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ARTICLE IN PRESS

Journal of the Franklin Institute 347 (2010) 358–376

Design of observer-based feedback control for

time-delay systems with application to automotive

powertrain control

Sun Yia,?, A. Galip Ulsoya, Patrick W. Nelsonb

aDepartment of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109, USA

bDepartment of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Received 18 May 2009; received in revised form 11 September 2009; accepted 15 September 2009

Abstract

A new approach for observer-based feedback control of time-delay systems is developed. Time-

delays in systems lead to characteristic equations of infinite dimension, making the systems difficult

to control with classical control methods. In this paper, a recently developed approach, based on the

Lambert W function, is used to address this difficulty by designing an observer-based state feedback

controller via assignment of eigenvalues. The designed observer provides estimation of the state,

which converges asymptotically to the actual state, and is then used for state feedback control. The

feedback controller and the observer take simple linear forms and, thus, are easy to implement when

compared to nonlinear methods. This new approach is applied, for illustration, to the control of a

diesel engine to achieve improvement in fuel efficiency and reduction in emissions. The simulation

results show excellent closed-loop performance.

& 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

PACS: 02.30.Yy; 07.05.Dz

Keywords: Time delays; Observer; Feedback; Eigenvalue assignment; Lambert W function

1. Introduction

As is well known, excellent closed-loop performance can be achieved using state feedback

control. In cases where all state variables are not directly measurable, the controller may have to

www.elsevier.com/locate/jfranklin

0016-0032/$32.00 & 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfranklin.2009.09.001

?Corresponding author. Tel.: +17347632227; fax: +17346473170.

E-mail addresses: syjo@umich.edu (S. Yi), ulsoy@umich.edu (A.G. Ulsoy), pwn@umich.edu (P.W. Nelson).

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be combined with a state observer, which estimates the state vector. For linear systems, the

design of such a controller with an observer is typically carried out based on eigenvalue

assignment [1]. Unlike systems of linear ordinary differential equations (ODEs), where the

methods for eigenvalue assignment are well-developed, the design procedure for linear systems

with time-delays in the state variables is not straightforward. In this paper, a new method for

design of observer-based feedback control of time-delay systems is presented, and illustrated

with a diesel engine control application.

Time-delays are inherent in many engineering systems, and such time-delays can limit and

degrade the achievable performance of controlled systems, and even induce instability. Delay

terms lead to infinite dimensionality in the characteristic equations, making time-delay systems

difficult to control with classical control methods. Thus, time-delay problems in engineering

systems are often solved indirectly by using approximation and/or prediction methods [2].

A widely used approximation method is the Pad? e approximation, which is a rational

approximation and results in a shortened fraction for the approximation of the delay term.

However, such approaches constitute a limitation in accuracy, can lead to instability of the

actual system and induce non-minimum phase and, thus, high-gain problems. Prediction-based

methods (e.g., Smith predictor, finite spectrum assignment, FSA [3], and adaptive Posicast [4])

have been used to stabilize time-delay systems by transforming the problem into a non-delay

system. Such methods require model-based calculations, which may cause unexpected errors

when applied to a real system. Furthermore, safe implementation of such methods is still an

open problem due to computational issues [2]. Controllers have also been designed using the

Lyapunov framework (e.g., linear matrix inequalities, LMIs or algebraic Riccati equations,

AREs) [5,6]. While these methods can be applied to quite general forms of systems (e.g., systems

with multiple time-delays), they require complex formulations and can lead to conservative

results and possibly redundant control. Using bifurcation analysis, stability conditions can be

obtained, and it is possible to design controllers, such as PID controllers [7]. If the controller has

a large number of unknown gains, however, it becomes more difficult to use such an approach.

Also, if one wants to assign the positions of the system poles, not only to stabilize the system, the

method may not be sufficient to find satisfactory control gains. The act-and-wait control concept

was introduced for systems with feedback delay in [8]. Using the approach, stability of the

closed-loop system can be described by a finite dimensional Floquet transition matrix and, thus,

the infinite-dimensional eigenvalue assignment problem is reduced to a finite-dimensional one

[9]. The method was verified by experiments in [10].

