related to the fuel consumption. An alleviation to this problem might be to form a platoon of vehicles. This will not only increase the ca- pacity of the traffic flow but also decrease the fuel consumption due to the slipstream effect that occurs behind traveling objects. Control has already been developed to handle fixed intermediate distances within a platoon of vehicles and is referred to as Automatic Intelligent Cruise Control in the vocabulary of the manufacturers. However none of these existing regulators are taking the slipstream effect into concern and is therefore not suitable for the purpose of fuel reduction by vehicle platooning. By paying regards to this particular effect the complexity of the dynamics is increased and new more sophisticated regulators are necessary. In this thesis a new distributed regulator is proposed which is able to take the slipstream effect into concern using linear quadratic regulation. This regulator maintains the present informational topology where each vehicle is able to measure the distance as well as the velocity to the vehicle ahead. Through simulations it is shown that the regulator is able to cope with realistic scenarios, a systematic decrease of error throughout the platoon has been observed and robustness has been proven.

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Acknowledgement

First of all we would like to thank Dr. Ather Gattami for introducing us to this interesting problem and for the regular supervision he gave us throughout this period of time. The weekly meetings were for us greatly inspiring where Dr. Gattami pointed us in direction, but still gave us the opportunities to discover the paths ourselves. We would also like to thank the Ph. D. student Assad Al Alam for sharing interest in our project and providing us with allot of technical information regarding heavy duty vehicles.

Finally it has been a great honor for us to receive the Talent Scholarship 2010 from the Department of Electrical Engineering at Royal Institute of Technology (KTH).

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Introduction

According to a survey, [1], made by the heavy duty vehicle manufacturer, Scania CV AB, the fuel cost constitutes approximately one third of their total operational cost. This together with the fact that the oil price have increased more than 50 %, in the period of 2001-2010, has made the fuel reduction for heavy duty vehicles, a commercial topic.

A terminology often used in sports is the slipstream effect which refers to the at- mospheric drag reduction behind a traveling vehicle. This region can therefore be used in advantage by vehicles traveling behind. In [4] the result of an experiment with two trucks traveling on a highway in 70 km/h presents that a fuel reduction of 4.7 − 7.7 % could be experimentally obtained with two identical trucks when taking advantage of vehicle platooning.

The fundamental problem in the control of coordinated vehicles is the integration of communication due to the decentralized architectures. The control objectives are often specified in terms of the entire formation, meanwhile the regulator controlling the engine can only be applied on each and every vehicle separately.

1.1 Related Work

During the last decades a large amount of work has been published about stability and control of vehicles in different formations. Heavy duty vehicles, (HDVs) in a vehicle platoon is one of the formations that have been in focus, until now mainly because of the environmental benefits and the safety reasons it result in. The introduction of the air drag reduction implies a new type of coupled dynamics and a new type of regulator than what are proposed in earlier studies, e.g. [6]. Since none of the existing proposed regulators take the air drag reduction into concern, they are neither suitable for the purpose of fuel reduction by vehicle platooning.

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CHAPTER 1. INTRODUCTION

1.2 Objective

By regulating a vehicle’s velocity, i.e. by acceleration or deceleration, the control strategy produces an increased energy consumption which in turn increases the fuel consumption. The brake-even point is when the energy consumption of the proposed regulator is less than the energy gain obtained by taking advantage of the slipstream effect. Therefore the objective of this thesis is to find a regulator that minimizes its energy consumption meanwhile it takes the slipstream effect into concern, is robust and is able to withstand realistic strenuous scenarios.

1.3 Cruise Control (CC) and Adaptive intelligent Cruise Control (AiCC)

Cruise Control, also referred to as speed control or autocruise control is a system that automatically maintains the speed of a vehicle according to a reference velocity.

