application of robust multivariable control on energy management of hybrid electric vehicles

7/27/2019 Application of Robust Multivariable Control on Energy Management of Hybrid Electric Vehicles

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Application of Robust Multivariable Control on

Energy Management of Hybrid Electric VehiclesSajjad Fekri and Francis Assadian

Automotive Mechatronics Centre, Department of Automotive Engineering,

School of Engineering, University of Cranfield, Bedfordshire MK43 0AL, United KingdomTel: +44 (0)1234 754 657, Fax: +44 (0)1234 750 425, Email: {s.fekriasl,f.assadian}@cranfield.ac.uk

Abstract

In this paper, the design and development of robust multivariable control design is presented to highlight

some of the automotive engineering practical issues with an emphasis on a non-trivial application of Hybrid

Electric Vehicle (HEV) Energy Management. Most of the existing energy management approaches are not

model-based dynamic systems and/or do not address the concepts of robust stability and performance.

By considering a multi-input multi-output (MIMO) dynamics of the HEV model, subject to uncertain

parameters and dynamics, it is demonstrated that the two subsystems of the HEV dynamics with strong

interactions (ie internal combustion engine and integrated DC motor/generator) are effectively controlled

over a defined drive cycle. The employed multivariable mixed-mu control synthesis provides guaranteed

stability- and performance robustness for the plant under investigation and is shown to yield satisfactory

performance subject to parametric uncertainties (such as combustion lag, rotational moment of inertia and

damping), unmodeled complex-valued uncertainty (due to fueling delay) and unmeasurable exogenous plant

disturbance (e.g. vehicle load). Simulation results verify that robust multivariable control systems design

could be a breakthrough to production of hybrid electric vehicles by yielding significant improvements on

the emission, fuel consumption, calibration time and cost.

Index Terms

Robust control; mixed- synthesis; multivariable feedback control systems; hybrid electric vehicle; diesel

engine modeling; uncertain systems; energy management.

I. Introduction

Hybrid Electric Vehicles (HEVs) have been received with great enthusiasm and attention in recent years

[1]. On the other hand, complexity of hybrid powertrain systems have been increased to meet end-user

demands and to provide enhancements to fuel efficiency as well as meeting new emission standards [2] which

accordingly require more complex (advanced) control system designs to achieve satisfactory performance for

the hybrid powertrain.

Prof. Assadian is also Head of Department of Automotive Engineering.


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While the main objective of hybrid electric vehicles is to reduce Carbon Dioxide (CO 2) emissions with

an optimised fuel consumption [3], a significant amount of research has been devoted to the field of energy

management for full Hybrid Electric Vehicles (HEVs) and Electric Vehicles(EV) [4].

There are a number of energy management methods in the literature of hybrid electric vehicles [5].

Among these energy management strategies, a number of heuristics techniques, say e.g. using Rule-based

or Fuzzy logic, have claimed to offer some improvements in the HEV energy efficiency [6], [7] where the

optimisation objective is, in a heuristic manner, a function of weighted fuel economy and driveability

variables integrated with a performance index, to obtain a desired closed-loop system response. However, such

heuristics based energy management approaches suffer from the fact that they guarantee neither an optimal

result in real vehicle operational conditions nor a robust performance if system parameters deviate from

their nominal operating points. Consequently, other strategies have emerged that are based on optimisation

techniques to search for sub-optimal solutions using programming concepts such as linear programming,

quadratic programming and dynamic programming [8], [9], [10]. Nonetheless, they may not lead to the

feasible casual solutions, since they assume that the future driving cycle is entirely known. Moreover, the

required burdensome calculations integrated with these approaches have imposed a high demand on practical

computational resources which prevent from being implemented on-line in a straightforward manner.

Two new HEV energy management approaches have been recently introduced in the literature. In the first

approach, instead of one specific driving cycle a set of driving cycles are considered for calculating an optimal

control law resulting in the stochastic optimisation approach [11]. A solution to this approach is calculated

off-line and stored in a state-dependent lookup table. Similar approach in this course employs Explicit Model

Predictive Control [12], [13]. In this approach, an entire control law is computed offline, whilst the controller

will be implemented online based on the lookup table, similar to the stochastic optimisation approach.

The lookup table provides a quasi-static control law which is directly applicable to the on-line vehicle

implementation. Although this method has potential to perform well for systems with fewer states, inputs,

constraints, and sufficiently short time-horizons [14], it cannot be utilised in a wide variety of applications

whose dynamics, cost function and/or constraints are time-varying due to e.g. parametric uncertainties

and/or unmeasurable exogenous disturbances. In other words, any lookup-table-based optimisation approach

may end up with severe difficulties in covering a real-world driving situation with a set of individual driving

cycles.

A recent approach has endeavored to decouple the optimal solution from a driving cycle in a game-theoretic

(GT) framework [15]. In this approach, effect of time-varying parameters (due to drive cycle) is represented

by the actions of the first player while the effect of operating strategy (energy management) is modeled by


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the actions of the second player. The first player (drive cycle) wishes to maximize the performance index

which reflects the optimisation objectives, say e.g. to minimise emission constraints and fuel consumption,

while the second player aims to minimize this performance index. Any solution to this approach is calculated

off-line and stored in a state-dependent lookup table. This lookup tables provides a quasi-static control law

which is directly employed for on-line implementation in vehicle. Similar to aforementioned methods, the

main drawbacks of the game-theoretic approach are the lack of robustness, and also due to there being a

quasi-static method, it cannot address the vehicle deriveability requirements.

If only the present state of the vehicle is considered, optimisation of the operating points of the each

component can still be beneficial. Towards this end, some methods define an optimisation criterion to

minimise the vehicle fuel consumption and reduce exhaust emissions [11]. A weighting factor is typically

included to prevent a drift in the battery from its nominal energy level and also to guarantee a charge

sustaining solution. This approach has been considered in the past, but it is still remained an arduous task

to select a weighting factor that is mathematically sound [16]. An alternative approach is to extend the

objective function with a fuel equivalent term. This term includes the corresponding fuel use for the energy

exchange with the battery in the optimisation criterion [17].

Two fundamental drawbacks of aforementioned strategies are firstly their consideration of driveability

being an afterthought and secondly the driveability issue is deemed in an ad-hoc fashion as these approaches

are not model-based dynamic. Currently applicable techniques, such as game-theoretic based optimisation,

utilise quasi-static models which are not sufficient to address driveability requirements [15].

Hybrid modeling tools have been recently developed to analyse and optimise a number of classes of hybrid

systems. Among these modeling tools developed to represent the hybrid systems, we shall refer to Mixed

Logical Dynamical (MLD) [18], HYbrid Systems Description Language (HYSDEL) [19], and Piecewise Affine

(PWA) models [20], [10], to name but a few. In addition, Hybrid Toolbox for Matlab [21] is developed for

modeling, simulation, and verifying hybrid dynamical models and also for designing hybrid model predictive

controllers. Almost all of these hybrid tools, however, have dealt with multivariable hybrid systems as

decoupled single-input single-output (SISO) loops and, thus, have not addressed the MIMO directional

properties quantified by the singular value decomposition (SVD), e.g., for the use of practical hybrid electric

vehicle energy management application. It should be self-evident that, in any MIMO case, transfer function

matrices describe all relevant dynamic systems. Thus, directionality issues predominate and we must use

singular value plots to understand the dynamic properties in the frequency domain.

