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OPTIMAL H-INFINITY CONTROLLER DESIGN AND
STRONG STABILIZATION FOR TIME-DELAY AND
MIMO SYSTEMS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Suat Gumussoy, B.S.E.E., M.S.* * * * *
The Ohio State University
2004
Ph.D. Examination Committee:Professor Hitay Ozbay, Adviser
Professor Andrea Serrani
Professor Hooshang Hemami
Approved by
Adviser
Department of ElectricalEngineering
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ABSTRACT
In this thesis, the problems of optimal H controller design and strong stabi-
lization for time-delay systems are studied. First, the optimal H controller design
problem is considered for time-delay plants with finitely many unstable zeros and
infinitely many unstable poles. It is shown that this problem is the dual version of
the same problem for the plants with finitely many unstable poles and infinitely many
unstable zeros, that is solved by the so-called Skew-Toeplitz approach. The optimal
H controller is obtained by a simple data transformation. Next, the solution of the
optimal H controller design problem is given for plants with finitely many unstable
poles or unstable zeros by using duality and the Skew-Toeplitz approach. Necessary
and sufficient conditions on time-delay systems are determined for applicability of
the Skew-Toeplitz method to find optimal H controllers. Internal unstable pole-
zero cancellations are eliminated and finite impulse response structure of the optimal
H controller is obtained. The problem of strong stabilization is studied for time
delay and MIMO finite dimensional systems. An indirect approach to design a stable
controller achieving a desiredH performance level for time delay systems is given.
This approach is based on stabilization of H controller by another H controller
in the feedback loop. In another approach, when the optimal controller is unstable
(with infinitely or finitely many unstable poles), two methods are given based on a
search algorithm to find a stable suboptimal controller. In this approach, the main
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idea is to search for a free parameter which comes from the parameterization of sub-
optimal H controller, such that it results in a stable H controller. Finally, the
strong stabilization problem and stable H controller design for finite dimensional
multi-input multi-output linear time invariant systems are studied. It is shown that
if a certain linear matrix inequality condition has a solution then a stable controller,
whose order is the same as the order of the generalized plant, can be constructed.
This result is applied to design stable H controller with the order twice of the order
of the generalized plant.
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To my mom and dad, Dilek and Kamil,
and my brother, Murat for ...
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ACKNOWLEDGMENTS
I would like to thank my advisor, Professor Hitay Ozbay, for his understand-
ing, support and help throughout this academic experience. He was the advisor
just I wished. I would also like to thank Professor Andrea Serrani and Professor
Hooshang Hemami for serving in my examination committee. I am thankful to Pro-
fessor Vadim I. Utkin for his valuable discussions.
I would also like to acknowledge the financial support from the National Science
Foundation, AFRL/VA and AFOSR.
My special thanks are for my parents, Kamil and Dilek Gumussoy, and my brother,
Murat Gumussoy who helped and assisted me at every stage in my life.
Many thanks to my office-mates, Pierre F.Quet and Xin Yuan for nice chats and
valuable discussions. They were just there, when I need to speak Hish.
I am thankful to my early-Ph.D.friends, Tankut Acarman, Veysel Gazi, Mehmet
Onder Efe, Umit Ogras, Oguz Dagc, Peng Yan and late-Ph.D friends, Cosku Kas-
nakoglu, Alvaro E. Gil, Nicanor Quijano, Jorge Finke. Since it is difficult to mention
all the names, I am also thankful to CRL graduate students and faculty.
I want to thank my non-departmental friends Cezmi Unal, Yelda Serinagaoglu,
Gokhan Korkmaz, Sahika Vatan, Tansu Demirbilek, Yucel and Derya Demirer for
spending their time with me.
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Although a graduate study in abroad causes to lost many friendship in your home
country, I am thankful to Kursad Ergut, Hatice Dinc, Ulas Bars Sarsoy, Ilke and
Unsal Soysal for everything they did in spite of distances.
While I were studying all night with no sleep, I were always happy to see my
friend Douglas Ellis, janitor, with smile on his face at 6:00a.m.
My last thanks are for my love, Ipek. Everything will be too much difficult without
you. When I started to doubt that there exist the one, you came...
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VITA
May 16, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Aksaray, Turkey
September, 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. in Electrical and Electronics Engi-neeringMiddle East Technical University
Ankara, TurkeySeptember, 1999 .... . . . . . . . . . . . . . . . . . . . . . . . .B.S. in Mathematics
Middle East Technical UniversityAnkara, Turkey
August, 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. in Electrical and Computer Engi-neeringThe Ohio State UniversityColumbus, Ohio
September, 2000 - Present . . . . . . . . . . . . . . . . . . Graduate Research AssociateThe Ohio State UniversityColumbus, Ohio
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PUBLICATIONS
Research Publications
Suat Gumussoy and Hitay Ozbay, On the Mixed Sensitivity Minimization for Sys-tems with Infinitely Many Unstable Modes, to appear in Systems and Control
Letters, 2004 (available on publishers web site since June 10, 2004).
Suat Gumussoy and Hitay Ozbay, Remarks on Strong Stabilization and Stable H-infinity Controller Design, to appear in Proceedings of 43rd IEEE Conference onDecision and Control, The Bahamas, December 2004.
Suat Gumussoy and Hitay Ozbay, On Stable H-infinity Controllers for Time Delay
Systems, Proceedings of the conference on Mathematical Theory of Network andSystems, July 2004.
Murat Saglam, Sami Ezercan, Suat Gumussoy and Hitay Ozbay, Controller tuning
for active queue management using a parameter space method, Proceedings of theconference on Mathematical Theory of Network and Systems, July 2004.
Suat Gumussoy and Hitay Ozbay, Control of Systems with Infinitely Many Unstable
Modes and Strong Stabilizing Controllers Achieving a Desired Sensitivity, Proceed-ings of the conference on Mathematical Theory of Network and Systems, August 2002.