In addition to a cruise control an Adaptive Cruise Control takes into account the distance to the vehicle in front. This means that the driver defines a reference velocity as well as a reference distance to the vehicle ahead. The distance is measured by radar or laser. The later is lower in cost but weather sensitive. Exactly as the CC, AiCC holds a reference velocity and in addition to that it also adapts its own velocity to the vehicle ahead by keeping a certain reference distance to it.

In some literature the Adaptive intelligent Cruse Control is simply known as ACC.

The introduction of the word intelligent to this abbreviation is simply adapted by some automobile manufacturers such as Scania AB to keep apart the AiCC from Air Climate Control, which is a totally different system.

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Chapter 2

Vehicle Modeling

2.1 Chapter outline

The purpose of this chapter is to derive a continuous linear vehicle model which describes the dynamics of a heavy duty truck and which is based on a specified platoon configuration. This is necessary in order to be able to produce a regulator as well as simulating the performance of a platoon utilizing the implemented regulator.

There are many ways of developing a linear vehicle model of this kind, in this thesis the model developed in [2] is used. The derivation of the model is done by first considering the forces acting upon a truck, thereafter expressing these forces as functions of vehicle parameters and finally linearizing the system in order to express the system as a regular linear time invariant, (LTI), system.

2.2 Platoon configuration

Figure 2.1 shows the configuration of the platoon. The platoon consists of i = {1, 2, ..., N} vehicles, where i = 1 denotes the leading vehicle. Each vehicle is able to measure the distance and the velocity of the vehicle in front of it, as well as its own velocity.

Figure 2.1. Platoon of N vehicles traveling on a flat road, each vehicle is able to measure the distance to and the velocity of the vehicle ahead.

For simplicity, the movement of the platoon is assumed to be longitudinal which means that these measurements are scalars. The three dimensional heading will instead implicitly be taken into concern through the forces acting on each vehicle.

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CHAPTER 2. VEHICLE MODELING Furthermore, the reference distance between the vehicles is defined by a minimum distance, based on a time gap, τ. This ensures that there is enough space between the vehicles in case of an emergency braking. The minimum distance is defined as:

d^τ_i,i−1 = τvi−1, v_i−1>0 (2.1)

∆d^τ_i,i−1 = τ∆vi−1, v_i−1>0 (2.2)

2.3 External forces acting on vehicles

Figure 2.2 shows the external longitudinal forces acting on a heavy vehicle in motion.

These forces can be decomposed through Newton’s second law:

m˙v = Fw− F_ad+ Fadr− F_roll− F_gravity (2.3)

Figure 2.2. Forces inflected upon a heavy duty vehicle traveling on a sloping road

The aerodynamic force produced by the air drag is described by F_ad = 1

2c_dA_aρ_αv², (2.4)

where Aa is the maximal cross sectional area of the vehicle, cω is the air drag coefficient and ρα is the air density. The reduction of the air drag produced by the vehicle in front is defined as

F_adr = 1 2

f(d)

100c_dAaραv², (2.5)

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2.3. EXTERNAL FORCES ACTING ON VEHICLES

where d is the intermediate distance between the vehicles and f(d) is a non-linear function for the air drag reduction due to one heavy duty vehicle in front, depicted in Figure 2.3. Since vehicles cannot be closer than 5 meters due to safety reasons this relationship between the air drag coefficient and the intermediate distance can be fairly described by a least square approximation.

f(d)lsq = −0.414d + 41.29 0 ≥ d ≤ 99

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70 80

c D reduction [%]

Relative Distance in Convoy [m]

Mapping of c

D reduction

Lead HDV One HDV Ahead Two HDVs Ahead

Figure 2.3. The change in the air drag coefficient and the distance vehicles in between, [4].

The rolling resistance is given by

F_roll= crmgcos(α), (2.6)

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CHAPTER 2. VEHICLE MODELING where cr is the roll coefficient. In turn the gravitational force affecting the vehicle is described by

F_gravity = mg sin(α), (2.7)

where g is the gravitational constant. The remaining force, Fw will be derived in the next section.