Towards a practical model-based dynamic controller approach, there are a number of model-based energy

management methods such as Model Predictive Controls (MPC). A recently developed package for the hybrid


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MPC design is referred to as Hybrid and Multi-Parametric Toolboxes [22] which is based on the traditional

model predictive control optimisation alternatives using generic optimisers. The main shortcoming of model

predictive control methods is that they can only be used in applications with sufficiently slow dynamics

[14], and hence are not suitable for many practical applications including HEV energy management problem.

For this reason the standard MPC algorithms have been retained away from modern production vehicles. In

fact, a number of inherent hardware constraints and limitations integrated with the vehicle electronic control

unit (ECU), such as processing speed and memory, have made on-line implementations of predictive control

approaches almost impossible. In a number of applications, MPC is currently applied off-line to generate

the required maps and then these maps are used on-line [ 11]. However, generation and utilisation of maps

defeat the original purpose of designing a dynamic compensator which maintains driveability. Therefore,

there is a vital need of model-based dynamic controllers with an increased processing speed, and robust to

uncertainties, for the real-time application of HEV energy management.

In this paper, we shall overview some of the theoretical and practical aspects of the robust multivariable

control using mixed- synthesis [23] for a practical problem of hybrid electric vehicle energy management.

The mixed- compensator design process in fact detunes an optimal H compensator [24], designed for

the nominal generalised plant, to hedge for the uncertain real parameters and the inevitable unmodeled

dynamics. The novelty of this work is indeed in the design and development of the robust multivariable

control system with practical significance of addressing dynamic vehicle driveability. The proposed control

design, to replace the existing non-dynamic and/or non-robust energy management design approaches,

is derived based on the dynamic models of the plant and hence driveability requirements are taken into

consideration as part of the controller design. After the design of the robust multivariable controller, which

makes up the main part of the energy management solution, the robust controller will be integrated into a

HEV system as the torque manager. By utilising this proposed design, one is capable to pose a quantified

robustness analysis in the presence of uncertainties due to, e.g., a change in the dynamics of the plant,

or lack of proper estimation of the vehicle load torque (plant disturbance). We shall highlight some of the

aspects of the multivariable robust control design by representing typical simulation results. We strongly

believe that the results of this work could make automotive engineers motivated to continue the design and

development of robust multivariable control design and, using its potential capability, to tackle a wide range

of applications in the automotive control system designs.

This paper is organized as follows. In Section II, we will overview the literature of the hybrid electric

vehicles. In Section III, we discuss briefly the characteristics of the robust multivariable control system

design. In Section IV, we provide in detail the mathematical description of the simplified dynamical model


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of the hybrid diesel electric vehicle (parallel type) along with analysing the open-loop dynamics. Simulation

results will be demonstrated in Section V. Section VI will present some of our conclusions.

II. Hybrid Electric Vehicles

Modern day automotive engineers must, among other objectives, maximize fuel economy and sustaina reasonably responsive car (i.e. maintain driveability) while still meeting increasingly stringent emission

constraints mandated by the government. Towards this end, Hybrid Electric Vehicles (HEVs) have been

introduced which typically combine two different sources of power, the traditional internal combustion

engine (ICE) with one (or more) electric motors, mainly for optimising fuel efficiency and reducing CO2

and greenhouse gases (GHG) [25].

Compared to the vehicles with conventional ICE, hybrid propulsion systems are potentially capable of

improving fuel efficiency for a number of reasons: they are able to recover some portion of vehicle kinetic

energy during braking and use this energy for charging the battery and hence, utilise the electric motor at

a later point in time, when required. Also, if the torque request (demanded by driver) is below a threshold

torque, the ICE can be switched off as well as during vehicle stop for avoiding engine idling. These are

some representative advantages of the hybrid vehicles compared to those of conventional vehicles. There are

other benefits offered by hybrid electric vehicles in general, e.g. engine downsizing and utilising the electric

motor/motors to make up for the lost torque. It turns out that the internal combustion engine of the hybrid

electric vehicle can be potentially designed with a smaller size and weight which results in a superior fuel

efficiency and reduced emissions [26].

The concept of sharing the requested power between the internal combustion engine and electric motor

for traction during vehicle operation is referred to as vehicle supervisory control or vehicle energy man-

agement [27]. The latter term, employed throughout this paper, is particularly referred to as a control

allocation for delivering the required wheel torque to maximize the average fuel economy and sustain the

battery state of charge (SoC) within a desired charging range [28].

The vehicle energy management development is a challenging practical control problem for which a

significant amount of research has been devoted in regard to the full HEVs and Electric Vehicles (EVs)

in the last decade [29]. To tackle such a challenging problem, there are currently extensive academic and

industrial research interests ongoing in the area of hybrid electric vehicles as these vehicles are expected to

make considerable contributions to the environmentally conscious requirements in the vehicle production

sector in the future see [30] and other references therein.

To make the above claim more legible, we have studied the number of IEEE publications published between


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1990 and 2010 see also [9]. Figure 1 depicts the number of publications recorded at the IEEE database1

whose abstract contains at least one of the strings hybrid vehicle(s) or hybrid electric vehicle(s).

1990 1995 2000 2005 20100

100

200

300

400

Year

NOP

Actual Data

Linear Fitting

Fig. 1. Hybrid vehicle research trend based on the number of publications of the IEEE over the period 1990 to 2010.

From Figure 1, it is obvious that the number of publications in the area of hybrid electric vehicles (HEVs)

has been drastically increased during this period, from only 3 papers in 1990 to 414 papers in 2010. Recall

that these are merely the IEEE database publications - obviously, there are other sources than those of the

IEEE which have not been taken into account in this study, including books, articles, conference papers,

theses, filed patents, and technical reports . Besides, a linear regression analysis of the IEEE publications

shown in Figure 1 indicates that research in the field of hybrid vehicles has been accelerated remarkably

since 2004. One may also predict that the number of publications in this area could be increased up to about

900 articles in 2015, that is more than twice as many as in 2010 - this is a clear evidence to acknowledge

that HEVs research and development is expected to make considerable contributions to both academia andindustry of production automotive sector in the future.

In regard to hybrid electric vehicles, here are the facts and regulations which must be taken into consid-

eration by automotive engineers:

Due to the ever increasing stringent regulations on fuel consumption and emissions, there are tremendous

mandates on Original Equipment Manufacturers (OEMs) to deliver fuel-efficient and less-polluting

vehicles at lower costs. Hence, the impact of advanced control design for the application of the hybrid

vehicle powertrain systems has become significantly important [3].

It is essential to meet end-user demands for increasingly complex new vehicles towards improving vehicle

performance and driveability [31], while still continuing to reduce costs and meeting new emission

standards.

There is a continuous increase in the gap between the theoretical control advancements and the practical

control strategies being applied to the existing production vehicles. This gap is due to a number of

missed opportunities in addressing fundamental functionalities, e.g. fuel economy, emissions, driveability,

1

See http://ieeexplore.ieee.org for more information.


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unification of control architecture and integration of the Automotive Mechatronics units on board

vehicle.

The robust MIMO control system design seem promising advanced tools to bridge this gap should they

could be developed, tested and implemented in the production hybrid vehicles.

III. Multivariable feedback control design

All linear time-invariant (LTI) models of real dynamic systems are subject to uncertainty. For each LTI

model, we must take into account both unmodeled dynamics and uncertain real parameters, as well as the

characteristics of unmeasurable exogenous plant disturbances and sensor noises. The design of a multi-input

multi-output (MIMO) dynamic compensator, and of the resulting robust feedback control system, must

possess guarantees of both stability- and performance-robustness with respect to the explicit performance

specifications posed by the control system designer [32].

