Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning

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UPTEC F 16059

Examensarbete 30 hp November 2016

Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning

Matilda Dahlqvist

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Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning

Matilda Dahlqvist

Heavy Duty Vehicle Platooning is seen as a possible way to reduce fuel consumption, in part obtained by reducing the airdrag. To apply this concept in reality there is need of a safe and fuel efficient way to automatically control the longitudinal velocity.

But as with all applications there are restrictions of what is really possible. This work focus on a few restrictions in the use of wireless communication, which is used to send information to the controllers.

This study determines a sparse optimal feedback gain matrix without loosing performance of the controller during HDV platooning. It determines with what other vehicles and how the vehicles in the platoon need to communicate. It is a theoretical study which is exploring the Linear Quadratic Sparsity Promoting Regulator (LQRSP) a method proposed by Fardad, Lin and Jovanovic(2011b). The main interest was to find a controller that is sparse structured and generated from a centralized controller with the same performance as the original controller. Another interest was also to determine a pattern of dependencies of information sent between vehicles, which should make it possible to draw conclusions on needed communication links during platooning.

A linearized vehicle model was derived from the forces acting on a driving vehicle and from the dynamics of the powertrain. This vehicle model is then used to derive a model for the whole platoon according to the states of interest. The performance of three different controllers was analyzed, one decentralized controller, one Linear Quadratic Tracking Controller and the Sparsity Promoting Linear Quadratic Regulator.

The comparison between the controller performances shows that the controller with sparser structure on the feedback gain matrix

keeps the same performance as the controller used as reference.

From the structure change of the feedback gain matrices a few overall conclusions can be made. For larger platoons the information regarding vehicles in front of the vehicle itself is most desirable in combination with information about the states of distance change from the two following vehicles and velocity change from the four following vehicles. The states representing the integrated error between the reference signal and the first vehicles velocity and the state which represents the distance change between the first and second vehicle are states that are important for all vehicles in the platoon.

ISSN: 1401-5757, UPTEC F 16059 Examinator: Tomas Nyberg Ämnesgranskare: Hans Rosth

Handledare: Assad Alam, Christian Larsson

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Populärvetenskaplig sammanfattning

Platooning, benämningen för att framföra fordon i en konvoj med ett bestämt relativt avstånd mellan fordonen, ses i dagsläget som en möjlig metod för att minska bränsleför- brukningen genom att luftmotståndet minskas. För att realisera detta koncept behövdes ett säkert och bränsleeektivt sätt att automatisk styra fordonens hastighet. Problemet när man applicerar teori i verkligheten är alla restriktioner som uppstår. Detta arbete fokuserar på restriktioner vid användning av trådlös kommunikation mellan fordonen, detta används för att skicka information mellan regulatorerna i fordonen.

Detta arbete bestämmer utan att förlora prestanda på regulatorn en gles återkopplingsmatris vid platooning med tunga fordon. Arbetet avgör med vilka andra fordon samt hur fordonen i konvojen ska kommunicera. Det är en teoretisk studie som undersöker 'The Linear Quadratic Sparsity Promoting Regulator' (LQRSP) en metod framtagen av Fardad, Lin, and Jovanovic (2011b). Huvudmålet är att nna en regulator med gles struktur i återkopplingsmatrisen som är genererad från en centraliserad regulator och behåller samma prestanda som den ursprungliga regulatorn. Ytterligare mål är att nna ett mönster av beroenden från informationen som sänds mellan fordonen, detta för att dra slutsatser om hur kommunikationen är länkad i en konvoj.

En linjäriserad fordonsmodell är beräknad från krafter som verkar vid framförandet av ett fordon samt utifrån dynamiken från fordonets drivlina. Denna fordonsmodell är sedan använd för att beräkna en modell för hela konvojen utifrån aktuella tillstånd. Tre olika regulatorer är analyserade, en decentraliserad regulator, en 'Linear Quadratic Tracking Controller' samt en 'Sparsity Promoting Linear Qudratic Regulator'.