2.4 System dynamics

It is natural to use the engine torque as the input to the system. In order to involve this variable in the dynamics the connections between the vehicle and the engine torque is considered. This series of connections is called the powertrain, Figure 2.4 shows a basic model defined in [2].

Figure 2.4. The powertrain of a heavy duty vehicle

Engine: An engine is on its own a very complicated system, in this model the clutch given by Newton’s second law is taken into account

Jeω˙e = Tu− T_c, (2.8)

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2.4. SYSTEM DYNAMICS

where ωe is the angular velocity of the shaft between the engine and the clutch.

Clutch: The clutch is considered to be stiff which implies that the clutch does not yield any changes of torque and angular velocity, namely:

T_t= Tc (2.9)

ωt= ωc (2.10)

Transmission: In this model the transmission is characteristic by two properties, namely the conversion ratio itbetween the input torque and output torque, as well as the efficiency of the gearbox nt. These two properties affect the propeller shaft according to:

T_p = itη_tT_t (2.11)

ω_p = itω_e (2.12)

Propeller shaft: The connection between the propeller shaft and the final drive is considered to be stiff, hence:

Tp = Tf (2.13)

ωp = ωf (2.14)

Final drive: Similar to the transmission, a conversion ratio and an efficiency constant characterizes the final drive. In the same manner these are taken into account as:

Td= ifηfTf (2.15)

ω_d= ifω_f (2.16)

Drive shaft: The connection between the wheels and the final drive is approxi- mated to be stiff.

T_w = Td (2.17)

ωw = ωd (2.18)

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CHAPTER 2. VEHICLE MODELING

Wheel: In this model the slip of the wheels will be neglected, hence the Newton’s second law provides

Jwω˙w = Tw− r_wFw, (2.19) where

r_w = v

ω_w = vi_ti_f

ω_e . (2.20)

Finally, equation (2.8) - (2.20) provides the expression of the force inflicted upon the wheels:

F_w = { ˙w = ˙v

r}= −J_w+ i²ti²_fη_tη_fJ_e

r_w² ˙v + i²_ti²_fη_tη_f rw

T_u (2.21)

By applying the specified forces, (2.4) - (2.7) together with (2.21) to equation (2.3) the following equation of dynamics is constructed

(J_w

r²_w + m + i²_ti²_fηtη_fJe

r_w² ) ˙vi = (iti_fηtη_f

r_ω T_u−1

2c_dA_αρ_αv²+ +1

2 f(d)

100cdAaραv²− c_rmgcos(α) − mg sin(α)), where the expression in the parenthesis on the left hand is known as the accelerated mass.

⇒ ˙vi= r²_w

Jw+ mr_w² + i²_ti²_fηtηfJe(iti_fηtη_f

r_ω T_u−1

2c_dA_αρ_αv²+ + 1

2 f(d)

100c_dA_aρ_αv²− c_rmgcos(α) − mg sin(α)) By utilizing the following four notations:

c_e= r_w²

Jω+ mrω² + i²ti²_fηtη_fJe

i_ti_fη_tη_f rω

cω= r_w²

J_ω+ mrω² + i²ti²_fη_tη_fJ_e 1

2Aaραc_d c_{f r} = r_w²

Jω+ mr_ω² + i²_ti²_fηtηfJe

c_rmg

cg = r_w²

J_ω+ mrω² + i²ti²_fη_tη_fJ_emg the dynamic of the vehicle can be simplified to:

˙v = ceT_u− c_ωφ(d)v²− c_{f r}cos(α) − cgsin(α) φ(d) = (1 − f(d)

100).

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2.5. LINEARIZED SYSTEM DYNAMICS

2.5 Linearized system dynamics

2.5.1 General linearization technique

Linearization of a function g : Rⁿ→ R around the point of interest p is performed by a first order Taylor expansion:

g(x) ≈ g(p) + ∇g|p(x − p)

assumed that g is continuous and first order differentiable in the vicinity of the point p.