Successful design of feedback controllers and observers hinges on the ability to check

stability and find stabilizing controller and observer gains. In this paper, a recently developed

method using the matrix Lambert W function [11] is applied to design of feedback controllers

and observers. The method is used to ensure asymptotic stability and dominant eigenvalues at

desired positions in the complex plane to achieve desired performance. Using the Lambert W

function-based approach, observer-based controllers for time-delay systems represented by

delay differential equations (DDEs) can be designed in a systematic way as for systems of

ODEs. That is, for a given time-delay system, the analytical free and forced solutions are

derived in terms of parameters of the system [12]. From the solution form, the eigenvalues are

obtained and used to determine stability of the system [13]. Furthermore, criteria for

controllability/observability and Gramians are derived from the solution form [14]. For a

controllable system, a linear feedback controller is designed by assigning dominant eigenvalues

to desired locations [15], and this can be done to achieve robust stability and to meet time-

domain specifications [16].

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Numerous methods have been developed for control of time-delay systems (e.g., see Table 1).

However, existing methods enable one to design either controllers or observers, yield nonlinear

forms of controllers, and/or do not assign eigenvalues exactly to the desired positions. The

approach presented here allows one to design linear feedback controllers and linear observers

via eigenvalue assignment (see Fig. 1). The observer-based feedback control designed in this

paper offers advantages of accuracy (i.e., no approximation of time-delays), ease of implemen-

tation (i.e., no estimates of relevant parameters or use of complex nonlinear controllers), and

robustness (i.e., not requiring model-based prediction). For illustration, the developed method is

applied to control of a diesel powertrain, where the controller design is challenging due to an

inherent time-delay, and the proposed approach can provide advantages in terms of ease of

design, as well as the performance of observer-based control. This paper is organized as follows.

In Section 2, a problem formulation and background are provided. The proposed method is

presented in Section 3, and the diesel engine control application is given in Section 4. In

Section 5, a summary and conclusions are presented, and topics for future research are noted.

2. Problem formulation

Consider a real linear time-invariant (LTI) system of DDEs with a single constant time-

delay, h, described by

_ xðtÞ ¼ AxðtÞ þ Adxðt ? hÞ þ BuðtÞ;

xðtÞ ¼ gðtÞ;

xðtÞ ¼ x0;

where xðtÞ 2 Rnis a state vector; A 2 Rn?n, Ad2 Rn?n, and B 2 Rn?rare system

matrices; uðtÞ 2 Rris a function representing the external excitation. A specified preshape

function, gðtÞ, and an initial state, x0, are defined in the Banach space of continuous

mappings [27].

With linear state feedback, combined with a reference input, rðtÞ 2 Rr,

uðtÞ ¼ rðtÞ ? KxðtÞ ? Kdxðt ? hÞ

t40

t 2 ½?h;0Þ

t ¼ 0

ð1Þ

ð2Þ

Table 1

Motivation for a new approach for design of observers: for time-delay systems, various studies devoted to

observer design are summarized in this table.

Description of approachReferences

Spectral decomposition-based

Observer: an integro-differential form

Observer: a linear form

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[8]

[25]

[26]

Lyapunov framework

Linear matrix inequality (LMI)

Algebraic Riccati equation (ARE)

Coordinate transformation

Infinite spectrum assignment

Continuous pole placement

Finite spectrum assignment (FSA)

For comparisons of approaches for feedback controllers, refer to Richard [2] and Yi et al. [15].

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one can stabilize, improve performance, and/or meet time-domain specifications for the

system (1) (e.g., see [15,22]), under the assumption that all the state variables, xðtÞ, can be

measured directly. This is achieved by choosing K and Kdð2 Rr?nÞ based on desired

rightmost closed-loop eigenvalues. Note that the Lambert W function-based approach is

applicable to systems with a single delay as in Eq. (1). For systems with multiple delays

caused by, e.g., additional feedback delays or delays in sensors, stability results introduced

in [28] can be applied.

In cases where direct access to all state variables is limited, use of a state observer

(estimator) is needed to obtain ^ xðtÞ, an estimate of the state variable, xðtÞ. Like systems of

ODEs, an asymptotically stable closed-loop system with a state observer (so-called

Luenberger observer where ^ xðtÞ converges asymptotically to xðtÞ as t goes to infinity) can

be achieved by placing eigenvalues for the observer dynamics at desired locations in the

complex plane (e.g., on the LHP). However, in contrast to ODEs, systems of DDEs, as in

Eq. (1), have an infinite number of eigenvalues (e.g., see Fig. 2) and, thus, calculation and

assignment of all of them is not feasible.