Jämförelsen mellan regulatorernas prestanda visar att en gles struktur på återkopplingsma- trisen håller samma prestanda som den ursprungliga regulatorn. Ur strukturändringen av återkopplingsmatrisen kan några övergripande slutsatser dras. För en konvoj med många fordon är information gällande fordonen framför respektive fordon i konvojen av stor vikt i kombination med information gällande tillstånden som representerar distansändringen från de två bakomvarande fordonen samt hastighetsändringen gällande de fyra efterföljande fordonen.

Tillstånden som representerar det integrerade felet mellan referenssignalen och första fordonets hastighet och tillståndet som representerar distansändringen mellan första och andra fordonet är tillstånd som är viktiga för alla fordon i konvojen.

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Acknowledgment

With this thesis Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning I complete the Master of Science degree in Engineering Physics at Uppsala Uni- versity. This thesis has been carried out at Scania CV AB in Södertälje at the department of Driver Assistance Controls (REVD) and was supervised at the Division of Systems and Control at Uppsala University.

At rst I want to express my gratitude to Scania Student Intro Program which gave me the opportunity to do my thesis at Scania. I want to express my further gratitude to my su- pervisors at Scania, Assad Alam and Christian Larsson for their valuable time and guidance.

I also want to thank my reviewer at Uppsala University, Hans Rosth for his important help and inputs. Thanks to Henrik Sandberg at Royal Institute of Technology (KTH) for making me aware of and proposing the use of the method LQRSP. And last but not least the group REVD deserves my thanks for making the time at Scania in Södertälje invaluable.

Matilda Dahlqvist Stockholm July 2016

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List of Figures

1 Forces acting on a vehicle. . . . 2

2 HDV Powertrain . . . . 3

3 Air drag reduction during platooning . . . . 5

4 Illustration of a platoon. . . . 7

5 Illustration centralized system. . . . 8

6 Illustration Decentralized System. . . . 9

7 Reference signal . . . . 15

8 Step response ve vehicles platoon LQT . . . . 16

9 Step response ten vehicle platoon LQT . . . . 16

10 Step response fteen vehicle platoon LQT . . . . 17

11 Accumulated error LQT . . . . 17

12 Step response ve vehicle platoon decentralized system . . . . 18

13 Step response ten vehicle platoon decentralized system . . . . 19

14 Step response fteen vehicle platoon decentralized system . . . . 19

15 Accumulated error decentralized system . . . . 20

16 Step response ve vehicle platoon LQRSP . . . . 21

17 Step response ten vehicle platoon LQRSP . . . . 21

18 Step response fteen vehicle platoon LQRSP . . . . 22

19 Accumulated error LQRSP . . . . 22

20 Diagonal structure in the sparse feedback gain matrix . . . . 23

21 The reference signals impact in the sparse feedback gain matrix . . . . 24

22 Distance depending states verses velocity depending states in the sparse feedback gain matrix . . . . 24

23 The feedback gain matrix with typical dependencies in the structure marked . . . 25 24 The feedback gain matrix for a ten vehicles platoon with structural patterns marked 26

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List of Tables

1 Values of cost-matrices LQT . . . . 15 2 Values of cost-matrices for decentralized controller . . . . 18 3 Data from sparsity-promoting algorithm . . . . 20

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1 Introduction

1.1 Background

Platooning, i.e when vehicles drive together with small intermediate distances between each other, has been a subject of high interest in the transportation business over the last couple of years.

The concept is seen as a possible way to reduce fuel consumption and thereby lead to less green house gas emissions from transport vehicles. This is in part obtained by reducing the airdrag during platooning. To apply this concept in reality there is need of a safe and fuel ecient way to automatically control the longitudinal velocity. But as with all applications there are restrictions for what is really possible. There are a few restrictions in the use of wireless communication, which is used to send information to the controllers. For example the range of the antenna, disturbances and delays. It is in that case important to know which information is most critical for the platoon to keep the performance of the controller.