For this specific case, g = ˙v, represents the change of velocity, or in other words acceleration of each vehicle in the platoon. Since the objective is to find a control law where each and every vehicle should act as smoothly as possible in the sense of velocity deviation, g|p is chosen to be g|p= 0.

2.5.2 Velocity and relative distance as states

The linearization points are chosen around a constant reference distance di = d0, a constant reference velocity vo= vref and the corresponding reference engine torque T_u⁰ = Tu^ref. The slope is considered to be a constant parameter α0 = αref, which implies that the states of the system is reduced to [ (∆Tuⁱ)^T (∆vi)^T (∆di)^T ]^T.

g(p) = g(Tu⁰, s⁰_i−1, s⁰_i, v⁰) = ceT_u⁰− c_ωφ(d0)v₀²− c_{f r}cos(α0) − cgsin(α0) = 0 ⇔ T_u⁰ = c_ωφ(d0)v0²+ cf rcos(α0) + cgsin(α0)

ce

Applying the Nabla operator

∇g|_p = ( ∂g

∂T_uⁱ, ∂g

∂d_i, ∂g

∂v_i)|p= (ce, −c_ω˜φv²_i, −2cωφ(di)vi)|p=

= (ce, −cω˜φv₀², −2cωφ(d0)v0), where ˜φ ≡ ^∂φ_∂d = 0.0414.

⇒ ˙vi≈ c_e∆T_uⁱ− c_ω˜φv²₀∆di−2cωφ(d0)v0∆vi

Notice that the leading vehicle in the platoon will obtain slightly different dynamics, therefore the following definitions are made:

• γ1 = −2cωv₀

• γi = −2cωφ(d0)v0 for i ∈ {2, ..., N}

• µi= − ˜φcωv₀² for i ∈ {2, ..., N}

• bi= ce for i ∈ {1, 2, ..., N}

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CHAPTER 2. VEHICLE MODELING

In turn the linearize system dynamics becomes:







∆v˙1

∆d˙2,1

∆v˙2

∆d˙3,2

∆v˙3

...˙

∆vN −1

∆dN −1,N˙

∆v˙N







=







γ₁ 0 0 0 0 · · · 0 0 0

1 0 −1 0 0 · · · 0 0 0

0 µ2 γ₂ 0 0 · · · 0 0 0

0 0 1 0 −1 · · · 0 0 0

0 0 0 µ3 γ3 · · · 0 0 0

... ... ... ... ... ... ... ... ...

0 0 0 0 0 · · · γ_{N −1} 0 0

0 0 0 0 0 · · · 1 0 −1

0 0 0 0 0 · · · 0 µ_N γ_N













∆v1

∆d2,1

∆v2

∆d3,2

∆v3

...

∆vN −1

∆dN,N −1

∆vN







+







b1 0 0 · · · 0 0 0 0 · · · 0 0 b2 0 · · · 0 0 0 0 · · · 0 0 0 b3 · · · 0 ... ... ... ... ...

0 0 0 · · · 0 0 0 0 · · · bN













∆Tu¹

∆T_u²

∆Tu³

∆T...u^N







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Chapter 3

Optimal control and Robustness

3.1 Chapter outline

This chapter provides the most necessary theory needed to be able to follow the reasoning in the sequent chapters. The theory of Optimal Control is the core in the derivation of the regulator while the section of Robustness provides the most vital theory of the analysis of the derived regulator.

3.2 Optimal control

Optimal control is the theory of how to solve dynamic optimization problems. More precise, it gives an answer to the question: How to find a control law for a certain system such that some optimality criteria are fulfilled? Due to the linearity of the vehicle model, the by far most adopted linear control theory is presented, namely the theory of linear quadratic control.