The state estimation problem for time-delay systems has been a topic of research interest

(e.g., see [22] and the references therein for a survey). The problem has been approached by

using methods based on spectral decomposition and state transformation developed in [17–19].

Such methods require extensive numerical computations to locate the eigenvalues of time-

delay systems. Prediction-based approaches (e.g., FSA) with a coordinate transformation have

been used to address this type of problem in [26]. Converting time-delay systems into non-

delay ones, the observer of an integro-differential form is designed to assign the eigenvalues of

finite dimensional systems. Based on the assumption that stabilizing feedback gains exist and

Fig. 1. The block diagram for observer-based state feedback control for time-delay systems is analogous to the

case for ODEs [1]. By choosing gain matrices K and L an asymptotically stable feedback controller and observer

can be designed so that the system has good closed-loop performance using the estimated state variables, ^ x,

obtained from the system output, y.

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are known, an observer can be designed based on a coordinate transformation [24]. For such

an approach, it is assumed that the system is stabilized by the memoryless linear state

feedback, uðtÞ ¼ ?KxðtÞ, and the gain matrix, K, is known. Also, Lyapunov-based approaches

have been used for development of design methods for observers and/or controllers (e.g., ARE

[23], LMI [21,22]). In [29], a numerical stabilization method was developed to obtain the

rightmost eigenvalues and move them to the left half plane (LHP) using sensitivities with

respect to changes in the control gain. This approach has been compared to the Lambert W

function-based approach in [15]. The approach can be applied to design observers [25]. See the

comparison of various developed approaches in Table 1.

2.1. Eigenvalue assignment

The system (1) with the controller (2) becomes

_ xðtÞ ¼ fA ? BKg

|fflfflfflfflfflffl{zfflfflfflfflfflffl}

Successful design of the feedback controller hinges on the ability to find the control gain

matrices, K and Kd, such that the closed-loop system (3) is stable and has desirable

performance. A method for eigenvalue assignment was developed in [15] based on the

?A0

xðtÞ þ fAd? BKdg

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

?Ad0

xðt ? hÞð3Þ

−5−4.5 −4 −3.5−3 −2.5 −2 −1.5 −1−0.5 0

−60

−40

−20

0

20

40

60

Imag (Sk: roots of the characteristic equation)

k = −2

k = −1

k = 1

k = 2

k = 3

k = 0

(principal branch)

. . .

. . .

Real (Sk: roots of the characteristic equation)

k = −3

Fig. 2. Eigenspectrum of the system (1) when A ¼ ?1, Ad¼ 0:5, and h ¼ 1: due to the delay term, Adxðt ? hÞ, in

the state equation and, thus, an exponential term in the characteristic equation, the number of eigenvalues is

infinite. Using methods developed for ODEs, it is not feasible to locate and assign them. The Lambert W function-

based approach provides a tool for analysis and control of time-delay systems mainly because each eigenvalue can

be expressed analytically in terms of parameters A, Ad, and h, and associated individually with a ‘branch’

(k ¼ ?1;...;?1;0;1;...;1) of the Lambert W function.

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solution to Eq. (1), which is given by

xðtÞ ¼

X

1

k¼?1

eSktCI

kþ

Zt

0

X

1

k¼?1

eSkðt?xÞCN

kBuðxÞdx

ð4Þ

where Skis the solution matrix expressed as

Sk¼1

hWkðAdhQkÞ þ A

ð5Þ

The coefficient CI

the initial condition, x0, while CN

x0. The numerical and analytical methods for computing CI

[12,30]. The following condition is used to solve for the unknown matrix Qk:

kin Eq. (4) is a function of A, Ad, h and the preshape function, gðtÞ, and

kis a function of A, Ad, h and does not depend on gðtÞ or

kand CN

kwere developed in

WkðAdhQkÞeWkðAdhQkÞþAh¼ Adh

and the matrix Lambert W function, WkðHkÞ, [30,31], which satisfies the definition:

WkðHkÞeWkðHkÞ¼ Hk

where k ¼ ?1;...;?1;0;1;...;1 denote the branches of the Lambert W function. The

solution, Qk, to Eq. (6) is obtained numerically, for example, by using the fsolve function in

Matlab; however, conditions for existence and uniqueness of such a solution to Eq. (6) are

still open problems [11].