In summary, the purpose of this paper is to highlight importance for analysis and concrete results related to

the performance tradeoffs that are always present in MIMO feedback designs with the guaranteed stability-

and performance-robustness. In particular, the following key engineering design issues of multivariable

feedback control system for the HEV energy management must be taken into consideration:

Controllers that are not robust, i.e. are designed without taking into account the uncertainties, could

potentially fail in providing stability, performance or both of them.

Controllers which are designed based on the unstructured H or even robust D-K iteration (complex

) [33], are very conservative and yield inferior setpoint tracking performance compared with the

compensators designed by the mixed- software that implements the full D, G K iteration [34].

In robust MIMO feedback designs, for fixed performance weights, changes in the level of uncertainty

associated with the real parameters can have a significant impact upon the directional properties of

the closed-loop system. Of course, these can be analyzed and evaluated by the SVD methodology.

In robust MIMO feedback designs with fixed uncertainty for the real parameters, designer-imposed

preferences for relative performance associated with different signals will also have a significant impact

on the directional properties of the closed-loop system. Again, these can be analyzed and evaluated

by the SVD concepts.

All unmodeled dynamics must be bounded by their maximum singular values and properly introduced

in the mixed- synthesis.

Examine and determine the actual closed-loop stability regions for different types of legal unmodeled

dynamics. These can lead to some unexpected behavior [35].


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Fortunately, in the past decade or so, the mixed- design methodology [36], [34] and Matlab software

[37], utilizing the so-called D, G K iteration, have been developed that can indeed be used to design robust

MIMO feedback control systems with the requisite stability- and performance-robustness guarantees.

The complex -synthesis (D K iteration) in its present form and as presently is implemented in

current numerical Matlab software package [38], [39], [37] has substantial disadvantages, that is, when real

parameter uncertainty is present in the system, the resulting compensator can be arbitrarily conservative.

The robust mixed- compensator design process detunes an optimal H compensator, designed for the

nominal generalised plant, to hedge for the worst uncertain real parameters and the inevitable unmodeled

dynamics. The greater the parametric uncertainty, the smaller the guaranteed performance. It is significantly

important to quantify the performance tradeoff of the reference tracking problem. Moreover, since we deal

with a MIMO design, directional properties quantified by the singular value decomposition (SVD) are

important, which illustrate the strong interactions of the subsystems. The theoretical foundation for the

mixed- analysis and synthesis is too copious for a comprehensive review in the framework of this paper.

Hence, we shall provide the key aspects of the mixed- design as discussed in the subsequent section. For a

more in-depth discussion the reader is referred to, e.g., Refs. [40], [41], [36], [42].

A. Mixed- Synthesis: D,G-K algorithm

In this section, a summary of the background for dealing with combined real and complex uncertainties,

so called mixed- synthesis is presented2.

Consider the robust performance problem of the mixed- setup [43] depicted in Figure 2.

The definition of the mixed structured singular value (mixed- ) depends upon the underlying block

structure of the uncertainties. The following types of uncertainties are considered in this setup: repeated

real scalars r, repeated complex scalars c, and full complex blocks C. The non-negative integers mr, mc,

and mC specify the number of uncertainty blocks of each type while the positive integers ki; i = 1, 2, . . . , mr+

mc + mC, define the dimensions of each block (i.e., the multiplicity of real or complex scalars should they

be repeated, or the size of a full complex block, respectively). Accordingly, the set of allowable uncertainties

can be defined as

X = {diag(

r1Ik1, . . . , rmrIkmr ,

c1Ikmr+1, . . . ,

cmcIkmr+mc ,

C1 , . . . ,

CMC

):

ri R, ci C,

Ci C

kmr+mc+ikmr+mc+i} (1)

2Although a recent version of Matlab Robust Control Toolbox [37] covers the mixed-design (D,G-K iteration), our robustmixed-design has been carried out using an advanced D,G-K iteration software provided to us by Prof. G.J. Balas [39] of theUniversity of Minnesota.


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1

2

1

2

0

0

( ) 0

( )

0 ( )

( )

p

n

P

I

I

I

s

s

s

s

O

O

0 0

0 0

0 0

=

pq

w z

u y

( , )P KM

( )P s

( )K s

Fig. 2. The mixed- setup. The uncertainty block, , contains the following blocks:

UNCERTAIN REAL PARAMETERS: All i are real scalars, |i| 1, i. Size of

identity matrix, I, depends on the number of times the same uncertain parameter

exists.

COMPLEX MODEL ERRORS: All j(s) reflect complex-valued unmodeled

dynamics at different places, j(s)

1, j.

PERFORMANCE: The P(s) block captures performance specifications in the

frequency-domain transformed into equivalent robustness problems, P(s)

1.

If the compensator K is absorbed into the plant P, the resulting stable transfer function matrix M

M(P, K) is used to evaluate the structured singular value with respect to the given uncertainty structure

(M) =

0; if det(I M) = 0, X

{ infX

[() : det(I M) = 0]}1; otherwise(2)

where denotes the maximum singular value. In mixed- synthesis, it is desired to design a compensator

K achieving

infKKS

supR+

(M(j)) (3)

where KS is the set of all real-rational, proper compensators that nominally stabilize P. Unfortunately,

the structured singular value cannot be calculated exactly. Thus, () is typically approximated by its upper

bound. Hopefully, the value of and its upper bound are close. A lower bound on () can also be calculated

to judge the tightness of the upper bound. In order to formulate an upper bound on the mixed- (M), the

following sets of block diagonal frequency-dependent scaling matrices are defined:

D = {diag(D1, . . . , Dmr+mc , d1Ikmr+mc+1, . . . , dmcIkmr+mc+mC ) :

Di Ckiki , det(Di) = 0, di C, di = 0} (4)


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and G(s) = GT(s). Augment D and Gh with identity matrices, and G with a zero matrix, of appropriate

dimensions so that D,G,Gh are all compatible with P. Form the new state-space system

PDG = (DP D1 G)Gh

3. Find the compensator K minimizing M(PDG, K) over all stabilizing, proper, real rational compen-

sators K.

4. Compute as

= supR

infD()D

G()G

inf()R()>0

{() : 1}

where

=

I +

G

2

()

1

4 1()

D()M(P, K)(j)

D

1

() j

G()

I +

G

2

()

1

45. Find D(), G() by solving the minimization problem

infD()D

G()G

I + G2()

14

1

D()M(P, K)(j)D1() jG()

I + G2()

14

pointwise across frequency.

6. Compare the new scaling matrices D(), G() with the previous estimates D(), G(). Stop, if they

are close, else replace D(), G() with D(), G() respectively and go to step 2.