3.2.1 Linear quadratic control

A typical setup when dealing with LQ problems is to minimize a quadratic cost function:

J(u) =^Z ^t^f

t0

x(t)^TQ(t)x(t) + u(t)^TR(t)u(t) dt + x(tf)^TSx(tf) (3.1) subject to dynamic constraints as well as subject to an initial condition. That is

J^∗= J(u^∗) = min_u ^Z ^t^f

t0

x(t)^TQ(t)x(t) + u(t)^TR(t)u(t) dt + x(tf)^TSx(tf) (3.2) Subject to:

˙x(t) = A(t)x(t) + B(t)u(t),

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CHAPTER 3. OPTIMAL CONTROL AND ROBUSTNESS

with the initial condition defined as:

x(t0) = x0.

The quadratic cost function introduces a way to minimize both the input, in this case the engine torque, and the error of the output, in this case the difference between the reference output and the actual output. This will be further clarified in Section 4.2.2. In order to ensure a physical meaning, the pair (A, B) needs to be controllable and the matrices Q and R needs to be positive definite.

The proof of the solution to the general LQ problem is shown by completion of squares, i.e. a certain optimal cost is assumed to be given by:

J^∗= x(t)^TP(t)x(t) := V (t). (3.3) In turn the derivate of (3.3) is:

˙V (t) = ˙x(t)^TP(t)x(t) + x(t)^T ˙P(t)x(t) + x(t)^TP(t) ˙x(t) =

= (Ax(t) + Bu(t))^TP(t)x(t) + x(t)^T ˙P(t)x(t) + x^T(t)P (t)(Ax(t) + Bu(t)) =

= x^T(t)A^TP(t)x(t) + u(t)^TB^TP(t)x(t)+ (3.4) + x(t)^T ˙P(t)x(t) + x^TP(t)Ax(t) + x^T(t)P (t)Bu(t) =

= x^T(A^TP(t)P (t) + P (t)A(t) + ˙P (t))x(t) + u(t)^TB^TP(t)x(t) + x^TP(t)Bu(t) (3.5) The integration of (3.5) provides the expression:

V(tf) − V (t0) =^Z ^t^f

t0

x^T(A^TP(t)P (t) + P (t)A(t) + ˙P (t))x(t)+

+ u(t)^TB^TP(t)x(t) + x^TP(t)Bu(t) dt

If J(u)−V (t0) is always greater than zero, then V (t0) is in fact the minimal solution.

J(u) − V (t0) = J(u) + V (tf) − V (t0) − V (tf)

=^Z ^t^f

t0

x(t)^TQ(t)x(t) + u(t)^TR(t)u(t) dt + x(tf)^TSx(tf)+

+^Z ^t^f

t0

x(t)^T(A^TP(t)P (t) + P (t)A(t) + ˙P (t))x(t)+ (3.6) + u(t)^TB^TP(t)x(t) + x^TP(t)Bu(t) dt−

− x(tf)^TP(tf)x(tf) =

=^Z ^t^f

t0

x(t)^T(Q + A^TP(t)P (t) + P (t)A(t) + ˙P (t))x(t) dt+

+^Z ^t^f

t0

u(t)^TR(t)u(t) + u(t)^TB^TP(t)x(t) + x(t)^TP(t)Bu(t) dt+ (3.7)

+ x(tf)^T(S − P (tf))x(tf) (3.8)

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3.2. OPTIMAL CONTROL

If P (t) is definied by:

˙P(t) = − A^TP(t) − P (t)A − Q + P (t)BR⁻¹B^TP(t) (3.9) P(tf) =S

(3.9) is the so called Riccati equation and has an unique solution that is positive semi-definite. With this definition (3.8) becomes:

J(u) − V (t0) =^Z ^t^f

t0

(u(t) + R⁻¹B^TP(t)x(t))^TR(u + R⁻¹B^TP(t)x(t)) dt ≥ 0

In turn, if u^∗ = −R⁻¹B^TP(t)x(t) then J(u^∗) = V (t0), hence u^∗ is the optimal regulator and the solution to (3.2) provided P (t) is the unique solution to (3.9).