The main idea of eigenvalue assignment is that with the new coefficients, A0? A ? BK

and Ad0? Ad? BKd, in Eq. (3) of the closed-loop system, the solution matrix Sk is

computed using Eq. (5) and the rightmost eigenvalues are assigned to the desired positions

li;desas

ð6Þ

ð7Þ

liðS0Þ ¼ li;des;

i ¼ 1;2;...;nð8Þ

by adjusting the gain matrices, K and Kd. Note that S0is the solution matrix obtained

using the principal branch (k ¼ 0) and liðS0Þ are the corresponding eigenvalues. Although

the system in Eq. (1) has an infinite number of eigenvalues, one advantage of the Lambert

W function approach is that the branches (k ¼ ?1;...;?1;0;1;...;1) of the function

distinguish all the eigenvalues individually (see Fig. 2), and the finite (n) eigenvalues of S0,

among all the Sk, are the rightmost ones and determine the stability of the system [13]. For

scalar cases, this has been proven in [32]. Such a proof can readily be extended to systems

of DDEs where A and Adcommute. Although such a proof is not currently available in the

case of the general matrix-vector DDEs, the same behavior has been observed in all cases

where Ad does not have repeated zero eigenvalues [15]. Note that while problems in

handling time-delay systems arise from the difficulty in: (1) checking the stability and (2)

finding stabilizing gains, Eq. (8) for eigenvalue assignment provides a explicit formulation

useful to address such problems, as shown in [15,16] with numerical examples. However, as

explained in Section 2.2, due to assignability issues, Eq. (8), which is solved by using

numerical nonlinear equation solvers (e.g., fsolve in Matlab), may not always yield a

solution for K and Kd. To resolve such a problem, instead of using Eq. (8), one can try with

fewer desired eigenvalues, or with just the real parts of the desired eigenvalues [15] to

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reduce the number of constraints, as

lrmðS0Þ ¼ ldes

RflrmðS0Þg ¼ Rfldesg

ðaÞ

ðbÞð9Þ

where lrmðS0Þ are the rightmost eigenvalues from among the n eigenvalues S0, and R

indicates the real part of its argument. In numerical computation one can use, for example,

functions in Matlab, such as max and real.

Previously, the method for eigenvalue assignment was used to design only full-state

feedback control as in Eq. (2) [15]. In this paper, it is now used to find the controller and

observer gain matrices (i.e., K and L in Eqs. (10) and (11) in the next section) for the

combined observer-based control in Fig. 1. This is described in Section 3. In other words,

using Eq. (8), or Eq. (9), it will be shown that one can assign both controller and observer

rightmost (i.e., dominant) poles for the infinite dimensional closed-loop eigenspectrum of

the observer-based controller for time-delay systems shown in Fig. 1.

2.2. Controllability, observability, and eigenvalue assignability

For systems of DDEs, controllability and observability have been studied extensively

(see, e.g., [2,14] and the references therein). Unlike systems of ODEs, there exist numerous

different definitions controllability and observability for systems of DDEs depending on

the nature of the problem under consideration (e.g., approximate, spectral, weak, strong,

point-wise and absolute controllability). Among the various notions, point-wise

controllability and point-wise observability were investigated to derive criteria and

Gramians for such properties using the solution form in Eq. (4) based on the Lambert W

function in [14].

For linear systems of ODEs, controllability (or observability) is equivalent to the

arbitrary assignability of the eigenvalues of the controller (observer) [1]. However,

conditions for such arbitrary assignment are still lacking for systems of DDEs. Even for

the scalar case of Eq. (1), limits in arbitrary assignment of eigenvalues exist and depend on

the values of the time-delay and the coefficients (see Appendix A). Although, for a simple

scalar DDE, study of the limits has been conducted in [33–35], generalization of such

results are challenging. The relationship between the derived criteria for controllability and

observability, and eigenvalue assignability by using a ‘linear feedback controller’ or a

‘linear observer’ was partially studied in [15] with examples, and is being further

investigated by the authors. Although extensive research during recent decades has been

reported in the literature, the relationship between eigenvalue assignment and con-

trollabilty/observabilty is still an open research problem.

3. Design of observer-based feedback controller

This section describes a systematic design approach for the combined controller-

observer for time-delay systems (see Fig. 1). The observer estimates the system states from

the output variables, while the control provides inputs to the system as a linear function of

the estimated system states.