1.2 Problem Formulation

This study aims to determine a sparse optimal feedback gain matrix without loosing performance of the controller during HDV platooning. The purpose is to determine with what other vehicles and how the vehicles in the platoon need to communicate. This will be done by exploring the Lin- ear Quadratic Sparsity Promoting Regulator (LQRSP), a method proposed by Fardad, Lin, and Jovanovic (2011b). The main interest lies in nding a controller that is sparse structured and generated from a centralized controller with the same performance as the original controller. Another interest is also to determine a pattern of dependencies of information sent between vehicles, which should make it possible to draw conclusions on needed communication links during platooning.

1.2.1 Limitations

To obtain a reasonable comparison, this study is restricted to the following conditions:

• The platoon will consist of a nite number, N, of vehicles

• The considered vehicles have identical dynamics

• We only consider a at and straight road

• No other trac is considered

• Disturbances on the system such as road condition and the amount wind are neglected.

These conditions are chosen to make the amount of parameters aecting the platoon as few as possible.

1.3 Contributions

The main contribution of this thesis is the theoretical analysis of the possibility to use feedback gain matrices with sparser structure than a centralized controller and still keep the same performance controlling a Heavy Duty Vehicle Platoon. The sparser structure should also form a pattern that will simplify the communication links between the vehicles in the platoon. The work has been done using the Linear Quadratic Sparse Promoting Regulator by Fardad, Lin, and Jovanovic (2011b).

1.4 Thesis Outline

This thesis is organized into seven chapters including this chapter. Chapter 2 presents the model used during this work, both the linear model of a HDV as well as the model used to describe the platoon. Chapter 3 includes all the control theory. That theory consist of three dierent controllers. First the system architecture is presented with two dierent architectures: centralized and decentralized control. Two control design methods Linear Quadratic Tracking Controller and the Sparse Promoting Linear Quadratic Regulator are then described. This chapter ends with the denitions of the cost matrices used in the control design. Chapter 4 includes results from the simulations, one decentralized controller and two centralized, one Linear Quadratic Tracking Controller and the Sparse Promoting Linear Quadratic Regulator. The simulations are done on

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three dierent sized platoons to get a wide perspective on the result. Chapter 5 includes an analysis of the result, followed by conclusions and thoughts about future work in Chapters 6 and 7.

2 Modeling

A way to describe the platoon as a system is needed to create a controller for a platoon of HDVs.

A model of a single vehicle needs to be derived from the vehicle dynamics. From this model a model of the whole system (the whole platoon) is derived as a state space representation.

2.1 Vehicle model

A vehicle in motion is aected by several external and internal forces. The main forces taken into consideration in this model and arrows representing the sign convention are shown in Figure 1.

F_g denotes the gravitational force component acting in the positive or negative direction of the vehicles moving direction due to road inclination. F_a is the force due to the airdrag of a moving vehicle and it is working opposite to the vehicles moving direction. F_ris the rolling resistance force acting between the tires and the road. F_bis the force between the wheels and the road when the vehicles is breaking. The propulsion force, Fp, is acting between wheel and road.

Figure 1: Forces acting on a moving vehicle vehicle, where Fp is the propulsion force moving the vehicle forward, F_g is the gravitational force, F_a the airdrag force, F_r the rolling resistance force and F_b the breaking force, α denotes the road inclination.(The picture is provided by courtesy of Alam (2014))

The fundamental parts for vehicle propulsion are the engine and driveline, together called the powertrain. The driveline consists of the components clutch, transmission, propeller shaft, nal drive, drive shafts and wheels. There are several ways to model the driveline depending on the purpose of the application. This model is a general model derived as in Eriksson and Nielsen (2014) and in Alam (2014).

2.1.1 Powertrain

An overview of the powertrain is visualized in Figure 2. The output torque from the engine is the input torque to the driveline. The component models interfaces are torque and angular velocities.

Due to elasticity, the torques and angular velocities of the components are dierent. To simplify the model the components are approximated as rigid. The transformation of the torque for the transmission and nal drive is depending on which gear that is engaged, modeled as a conversion ratio. Automatic transmission uses a torque converter which has lower eciency than manual transmission which is taken into account when modeling.