Assuming perfect state-space realizations D(s), G(s) for D() and jG(), respectively, it is shown in [36]

that this iteration is monotonically non-increasing. The resulting mixed- synthesis compensator K from

step 3 satisfies

supR

M(P, K)(j)

(9)

In order to augment the state-space realizations of D(s) and G(s) to the generalized plant, a spectral

factorization is used to determine a transfer function matrix for the expression (I + G2(s))1/2:

GhGh = (I + G

G)1 (10)

where G(s) = GT(s) (step 2). Assuming a minimal realization for the square transfer function matrix

P

G

A B

C D

(11)


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*Gb-

hG1D- D

P

K

DGP

w z

u y

Fig. 3. Scaled system PDG for H compensator design step in the mixed- (D,G-K iteration).

and defining positive definite R, Q

R = I + DTD (12)

Q = I + DDT (13)

then the spectral factor Gh is given as

Gh

A BR1DTC BR1BTX BR

12

R1(DTC+ BTX) R1

2

(14)

where X > 0 satisfies the Riccati equation

(A BR1DTC)TX + X(A BR1DTC) XBR1BTX+ CTQ1C = 0 (15)

Accordingly,

P Gh A BR1DTC BR1BTX BR

1

2

Q1(C DBTX) DR1

2

(16)

Both Gh and P Gh are stable realizations. With these state-space representations the plant can be aug-

mented as illustrated in Figure 3. The resulting system PDG fits into the standard H optimal control

framework.

Similar to complex- synthesis, the joint problem of optimizing D, G and K is not convex. Thus the given

scheme will not guarantee finding the global optimum of Equation ( 3). However, the iterative procedure

in complex- synthesis, where alternately the D-scales are optimized for a fixed compensator and then the

compensator is optimized for fixed D-scales, has proved to be an efficient design tool for a variety of problems

[44], [37]. This could be the case also in the mixed- synthesis.

In this paper, we design a two-input two-output (TITO) control system using mixed- synthesis for the

application of energy management of an uncertain hybrid electric vehicle and demonstrate the very important

tradeoffs between performance and uncertainty; it could be widely adopted in new problems of the automotive

industry to be addressed by robust multivariable control design.

We have employed the mixed- software [39] which fully implements the D, G K iteration leading to


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the best possible robust compensator. The problem of synthesizing a controller with existing control theory

and computational tools, which is (optimally) robust to structured mixed uncertainty is very difficult, since

the formulated D, G K optimization problem is not convex. Because of this non-convexity, in our studies,

we used a variety of initial conditions in order to avoid finding a local minimum. The complex- synthesis

procedure as an efficient robust design tool, however, has been successfully applied to a broad variety of

problems (see, for example, [45]). The mixed- synthesis problem extends the above procedure to the mixed

real/complex uncertainties case, by exploiting some new analysis tools recently developed for the mixed-

upper bound [41].

IV. Diesel HEV model

In Section II, we overviewed the literature of the hybrid electric vehicles (HEVs) which could, in some

extent, clarify the importance of our work carried out in the field of advanced energy management for the

HEV systems. In this section, we shall investigate how to model a simplified hybrid electric vehicle by

replacing the sophisticated nonlinear dynamic of the diesel internal combustion engine. We shall integrate

this tractable uncertain HEV model (subject to real and complex uncertainties), for the first time, with

recent advances on robust multivariable control design using mixed- synthesis.

Before describing the structure of the simplified diesel hybrid electric vehicle, let us first overview a

generic HEV structure. A representative configuration of an advanced 4 4 parallel hybrid electric vehicle

configuration is shown in Figure 4.

S e t p o i n t T o r q u e

C o m m a n d s

( h i g h - l e v e l s t a t i c

o p t i m i s a t i o n )

M I M O

R o b u s t

C o n t r o l l e r

G e a r C I M G C l u t c h

D i e s e l I C E

Fig. 4. Schematic structure of a parallel 4 4 Hybrid Electric Vehicle (HEV). Low-level control components such as high voltageelectric battery, electric rear axle drive etc are excluded in this high-level energy management configuration.

The hybrid electric vehicle structure shown in Figure 4 is equipped with a turbocharged diesel engine

and a crankshaft integrated motor/generator (CIMG) which is directly mounted on the engine crankshaft.

The CIMG is used for starting and assisting the engine in motoring-mode, and also for generating electric

energy via charging a high-voltage battery (not shown in the figure). As our intention in this study is to

investigate the full-hybrid mode, we shall assume that the integrated ICE-CIMG clutch is fully engaged


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and hence our descriptive HEV dynamical model (see IV-C) excludes a clutch dynamics as it is shown in

Figure 4. Likewise, the gearbox is shown in Figure 4 but no gear setting was considered in our simplified

HEV demonstration. This is due to the fact that our empirical diesel engine model is derived with engine

speed range of = [1200, 2000]rpm running at the first gear.

It is also worthwhile to emphasise that our design methodology on the development of the HEV energy

management is a high-level design strategy. For this reason, most of the common low-level subsystems, inte-

grated within the typical HEV dynamics, are not considered in the HEV configuration as shown in Figure 4.

These low-level subsystems include CIMG low-level motor control, high voltage battery management, low

level clutch control, low level transmission control, and electrical distribution including DC-DC converter,

to name just a few. Furthermore, the dynamics of the engine model includes the average torque responses

of both diesel engine and CIMG over all four cylinders, which are the quantities of interest.

For designing a well balanced feedback control law, control engineers should possess a good comprehension

of the physics of the plant under investigation. One of the challenging aspect of any model based engine

control development is the derivation of a simplified yet insightful model. For instance, the frequency range of

the engine system, the nonlinearities associated with the internal engine processes (i.e. combustion process,

heat distribution, air flow), the severe cross coupling, the inherent sampling nature of the four cycle internal

combustion engine, and the limited output sensor dynamic capabilities all contribute to make this modelling

step a most arduous task [46].

There are two main reasons to highlight the importance of simplified HEV dynamical models: First, it is not

usually possible to obtain a detailed diesel engine data (or model) from the production vehicle manufacturer

[47]. Secondly, obtaining a precise mathematical model of a HEV powertrain is a very challenging task

particularly due to multi-energetic nature and switching dynamics of a hybrid powertrain.

For the above reasons, and for the ease of development of an advanced HEV energy management system,

it is essential to obtain a straightforward and realistic model of the propulsion system to which an efficient

control strategy, such as our proposed robust mixed- design methodology, could be applied. Generally

speaking, this model shall be used for the simulation of the overall vehicle motion (at longitudinal direction).

Therefore, we do not intend to utilise any detailed model of the internal engine processes, but rather a high-

level torque manager model that will generate control efforts based on a given set-point torque commands.

Recall that this torque management structure could be easily adopted to other engine configurations in a

straightforward manner.

As stated earlier, our developments towards a simplified hybrid model are based on a high fidelity

simulation model of the overall diesel hybrid electric vehicle. This HEV dynamical model is modeled using


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TB

1200 50 100 1441200 100 140 1421600 50 84 1401600 100 96 1372000 50 80 140

2000 100 72 134TABLE I

Experimental results of fueling delay, [msecs], and combustion lag, [msecs], as functions of diesel enginespeed [rpm] and brake torque [NM]. These results are captured by measuring the step response of the engine

to a step change in the engine brake torque.

two subsystems, a diesel internal combustion engine (ICE) and an armature-controller DC electric motor.

The mathematical modeling of these two subsystems will be discussed in the remainder of this section.

A. Simplified diesel engine model

In this section, we shall present a simplified dynamical model of a turbo-charged diesel engine. This

simplified model is based upon the nonlinear diesel engine dynamics and the fact that it must capture both

the transient and steady-state dominant modes of the diesel engine during operational conditions.

The ICE indicated torque Tind is assumed to be mapped from the delayed fueling input proportionally,

and has limited bandwidth due to internal combustion dynamic effects, arising e.g. due to combustion and

turbo lag. In a mathematical representation, we will have

Tind(t) =1

()s + 1TdemB (t ()) (17)

where is the speed-dependant time constant due to combustion lag, is the speed-dependant time-delay

due to fueling course and TdemB is the mapped fueling input representing the required ICE crankshaft (brake)

torque.