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Step 1: Obtain the equation for the closed-loop system with K and Kd:

_ x ¼ Ax þ Adxdþ Bu

u ¼ ?Kx ? Kdxdþ r

) _ x ¼ ðA ? BKÞx þ ðAd? BKdÞxdþ Br

where xd? xðt ? hÞ. Then, choose the desired positions of the rightmost eigenvalues of the

feedback controller dynamics. They can be selected, for example, to meet design

specifications in the time domain with desired damping ratio, z, desired natural frequency,

on, of the closed loop response and as ldes¼ s7jod¼ ?zon7jon

Step 2: Using the desired eigenvalues, Eqs. (8) and (5), the gain matrices, K and Kd, are

obtained numerically for a variety of initial conditions by an iterative trial and error

procedure with the coefficients of the closed-loop system in Eq. (10), A0? A ? BK and

Ad0? Ad? BKd. As explained in Section 2.1, if solutions cannot be found, one can try with

fewer desired eigenvalues or with just the real parts of the desired rightmost eigenvalues

(i.e., using Eq. (9) instead of Eq. (8)), to find the control gains.

Step 3: Consider an observer with a gain matrix L:

_ x ¼ Ax þ Adxdþ Bu

y ¼ CxðtÞ

_^ x ¼ A^ x þ Ad^ xdþ Lðy ? C^ xÞ þ Bu

_ x ?_^ x ¼ Aðx ? ^ xÞ þ Adðxd? ^ xdÞ ? Lðy ? C^ xÞ

) _ e ¼ ðA ? LCÞe þ Aded

where e ? x ? ^ x. Then, choose the desired positions of the rightmost eigenvalues of the

observer dynamics. A reasonable choice of desired positions of observer rightmost

eigenvalues mainly depends on the amount of measurement noise and the size of modeling

inaccuracies. While fast eigenvalues amplify measurement noise, slow eigenvalues lead to

slow convergence of the estimates of the state variables. A typical ‘rule of thumb’ is that

the magnitudes of the negative real parts of the rightmost eigenvalues of Eq. (11) should be

1:5?2 times larger than those of Eq. (10) to guarantee fast response [1].

Step 4: Using the desired eigenvalues, Eqs. (8) and (5), find the observer gain matrix, L,

with new coefficients in Eq. (11), A0? A ? LC and Ad0? Ad. As in Step 2, if solutions

cannot be found, one can try with fewer desired eigenvalues or with just the real parts of

the desired rightmost eigenvalues (i.e., using Eq. (9) instead of Eq. (8)).

Note that, as mentioned in Section 2.2, unlike ODEs, conditions for assignability are

still lacking for systems of DDEs. Even for the scalar case of Eq. (1), limits in arbitrary

assignment of eigenvalues exist (e.g., see Appendix A of this paper). For systems of DDEs,

depending on the structure or parameters of the given system, there exists limitations on

the rightmost eigenvalues. In that case, the above approach does not yield any solution for

the controller and observer gains. To resolve the problem, one can find the gains by using a

trial and error method with fewer desired eigenvalues (or with just the real parts of the

desired eigenvalues), or different values of the desired rightmost eigenvalues as explained

in Steps 2 and 4.

Although the Kalman filter for time-delay systems renders an observer optimal in the

sense of minimizing the estimation error in the presence of noise and model uncertainty

(see, e.g., [36,37] and the references therein), such an approach requires the selection of

covariance matrices for process and measurement noise by trial-and-error to obtain the

desired performance of the filter/estimator. On the other hand, the design of observers via

ð10Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 ? z2

p

.

ð11Þ

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eigenvalue assignment may be sub-optimal, but can achieve a similar performance by

adjusting desired location for eigenvalues [1].

3.1. Separation principle

For systems of ODEs, it is shown that the eigenvalues of the state estimator are not

affected by the feedback and, consequently, the design of state feedback and the design of

the state estimator can be carried out independently (i.e., the so-called separation principle).

For the time-delay system in Eqs. (10) and (11), we can show in a straightforward manner

that the separation principle holds. The two equations can be combined into

?

Then, the eigenvalues are roots of the characteristic equation given by

"

_ x

_ e

?

¼

ðA ? BK ? BKdÞ

0

BK

ðA ? LCÞ

"#

x

e

? ?