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Figure 2: The overview of a HDV powertrain consisting of engine, clutch, gearbox, propeller shaft, drive shaft, nal drive and wheels. (The picture is provided through courtesy of Alam (2014)) Engine

The model is based on a diesel fuel combustion engine which produces a torque as output. The internal dynamics of the engine are not modeled. It is assumed that the engine inertia, Ie, is known. Using Eulers's second law the engine torque output, Te , is given by

I_eω˙_e = T_e− T_c, (2.1)

where ˙ω_e denotes the angular acceleration of the engine and T_c denotes the external torque from the clutch.

Clutch

The clutch in vehicles with manual transmission connects the engines ywheel with the transmission input shaft pressing frictional discs together. Since the components are assumed to be rigid the internal friction is neglected. An engaged clutch is modeled as

T_c= T_t,

ω_c= ω_t, (2.2)

where T_tdenotes the transmission torque input and ω_trespectively ω_cthe output angular velocities from the transmission and clutch.

Transmission

The transmission, also called the gearbox, consists of a set of gears and is the connection between the clutch and the propeller shaft. Each gear has a conversion ratio γ_t. By neglecting damping losses and the rotation inertia the relation between input and output torque can be modeled as

T_tγ_tη_t= T_p,

γ_tω_p= ω_t, (2.3)

where T_pdenotes the output torque for the propeller shaft, η_tthe eciency of the gearbox and ω_p denotes the angular velocity of the propeller shaft.

Propeller shaft

The propeller shaft connects the transmission with the nal drive. Approximated as sti the torque for the nal drive T_f is given as

T_p= T_f,

ω_p= ω_f, (2.4)

where ωfdenotes the angular velocity of the nal drive.

Final Drive

By the same assumptions as for the transmission the equation for nal drive is T_fγ_fη_f= T_d,

γ_fω_d= ω_f, (2.5)

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where γ_fdenotes the conversion ratio for the nal drive, η_fthe eciency of the nal drive transformation, Td the output torque for the drive shaft and ωddenotes the angular velocity of the drive shaft.

Drive shaft

By the same assumptions as for the propeller shaft the equation for the drive shaft is T_d= T_w,

ω_d= ω_w, (2.6)

where T_w denotes the wheels torque and ω_wthe angular velocity of the wheels.

Wheels

The torque on the wheels depends on the dynamics of the vehicle I_wω˙_w= T_w− r_wF_p,

˙v = r_wω˙_w= r_w

γ_tγ_fω˙_e, (2.7)

where Iwdenotes the moment of inertia of the wheels, ˙ωwthe angular acceleration of the wheels, r_wthe radius of the wheels, ˙v the acceleration of the vehicle and Fpdenotes the vehicle propulsion force.

2.1.2 External Forces

The external forces acting on the vehicle in this model are the aerodynamic drag, the rolling resistance between the wheels and the road and the longitudinal component of the gravitational force.

Aerodynamic drag

A force in a moving vehicles negative driving direction occur due to the aerodynamic drag and can be approximated by the equation

F_a,body=1

2c_DA_aρ_a(v − v_wind)². (2.8) Here cD denotes the air drag coecient, Aa the maximum cross-sectional area of the vehicle, ρa

the air density, v the vehicles velocity and vwind denotes the velocity of the wind.

Arranging HDVs in a platoon reduces air drag for all participating vehicles Alam, Gattami, and Johansson (2010). Several empirical studies have been conducted determining which eect the distance, d, between the vehicles has on the air drag coecient, for example Davila, Aramburu, and Freixas (2013). The air drag coecient for a vehicle in a platoon is reduced with a function Φ(d) ∈ [0, 1], (c_D = c_dΦ(d) where c_d denotes the air drag coecient for a single vehicle traveling alone). The total aerodynamic drag force will then be represented by

F_a=1

2c_dΦ(d)A_aρ_av² (2.9)

when approximating vwind≈ 0. The function Φ(d) are dependent on the vehicles position in the platoon and can be represented as

Φ_i(d) = 1 −f_i(d)

100 (2.10)

where

f_i(d) = a_i_,dd + b_i_,d (2.11)

which is a linearization of the non-linear functions from an empirical studies depicted in Figure 3 where i denotes the ith vehicle in the platoon.