FIELDS OF STUDY
Major Field: Electrical Engineering
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TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapters:
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Literature Review on Optimal H Controller Design for Time-DelaySystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Literature Review on Stable H Controller Design . . . . . . . . . 41.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. On the Mixed Sensitivity Minimization for Systems with Infinitely Many
Unstable Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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3. H Controller Design for Neutral and Retarded Delay Systems . . . . . 163.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Neutral and Retarded Time Delay Systems with Finitely
Many Unstable Zeros . . . . . . . . . . . . . . . . . . . . . . 213.1.2 FIR Structure of Neutral and Retarded Time Delay Systems 24
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Factorization of the Plants . . . . . . . . . . . . . . . . . . . 27
3.2.2 Optimal H Controller Design . . . . . . . . . . . . . . . . 293.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1 IF Plant Example . . . . . . . . . . . . . . . . . . . . . . . 333.3.2 FF Plant Example . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4. Stable H Controllers for Delay Systems: Suboptimal Sensitivity . . . . 404.1 Optimal Sensitivity Problem for Delay Systems . . . . . . . . . . . 414.2 Sensitivity Deviation Problem . . . . . . . . . . . . . . . . . . . . . 43
4.3 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5. On Stable H Controllers for Time-Delay Systems . . . . . . . . . . . . 495.1 Structure ofH Controllers . . . . . . . . . . . . . . . . . . . . . . 505.2 Stable suboptimal H controllers, when the optimal controller has
infinitely many unstable poles . . . . . . . . . . . . . . . . . . . . . 52
5.3 Stable suboptimal H controllers, when the optimal controller hasfinitely many unstable poles . . . . . . . . . . . . . . . . . . . . . . 56
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6. Remarks on Strong Stabilization and Stable H Controller Design . . . 706.1 Strong stabilization of MIMO systems . . . . . . . . . . . . . . . . 71
6.2 Stable H controller design for MIMO systems . . . . . . . . . . . 746.3 Numerical examples and comparisons . . . . . . . . . . . . . . . . . 75
6.3.1 Strong stabilization . . . . . . . . . . . . . . . . . . . . . . 75
6.3.2 Stable H controllers . . . . . . . . . . . . . . . . . . . . . 766.4 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Appendices:
A. Skew-Toeplitz Approach for Mixed Sensitivity Problem . . . . . . . . . . 85
B. IF and FI Plant Calculations . . . . . . . . . . . . . . . . . . . . . . . . 88
B.1 Example: IF Plant Case . . . . . . . . . . . . . . . . . . . . . . . . 88
B.2 Example: FF Plant Case . . . . . . . . . . . . . . . . . . . . . . . . 89
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
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LIST OF FIGURES
Figure Page
3.1 Optimal H Controller for IF plants . . . . . . . . . . . . . . . . . . 30
3.2 Optimal H Controller for FI plants . . . . . . . . . . . . . . . . . . 31
3.3 fT(t) for IF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 fopt(t) for IF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 fT(t) for FF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 fopt(t) for FF plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 (i) F = 0: the weighted sensitivity is H optimal; (ii) F = 0: thecontroller K is stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1 wmax and max versus u . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Z(s) plot for right half plane . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 min versus n2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.4 Magnitude plot ofU(j ) . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.5 u values resulting stable H controller . . . . . . . . . . . . . . . . 696.1 Standard Feedback System . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Comparison for plant G1 . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Comparison for plant G2 . . . . . . . . . . . . . . . . . . . . . . . . . 76
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LIST OF TABLES
Table Page
6.1 Stable H controller design for combustion chamber . . . . . . . . . 77
6.2 Stable H controller design for Example in [1] . . . . . . . . . . . . . 79
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CHAPTER 1
INTRODUCTION
Mathematical modeling of physical systems is typically based on certain simpli-
fying assumptions and physical laws of nature. Hence it always introduces modeling
errors, i.e., the plant model from which a controller is to be designed is just an ap-
proximation of the actual plant. Therefore, it is important to use design techniques
which guarantee stability and performance against such uncertainties. Robust control
is an important field of feedback control theory, that is concerned with stability and
performance of systems with uncertainties.
One of the most important tools of robust control is H control. Since the
H norm can interpreted as the worst-case gain of the system, it is an appropriate
criterion to pose the disturbance minimization problems in this setting. There are
three major types of plant uncertainties, i.e., additive, multiplicative, coprime factor
uncertainties. Considering these types of uncertainties of the plant, different types of
H
controllers are designed to achieve the closed loop stability (Robust Stabilization
Problem) and pre-specified performance level (Robust Performance Problem) [2].
Stable controller design for a given plant is called strong stabilization. It is de-
sirable to have a stable controller for practical and theoretical reasons, ([3, 4]). For
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example it is well known that the unstable poles of the controller degrades the track-
ing performance and/or disturbance rejection capabilities of the closed loop system
([5, 6, 7]). Moreover, unstable controllers are highly sensitive and their response to
sensor-faults and plant uncertainties/nonlinearities are unpredictable ([8]). It is also
well-known, [3], that the strong stabilization problem has a close connection with
simultaneous stabilization of two or more plants by a single controller. Also, stable
controllers can be tested off-line to check some faults in the implementation and to
compare with the theoretical design specifications. Therefore, it is desirable to use
strongly stabilizing controllers in the feedback loop whenever possible.
In this thesis, a fundamental tool in robust control, H control theory, will be used
to design optimal H controllers for time-delay systems and stable H controllers for
time-delay and multi-input multi-output (MIMO) systems. Unless otherwise speci-
fied, the discussion will be primarily be confined to single-input single-output (SISO)
systems. For the rest of this Chapter, a brief overview of the literature on optimal
H
controller and stable H
controller design will be given and the outline of the
dissertation can be found in the last section of this Chapter.
1.1 Literature Review on Optimal H Controller Design forTime-Delay Systems
It is well known that H controllers for linear time invariant systems with finitely
many unstable modes can be determined by various methods, see e.g. [9, 10, 11,
12, 13, 14, 15]. In particular, H control problem for time-delay systems is studied
by many researchers. When the plant is a dead-time system in the form ehsP0(s),
where P0 is a rational transfer function, optimal H controller design problem is
solved in [16, 17, 18, 19, 20]. A general class of infinite dimensional H control
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problems are solved by operator theoretic methods. In this thesis we will refer to one
of these methods known as the Skew-Teoplitz approach. In [2], a mixed sensitivity
minimization problem is converted into two-block, and then to an one-block H
problem after a series of factorizations and transformations, and the resulting Nehari
problem is solved using the Commutant Lifting Theorem. In [15] the overall algorithm
is significantly simplified. The optimal H controller can be obtained by this method
for SISO dead-time systems.
State-space methods for H control of dead-time systems are given in [21, 22].
In these papers, the infinite dimensional problem is reduced to finite dimensional
problems and the corresponding problems are solved. Another solution to same prob-
lem is given in [23] by using the J-spectral factorization approach. Moreover, in the
same work, a special structure of the optimal and suboptimal H controllers are
shown (i.e., infinite dimensional part of the controller can be represented in finite
impulse response filter (FIR)). Optimal H controller design for MIMO plants with
input/output delays is solved in [24]. In this paper, multiple input/output delays
are decomposed into dead-time systems (so-called adobe plant problems), then opti-
mal controller is constructed from these systems. The FIR structure of the infinite
dimensional part of the optimal H controller is also shown for MIMO plants with
input/output delays.
Although the Skew-Toeplitz approach [2] solves the mixed sensitivity problem for
general infinite dimensional systems, connections of these systems with time-delay
systems are little known. In this thesis, the link between Skew-Toeplitz approach and
time-delay systems will be established. Moreover, general time-delay systems will be
considered including dead-time systems.
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1.2 Literature Review on Stable H Controller Design
A necessary and sufficient condition for the existence of a stable controller is the
parity interlacing property (p.i.p.), [25]. A plant satisfies the p.i.p. if the number of
poles of the plant (counted according to their McMillan degrees) between any pair
of real right half plane blocking zeros is even. There are several design methods for
constructing strongly stabilizing controllers, see e.g. [3, 25]. In general, the procedures
involve construction of a unit in H satisfying certain interpolation conditions. A
parameterization of all stabilizing controllers for SISO case is obtained in [3]. For
stable controller design with rational controllers see [26, 27]. It is known that if the
plant is arbitrarily close to violating p.i.p., then the order of the strongly stabilizing
controller can be unbounded [28]. Certain bounds on the norm and the degree of
unit interpolants are given in [29] by using Nevanlinna-Pick interpolation theory. A
conservative bound on controller order of strongly stabilizing controller is obtained in
[30]. Stable controller design for MIMO systems is considered in [7]. The necessary
and sufficient condition for strong stabilization, parity interlacing property, is shown
in [31] for SISO delay systems. A design method to find strongly stabilizing controller
for SISO systems with time delays is given [32] in which the stable controller is
constructed by using the unit satisfying some interpolation conditions.