þ

Ad

0

0

Ad

"#

xd

ed

"#

ð12Þ

det

sI ? A þ BK þ BKd? Ade?sh

0

?BK

sI ? A þ LC ? Ade?sh

#

¼ 0

) det½sI ? A þ BK þ BKd? Ade?sh? ? det½sI ? A þ LC ? Ade?sh? ¼ 0

Therefore, the two sets of eigenvalues can be specified separately and the introduction of the

observer does not affect the eigenvalues of the controller. Hence, selection of gains

K (and/or Kd) and L can be performed independently.

ð13Þ

4. Application to diesel engine control

In this section, control of a diesel engine is considered to illustrate the advantages and

potential of the method proposed in Section 3. Specifically, a feedback controller and

observer are designed via eigenvalue assignment using the Lambert W function-based

approach. A diesel engine with an exhaust gas recirculation (EGR) valve and a turbo-

compressor with a variable geometry turbine (VGT) was modeled in [38] with three state

variables, xðtÞ ? fx1 x2 x3gT: intake manifold pressure (x1), exhaust manifold pressure

(x2), and compressor power (x3). The model includes intake-to-exhaust transport delay

(h ¼ 60ms when engine speed, N, is 1500 RPM). Thus, a linearized system of DDEs was

introduced in [26], for a specific operating point (N ¼ 1500 RPM):

?27

9:6

?12:5

09

?5

C ¼ ½0

Because of the time-delay, which is caused by the fact that the gas in the intake manifold

enters the exhaust manifold after transport time, h, the system can be represented by a

system of DDEs as in Eq. (1) with the coefficients in Eq. (14). The number of eigenvalues is

infinite and one of them is positive real. Thus, the response of this linearized system shows

unstable behavior (see the eigenspectrum and the response in Fig. 3 for the diesel engine

model in Fig. 4). The system with the coefficients in Eq. (14) satisfies the conditions for

point-wise controllability and observability. That is, all rows of Eq. (B.1) and all columns of

A ¼

3:66

0

2

64

3

75;

Ad¼

000

2100

000

2

64

3

75B ¼

0:26

?0:9

0

0

?0:8

0:18

2

64

3

75;

10?

ð14Þ

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Eq. (B.2) with the coefficients in Eq. (14) are linearly independent. Therefore, the system of

Eq. (14) is point-wise controllable and point-wise observable [14] (see Appendix B).

For a non-delay model, which is also unstable, by constructing the control Lyapunov

function (CLF) a feedback control law was designed in [38]. A feedback controller for

−120−100 −80−60

Real (s)

−40−200

−300

−200

−100

0

100

200

300

Imaginary (s)

unstable eigen value

+0.9224

012345678910

−500

0

500

1000

1500

2000

State Variables

intake manifold pressure (p1)

exhaust pressure (p2)

compressssor power (Pc)

Time, t

Fig. 3. Eigenspectrum of the linearized diesel engine system with the coefficients in Eq. (14). Due to the inherent

transport time-delay the number of eigenvalues is infinite and without any control the system has an unstable

eigenvalue (left). Thus, the response of the system is unstable (right) [26].

Fig. 4. Diagram of a diesel engine system in Eq. (14). With limited measurement y ¼ x2, all the state variables are

estimated by using an observer, and then used for state feedback control [26].

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the diesel engine systems with the time-delay was developed in [39] using the concept of

control Lyapunov–Krasovsky functionals (so-called CLKF). In [26], FSA combined

with a coordinate transformation (FSA cannot be applied directly to the system where

time-delays are not in actuation) was used for design of the observer for the system,

which takes an integro-differential form.

As shown in Section 3.1, the separation principle holds, and, thus we can

independently design the controller and observer. First consider the case of linear full-

state feedback. The control input, uðtÞ ¼ fu1ðtÞ u2ðtÞgT, where u1ðtÞ is a control input for