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Figure 3: Empirical result of air drag reduction of HDV platooning.(The picture is provided through courtesy of Alam et al. (2010))

As in Liang (2011) the functions is approximated with a rst order least square approximation resulting in

f1(d) = −0.9379d + 12.8966 0 ≤ d ≤ 15, f₂(d) = −0.4502d + 43.0046 0 ≤ d ≤ 80, f3(d) = −0.4735d + 51.5027 0 ≤ d ≤ 80.

(2.12)

The platoon in this study have an amount of vehicles participating which exceeds three, which is the case in the empirical study from which the result i gathered. The air drag coecients for vehicles i > 3 are approximated to be the same as for the third vehicle, fi>3(d) = f3(d).

Rolling resistance

A frictional force between the tires and the road emerge during rolling. This force acts on the vehicle according to

F_r= c_rmg cos(α), (2.13)

where m denotes the vehicles mass, g the gravitational constant, α the road inclination and cr

denotes the rolling resistance coecient which depends on the tires and surface of the road.

Gravitational Force

The longitudinal component of the gravitational force is

F_g= mg sin(α), (2.14)

where m denotes the vehicles mass, g is the gravitational constant and α denotes the road inclination.

2.1.3 Nonlinear vehicle model

To conclude the rst part of the nonlinear vehicle model an expression is derived from the Equations (2.1)-(2.7) describing the components of the powertrain. The propulsion force can be expressed in two terms as

F_p= γ_tγ_fη_tη_f

r_w T_e−I_w+ γ_t²γ_f²η_tη_fI_e

r_w² ˙v (2.15)

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where the rst term denotes the propulsion force, F_e, generated from the engine and the second term denotes the internal inertia, mi˙v. That is

F_e= γ_tγ_fη_tη_f

r_w , mi= I_w+ γ²_tγ_f²η_tη_fI_e

r²_w . (2.16)

Applying Newton's second law on the forces in Figure 1 a nonlinear vehicle model can be derived as

m ˙v = F_p− F_b− F_a− F_r− F_g. (2.17) The braking force F_b is often omitted in vehicle modeling due to diculty in modeling brake control logic and other brake characteristics. This gives the nonlinear vehicle model

m_a˙v = F_e− F_a− F_r− F_g

= γ_tγ_fη_tη_f r_w T_e−1

2c_dΦ(d)A_aρ_av²− c_rmg cos(α) − mg sin(α), (2.18) where ma, here called the vehicles accelerated mass, is dened by

m_a= m + m_i, (2.19)

where m is the vehicles physical mass.

The acceleration of the vehicle can be expressed as

˙v = f (v, d, T ) = k_eT − k_dΦ(d)v²− k_rcos(α) − k_gsin(α) (2.20) where T = T_e to simplify the notation and with the constants

k_e = r_wγ_tγ_fη_tη_f

I_w+ mr_w² + γ²_tγ_f²η_tη_fI_e, k_d=

1

2r_w²A_aρ_ac_d I_w+ mr_w² + γ²_tγ_f²η_tη_fI_e, k_r= c_rr_w²mg

I_w+ mr_w² + γ²_tγ_f²η_tη_fI_e, k_g= r_w²mg

I_w+ mr_w² + γ²_tγ_f²η_tη_fI_e.

(2.21)

2.1.4 Linear vehicle model

Nonlinear models can be dicult to use as basis for control design. A vehicle model can in a satisfying way be linearized, still fullling the purpose of being a useful approximation of the real system. The model is linearized with respect to an equilibrium point by applying a Taylor approximation to Equation (2.20). The equilibrium point corresponds to a set reference velocity v0, the torque which maintain the velocity T0 and the distance between vehicles d0. Note that due to the limitations of this project α = 0. The rst order Taylor approximation around the equilibrium points is given by

∆ ˙v ≈ f (v0, T0, d0) + f_v⁰(v0, T0, d0)∆v + f_d⁰(v0, T0, d0)∆d + f_T⁰(v0, T0, d0)∆T

= −2k_dΦ(d0)v0∆v − Φ⁰(d0)v²₀k_d∆d + k_e∆T (2.22) where ∆v = v − v0, ∆d = d − d0 and ∆T = T − T0. The equation can for each individual vehicle in the platoon be expressed as

∆ ˙v_i= a_i∆v_i+ b_i∆d_i+ c_i∆T_i (2.23) where i is the vehicles position in the platoon, ai = −2k_dΦ_i(d₀)v₀, bi= −Φ⁰_i(d₀)v₀²k_dand ci= k_e.