The H performance cost minimization with stable controller has also been stud-
ied in the literature. For finite dimensional SISO plants, optimal stable
H con-
troller for weighted sensitivity minimization is given in [33]. Using the interpolation
conditions of sensitivity function and Nevanlinna-Pick approach, optimal stable H
controller is obtained. An explicit formula for this optimal controller is given in [34]
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where certain parameters of the controller are found from the solution of a set of non-
linear equations. The SISO discrete-time version of the weighted sensitivity problem
is solved in [35] by using a convex integer programming approach. For the mixed
sensitivity problem, certain conditions on the weighting functions and the plant are
determined in [36] for stability of the optimal controller. In [37] a sufficient condi-
tion is obtained for the synthesis of SISO finite dimensional suboptimal stable H
controllers, by converting the problem into a Nevanlinna-Pick interpolation problem.
For the same problem with finite dimensional MIMO systems, a design method
has been developed in [38]. The problem of weighted sensitivity minimization with
stable controllers is reduced into finite dimensional optimization problem of finding
a set of integers and some admissible matrices. In [39], the stability of the controller
is guaranteed if nonnegative definite solutions exist to the design equations; further-
more, under this condition, the controller order is less than or equal to that of the
plant. Similarly, in [40], a sufficient condition is obtained for the existence of a stable
H
controller. This condition is expressed in terms of an Algebraic Riccati Equation
(ARE) derived from the state-space realization of the central controller. The dimen-
sion of the stable H controller determined in [40] is less than or equal to 2n, where
n is the dimension of the generalized plant. In [41] strong stabilization problem is
considered for MIMO finite dimensional linear time invariant systems. It is shown
that if a two block H problem is solvable, then a strongly stabilizing controller can
be derived. The controller is of the same order as the plant. Moreover, under the
given sufficient conditions finite dimensional characterizations of strongly stabilizing
controllers are obtained. Another sufficient condition is presented for the existence
of a stable H controller in [42] using chain-scattering approach where the controller
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design algorithm requires solution of an ARE. An improved H control design al-
gorithm with strong stability condition is presented in [8]. By choosing a weighting
function, the conservatism in two-block problem for stabilizing controller design is
reduced. However, the weight function causes a substantial increase in the order of
the controller.
The stable H controller design for infinite dimensional systems is still an open
research problem. In this thesis, this problem will be addressed and stable H
controller design methodologies will be given for time-delay systems as well as MIMO
systems.
1.3 Dissertation Outline
In Chapter 2, optimal H controller design for time-delay systems with finitely
many zeros and infinitely many unstable poles are considered. It is assumed that the
unstable zeros and unstable poles of the plant are captured in inner terms and the
stable part of the plant is represented as an outer term. It is shown that this problem
is the dual case of the usual Skew-Toeplitz approach [2, 15], and the optimal H
controller is given for this plant. In Chapter 3, using this result and the Skew-Toeplitz
method, the optimal H controller design is extended for the time-delay plants with
finitely many zeros or poles. Necessary and sufficient conditions are obtained for
the applicability of the Skew-Toeplitz approach on generalized time delay systems.
Moreover, the controller is constructed such that there are no internal unstable pole-
zero cancellations and the FIR structure of the H controllers are shown for the same
class of plants.
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An indirect approach to design stable controller achieving a desired H perfor-
mance level for dead time systems is given in Chapter 4. This approach is based on
stabilization of H controller by another H controller in the feedback loop. Sta-
bilization is achieved and the sensitivity deviation is minimized. Then in Chapter
5, two other design methods are given for the same problem. Moreover in Chapter
6, stable H controller design for finite dimensional linear invariant MIMO plants is
considered. The design method in [41] is generalized using linear matrix inequalities.
The links between the Chapters of thesis are as follows: Chapters 2 and 3 are
concerned with optimalH controller design for general time-delay systems. In
Chapters 4, 5 and 6, the strong stabilization problem is considered. More specifically,
Chapters 4 and 5 give stable H controller design methods for time-delay systems,
and Chapter 6 deals with the same problem for finite dimensional MIMO systems.
Finally, in Chapter 7 concluding remarks are given.
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CHAPTER 2
ON THE MIXED SENSITIVITY MINIMIZATION FORSYSTEMS WITH INFINITELY MANY UNSTABLE
MODES
The main purpose of this Chapter is to show that H controllers for systems with
infinitely many unstable modes can be obtained by the Skew-Toeplitz approach [2],
using a simple data transformation. An example of such a plant is a high gain system
with delayed feedback (see Section 2.2). Undamped flexible beam models, [43], may
also be considered as a system with infinitely many unstable modes.
In earlier studies, e.g. [15],
H controllers are computed for weighted sensitivity
minimization involving plants in the form
P(s) =Mn(s)
Md(s)No(s) (2.1)
where Mn(s) is inner and infinite dimensional, Md(s) is inner and finite dimensional,
and No(s) is the outer part of the plant, that is possibly infinite dimensional. In
the weighted sensitivity minimization problem, the optimal controller achieves the
minimum H cost, opt, defined as
opt = inf C stabilizing P
W1(1 + P C)1W2P C(1 + P C)1
, (2.2)
where W1 and W2 are given finite dimensional weights. Note that in the above
formulation, the plant has finitely many unstable modes, because Md(s) is finite
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dimensional, whereas it may have infinitely many zeros in Mn(s). In this section, by
using duality, the mixed sensitivity minimization problem will be solved for plants
with finitely many right half plane zeros and infinitely many unstable modes.
2.1 Main Result
Assume that the plant to be controlled has infinitely many unstable modes, finitely
many right half plane zeros and no direct transmission delay. Then, its transfer func-
tion is in the form P = NM
, where M is inner and infinite dimensional (it has infinitely
many zeros in C+, that are unstable poles ofP), N = NiNo with Ni being inner finite
dimensional, and No is the outer part of the plant, possibly infinite dimensional. For
simplicity of the presentation we further assume that No, N1o H.
To use the controller parameterization of Smith, [44], we first solve for X, Y H
satisfying
N X + M Y = 1 i.e. X(s) =
1 M(s)Y(s)
Ni(s)
N1o (s). (2.3)
Let z1,...,zn be the zeros of Ni(s) in C+, and again for simplicity assume that they
are distinct. Then, there are finitely many interpolation conditions on Y(s) for X(s)
to be stable, i.e.
Y(zi) =1
M(zi)i = 1, . . . , n .
Thus by Lagrange interpolation, we can find a finite dimensional Y
H and infinite
dimensional X H satisfying (2.3), and all controllers stabilizing the feedback
system formed by the plant P and the controller C are parameterized as follows, [44],
C(s) =X(s) + M(s)Q(s)
Y(s) N(s)Q(s) (2.4)
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where Q(s) H and Y(s) N(s)Q(s)) = 0.
Now we use the above parameterization in the sensitivity minimization problem.