Our simplified diesel engine model is empirically derived using a turbo-charged diesel engine at speed range

of = [1200, 2000]rpm with operational brake torque acting at TB = [50, 100]NM. The speed-dependantfueling delay () and combustion lag () are given in Table I.

Our diesel model also contains a speed-dependant torque loss TLoss arising due to friction torque, ancillary

torque and pumping loss. Such a total torque loss is typically a nonlinear function of the engine speed and

could be represented by table mapping. However, for the purpose of the studied operating range of engine

speed and brake torque, namely = [1200, 2000]rpm and TB = [50, 100]NM, the total engine torque loss is

well approximated by a linear function of modeled as TLoss = m where m = 0.12 ( in [rad/sec]).


16/29

The simplified diesel engine model can be described as the following state-space equations:

x1 = 1x1 +

1T

demB (t )

Tind = x1

TB = Tind TLoss

(18)

where x1 is the state associated with the combustion lag dynamics.

The diesel dynamic shown in Equation (18) will be used in the overall configuration of the HEV dynamics.

B. Simplified CIMG Model

Assuming that the hybrid electric drivetrain includes an armature-controlled CIMG (DC motor), the

applied voltage va controls the motor torque (TM) as well as the angular velocity of the shaft. The

mathematical dynamics of the CIMG could be represented as

Ia =1

Las + Ra(vdema vemf) (19)

vemf = kb

TM = kmIa

where km and kb are torque and back emf constants, vdema is control effort as of armature voltage, vemf is

the back emf voltage, Ia is armature current, and La and Ra are inductance and resistance of the armature,

respectively.

Regarding the fact that the engine speed is synchronised with that of the CIMG in the full-hybrid mode,

the rotational dynamics of the hybrid drivetrain (including inertial crankshaft and motor) is given as follows:

J + b = TB + TM TL (20)

where is the drivetrain speed, J is the effective combined moment of rotational inertia (including CIMG

rotor, crankshaft, flywheel, driveline, driveshaft and wheels), b is the effective joint damping coefficient, and

TL is the vehicle load torque representing the plant disturbance.

The armature-controlled CIMG model as of Equation (19) along with the rotational dynamics of Equa-

tions (20) could be integrated into the following state-space equations:

x2 =1La

vdema Rax2 kbLa

x3

x3 =1Jx1 +

kmJ x2

b+mJ x3

1JTL

TM = kmx2

(21)


17/29

where x2 and x3 are the states associated with the armature current (Ia), and drivetrain speed (), respec-

tively.

From Equations (18) and (21), a simplified but realistic simulation HEV model with a detailed component

representation of diesel ICE and DC electric motor (CIMG) will be used as a basis for deriving the HEV

model in the subsequent section.

C. Simplified diesel hybrid electric vehicle model

Based on the state-space representation of both the diesel ICE and electric CIMG, given in Equation ( 18)

and Equation (21), respectively, we can now build the proposed simplified 3-state HEV model.

A schematic representation of the simplified parallel diesel hybrid electric vehicle model is shown in

Figure 5.

aa

b

RsL

k

+

bk

emfv

dem

av

bJs +

1

LT

indT

LossT

MT

dem

BT

E n g i n e t o r q u e l o s s v s

d r i v e l i n e s p e e d m a p p i n g

BT

dem

MT

L o a d ( i n c l u d i n g m o t o r

r o t o r , c r a n k s h a f t ,

f l y w h e e l , d r i v e l i n e ,

d r i v e s h a f t a n d w h e e l s )

C o m b u s t i o n / t u r b o l a g F u e l i n g d e l a y

A r m a t u r e

M o t o r t o r q u e v s a r m a t u r e

v o l t a g e m a p p i n g

e-

( ) s

1

1

+ ( ) s

Fig. 5. Simplified model of the parallel Diesel Hybrid Electric Vehicle.

Recall that the setpoint torque commands (TreqB and TreqM ) are provided to the controller by a high-level

static optimisation algorithm, not discussed in this study (see also Figure 4); see [15] for more details. Also,

in Figure 5 the engine brake torque TB and the CIMG torque TM are estimated torque feedback signals.

However, the details of estimation approach are not discussed here. For the sake of simplicity, one could

assume that both ICE and CIMG output torque signals are measured with inherent sensor noises to represent

the estimation errors.

In addition, due to there being in hybrid mode, it is assumed that the ICE/CIMG clutch is fully engaged

and hence the clutch model is excluded from the main HEV dynamics - it was previously shown in HEV

schematic diagram of Figure 4. Also, the gear setting is disregarded at this simplified model, as discussed

earlier. Furthermore, without loss of generality, the look-up mapping table of CIMG torque request ( TdemM )

vs armature voltage request (vdema ) is simply assumed to be an unit gain in this model.


18/29

By omitting the dependance on time-delay for the sake of simplicity, the overall state-space equations of

the simplified HEV model is represented by

x =

1 0 0

0 Ra kbLa

1J

kmJ

m+bJ

x +

1 0

0 1La

0 0

u +

0

0

1J

TL

y =

1 0 m

0 km 0

x

(22)

where x R3 is the state of the HEV system obtained from Equations (18) and (21), u = [TdemB (t

) TdemM ]T and y = [TB TM]

T are control signals and HEV torque outputs, respectively.

As shown in Equation 22, the plant includes one actuator unmodeled dynamics described by the uncertain

value in the fueling control time delay, given by the delay time , three uncertain real parameters associatedwith the diesel engine combustion lag , the hybrid inertial load J, and the effective damping coefficient b.

These are modeled according to the mixed- synthesis framework, using either structured or unstructured

uncertainty models.

1) Real parameter uncertainty: The real uncertain parameter , J and b are described by

= +

J = J + JJ (23)

b = b + bb

where , J and b are the nominal values, , J and b are the uncertain ranges and (,J,b) R determine

the structured uncertainty values and satisfy || 1, |J| 1 and |b| 1.

It is worthwhile to emphasise that combustion lag is not a constant uncertain parameter - it is rather a

time-varying speed-dependant parameter. The robust LTI MIMO control synthesis cannot deal with slowly-

varying parameters, from a theoretical perspective. However, in practice, as also evidenced through our

simulations presented in Section V, sufficient slowly-varying parameters could be dealt as constant uncertain

parameters [48]. We shall assume the similar case for combustion lag and fueling delay to be modeled

as constant uncertain parameters in the robust mixed- design.

By considering the linear fractional transformation (LFT) for the above three real parametric uncertainties,

it turns out that the parameters and b appear in multiplicative form, while the J uncertainty occurs in

the quotient form 1/J.

For the mixed- synthesis modeling, the real uncertainties , 1/J and b are regarded as scalar gain blocks,


19/29

hence, described by the lower-loop LFT transfer functions [37] as

= FL

1 0

,

1J = FL

1J

1

J1 J

, J

b = FL

b b

1 0

, b

(24)

The real uncertain parameters values for our case study are

[0.13, 0.15]N/m = 0.14N/m, = 0.1

J [0.5, 1.5]N m/(Rad.s

2

)

J = 1N m/(Rad.s

2

),

J = 0.5N m/(Rad.s

2

)b [0.05, 0.25]N m/(Rad.s) b = 0.15N m/(Rad.s), b = 0.1N m/(Rad.s)

(25)

2) Complex-valued unmodeled dynamics: In addition to the three real uncertainties, there is a time-delay

in the ICE fueling control channel which represents an unmodeled complex-valued dynamics. In order to

model such a pure ICE fueling time-delay, which is an indefinite dimension block, by a complex-valued

uncertainty, we shall consider a frequency upper-bound to be represented as a multiplicative error

eM(s) = es 1 (26)

as shown in Figure 6.