2.2 Platoon model

The linear vehicle model is used to derive a model for the whole system according to the states of interest. Since the platoon is moving around the equilibrium point the states will represent the change from this equilibrium point for the vehicles torque, velocity and the distance between two vehicles. The platoon model is a reference tracking model, which in this case means that the rst vehicle will follow a set velocity reference. The model is derived in accordance to earlier work of controlling a platoon done by Liang (2011).

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2.2.1 Reference Tracking model

A state space model for a vehicle platoon consisting of N number of vehicles is given as

∆x = [∆e ∆v₁∆d_1,2 ∆v₂ ∆d_2,3.... ∆v_{N −1}∆d_{N −1,N} ∆v_N]^T,

∆u = [∆T₁∆T₂ ...∆T_{N −1}∆T_N]^T, (2.24)

where

∆ ˙e = r − ∆v1,

∆ ˙d_i−1,i= ∆v_i−1− ∆v_i,

∆ ˙v1= a1∆v1+ b1∆d1,2+ c1∆T1,

∆ ˙vi= ai∆vi+ bi∆di−1,i+ ci∆Ti , ∀ 1 < i ≤ N,

(2.25)

in accordance to variables shown in Figure 4.

Figure 4: Illustration of a platoon.

The rst state in the system makes it possible to use the concept of integral action to eliminate the steady state error of the system using ∆e = R (r − ∆v1)dt. The platoon model for a N vehicle platoon can be written in a state-space equation

˙

x =Ax + Bu + Dr

y =Cx (2.26)

where x is the states space vector, u the input signals and y the measured output signal. Where the matrices are represented as

A =





 0 −1 0 a1 b1

1 0 −1

b₂ a₂ 0

... ... ...

b_{N −1} a_{N −1} 0

1 0 −1

bN aN





 , B =







0 0

c1 0

0 0

0 c₂ ...

c_{N −1} 0

0 0

0 cN





 ,

C =





 1 0

0 1 0

... ... ...

0 1 0

0 1





 , D =





 1 0 0 0...

0 0 0





 .

(2.27)

3 Control Theory

Controllers are used in part to make a system reach a desired behavior. If the performance of the system can be expressed as a mathematical criterion, the control problem can be to solve the mathematical optimization problem. Linear Quadratic control is a method based on a linear model and a quadratic cost function is to be minimized or maximized. In this chapter the proposed optimal control will be presented.

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3.1 System architecture

In this section two dierent architectures are presented. The optimal solution is a so called centralized system where the system is seen as one unit. This solution is also the most critical since realizations of such systems are complex due to limitations in hardware. For example, an implemented controller has a set sampling time which makes the size of the platoon a critical moment since the information between the vehicles need to be sent during that time. With large distances between the platooning vehicles that may not be possible. Another aspect to take into consideration is disturbances in the wireless communication which will result in a system needed for verication of transmissions. That is why often other architectures are used as an approximation of the system.

3.1.1 Centralized System

A centralized system is a system where the platoon can be seen as one unit. All the vehicles will send data to a central unit where the control strategy for each vehicle will be calculated taking into consideration all the information from all vehicles. An illustration of this system can be seen in Figure 5.

Figure 5: Illustration of a platoon as a centralized system.

The systems state space model can be described in (3.1) where N is the number of vehicles.







˙e

˙v1

d˙1,2

...

d˙_{N −1,N}

˙v_N







=







A_1,1 A_1,2 A_2,1 A_2,2 A_2,3

A3,1 A3,3 A3,3

... ... ...

A2N −1,2N −2 A2N −1,2N −1 A2N −1,2N

A2N,2N −1 A2N,2N











 e v₁ d1,2

...

dN −1,N

vN







+







0 0

B2,1 0

0 0

... ...