First note that,
(1 + P(s)C(s))1 = M(s)(Y(s) N(s)Q(s)),
P(s)C(s)(1 + P(s)C(s))1 = N(s)(X(s) + M(s)Q(s)). (2.5)
Then,
infC stab. P
W1(1 + P C)1
W2P C(1 + P C)1
= inf QH
W1(Y N Q)W2N(X + M Q)
(2.6)
where Y(s) N(s)Q(s) = 0, W1 and W2 are given finite dimensional (rational)weights. From (2.3) equation, we have W1Y W1N QW2N1MYN + W2M N Q
=
W1(Y Ni(NoQ))W2(1 M(Y Ni(NoQ)))
.(2.7)
Thus, the H optimization problem reduces to
opt = inf Q1H and YNiQ1=0
W1(Y NiQ1)W2(1 M(Y NiQ1))
(2.8)
where Q1 = NoQ, and note that W1(s), W2(s), Ni(s), Y(s) are rational functions, and
M(s) is inner infinite dimensional.
The problem defined in (2.8) has the same structure as the problem dealt in
Chapter 5 of the book by Foias, Ozbay and Tannenbaum [2], where Skew-Toeplitz
approach has been used for computing H optimal controllers for infinite dimensional
systems with finitely many right half plane poles. Our case is the dual of the problem
solved in [2, 45], i.e., there are infinitely many poles in C+, but the number of zeros in
C+ is finite. Thus, by mapping the variables as shown below, we can use the results
of [2, 45] to solve our problem:
WFOT1 (s) = W2(s) (2.9)
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WFOT2 (s) = W1(s)
XFOT(s) = Y(s)
YFOT(s) = X(s)
MFOTd = Ni(s)
MFOTn (s) = M(s)
NFOTo (s) = N1o (s),
and the optimal controller, C, for the two block problem (2.6) is the inverse of optimal
controller for the dual problem in [2], i.e.,
CFOTopt
1.
If we only consider the one block problem case, with W2 = 0, then the minimiza-
tion of
W1(Y NiQ1)
is simply a finite dimensional problem. On the other hand, minimizing
W2(1 M(Y NiQ1))
is an infinite dimensional problem.
2.2 Example
In this section, we illustrate the computation ofH controllers for systems withinfinitely many right half plane poles. The example is a plant containing an internal
delayed feedback:
P(s) =R(s)
1 + ehsR(s)
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where R(s) = k
sas+b
with k > 1, a > b > 0 and h > 0. Note that the denominator
term (1 + ehsR(s)) has infinitely many zeros n jn, where n o = ln(k)h > 0,
and n (2n + 1), as n . Clearly, P(s) has only one right half plane zero at
s = a.
The plant can be written as,
P(s) =Ni(s)
M(s)No(s) (2.10)
where
Ni(s) = s as + a
No(s) =1
1 + (sb)k(s+a)
ehs
M(s) =(s + b) + k(s a)ehs(s b)ehs + k(s + a)
It is clear that No is invertible in H, because sb
k(s+a) < 1. By the same argument,
M is stable. To see that M is inner, we write it as
M(
s) =
m(s) + f(s)
1 + m(s)f(s)with m(s) =
sas+a
ehs, and f(s) = s+b
k(s+a). Note that m(s) is inner, m(s)f(s) is
stable, and M(s)M(s) = 1. Thus M is inner, and it has infinitely many zeros in
the right half plane.
The optimal H controller can be designed for weighted sensitivity minimization
problem in (2.2) where P is defined in (2.10) and weight functions are chosen as
W1(s) = , > 0 and W2(s) = 1+s+s , > 0, > 0, < 1. As explained before,
this problem can be solved by the method in [2] (see Appendix A for details) after
necessary assignments are done, WFOT1 (s) =1+s+s
, WFOT2 (s) = , MFOTd =
sas+a
,
MFOTn (s) =(s + b) + k(s a)ehs(s b)ehs + k(s + a)
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NFOTo (s) =(s b)ehs + k(s + a)
k(s + a).
We will briefly outline the procedure to find the optimal H controller.
1) Define the functions,
F(s) =
s
a + bs
, =
1 222 2 for > 0
where a =
1 + 22 22, and b =
(1 22)2 + 2.
2) Calculate the minimum singular value of the matrix,
M =
1 j M F(j) jM F(j)1 a M F(a) aM F(a)
M F(j) jM F(j) 1 jM F(a) aM F(a) 1 a
for all values of (max{,
1+22}, 1
) and M F(s) = M(s)F(s). The
optimal gamma value, opt, is the largest gamma which makes the matrix M
singular.
3) Find the eigenvector l = [l10, l11, l20, l21]T such that Moptl = 0.
4) The optimal H controller can be written as,
Copt(s) =kf + K2,FIR(s)
K1(s)
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where kf is constant, K1(s) is finite dimensional, and K2,FIR(s) is a filter whose
impulse response is of finite duration
K1(s) =
k(l21s + l20)
opt(+ s) ,
kf =
kboptl11 optl21
2opt 2
,
K2,FIR(s) = A(s) + B(s)ehs,
kf + A(s) =k(s + a)c(s) + (s + b)d(s)
((1 2opt2) + (2opt 2)s2)(s a),
B(s) =(s b)c(s) + k(s a)d(s)
((1 2opt2) + (2opt 2)s2)(s a)
where c(s) = (aopt + bopts)(l11s + l10) and d(s) = opt( s)(l21s + l20).
As a numerical example, if we choose the plant as P(s) =2( s3s+1)
1+2( s3s+1)e0.5sand
the weight functions as W1(s) = 0.5, W2(s) =1+0.1s0.4+s
, then the optimal H cost is
opt = 0.5584, and the corresponding controller is
Copt(s) =
0.558s + 0.223
2s + 3.725 (1.477 + K2,FIR(s))
where K2,FIR(s) is equal to
(2.081s2 6.302s 0.826) (0.615s3 0.768s2 5.269s + 1.587)e0.5s(0.302s3 0.905s2 + 0.950s 2.850)
and its impulse response is of finite duration, k2,FIR(t) = L1(K2,FIR(s)) can be
written as
0.27e3t + 7.16 cos(1.77t) + 0.36 sin(1.77t) 2.037(t 0.5) 0 t 0.5
0 t > 0.5.
2.3 Summary of Results
In this Chapter, we have considered H control of a class of systems with in-
finitely many right half plane poles and infinitely many right half plane zeros. We
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have demonstrated that the problem can be solved by using the existing H control
techniques [2] for infinite dimensional systems with finitely many right half plane
poles. An example from delay systems is given to illustrate the computational tech-
nique. This result is used in the next Chapter to design optimal H controller for
general time-delay systems, including dead-time systems.
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CHAPTER 3
H CONTROLLER DESIGN FOR NEUTRAL ANDRETARDED DELAY SYSTEMS
In this Chapter, the mixed sensitivity minimization problem is solved for Neutral
and Retarded Delay Systems. Assume that p(s) is a quasi-polynomial which can be
written as
p(s) = p1(s)eh1s + . . . + pn(s)e
hns
where h1 < h2 < .. . < hn and p1(s), p2(s), . . . , pn(s) are polynomials. If the degree of
the polynomial p1(s) is larger or equal to p2(s), . . . , pn(s), then p(s) is called as neutral
quasi-polynomial and if the degree of the polynomial p1(s) is strictly larger than all
the other polynomials, then p(s) is called as retarded quasi-polynomial. Assume that
q(s) is a stable polynomial and the degree of q(s) is equal to that of p1(s) and define
R(s) =p(s)
q(s),
Ri(s) =
pi(s)
q(s) , i = 1, . . . , n .