EGR valve opening and u2ðtÞ is a control input for the turbine (VGT) mass flow rate (for a

detailed explanation, the reader may refer to [38]), is given by

uðtÞ ¼ KxðtÞ þ rðtÞ

where K is the 2 ? 3 feedback gain matrix,

k1

k2

k3

k4

k5

k6

ð15Þ

K ¼

"#

ð16Þ

Then, the system matrices of the closed-loop system become

A0? A þ BK

2

¼

?27 þ 0:26k1

9:6 ? 0:9k1? 0:8k4

0:18k4

3:6 þ 0:26k2

?12:5 ? 0:9k2? 0:8k5

9 þ 0:18k5

6 þ 0:26k3

0 ? 0:9k3? 0:8k6

?5 þ 0:18k6

64

3

75;Ad0

? Ad

ð17Þ

Following the Steps 1 and 2 introduced in Section 3, based on the Lambert W function,

the gain matrix, K, is selected so that the system can have improved performance as well as

be stabilized. For the system in Eq. (14), it was not possible to assign all eigenvalues of S0

by using Eq. (8). Instead, by reducing the number of eigenvalues specified to one, one can

find the feedback gain and assign the rightmost eigenvalue of the system with Eq. (9a). For

example, assuming that the desired rightmost eigenvalue, ldes¼ ?10, which is chosen by

considering the desired speed of the closed-loop system [26], the resulting feedback gain is

obtained as

?

K ¼

0:0001

0:0000

?13:8835

50:3377

0:0000

50:8222

?

ð18Þ

The eigenspectrum is shown in Fig. 5. Among an infinite number of eigenvalues, the

rightmost (i.e., dominant) subset is computed by using the principal branch (k ¼ 0)

and all the others are located to the left, which is one of the prominent advantages

of the Lambert W function-based approach. Note that the rightmost eigenvalue is

placed exactly at the desired position, ldes¼ ?10, and the unstable system is stabilized

(compare to the eigenspectrum in Fig. 3).

Next we consider the design of an observer to estimate the unmeasured states.

The observer gain matrix, L, is obtained in a similar way, following Steps 3 and 4 in

Section 3. Considering available sensors [26], only the exhaust manifold pressure is

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measured. Thus, y ¼ x2and the output matrix and observer gain are given by

L1

L2

L3

C ¼ ½010?;

L ¼

2

64

3

75

ð19Þ

Then, the new coefficients of the dynamical equation of the state observer become

2

A ? LC ¼

?27

9:6

0

3:6 ? L1

?12:5 ? L2

9 ? L3

6

0

?5

64

3

75;

Ad¼

000

21 00

000

2

64

3

75

ð20Þ

−200 −150−100

Real (s)

−50 0

−800

−600

−400

−200

0

200

400

600

800

Imaginary (s)

eigenvalues obtained using

the principal branch (k=0), S0

zoomed in

zoomed in

−26−24−22−20−18−16 −14 −12−10 −8

−40

−20

0

20

40

60

Real (s)

Imaginary (s)

Fig. 5. Eigenvalues of the closed-loop systems with coefficients in Eqs. (17) and (18). Among an infinite number of

eigenvalues, the rightmost (i.e., dominant) subset is computed by using the principal branch (k ¼ 0) and all the

others are located to the left. Note that the rightmost eigenvalue is placed exactly at the desired position,

ldes¼ ?10, and the open-loop unstable system is stabilized (compare to Fig. 3).

−140−120−100−80−60−40−20 0

−500

−400

−300

−200

−100

0

100

200

300

400

500

Real (s)

Imaginary (s)

zoomed in

zoomed in

eigenvalues obtained using

the principal branch (k = 0), S0

−16−15.8−15.6−15.4

Real (s)

−15.2−15

−8

−6

−4

−2

0

2

4

6

8

Imaginary (s)

λdes = −15

Fig. 6. Eigenvalues of the closed-loop systems with coefficients in Eqs. (20) and (21): among an infinite number of

them, the rightmost (i.e., dominant) subset is computed by using the principal branch (k ¼ 0) and all the others

are located to the left. Note that the rightmost eigenvalue is placed exactly at the desired position, ldes¼ ?15, and

the unstable system is stabilized (compare to Fig. 3).

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For example, for a chosen desired eigenvalue ldes¼ ?15 so that the dynamics of

the observer is well damped and faster than the controller dynamics, the gain matrix

obtained is (see the eigenspectrum in Fig. 6)

2

L ¼

6:4729

9:5671

16:0959

64

3

75

ð21Þ

Similarly to the controller case, one can find the observer gain matrix, L, just with one

rightmost eigenvalue specified (i.e., by using Eq. (9a)). As mentioned in Sections 2 and 3

and Appendix A, there exist limits in assignment of eigenvalues with linear controllers or

observers. For the system with coefficients in Eq. (14), the rightmost eigenvalues can be

moved as far to the left as ldes¼ ?25:0 for the controller and ldes¼ ?15:3 for the

observer, respectively, and the corresponding limiting gains, as determined numerically for

this example, are

K ¼

?0:0004

0:0020

?3:3044

45:9131

?0:0006

84:0699

??