That is:

u^∗ = −Kx(t) K= −R⁻¹B^TP(t) where P(t) is the solution to the Riccati equation:

˙P(t) = −A^TP(t) − P (t)A − Q + P (t)BR⁻¹B^TP(t) The minimal cost is therefore:

J(u^∗) = x^T₀P(t)x0

A more simple set up than above is obtained when all the matrices (A, B, Q, R) are constant and when studying the infinite horizon problem:

J^∗ = min_u ^Z ^∞

0

x(t)^TQx(t) + u(t)^TRu(t) dt Subject to:

˙x(t) = Ax(t) + Bu(t) Initial condition:

x(0) = x0

This setup is the so called Linear Quadratic Regulator, (LQR), problem which im- plies that P (t) = P = constant is the solution to the steady state Riccati Equation, also known as the Algebraic Riccati Equation, (ARE):

0 = −A^TP − P A − Q+ P BR⁻¹B^TP (3.10)

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CHAPTER 3. OPTIMAL CONTROL AND ROBUSTNESS

3.3 Robustness

Robust control is a branch of control theory that aims to achieve a robust performance and/or stability of the system.

A mathematical model of a plant provides a map from inputs to response. The quality of the model depends on how closely its response matches the response of the true plant. In practice, since no single model can respond exactly as the true plant one needs to find a model simple enough to facilitate the design, yet complex enough to give confidence that design based on the true model will work on the real plant. Methods have therefore been developed to handle these "unstructured uncertainties" as generic errors.

3.3.1 Identification of errors and disturbances

When it comes to a platoon of vehicles we have chosen to focus on:

• Model uncertainty - The most obvious uncertainties are that the model of the system is derived under assumptions that slope of the road is constantly zero, α_i = 0 and the masses of the vehicles mi are exact and constant with respect to time. In the real world, the vehicles are affected by precipitation, wind, friction change of the road and leaning deviation, together with many other factors. Many of those disturbances can fairly be projected on the uncertainty in mass.

• Measurement disturbance - white noise in the measurement of the preceding vehicle’s states. It has been showed in [8] that white noise do not affect stability of heavy vehicles traveling in platoons. This is due to the fact that heavy vehicles simply do not react fast enough to the frequent changes in measurement due to their inertia.

• Delays - there is always latency in communication and information processing together with other durations. For a longitudinal control system the delays can be split up in two main components: one is the time used for recognizing a hard brake in the regulator, in a typical system, the duration of this time delay is about 60 ms. The other is the time delay in the brake hardware, it ranges from 10 to 100 ms, therefore the total time delay can range from 70 to 160 ms, [5].

• Package loss - some of the measurements do not reach the processing stage.

• State deviation robustness - a vehicle measures the states of the preceding one. It lacks control of the vehicle in front and can only react to its actions.

Therefore a robust analysis has been conducted to see how the deviation in the states of the preceding vehicle affects its own states.

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Chapter 4

Methods

4.1 Chapter outline

In this chapter the derivation of the regulator and its subsequent results is presented.

In accordance to the objective of this thesis the proposed algorithm, Algorithm 1, is based on the special type of communication topology where each vehicle only has the knowledge of the states of the preceding vehicle. In turn, this means in accordance with the platoon configuration that each vehicle only is able to measure the distance to and the velocity of the vehicle ahead. The following algorithm is therefore addressed to find a regulator that has the corresponding structural appearance.

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CHAPTER 4. METHODS

4.2 LQR by a structural-decomposition algorithm

Consider the vehicle model with velocity and intermediate distance as state.







∆ ˙v1

∆ ˙d2,1

∆ ˙v2

∆ ˙d3,2

∆ ˙v3

∆ ˙vN −1...