Similarly, R(s) is called neutral delay system if p(s) is neutral quasi-polynomial and
R(s) is called retarded delay system if p(s) is retarded quasi-polynomial. The plant
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is assumed to have a transfer function whose numerator and denominator can be
expressed as Neutral or Retarded Delay System, i.e.,
P(
s) =
R(s)
T(s) = ni=1 Ri(s)e
his
mj=1 Tj(s)ejs .In this Chapter, optimal H controllers are designed for neutral/retarded delay
systems. First, a necessary and sufficient condition is derived for neutral and retarded
delay systems to have finitely many right half plane zeros. Optimal controllers are
designed for the time-delay systems with finitely many unstable zeros or poles. Spe-
cial structures of the corresponding controllers for these types of plants are studied.
Moreover, in this Chapter, the largest class of neutral/retarded delay systems (see
equation (3.5)) are determined for which the techniques of [2] and [15] are applicable.
The link between [15] and [23] is established for neutral/retarded plants, i.e., the
finite impulse response structure in the controller exists not only for plants in the
form ehsP0(s), but also for general retarded/neutral delay systems.
In [15], it is assumed that the plant is single input single output (SISO) and admits
the representation as,
P(s) =mn(s)No(s)
md(s)(3.1)
where mn(s) is inner, infinite dimensional and md(s) is inner, finite dimensional and
No(s) is outer, possibly infinite dimensional. The optimal H controller, Copt, stabi-
lizes the feedback system and achieves the minimumH cost, opt:
opt =
W1(1 + PCopt)1W2PCopt(1 + PCopt)1
(3.2)
where W1 and W2 are finite dimensional weights for the mixed sensitivity minimization
problem.
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In Chapter 2, the optimal H controller design is given for infinite dimensional
systems with finitely many unstable zeros by using the duality with the problem (3.2).
The plant has a factorization as,
P(s) =md(s)No(s)
mn(s)(3.3)
where mn is inner, infinite dimensional, md(s) is finite dimensional, inner, and No(s)
is outer, possibly infinite dimensional. For this dual problem, the optimal controller,
Copt, and minimum H cost, opt, are found for the mixed sensitivity minimization
problem which can be defined as,
opt = W1(1 + PCopt)1W2PCopt(1 + PCopt)1 . (3.4)
Optimal controller designs for problems (3.2) and (3.4) are outlined in Appendix A.
In this Chapter, we will design optimal H controller for a class of neutral and
retarded time-delay systems,
P(s) =R(s)
T(s)=
ni=1 Ri(s)e
his
mj=1 Tj(s)e
js.
The assumptions on the plant is given in Section 3.1 as A.1 A.7. We apply the
design methods in [15] and Chapter 2 for this plant. Note that P is the generalized
version of time-delay systems for SISO case. We assume that P has finitely many
right half plane zeros or poles (assumption A.7 in Section 3.1). We give the necessary
and sufficient conditions when this assumption is valid in Section 3.1.1. The special
structure ofH controllers is given in this section. Note that single delay plant with
finite dimensional plant is a special case of the plant P. By showing finite impulse
response filter structure of neutral and retarded time-delay systems, we eliminate
right half plane pole-zero cancellation in the controller. The resulting controller in
this structure is numerically stable and easy for implementation.
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3.1 Preliminary Work
We consider the plant, P, which has a transfer function,
P(s) = R(s)T(s)
= ni=1 Ri(s)ehismj=1 Tj(s)e
js. (3.5)
satisfying the following assumptions,
A.1 Ri and Tj are finite dimensional, stable, proper transfer functions,
A.2 Time delays are nonnegative rational numbers shown as hi and j with hi < hi+1
for i = 1, . . . , n
1 and j < j+1 for j = 1, . . . , m
1,
A.3 h1 1,
A.4 Relative degree of R1 is smaller than or equal to that of Ri, j = 2, . . . , n and
the relative degree of T1 is smaller than or equal to that of Ti, i = 2, . . . , m,
A.5 Relative degree of T1 is smaller than or equal to that of R1,
A.6 P has no imaginary axis zeros or poles,
A.7 R or T has finitely many zeros in the right half plane.
These assumptions are not restrictive. It is always possible to obtain a representa-
tion such that A.1 A.2 are satisfied. The first assumption results in stable transfer
functions in the controller which is desired for implementation purposes. The assump-
tion A.3 guarantees that the plant is causal. The numerator and denominator of the
plant is neutral or retarded time delay system by the assumption A.4. The plant is
proper transfer function if the assumption A.5 is valid. Note that the assumptions
A.1A.5 are necessary by definition of neutral and retarded time delay systems. The
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condition A.6 is a technical assumption which can be removed, but it is assumed for
simplicity in derivation. The last condition comes from the limitation of standard the
skew-toeplitz approach [2] and its dual approach given in Chapter 2.
For example, consider the plant whose dynamical behavior is described in time
domain by the following set of equations:
x1(t) = x1(t 0.2) x2(t) + u(t) + 2u(t 0.4),
x2(t) = 5x1(t 0.5) 3u(t) + 2u(t 0.4),
y(t) = x1(t). (3.6)
Its transfer function is
P(s) =s + 3 + 2(s 1)e0.4ss2 + se0.2s + 5e0.5s
, (3.7)
and it can be re-written in form of (3.5) as,
P =R
T=
R1eh1s + R2e
h2s
T1 + T2e2s + T3e3s(3.8)
where
R1 =s+3
(s+1)2, R2 =
2(s1)(s+1)2
,
T1 =s2
(s+1)2, T2 =
s(s+1)2
, T3 =5
(s+1)2,
are stable proper finite dimensional transfer functions and the delays are
h1 = 0, h2 = 0.4,
1 = 0, 2 = 0.2, 3 = 0.5.It is clear that A.1 A.3 are satisfied. The relative degree of R1 is equal to that of
R2 which is 1 and the relative degree of T1, which is 0, is smaller than that of T2 and
T3, which are 1 and 2, respectively. Also, the relative degree of T1 is smaller than
that of R1. Therefore assumptions A.4 A.5 hold. The plant P has no imaginary
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axis zeros. By a simple computation, T has finitely many right half plane zeros at
0.4672 1.8890j, whereas R has infinitely many unstable zeros converging to the
asymptote 1.7329 (5k + 2.5)j as k . In conclusion, this system satisfies all
the assumptions, A.1 A.7.
We define the conjugate of R(s) =n
i=1 Ri(s)ehis as R(s) := ehnsR(s)MC(s)
where MC is inner, finite dimensional whose poles are poles of R. For example, we
can calculate the conjugate of R(s) in the previous example,
R(s) =s + 3
(s + 1)2+
2(s 1)(s + 1)2
e0.4s
where h2 = 0.4 and MC(s) = s1
s+12. The conjugate of R(s) can be written as,
R(s) = eh2sR(s)MC(s),
=
2
(s + 1)+
(s 3)(s + 1)2
e0.4s
.