0 0

0 B2N,N











 T1

T2

...

TN −1

TN





 +





 D1

0...

0





 r.

(3.1)

3.1.2 Decentralized System

A decentralized system is a system which can be seen as several smaller systems where each system has its own controller and its own information depending on the connection to the other systems.

In this case each vehicle receives information from the vehicle in front of it. Each vehicle will see the velocity of the vehicles in front as a disturbance to its own system. Figure 6 represents an illustration of a decentralized system.

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Figure 6: Illustration of a platoon as a decentralized system.

The systems state space model is described in (3.2) where N is the number of vehicles.

˙e

˙v1

=0 A1,2

0 A2,2

e v1

+

0 B2,1

T₁+D1

0

r,

d˙_1,2

˙v₂

=A_3,3 A3,4

A4,3 A4,4

d_1,2 v2

+

0 B4,2

T2+A_3,2 0

v1, ... = ...

d˙_{N −1,N}

˙v_N

=A2N −1,2N −1 A2N −1,2N

A2N,2N −1 A2N,2N

dN −1,N

vN

+

0

B_{2N −1,N}

T_N +A2N −1,2N −2

0

v_{N −1}.

(3.2)

3.2 Linear Quadratic Tracking Control

The linear quadratic tracking system (LQT) maintains the output as close as possible to the reference signal with minimal control energy. The system in Equation (2.26) is given as

˙

x =Ax + Bu + Dr,

y =Cx. (3.3)

The quadratic performance criteria to be minimized is dened as

J = 1 2

Z ∞ 0

x^T(t)Qx(t) + u^T(t)Ru(t) dt (3.4) where the matrices A, B, C, D, Q and R are constant in time. For this problem the matrices A and Bhave to be controllable and the matrix Q has to be positive semi denite and the matrix R has to be positive denite. The optimal control is obtained by

u^∗(t) = −Lx^∗(t). (3.5)

Where the matrix L is dened as

L = R⁻¹B^TP (3.6)

and has a full matrix structure, given as







L_1,1 L_1,2 L_1,3 L_1,4

L_2,1 L_2,2 L_2,3

L_3,1 L_3,2 L_{N −3,2N}

L4,1 ... LN −2,2N −1 LN −2,2N

LN −1,2N −2 LN −1,2N −1 LN −1,2N

LN,2N −3 LN,2N −2 LN,2N −1 LN,2N







. (3.7)

The matrix P is the solution to the algebraic Riccati equation (ARE)

P A + A^TP + Q − P BR⁻¹B^TP = 0, (3.8)

(Naidu 2003).

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3.3 Sparse Promoting Linear Quadratic Regulator

A sparse promoting linear quadratic regulator (LQRSP) can be described as an LQR where the feedback gain has been designed to be sparse in addition to minimizing the variance amplication (also called H2-norm) of a distributed system. The LQRSP method was introduced by Fardad, Lin, and Jovanovic (2011b). The method consists of two steps, which due to the separability of the penalty-functions can be divided into sub-problems. The optimal control problem is incorporated with sparsity-promoting penalty functions which penalize the number of communication links in the distributed controller. This part in the method alternates between sparsity-promoting and nding the optimal performance of the closed-loop system. It is solved using the alternating direction method of multipliers (ADMM). In the second part the identied sparsity patterns is used to nd the optimal feedback gain. The sparse promoting linear quadratic regulator is used to analyze dierent structures of the feedback gain to determine the desired communication architecture.

The regulator is implemented as in Lin, Fardad, and Jovanovic (2013). This method does not always nd the global minimum due to the non-convexity of the problems. A reformulation of the problem using sequential convex programming where the nonconvex constraints via linearizations are satised upon convergence is presented in Fardad and Jovanovic (2014).