∆ ˙dN −1,N

∆ ˙vN







=







γ₁ 0 0 0 0 · · · 0 0 0

1 0 −1 0 0 · · · 0 0 0

0 µ2 γ₂ 0 0 · · · 0 0 0

0 0 1 0 −1 · · · 0 0 0

0 0 0 µ3 γ3 · · · 0 0 0

... ... ... ... ... ... ... ... ...

0 0 0 0 0 · · · γ_{N −1} 0 0

0 0 0 0 0 · · · 1 0 −1

0 0 0 0 0 · · · 0 µ_N γ_N













∆v1

∆d2,1

∆v2

∆d3,2

∆v3

...

∆vN −1

∆dN,N −1

∆vN







+







b₁ 0 0 · · · 0 0 0 0 · · · 0 0 b2 0 · · · 0 0 0 0 · · · 0 0 0 b3 · · · 0 ... ... ... ... ...

0 0 0 · · · 0 0 0 0 · · · bN













∆Te¹

∆Te²

∆T_e³

∆T..._e^N







= A¯x + B∆ ¯Te

Each and every vehicle can measure only the preceding vehicle’s velocity and relative distance to it, with an exception for vehicle number one. The problem can be decomposed into smaller subproblems connected through the velocity of the preceding vehicle.

• Subsystem 1

∆ ˙v1 = A1∆v1+ B1∆Te¹,

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4.2. LQR BY A STRUCTURAL-DECOMPOSITION ALGORITHM

where A1 = γ1 and B1 = b1. The optimal state feedback K1 = (?) ∈ R^1×1 can simply be found for this problem by solving (3.10), the linear quadratic cost function is defined by:

J₁^∗ = min

∆T_e¹(^Z ^∞

0 ∆v1TQ₁∆v1+ ∆(Te¹)^TR₁∆Te¹dt)

• Subsystem 2

The second vehicle has full knowledge of the dynamics of the preceding vehicle’s closed loop system. It is therefore fair to take this into concern when calculating the optimal feedback for Subsystem 2. Hence, the dynamics of the second subsystem become







∆ ˙v1

∆ ˙d2,1

∆ ˙v2





= ˆA₂







∆v1

∆d2,1

∆v2





+ B2∆Te²

where Aˆ2 = A2−





 b₁

0 0





 h

K1 0 0 ⁱ=







γ₁ 0 0

1 0 −1

0 µ2 γ₂





−





 b₁

0 0





 h

K1 0 0 ⁱ,

and

B₂=





 00 b₂





.

The optimal state feedback K2= (? ? ?) ∈ R^1×3is once again calculated from (3.10), where the linear quadratic cost function is defined as:

J₂^∗= min

∆T_e²(^Z ^∞

0

x^T₂Q2x2+ ∆(Te)^2TR2∆Te²dt) x₂ =^h ∆v1 ∆d2,1 ∆v2

iT

• Subsystem 3

In contrary to the second vehicle, the third vehicle has not full information of the dynamics of the preceding vehicle. This is due to the fact that it only can measure the states of the second vehicle and not of the first vehicle. In this case one need to estimate the closed loop dynamics of the coupled state,

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CHAPTER 4. METHODS

∆v2. This is done by the estimated regulator ˜K₂ which implies the estimated dynamic of Subsystem 3 as







∆ ˙v2

∆ ˙d3,2

∆ ˙v3





= ˆA3







∆v2

∆d3,2

∆v3





+ B3∆Te³

where

Aˆ₃= A3−





 b2

00





K˜₂=







γ2 0 0

1 0 −1

0 µ3 γ₃





−





 b2

00





K˜₂, and

B3=





 00 b3





.

The optimal state feedback K3 = (? ? ?) ∈ R^1×3is attained by solving (3.10), the linear quadratic cost function is defined as:

J₃^∗ = min

∆T_e³(^Z ^∞

0

x^T₃Q3x3+ ∆T_e³^TR3∆T_e³dt)

x3=^h ∆v2 ∆d3,2 ∆v3

iT

...