Fig. 6. Block diagram of the fueling time-delay error model.

The multiplicative error magnitude, eM(s), can then be approximated by a high-pass transfer function

Wun(s) with a real pole. The transfer function Wun(s) is multiplied by a delta block un(s) C that satisfies

||un(s)|| 1, which introduces a phase uncertainty in the range of 180.

The maximum value of the fueling time-delay uncertainty is max = 140 msecs, so the pole of the

Wun(s) transfer function is set near the maximum time-delay frequency of 15rad/secs likewise, the gain is

adjusted until Wun forms a tight upper bound for the magnitude of the eM(s) transfer function.

The frequency-domain upper bound for the unmodeled time-delay is required for mixed- synthesis design


20/29

and is the magnitude of the transfer function

Wun(s) =2.15s

s + 15(27)

which is shown in Figure 7 along with the multiplicative error magnitude due to unmodeled dynamics

arising from ICE fueling control channel delay.

101

100

101

102

103

50

40

30

20

10

0

10

Frequency (rad/sec)

Magnitude(dB)

eM

(j)

w

(j)

Fig. 7. Frequency response of unmodeled fueling time-delay dynamics for max = 140 msecs and its frequency-domain upperbound |Wun(j)| of Equation (27).

An actual time-delay of () mapped from engine speed and brake torque (see Table I) will be used in

our simulation.

To carry out the mixed- design, we must also select frequency-dependent weight matrices upon the

disturbances, sensor noises, outputs and control signals as follows subsequently.

D. Frequency-dependant weights

Our design goal is to have the true MIMO HEV respond effectively to the ICE and CIMG setpoint

torque requests. These performance specifications include:

Decoupled responses from ICE setpoint torque to the ICE brake torque TB and from the CIMG setpoint

torque to the CIMG output torque TM.

The ICE desired torque (handling quality (HQ)) response from the ICE setpoint torque to the ICE

brake torque TB should match the damped second-order response HQICE =100

s2+16s+100 .

The CIMG handling quality response from the CIMG torque request input to the CIMG torque output

should match the damped second-order response HQCIMG =144

s2+16.8s+144 .

In order to design the robust feedback controller the following type of performance weight upon the output

vector is used:

Wp(s) =1

s +

A1p 0

0 A2p

(28)


21/29

which reflects our performance specification (setpoint tracking) for the frequency range rad/sec where

the vehicle load disturbance TL(t) has most of its power. Ajp are the performance parameters see Table II.

The greater the performance parameter, the superior the torque reference tracking performance.

To capture the limits on the actuator deflection rates, the following control rate weights are used in the

robust mixed- design:

AE =

1

s1+ s

N

AM =

1

s1+ s

N

(29)

where N specifies the derivative coefficient, which determines the pole location of the filter in the derivative

action.

The ICE torque control and CIMG torque control actuators have 100NM/sec and 300NM/sec limits on

their deflection rates. Also, they have 400NM and 200NM limits on their deflection which are considered

in designing the frequency-domain control design weights.

We select a control frequency weight, denoted by Wact(s), in order to penalize the control actions differently

in different frequency regions. This is used to limit the bandwidth of the closed-loop system. The form of

Wact(s) used is

Wact(s) = diag

0.1(s + 60)(s + 600)

, 0.01, 0.2(s + 100)(s + 1000)

, 0.0033

(30)

so that we allow larger controls in lower frequencies and we penalize for large controls at much higher

frequencies.

Also, the (unmeasured) plant disturbance (vehicle load TL(t) in our case study), is modeled by a stationary

stochastic process generated by driving a low-pass filter, Wd(s), with continuous-time white noise (t), with

zero mean and unit intensity, as follows:

TL(s) = Wd(s) (s) (31)

with

Wd(s) =

s + (32)

where the frequency range is where the vehicle load has most of its power.

There are also frequency-domain weights on the measurement noise Wn. We have assumed that perfor-

mance variables (ICE torque and CIMG torque outputs) are measurable. In reality, this might be infeasible


22/29

and, thus, an estimation algorithm is required to estimate the HEV output torques. We do not intend to

discuss details of the designed estimators in this paper for the sake of readability. Nonetheless, without loss

of generality, the selected measurement noise can be translated into modeling of the torque estimation errors.

In either case, we have chosen constant measurement noise of Wn =diag(0.05, 0.05) indicating of %5 error

due to the estimation (or measurement noise) on the torque output signals.

The desired setpoint tracking problem requires that the effects of plant disturbance TL(t) (primarily) and

also sensor noise (estimation errors) be minimized.

The frequency-dependant design weights of Equations (27)-(32) together with the state-space equations

of Equation (22) are shown in Figure 8 which will be used in designing the robust multivariable feedback

control using mixed- synthesis described in Section III. Some representative simulation results of HEV

energy management case study will be shown in Section V to highlight some advances of our proposed

MIMO control system design.

actW acte

I C E T o r q u e S e t p o i n t

C I M G T o r q u e

S e t p o i n t

pe

KHEVEA

MA

I C E T o r q u e

D e m a n d

C I M G T o r q u e

D e m a n d

ICEHQ

pW

CIMGHQ

dem

BTdem

MT MT

BT

nW

S e n s o r

n o i s e

LT

w z

b

un0

02 2JI

Fig. 8. Robust Control Design for HEV energy management problem.

E. Frequency-domain analysis

The open-loop singular value decomposition of the uncertain MIMO HEV model from the input u(t) =

[TL, TdemB , T

demM ]

T to the output y(t) = [TB , TM]T are shown in Figure 9 for different values of real

uncertainties J, b, and complex-valued unmodeled dynamics and un.

Note that the worst case uncertainties have most effect over the frequency range 0.1 rad/sec. Also,

there are large variations in the resonant peaks over the frequency range [2, 20] rad/sec. Moreover, there is

a huge gap between the maximum and the minimum singular values of the worst case uncertain parameters


23/29

102

100

102

104

60

50

40

30

20

10

0

10

20

30

Frequency (rad/sec)

Singu

larValues(dB)

Fig. 9. Singular values decomposition of the open-loop uncertain hybrid electric vehicle dynamic.

(around 3 dB) which will create significant challenges for control system design, even for the simplified LTI

HEV system.

A mixed- synthesis controller was designed for this system using the frequency-dependant weighting

functions integrated into the generalised plant P(s), along with the HEV model uncertainty and performance

objectives, as shown in Figure 10.

)(sP

Req

BT

ReqMT

LT

BnMn

pe

Me

acte

demT

)(sK

Fig. 10. Robust control design structure using mixed- synthesis for the HEV energy management problem. nB and nM aretorque sensor noises when the torque output signals are available to measure (e.g. when using dyno). If these torque outputsignals are not measurable, nB and nM can also be translated as appropriate torque estimation errors.

The robust control diagram shown in Figure 10 includes the generalised HEV model P(s), the uncertainty

block = diag(JI22, b, , un) and the robust TITO controller K(s) to be designed using mixed-

synthesis.

V. Simulation results

In this section, we shall present the results of robust multivariable control design described in Section III

for the application of the simplified HEV energy management system discussed in Section IV.