We will design an optimal H controller for the plant in (3.5) which satisfies the
assumptions A.1
A.7. In the next sections, we will give an equivalent condition
to check whether the assumption A.7 is valid or not and standard Skew-Toeplitz
approach to design H controller for infinite dimensional systems will be outlined.
3.1.1 Neutral and Retarded Time Delay Systems with FinitelyMany Unstable Zeros
We begin with preliminary observations (see also [46] for further details). First,
recall that R1(s), . . . , Rn(s) are stable and proper transfer functions. Therefore,
R(s) =n
i=1 Ri(s)ehis has (infinitely) finitely many right half plane zeros if and
only if
RG(s) = 1 +R2(s)
R1(s)eh1s + . . . +
Rn(s)
R1(s)ehns
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has (infinitely) finitely many unstable zeros respectively.
The following fact will be used later in this Chapter and in Chapter 5.
Claim: RG(s) has no unstable zeros with real part extending to infinity.
Proof: Assume that s = 0 +j . For sufficiently large o, we can write the following
inequalities,
i Ri(0 + j)R1(0 + j)
+i i = 2, . . . , nbecause all Ri(s) terms are finite dimensional. Following holds by the triangle in-
equality,
1 2 eh20 . . . n ehn0 |RG(0 + j)| 1 + +2 eh20 + . . . + +n ehn0
This inequality tells us that RG(s) has no unstable zeros with real part extending to
infinity because lim0 |RG( + j)| = 1. Therefore, R(s) has no unstable zeros
with real part extending to infinity.
Lemma 3.1.1. Assume that R(s) =
ni=1 Ri(s)e
his is a neutral or retarded time
delay system with no imaginary axis zeros and poles, then the system R has finitely
many right half plane zeros if and only if all the roots of the polynomial, 1+ 2rh2h1 +
+ nrhnh1 has magnitude greater than 1 where
i = lim
Ri(j)R11 (j) i = 2, . . . , n ,
hi =hi
N, N, hi Z+, i = 1, . . . , n .
If the system R satisfies this condition, it is called as F-system where F is an abbre-
viation for finitely many unstable zeros.
Proof. Since the system is neutral or retarded time delay system, it may have infinitely
many right half plane zeros extending to infinity in imaginary part with fixed positive
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real part [46]. R has infinitely many right half plane zeros if and only if for fixed
o > 0, lim R(o +j) has infinitely many unstable zeros. Define r = e(o+j)/N,
then
lim
R(o + j)
R1(o + j )= rh1 + 2r
h2 + + nrhn. (3.9)
Let r0 is the root of (3.9) which satisfies |ro| > 1. Then the system R do not have
infinitely many right half plane zeros, since
|ro| = eo/N,
o =
Nln
|ro
|< 0
which is a contradiction. Therefore, if all the roots of the polynomial (3.9) has
magnitude less than 1, then the system, R, does not have infinitely many unstable
zeros. Assume that there exist a root, ro, of (3.9) with magnitude smaller than 1.
Then, there are infinitely many right half plane zeros ofR converging to the asymptote
ro,k = ln |ro|N jN(ro + 2k) as k where k Z and ro is the phase of the
complex number ro. The lemma is also valid when delays are real numbers. If the
delays are rational numbers, the resulting function is polynomial which is easy to
check its roots. When the delays are real numbers, it will be difficult to find the
roots of the function. Therefore, the delays are assumed as rational numbers for
convenience.
We define R(s) as I-system (where I is an abbreviation for infinitely many unstable
zeros) if R is F-system. In order to find whether R is an I-system, we can check the
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magnitude of all the roots of the polynomial, 1 + 2rh2h1 + + nrhnh1 where
i = lim
Ri(j)R11 (j) i = 2, . . . , n ,
hi =
hi
N, N, hi Z+, i = 1, . . . , n .
If the magnitude of all the roots is smaller than 1, then R is an I-system.
In this section, the plant in (3.5) has finitely many unstable zeros or poles. This
is guaranteed by assumption A.7. By Lemma 3.1.1, it is easier to check that the
assumption A.7 is valid by finding the magnitude of all roots of the given polynomial.
We can conclude that the assumption A.7 is satisfied if and only if either R or T is a
F-system (or both).
3.1.2 FIR Structure of Neutral and Retarded Time DelaySystems
In this section, we will show special structure of neutral and retarded time delay
systems. This is a key lemma to be used in the next section for the proofs of main
results.
Lemma 3.1.2. LetR be as in Lemma 3.1.1 and MR be a finite dimensional system.
Define S+z be the set of common C+ zeros of R and MR. Then RMR , can always be
decomposed as,
R(s)
MR(s)= HR(s) + FR(s) (3.10)
where HR is an infinite dimensional system which does not have any poles in S+z and
FR is a finite impulse filter.
Proof. For clarity of the presentation we give the proof for the case where the entries
z1, z2, . . . , z nz of S+z are distinct. For the general case main idea is the same, except
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that some of the expressions below may become more complicated (notation needs to
be modified to include the multiplicity of each common zero, and the partial fraction
expansion and inverse Laplace transformation needs to consider these multiplicities).
So, we assume R(zk) = MR(zk) = 0, k = 1, . . . , nz. We can rewrite the expressionR
MRas
R
MR=
ni=1
Ri
MRehis.
We can decompose each term by partial fraction, RiMR
= Hi + Fi where the poles ofFi
are elements ofS+z and define the terms HR and FR as
HR(s) =n
i=1
Hi(s)ehis,
FR(s) =n
i=1
Fi(s)ehis.
Note that each term, Fk is strictly proper and FR(zk) is finite k = 1, . . . , nz. The
lemma ends if we can show that FR is a FIR filter. Inverse Laplace transform of FRcan be written as
fR(t) =nz
k=1
ni=1
Res{Fi(s)}
s=zkezk(thi)uhi(t)
where Res(.) is the residue of the function and uhi(t) is the unit step function delayed
by hi. The impulse response is equivalent to
fR(t) =nz
k=1
ezkt
n
i=1
Res{Fi(s)}
s=zkehizk
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for t > hn. It is clear from partial fraction that Res{Fi(s)}
s=zk= Ri(zk) which
results fR(t) is FIR filter,
fR(t) =nz
k=1 ezktn
i=1 Ri(zk) ehizk ,=
nzk=1
ezktR(zk) 0 for t > hn,
since zk is the zero of R, i.e., R(zk) = 0. Therefore, we can conclude that FR is a
FIR filter with support [0, hn].
Note that this decomposition eliminates right half plane pole-zero cancellation in
RMR and brings into form which is easy for numerical implementation. Lemma 3.1.2
explains the FIR structure of H controllers. Since these controllers satisfy inter-
polation conditions, right half plane pole-zero cancellations occur. By Lemma 3.1.2,
we can see that it is always possible to form an FIR structure in H controllers for
time-delay systems.
Assume that P is in the form of P(s) =
nk=1 Pk(s)e
hks where Pk are stable,
proper, finite dimensional transfer functions. Also, P0 is a bi-proper, finite dimen-
sional system. By partial fraction, we can write PkP0
as
Pk(s)
P0(s)= Pk,p(s) + Pk,0 k = 1, . . . , n
where the Pk,0 is a proper transfer function whose poles are same as the zeros of P0.