• Subsystem N

In the same manner as subsystem 3, the dynamics of the N : th subsystem become







∆ ˙vN −1

∆ ˙dN,N −1

∆ ˙vN





= ˆA_N







∆vN −1

∆dN,N −1

∆vN





+ BN∆T_e^N where

AˆN = AN −





 b_{N −1}

00





K˜N −1=







γ_{N −1} 0 0

1 0 −1

0 µ_N γ_N





−





 b_{N −1}

00





K˜N −1,

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4.2. LQR BY A STRUCTURAL-DECOMPOSITION ALGORITHM

and

BN =





 00 bN





.

The optimal state feedback KN = (? ? ?) ∈ R^1×3 is as previously attained by solving (3.10), where the linear quadratic cost function is defined as:

J_N^∗ = min

∆T_e^N(^Z ^∞

0

x^T_NQNxN + ∆(Te^N)^TRN∆Te^Ndt) x_N =^h ∆vN ∆dN,N −1 ∆vN

iT

Finally, the state feedback for the collected dynamics, with the desired sparse struc- ture can be composed from the produced regulators, Ki = (? ? ?)|Ki:

K=







?|_K₁ 0 0 0 0 · · · 0 0 0

? ? ?|K2 0 0 · · · 0 0 0 0 0 ? ? ?|_K₃ · · · 0 0 0 ... ... ... ... ... ... ... ... ...

0 0 0 0 0 · · · ? ? ?|_K_N







This procedure can be summarized in the following algorithm:

Algorithm 1. LQR by a structural-decomposition 1. Calculate the CC for Subsystem 1, K1. 2. For i=2

• Extract the corresponding subsystem dynamics from the collected dynam- ics, A.

A₂ =







γ₁ 0 0

1 0 −1

0 µ2 γ2







• Take into concern the dynamics of the leading vehicle by subtraction of the CC regulator, K1

Aˆ2= A2−





 b₁

0 0





K1

• Calculate the optimal AiCC regulator, K2

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CHAPTER 4. METHODS

3. For vehicle i=3 to i=N:

• Extract the corresponding subsystem dynamics from the collected dynam- ics, A.

A_i =







γ_i−1 0 0

1 0 −1

0 µi γi







• Estimate the dynamics of the preceding vehicle by subtraction of the es- timated regulator, ˜Ki−1

Aˆi= Ai−





 b_i−1

00





K˜i−1

• Calculate the optimal AiCC regulator, Ki

4.2.1 The choice of ˜K_i

The estimation of the preceding vehicle’s closed loop dynamics,

∆ ˙vi−1=^h 0 µi−1 γi−1

i







∆vi−2

∆di−1,i−2

∆vi−1





− b_i−1^h K_i−1¹ K_i−1² K_i−1³ ⁱ







∆vi−2

∆di−1,i−2

∆vi−1





≈

≈^h γ_i−1 0 0 ⁱ







∆vi−1

∆di,i−1

∆vi





− b_i−1K˜_i−1







∆vi−1

∆di,i−1

∆vi







can be done in many ways for instance by:

• Approximating ∆vi−2 with ∆vi and ∆di−1,i−2 with ∆di,i−1, this yields the regulator:

K˜_i−1=^h K_i−1³ , −^µ_bⁱ⁻¹

i−1 + Ki−1² , K_i−1¹ ⁱ (4.1)

• Approximating ∆vi−2 with ∆vi−1 and ∆di−1,i−2 with ∆di,i−1 which implies the regulator:

K˜_i−1=^h K_i−1¹ + Ki−1³ , −^µ_bⁱ⁻¹

i−1 + Ki−1² , 0 ⁱ (4.2)

• Truncating ∆vi−2 and ∆di−1,i−2 which corresponds to the the regulator:

K˜i−1=^h K_i−1³ , 0, 0 ⁱ (4.3)

Structural Intelligent Platooning by a Systematic LQR Algorithm