For the uncertain HEV energy management application an equated LTI discrete-time system of the

continuous-time state-space dynamics described in Equation (22) is obtained using a sampling interval ts


24/29

(see Table II). The plant initial condition x0 R3 is assumed zero in the simulations.

The parameters used in the control system design along with other physical constants of the HEV model

dynamics are provided in Table II.

Parameter Value Unit

Sampling time (ts) 8 msecsMaximum ICE fueling delay () 140 msecsICE combustion lag () [130-150] msecsMotor armature resistance (Ra) 1 OhmsMotor armature inductance (La) 0.3 HenrysMotor torque constant (km) 0.25 NM.Amp

1

Motor back emf constant (kb) 0.25 Volts.secs.rad1

Effective drivetrain inertia (J) [0.5-1.5] kg.m2/s2

Effective drivetrain damping (b) [0.05,0.25] NmsBandwidth design frequency () 0.01 rad.sec1

ICE torque performance parameter (A1p) 20

CIMG torque performance parameter (A2p) 190 Derivative filter coefficient (N) 50

TABLE IIPhysical constants of the HEV model and control system design parameters.

We shall analyse and evaluate the closed-loop behavior of the HEV energy management system using

the robust feedback control design applied to the high-fidelity simplified model of the HEV described in

Section IV.

Our simulations have been carried out in Simulink and implemented in discrete-time using a zero-order

hold with a sampling time of ts = 8 msecs see Table II.

We shall emphasis that the developed robust mixed- synthesis, due to there being a dynamic control

system, does not require knowledge about future driving conditions. Such future driving conditions in our

case study include setpoint torque commands (requested by driver) and vehicle load torque. This fact will

make implementation of all sort of H optimisation based algorithms more tractable to be applied in real

dynamical applications.

Figure 11 shows typical simulation results for the period of20 secs in tracking the HEV requested setpoint

torques. During this simulation period, the system is in hybrid mode as both ICE torque and CIMG torque

are requested.

As shown in Figure 11, at times t = 5 secs and t = 15 secs , the TITO controller is requested to follow the

commands for an increased and decreased ICE torques, respectively, as illustrated in Figure 11(a). Similarly,

there was an increased request for the CIMG torque (from 20 NM to 40 NM) at time t = 10 secs, and the

controller has successfully delivered this torque request, as depicted in Figure 11(b).


25/29

0 5 10 15 2020

0

20

40

60

Time (Secs)

TB

[NM]

TBreq

TB

(a) Engine Brake Torque.

0 5 10 15 2015

20

25

30

35

40

45

Time (secs)

TM

[NM]

TM

req

TM

(b) CIMG Torque.

Fig. 11. Simulation results of the HEV Torque setpoints and outputs using our proposed robust control design.

Despite the fact that the HEV energy management is a highly-coupled MIMO dynamical system, both the

diesel ICE and the DC electric motor could satisfactorily track the requested torque setpoints. Recall that,

since we deal with a MIMO design, directional properties quantified by the singular value decomposition

(SVD) are significantly important, which illustrate the strong interactions of the two subsystems.

Recall that our torque manager structure, as discussed earlier, assumes that setpoint torque commands

are provided by some sort of static optimisation algorithms. The designed robust controller is then enquired

to optimise control efforts so as to track the requested torque references at the HEV model outputs.

Figure 12 shows the vehicle load torque used for the purpose of our simulations, ICE torque loss and control

efforts generated by the robust mixed- controller. We shall assume that the plant disturbance (vehicle load)

is unknown while robust controller attempts to minimise its effects on the performance outputs. Figure 12(c)

shows that the robust MIMO mixed- fully satisfies the required optimisation constraints due to actuator

limitations.

Figure 13 shows simulation results in regard to drivetrain speed and vehicle speed. It is worthwhile to

point out that as illustrated in Figure 13(a), by requesting large torque commands, we have in fact violated

our empirical HEV modeling assumption in that drivetrain speed is limited to = [1200, 2000]rpm. However,

it can be seen that the robust controller could satisfactorily control the HEV energy management dynamicsin real-time. The vehicle speed shown in Figure 13(b) has been calculated using a dynamic model of the

vehicle as a function of the drivetrain speed which is not discussed here.

It is also important to mention that fueling delay and combustion lag are functions of engine speed and

brake torque see Table I. However, in designing the robust MIMO control system we require to utilise an LTI

model of the HEV energy management plant hedging on worst case of the ICE speed-dependant parameters.

Towards this end, we assume that and are sufficiently slow, which are accordingly considered as constant

uncertain parameters in our design. However, in the simulation, we have utilised the actual time-varying


26/29

0 5 10 15 200

10

20

30

40

50

Time (Secs)

TLoad

[NM]

(a) Vehicle (load) Torque.

0 5 10 15 200

10

20

30

40

Time (Secs)

TLoss

[NM]

Ancilary Torque

FrictionPumping loss

Total losses

(b) Torque Loss.

0 5 10 15 200

100

200

300

400

Time (Secs)

u(t)

TdemBT demM

(c) Control signals.

Fig. 12. Simulation results of vehicle load, Torque loss, and Control efforts.

0 5 10 15 20500

1000

1500

2000

2500

3000

Time (Secs)

w

(rpm)

(a) Drivetrain Speed.

0 5 10 15 200

5

10

15

20

25

Time (Secs)

v(mph)

(b) Vehicle Speed.

Fig. 13. Simulation results of parallel diesel HEV drivetrain speed and vehicle speed.

(speed-dependant) parameters.

Regarding the real-time simulations in Simulink (fixed-step) using a sampling time of ts, the simulation

time required for a single run of20 secs was approximately 20 times faster than real-time running a Toshiba

Portege laptop with an Intel(R) Core(TM) i5 processor, at 2.4GHz under Windows 7 Pro platform.


27/29

VI. Conclusions

The aim of this paper was to present a robust MIMO feedback control design with an application for

the energy management of hybrid electric vehicles (HEVs). The main goal of energy management in hybrid

electric vehicles is to reduce the CO2 emissions with an enhanced fuel consumption for a hybrid powertrain

control system. The applicability of existing controllers in the energy management setting, however, has

shown a main drawback of these algorithms as most of them do not address the problems of stability-

and performance-robustness tradeoff and are not dynamic model-based approaches this turns out that the

driveability aspect in the existing control designs is an afterthought. The proposed robust design architecture

could resolve shortcomings of the existing non-robust control design methodologies. Moreover, the dynamical

robust multivariable control design does not explicitly utilise any knowledge in regard to the future driving

cycle. The singular value decomposition analysis were shown to address directionality so as to understand

the dynamic properties in the frequency domain. Simulation results illustrated that the robust multivariable

control design could be a very promising control system design methodology while it could play a key role

in a wide variety of challenging complex automotive applications.

References

[1] Curtis D. Anderson and Judy Anderson. Electric and Hybrid Cars: A History. McFarland & Co Inc, 2nd edition, Dec. 2009.

[2] Iqbal Husain. Electric and Hybrid Vehicles: Design Fundamentals. CRC Press, 1st edition, 2003.

[3] S. Fekri and F. Assadian. Model Predictive Control, chapter Fast Model Predictive Control and its Application to EnergyManagement of Hybrid Electric Vehicles. InTech Publisher, June 2011.

[4] H. Khayyam et al. Energy Management, chapter Intelligent Energy Management in Hybrid Electric Vehicles. InTech

Publisher, March 2010.

[5] V.H. Johnson et al. Hev control strategy for real-time optimization of fuel economy and emissions. In In Proc. of the Future

Car Congress, Washington DC, USA, April 2000.