3.3.1 Optimal control problem

The control problem given on a state space representation can be formulated as

˙

x = Ax + Bu + Dr, z =Q^1/2

0

x +

0 R^1/2

u, u = −Lx,

(3.9)

where z is the performance output and Q = Q^T ≥ 0 and R = R^T > 0are the performance cost matrices. Assume that (A, B) is stabilizable and (A, Q) detectable. The closed-loop system is given by

˙

x = (A − BL)x + Dr, z =

Q^1/2

−R^1/2L

x. (3.10)

In subject to structural constraints on the design of the optimal state feedback gain L, assume that there exists a subspace S taking into consideration these constraints and assume there exists a L ∈ S which is stabilizing. The structured minimization problem is then given by

minimize J(L),

subject to L ∈ S, (3.11)

where J(L) is the square of the H2norm of the transfer function from r to z. This can be expressed as

J (L) =(trace(D^TP (L)D), Lis stabilizing,

∞, otherwise. (3.12)

The matrix P (L) on the form

P (L) = Z ∞

0

e^(A−BL)^T^t Q + L^TRL e^(A−BL)tdt (3.13) denotes the observability Gramian for the closed-loop system, and is the solution to the Lyapunov equation

(A − BL)^TP + P (A − BL) + (Q + L^TRL) = 0. (3.14) Note that with L dened as in (3.6) the Equations (3.14) and (3.8) will be identical.

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3.3.2 Sparsity-Promoting Penalty Functions

The problem expressed in Equation (3.11) assume a specic structure of the feedback gain L.

According to the aim of this thesis the communication structure should be found without any prior assumptions on the sparsity pattern on L, which is why the optimization problem instead can be given as

minimize J(L) + γ card(L). (3.15)

Here the sparsity of L is incorporated in the objective function. card(L) denotes the cardinality function, which represents the number of non-zero elements of a matrix. The scalar γ varies over [0, +∞)and represents the trade-o between the sparsity of the feedback gain and the performance J (L). γ = 0 will generate a centralized feedback gain matrix Lc in (3.16) (the same as the solution to the standard LQT problem) and as γ increases a sparser L is promoted, with the aim of a diagonal structure like in (3.17) - (3.18)







L1,1 L1,2 L1,3 L1,4

L2,1 L2,2 L2,3

L3,1 L3,2 LN −3,2N

L_4,1 ... L_{N −2,2N −1} L_{N −2,2N}

L_{N −1,2N −2} L_{N −1,2N −1} L_{N −1,2N} L_{N,2N −3} L_{N,2N −2} L_{N,2N −1} L_N,2N







, (3.16)







L_1,1 L_1,2

L2,1 L2,2 L2,3

...

LN −1,2N −2 LN −1,2N −1 LN −1,2N

LN,2N −1 LN,2N







, (3.17)





 L1,1

L2,2

...

L_{N −1,2N −1}

L_N,2N







. (3.18)

Due to inconvenience working with the cardinality function it is often approximated with other sparsity-promoting functions. Lin, Fardad, and Jovanovic (2013) propose dierent variations of sparsity-promoting functions, all consisting of the `1 norm, which typically replaces cardinality functions when sparsity is desired and is widely used for this purpose in applied statistics, machine learning and signal processing. In line with the aim of this thesis the weighted `1 norm will be used in this case,

g(L) =X

i,j

W_i,j|L_i,j|, (3.19)

where Wi,j are non-negative weights. This function was used to enhance sparsity in signal recovery in Candés, Wakin, and Boyd (2008) where it is stated that this method outperforms the `1

minimization since fewer measurements is needed for exact recovery. This weighted algorithm is implemented by using iterations where the weights are determined by the solutions of the weighted

`1 problem in the previous iteration. This method was later used to design sparse feedback gain for a class of distributed systems by Fardad, Lin, and Jovanovic (2011b). The relaxed optimal control problem is then given by

minimize J(L) + γg(L). (3.20)

3.3.3 Alternating Direction Method of Multipliers

The alternating direction method of multipliers (ADMM) is in this case used to alternate between optimizing the closed-loop system and to promote the sparsity of the feedback gain using partial

Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning

Examensarbete 30 hp November 2016

Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning

Matilda Dahlqvist

Abstract

Sparsity Promoting Linear Quadratic Regulator for Heavy Duty Vehicle Platooning

Contents

List of Figures

List of Tables

1 Introduction

2 Modeling

3 Control Theory