Design of observerbased feedback control for timedelay systems with application to automotive powertrain control
ABSTRACT A new approach for observerbased feedback control of timedelay systems is developed. Timedelays 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 observerbased 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 closedloop performance.

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ABSTRACT: This paper presents an algebraic approach to design the control law of a LTI observer used to stabilize a LTI plant with an output delay. Different than the existing work, we use the observer gains to influence the plant stability. This becomes possible simply by removing the delay terms from the observer part. Given the plant controller gains, our approach can find the parametric regions with respect to the observer controller gains so that gains selected from these regions make the combined plantobserver system asymptotically stable independent of the amount of delay in the plant. An example with simulations is provided to demonstrate the advantages of the proposed observer design.American Control Conference (ACC), 2011; 06/2011  SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments.06/2014;
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Journal of the Franklin Institute 347 (2010) 358–376
Design of observerbased feedback control for
timedelay 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 observerbased feedback control of timedelay 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 observerbased 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 closedloop 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 closedloop 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
00160032/$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.
Email 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 welldeveloped, the design procedure for linear systems
with timedelays in the state variables is not straightforward. In this paper, a new method for
design of observerbased feedback control of timedelay systems is presented, and illustrated
with a diesel engine control application.
Timedelays are inherent in many engineering systems, and such timedelays 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 timedelay systems
difficult to control with classical control methods. Thus, timedelay 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 nonminimum phase and, thus, highgain problems. Predictionbased
methods (e.g., Smith predictor, finite spectrum assignment, FSA [3], and adaptive Posicast [4])
have been used to stabilize timedelay systems by transforming the problem into a nondelay
system. Such methods require modelbased 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 timedelays), 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 actandwait control concept
was introduced for systems with feedback delay in [8]. Using the approach, stability of the
closedloop system can be described by a finite dimensional Floquet transition matrix and, thus,
the infinitedimensional eigenvalue assignment problem is reduced to a finitedimensional 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
functionbased approach, observerbased controllers for timedelay systems represented by
delay differential equations (DDEs) can be designed in a systematic way as for systems of
ODEs. That is, for a given timedelay 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 timedelay 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 observerbased feedback control designed in this
paper offers advantages of accuracy (i.e., no approximation of timedelays), ease of implemen
tation (i.e., no estimates of relevant parameters or use of complex nonlinear controllers), and
robustness (i.e., not requiring modelbased prediction). For illustration, the developed method is
applied to control of a diesel powertrain, where the controller design is challenging due to an
inherent timedelay, and the proposed approach can provide advantages in terms of ease of
design, as well as the performance of observerbased 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 timeinvariant (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 timedelay systems, various studies devoted to
observer design are summarized in this table.
Description of approachReferences
Spectral decompositionbased
Observer: an integrodifferential 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 timedomain 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 closedloop eigenvalues. Note that the Lambert W functionbased 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 closedloop system with a state observer (socalled
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 timedelay 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. Predictionbased approaches (e.g., FSA) with a coordinate transformation have
been used to address this type of problem in [26]. Converting timedelay systems into non
delay ones, the observer of an integrodifferential 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 observerbased state feedback control for timedelay 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 closedloop 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, Lyapunovbased 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
functionbased 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 closedloop 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 timedelay 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 closedloop 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 matrixvector 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 timedelay 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 fullstate
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 observerbased 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 closedloop eigenspectrum of
the observerbased controller for timedelay 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,
pointwise and absolute controllability). Among the various notions, pointwise
controllability and pointwise 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 timedelay 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 observerbased feedback controller
This section describes a systematic design approach for the combined controller
observer for timedelay 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 closedloop 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 closedloop 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 timedelay 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 trialanderror 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 suboptimal, 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 socalled separation principle).
For the timedelay 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 functionbased
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 intaketoexhaust 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 timedelay, 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
pointwise 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 pointwise controllable and pointwise observable [14] (see Appendix B).
For a nondelay 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 timedelay 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 timedelay was developed in [39] using the concept of
control Lyapunov–Krasovsky functionals (socalled CLKF). In [26], FSA combined
with a coordinate transformation (FSA cannot be applied directly to the system where
timedelays are not in actuation) was used for design of the observer for the system,
which takes an integrodifferential 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 closedloop 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 closedloop 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 functionbased 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 closedloop 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 openloop 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 closedloop 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 observerbased state feedback control for
timedelay 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 functionbased 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 observerbased controller
design and offers several advantages. The designed observerbased 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 timedelays, ease of implementation compared to nonlinear
forms of controllers, and robustness since it does not use modelbased prediction.
To make the proposed approach more effective, the relation between controllability (or
observability) and eigenvalue assignability for timedelay 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, airtofuel ratio control, where time
delays exist due to the time between fuel injection and sensor measurement for exhaust,
and idle speed control, where timedelays exist due to the time between the intake stroke of
the engine and torque production, are also being studied.
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