[6] S.R. Cikanek and K.E. Bailey. Regenerative braking system for a hybrid electric vehicle. In In Proc. Of the American

Control Conf., pages 31293134, Anchorage, AK, May 2002.

[7] N.J. Schouten et al. Fuzzy logic control for parallel hybrid vehicles. IEEE Trans. On Control Systems Technology, 10(3):460

468, May 2002.

[8] M. Ramsbottom and F. Assadian. Use of approximate dynamic programming for the control of a mild hybrid. In In Proc.

of WMG Hybrid Conference, University of Warwick, Warwick, UK, Dec. 2006.

[9] A. Sciarretta and L. Guzzella. Control of hybrid electric vehicles. IEEE Control Systems Magazine, pages 6070, April 2007.

[10] G. Ripaccioli et al. Proc. of the 12th International Conference on Hybrid Systems Computation and Control (HSCC

2009), chapter Hybrid Modeling, Identification, and Predictive Control: An Application to Hybrid Electric Vehicle Energy

Management. Springer-Verlag, April 2009.

[11] I. Kolmanovsky et al. Optimization of powertrain operating policy for feasibility assessment and calibration: Stochastic

dynamic programming approach. In In Proc. of the American Control Conf., pages 14251430, Anchorage, AK, May 2002.


28/29

[12] D.M. De la Pena et al. Robust explicit mpc based on approximate multiparametric convex programming. IEEE Transactions

on Automatic Control, 51:13991403, 2006.

[13] A.G. Beccuti et al. Explicit Hybrid Model Predictive Control of the dc-dc Boost Converter. In IEEE PESC, Orlando,

Florida, USA, June 2007.

[14] Wang and Boyd. Fast model predictive control using online optimization. In Proceedings of the 17th IFAC World Congress,

pages 69746979, Seoul, Korea, July 2008.

[15] C. Dextreit et al. Hybrid electric vehicle energy management using game theory. In SAE World Congress & Exhibition,

Apr. 2008.

[16] G. Rousseau et al. Design optimisation and optimal control for hybrid vehicles. In In Proc. of the International Conference

on Engineering Optimization, EngOpt 2008, Rio de Janeiro, Brazil, June 2008.

[17] J. Kessels. Energy Management for Automotive Power Nets. PhD thesis, Faculteit Electrical Engineering, Technische

Universiteit Eindhoven, The Netherlands, 2007.

[18] A. Bemporad and M. Morary. Control of systems integrating logic, dynamics, and constraints. Automatica, 35:407427,

1999.

[19] F.D. Torrisi and A. Bemporad. Hysdel a tool for generating computational hybrid models. IEEE Trans. on Control

Systems Technology, 12(2):235249, 2004.

[20] E.D. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Trans. on Automatic Control, 26(2):346358, 1981.

[21] A. Bemporad. Hybrid Toolbox - Users Guide. http://www.dii.unisi.it/hybrid/toolbox, Jan. 2004.

[22] D.A.C. Narciso et al. A framework for multi-parametric programming and control an overview. In Proc. of IEEE

International Engineering Management Conference, IEMC Europe 2008, pages 1 5, June 2008.

[23] P.M. Young and J.C. Doyle. Computation ofwith real and complex uncertainties. In Proc. of the IEEE Conf. on Decision

and Control, pages 12301235, 1990.

[24] J.C. Doyle et al. State-space solutions to standard H2 and H control problems. IEEE Trans. on Automatic Control,

34(8):831847, Aug. 1989.

[25] Allen Fuhs. Hybrid Vehicles. CRC Press, 1st edition, 2008.

[26] G. Steinmaurer and L. Del Re. Optimal energy management for mild hybrid operation of vehicles with an integrated starter

generator. In In Proc. of the SAE World Congress, Detroit, Michigan, April 2005.

[27] T. Hofman and R. Van Druten. Energy analysis of hybrid vehicle powertrains. In In Proc. Of the IEEE Int. Symp. On

Vehicular Power and Propulsion, Paris, France, Oct. 2004.

[28] Bemporad et al. The explicit linear quadratic regulator for constrained systems. Automatica, 38:320, 2002.

[29] D. Cundev. Hybrid-electric Cars and Control Systems: Control strategy of car hybrid system and its experimental confirmation.

LAP LAMBERT Academic Publishing, Aug. 2010.

[30] B.M. Baumann et al. Mechatronic design and control of hybrid electric vehicles. In IEEE/ASME Trans. On Mechatronics,

volume 5, pages 5872, March 2000.

[31] E. Cacciatori et al. Evaluating the impact of driveability requirements on the performance of an energy management control

architecture for a hybrid electric vehicle. In In the 2nd International IEE Conference on Automotive Electronics, IEE Savoy

Place, UK, 2006.

[32] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control Analysis and Design. John Wiley, 2nd edition, Sep.

2005.

[33] K. Zhou et al. Robust and Optimal Control. Englewood Cliffs, Prentice-Hall, NJ, USA, 1996.


29/29

[34] P. M. Young. Structured singular value approach for systems with parametric uncertainty. Int. J. of Robust and Nonlinear

Control, 11:653680, 2001.

[35] M. Athans J. Vasconcelos and S. Fekri. Stability- and performance-robustness tradeoffs: Mimo mixed- vs complex-design.

Int. J. of Robust and Nonlinear Control, Feb. 2008.

[36] P.M. Young. Controller design with mixed uncertainties. In Proc. of the American Control Conf., pages 23332337, Baltimore,

MD, June 1994.

[37] G.J. Balas et al. -Analysis and Synthesis Toolbox of Matlab, Users Guide. The MathWorks Inc., June 2004.

[38] G. Stein and J.C. Doyle. Beyond singular values and loopshapes. AIAA Journal of Guidance and Control, 14(1):516, Jan.

1991.

[39] G.J. Balas. Private communication re mixed- software, 2003.

[40] M.K.H. Fan et al. Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Trans. on

Automatic Control, 36(1):2538, Jan. 1991.

[41] P.M. Young et al. analysis with real parametric uncertainty. In Proc. of the IEEE Conf. on Decision and Control, pages

12511256, 1991.

[42] K. Zhou and J.C. Doyle. Essentials of Robust Control. Englewood Cliffs, Prentice-Hall, NJ, USA, 1998.

[43] Michael Athans. Lecture notes on design of robust multivariable feedback control systems. Instituto Superior Tecnico (IST),

Technical University of Lisbon, 2002.

[44] A. Packard and J.C. Doyle. The complex structured singular value. Automatica, 29(1):71109, 1993.

[45] G.J. Balas and A. Packard. The Structured Singular Value ( ) Framework, pages 671687. The Control Handbook,

(W.S. Levine, ed.). CRC Press Inc, 1996.

[46] J. B. Lewis. Automotive engine control: A linear quadratic approach. PhD thesis, Dept. of Electrical Engineering and

Computer Science, Massachusetts Institute of Technology, MA, USA, 1980.

[47] M. Kaszynski and O. Sawodny. Modeling and identification of a built-in turbocharged diesel engine using standardized

on-board measurement signals. In Proc. of IEEE International Conference on Control Applications (CCA 2008), pages

712, Sept. 2008.

[48] S. Fekri. Robust adaptive MIMO control using multiple-model hypothesis testing and mixed- synthesis. PhD thesis, Instituto

Superior Tecnico (IST), Lisbon, Portugal, 2005.

application of robust multivariable control on energy management of hybrid electric vehicles

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