Then, the decomposition operator, , can be defined as,
(P(s), P0(s)) = HP(s) + FP(s)
where HP =n
k=1 Pk,p(s)ehks and FP =
nk=1 Pk,0(s)e
hks are infinite dimensional
systems. Note that if the zeros of P0 are also unstable zeros of P, then FP is an FIR
filter shown in Lemma 3.1.2.
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3.2 Main Results
In this section, we will construct the optimal H controller for the plant, P, with
a transfer function in the form of (3.5) satisfying assumptions A.1 A.7. The plant,P = R
T, is assumed to be one of the following:
i) R is I-system and T is F-system (IF plant),
ii) R is F-system and T is I-system (FI plant),
iii) R is F-system and T is F-system (FF plant).
For each case, we will find an optimal H controller and obtain a structure such that
there is no internal pole-zero cancellations in the controller.
3.2.1 Factorization of the Plants
In order to apply skew-toeplitz approach [2], we need to factorize the plants as in
(3.1) or (3.3).
IF Plant Factorization
Assume that the plant in (3.5) satisfies the assumptions A.1 A.7, R is I-system
and T is F-system, then we can factorize the plant as
mn = e(h11)sMR(s)
{eh1sR(s)}R(s)
,
md = MT(s),
No =
R(s)MR(s)
{e1sT(s)}MT(s)
(3.11)
where MR is an inner function whose zeros are the right half plane zeros of R(s).
Since R is an I-system, the conjugate of R has finitely many right half plane zeros,
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therefore MR is well defined. Similarly, zeros of MT are right half plane zeros of T.
Note that mn and md are inner functions, finite and infinite dimensional respectively.
No is an outer term. Although there are right half plane pole-zero cancellations in
md and No, this problem will be solved in Section (3.2.2) by using the technique in
Section 3.1.2.
FI Plant Factorization
The plant satisfies the assumptions A.1 A.7, R is F-system and T is I-system.
Since the design method in Chapter 2 requires the plant and its inverse to be causal
proper transfer functions, we restrict P in (3.5) to satisfy the following additional
assumptions:
B.3 h1 = 1 = 0,
B.5 Relative degree ofT1 is equal to that of R1.
Then the plant P can be factorized as in (3.3),
mn = MT(s)T(s)
T(s),
md = MR(s),
No =
R(s)MR(s)
T(s)MT(s)
. (3.12)
The zeros of inner function MR are right half plane zeros of R. The unstable zeros
of
T(s) are the same as the zeros of inner function MT. Similar to previous section,
conjugate ofT has finitely many right half plane zeros since T is a I-system. The right
half plane pole-zero cancellations in mn and No will be eliminated in Section 3.2.2.
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FF Plant Factorization
For FF plants, assumptions for IF plants are valid (i.e., assumptions A.1 A.7),
but R and T are F-systems. The plant P has a factorization as in (3.1),
mn = e(h11)sMR(s),
md = MT(s),
No =
{eh1sR(s)}MR(s)
{e1sT(s)}MT(s)
(3.13)
where MR and MT are inner functions whose zeros are right half plane zeros of R and
T respectively. Note that when h1 = 1, mn is finite dimensional term. Then exact
internal pole-zero cancellations are possible for this case (except the ones in No).
3.2.2 Optimal H Controller Design
By using the methods in [15] and Chapter 2, given the appropriately chosen weight
functions W1 and W2, the optimal H cost, opt can be found. After opt is calculated,
it is easy to obtainE
opt,F
opt andL
. We will give the structure of optimal H
controllers for each type of plant.
Controller Structure of IF Plants
By using the method in [15] and Chapter 2, we can obtain opt, Eopt, Fopt, L.
The optimal controller can be written as,
Copt =
Kopt(s)e1sT(s)
MT(s) R(s)MR(s)
+ e1sR(s)Fopt(s)L(s)(3.14)
where Kopt(s) = Eopt(s)Fopt(s)MT(s)L(s). In order to obtain the structure of
controller:
1. Do the necessary cancellations in Kopt,
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(Hopt 1) + Fopt
HT + FT- - -d
6
+
Figure 3.1: Optimal H Controller for IF plants
2. Partition Kopt as Kopt(s) = opt(s)T(s), where opt is a bi-proper transfer
function. The zeros of opt are right half plane zeros of EoptMT,
3. Obtain (HT, FT), (HR,1,FR1) and (H2, FR2) by using the partitioning operator
HT + FT = (e1sT T, MT),
HR1 + FR1 = (R, MRopt),
HR2 +F
R2 = (e1sRFoptL, opt),
as explained in Section 3.1.2.
Then, the optimal controller has the form
Copt =HT + FT
Hopt + Foptwhere HT, Hopt = HR1 + HR2 are neutral/retarded time-delay systems and
FT,
Fopt = FR1 + FR2 are FIR filters. The controller has no right half plane pole-zero
cancellations and its structure is shown in Figure 3.1.
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(HR 1) + FR
Hopt + Fopt- - -d
6
+
Figure 3.2: Optimal H Controller for FI plants
Controller Structure of FI Plants
After the data transformation is done as shown in (2.9) in Chapter 2, we can
find opt, Eopt, Fopt, L as in IF plant case. We can write the inverse of the optimal
controller similar to (3.14):
C1opt =Kopt(s)
R(s)
MR(s)
T(s)
MT(s)+ T(s)Fopt(s)L(s)
(3.15)
where Kopt
(s) = Eopt
(s)Fopt
(s)MR
(s)L(s). We can obtain the optimal controller
following exactly same steps for IF plant case (note that R and T are interchanged,
and h1 = 1 = 0 from assumptions):
Copt =Hopt + Fopt
HR + FR .
The optimal controller is the dual of the controller for the IF plants, as expected.
There is no internal right half plane pole-zero cancellations and its structure is shownin Figure 3.2.
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Controller Structure of FF Plants
Structure of FF plants is similar to that of IF plants. We can calculate opt, Eopt,
Fopt, L by the method in [15], Chapter 2 and then write the optimal controller as:
Copt =Kopt(s)
{e1T(s)}MT(s)
{eh1sR(s)}MR(s)
+ e1sR(s)Fopt(s)L(s)(3.16)
where Kopt(s) = Eopt(s)Fopt(s)MT(s)L(s). We can find controller structure follow-
ing similar steps as in IF plants case:
1. Do the cancellations in Kopt,
2. Factorize Kopt and find opt, T,
3. Obtain (HT, FT), (HR1, FR1)and (HR2, FR2) by
HT + FT = (e1sT T, MT),
HR1 + FR1 = (eh1sR, MRopt),
HR2 + FR2 = (e1sRFoptL, opt).
The controller structure can be written as:
Copt =HT + FT
Hopt + Foptwhere Hopt = HR1 + HR2 and Fopt = FR1 + FR2. Note that the resulting structure
is same as IF plant case. It is possible to cancel the zeros of opt with denominator
when h1 = 1 = 0 which is considered in Example 3.3.2.
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3.3 Examples
We will show the optimal H controller design for IF and FF plants. Since FI
plant case is the dual of IF plant case, example of this section is omitted. The plants
in the examples are SISO plants. Sensitivity and complementary sensitivity weight
functions are chosen as,
W1(s) = 2
s + 1
10s + 1
,
W2(s) = 0.2(s + 1.1)
which are the same weight functions taken in [2].