;

L ¼

6:4650

9:5660

16:0991

2

64

3

75

ð22Þ

00.20.4 0.60.811.21.4 1.61.82

−0.1

0

0.1

0.2

0.3

intake manifold

pressure

state

estimated state

0 0.20.40.60.81 1.21.41.61.82

−0.5

0

0.5

exhaust

pressure

0 0.20.4 0.6 0.811.21.41.6 1.82

−0.4

−0.2

0

0.2

0.4

compressor

power

Time, t

Time, t

Time, t

Fig. 7. Responses with each controller: the reference inputs for this simulation run are selected randomly: r1is a

step input with amplitude 0.5 and r2 is a sine wave with amplitude 20 and frequency 0.7Hz. The rightmost

eigenvalues of feedback control and observer are ?10 and ?15, respectively. The state variables estimated by

using observer (dashed line) converge into those of the plant (solid line), which are stabilized by state feedback

control.

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Overall performance of the controlled system in Fig. 1 with the parameters in Eqs. (14),

(18), (19) and (21) is shown in Fig. 7. The reference inputs for this simulation run are

selected arbitrarily: r1 is a step input with amplitude 0.5 and r2 is a sine wave with

amplitude 20 and frequency 0.7Hz. The rightmost eigenvalues of the feedback controller

and observer are ?10 and ?15, respectively. The state variables estimated using the

observer (dashed line) converge to those of the plant (solid line), which are stabilized by

state feedback control.

As shown in Fig. 1, the asymptotically stable feedback controller with observer takes a

simple form similar to that for linear systems of ODEs [1]. They do not require, as with

previous approaches, the approximate integration of state variables during finite intervals,

nor construction of a cost function or inequalities. This can lead to ease of design, analysis,

and implementation, which is one of the main advantages of the proposed approach.

5. Summary, conclusions and future work

In this paper, a new approach for the design of observer-based state feedback control for

time-delay systems was developed. The separation principle is shown to hold, thus, the

controller gain and the observer gain can be independently selected so that the two

dynamics are simultaneously asymptotically stable. The design hinges on eigenvalue

assignment to desired locations that are stable, or ensure a desired damping ratio and

natural frequency. The main difficulty, which is addressed in this paper, is caused by the

fact that systems of DDEs have an infinite number of eigenvalues, unlike systems of ODEs.

Thus, to locate them all to desired positions in the complex plane is not feasible. To find

the dominant subset of eigenvalues and achieve desired eigenvalue assignment, the

Lambert W function-based approach, developed recently by the authors has been used.

Using the proposed approach, the feedback and the observer gains are obtained by placing

the rightmost, or dominant, eigenvalues at desired values. The designed observer provides

an estimate of the state variables, which converges asymptotically to the actual state and is

then used for state feedback to improve system performance. The technique developed is

applied to a model for control of a diesel engine, and the simulation results show excellent

performance of the designed controller and observer.

The proposed method complements existing methods for observer-based controller

design and offers several advantages. The designed observer-based controller for DDEs

has a linear form analogous to the usual case for ODEs. The rightmost (i.e., dominant)

eigenvalues, for both observer and controller, are assigned exactly to desired feasible

positions in the complex plane. The designed control can have improved accuracy by not

ignoring or approximating time-delays, ease of implementation compared to nonlinear

forms of controllers, and robustness since it does not use model-based prediction.

To make the proposed approach more effective, the relation between controllability (or

observability) and eigenvalue assignability for time-delay systems needs to be investigated

further. Specifically, research is needed to generalize the results for the scalar case (see

Appendix A). As new engine technologies are continuously developed, the proposed design

approach can play a role in handling delay problems for automotive powertrain systems.

Other than the presented diesel control, for example, air-to-fuel ratio control, where time-

delays exist due to the time between fuel injection and sensor measurement for exhaust,

and idle speed control, where time-delays exist due to the time between the intake stroke of

the engine and torque production, are also being studied.

371