3.3.1 IF Plant Example
We will design optimal H controller for the plant whose time domain represen-
tation is given by (3.6) in Section 3.1. The transfer function for this plant is as in
(3.7). The plant can be written in the form of (3.5) as demonstrated in (3.8). Note
that R is a I-system, since its corresponding polynomial (see Lemma 3.1.1) is 1 + 2 r
and the magnitude of root is smaller than 1. It is clear that T is a F-system, since
its corresponding polynomial is a constant, 1. Therefore it has no roots and satisfies
the condition trivially. Therefore, the plant is a IF plant with h1 = 1 = 0. In order
to find optimal controller, the plant is factorized as in (3.1),
mn = MR(s)R(s)
R(s)
,
md = MT(s),
No =
R(s)MR(s)
T(s)MT(s)
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where
MR =s 0.2470s + 0.2470
,
MT =
s2
0.9344s + 3.7866
s2 + 0.9344s + 3.7866 ,
R =2(s + 1) + (s 3)e0.4s
(s + 1)2.
For the given plant and weight functions, the optimal H cost (3.2) is opt = 0.7203
and the corresponding optimal controller is [2] (see A.3 at Appendix A),
Copt = Eopt(s)md(s)N1o (s)Fopt(s)L(s)
1 + mn(s)Fopt(s)L(s)
where
Eopt =3.4812 + 47.8803s2
0.5188(1 100s2) ,
Fopt =0.5188(1 10s)
1.384s2 + 2.842s + 1.381,
L =0.589s2 0.3564s + 0.16160.589s2 + 0.3564s + 0.1616
.
The optimal controller,
Copt, has internal unstable pole-zero cancellations (i.e., thezeros of Eopt and md are cancelled by the denominator of the controller). In order
to eliminate this, we will follow the procedure given in Section 3.2.2:
steps 1-2 After cancellation, Kopt = EoptFoptMTL can be factorized as
Kopt = opt(s)T(s),
where
opt =3.4812 + 47.8803s2
(1.384s2 + 2.842s + 1.381)MT(s),
T =L(s)
(1 + 10s).
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Impulse response of FT(s)
time
Figure 3.3: fT(t) for IF plant
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Impulse response of F(s)
time
Figure 3.4: fopt(t) for IF plant
step 3 We can find (HT, FT), (HR,1,FR1) and (HR2, FR2) by the partitioning opera-
tor, . The expressions can be found in Appendix B (see equations (B.1-B.6)).
Then, the optimal controller can be found from,
Copt = HT + FTHopt + Fopt
where Hopt = HR1 + HR2 and Fopt = FR1 + FR2 . The impulse responses ofFopt and
FT are shown in Figures 3.3 and 3.4.
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3.3.2 FF Plant Example
We will find the optimal H controller for the plant whose time-domain repre-
sentation is
x1(t) = x1(t) + 3x2(t 0.2),
x2(t) = x2(t) + 2x3(t 0.5),
x3(t) = x3(t) + 2x1(t 0.1) + u(t),
y(t) = x1(t) + x2(t) + x3(t). (3.17)
The corresponding transfer function for this plant is
P(s) =(s 1)2 + 2(s 1)e0.5s + 6e0.7s
(s 1)3 12e0.8s
and it can be rewritten as
P(s) =R(s)
T(s)=
(s1)2
(s+1)3+ 2(s1)
(s+1)3e0.5s + 6
(s+1)3e0.7s
(s1)3
(s+1)3 12
(s+1)3e0.8s
.
It is clear that both R and T are F-system. Since h1 = 1 = 0, exact cancellation inthe controller is possible, otherwise optimal controller design is the same as previous
example. We can factorize the plant as,
mn = MR(s),
md = MT(s),
No =
R(s)MR(s)
T(s)MT(s)
(3.18)
where
MR =s2 2.757s + 4.455s2 + 2.757s + 4.455
,
MT =s3 4.09s2 + 8.185s 9.128s3 + 4.09s2 + 8.185s + 9.128
.
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Given the weight functions of the previous example, the optimal H cost is computed
as opt = 0.7031. The optimal controller is as in (A.3) where
Eopt =3.5056 + 45.4406s2
0.4944(1 100s2),
Fopt =0.4944(1 10s)
1.3482s2 + 2.7681s + 1.3446,
L =0.1364s3 + 0.3852s2 + 0.5654s + 0.1154
0.1364s3 0.3852s2 + 0.5654s 0.1154 .
Since mn is finite dimensional, 1 + mn(s)Fopt(s)L(s) is finite dimensional. Therefore,
the exact cancellation between opt and 1 + mn(s)Fopt(s)L(s) is possible. We will
follow the procedure of Section 3.2.2 with a slight modification:
steps 1-2 After the cancellation, Kopt = EoptFoptMTL can be factorized as
Kopt = opt(s)T(s),
where
opt =3.5056 + 45.4406s2
1.3482s2
+ 2.7681s + 1.3446
MT(s),
T =L(s)
(1 + 10s).
Then, the controller can be written as,
Copt =
T TMT
R
MR
1+MRFoptL
opt
step 3 We perform the cancellations in K = 1+MRFoptLopt as shown in Appendix B(see equations (B.7-B.8)),
step 4 We can find the controller parameters (HT, FT) from
HT + FT = (T T, MT)
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
Impulse response of FT(s)
time
Figure 3.5: fT(t) for FF plant
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4
0.3
0.2
0.1
0
0.1
0.2
0.3
Impulse response of F(s)
time
Figure 3.6: fopt(t) for FF plant
and (Hopt,Fopt) from
Hopt + Fopt = (RK, MR).
See Appendix B for these partitions (equations (B.9-B.12)). In Figure 3.5 and
3.6, impulse responses ofFT and Fopt are given.
Note that the structure of controller is same, however exact cancellation simplifies
the controller expression.
3.4 Summary of Results
In this Chapter, we determined a necessary and sufficient condition for general
time-delay systems for applicability of the Skew-Toeplitz approach [2]. Note that
the neutral and retarded delay systems are the generalized versions of the time-
delay systems. The optimal H controllers for these types of generalized time delay
systems can be obtained by using the Skew-Toeplitz method. Moreover, the internal
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unstable pole-zero cancellation problem is eliminated by using special structure of the
controller.
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CHAPTER 4
STABLE H CONTROLLERS FOR DELAY SYSTEMS:SUBOPTIMAL SENSITIVITY
In this Chapter, an indirect method for strongly stabilizing H controller designfor systems with time delays is given. It is known that for a certain class of time
delay systems the optimal H controllers designed for sensitivity minimization lead
to controllers with infinitely many unstable modes, [47, 48]. An indirect way to
obtain a strongly stabilizing controller, in this case, is to internally stabilize the
optimal sensitivity minimizing H controller, while keeping the sensitivity deviation
from the optimum within a desired bound. The proposed scheme is illustrated in
Figure 4.1: the objective is to have a stable feedback system, and to minimize the
weighted sensitivity function W S := W(1 + P K)1, with a stable K. We will assume
that for given W and P, the optimal H controller Copt is determined. Then, F will
be designed to yield a stable K, such that the feedback system remains stable, and
W S is relatively close to the optimal weighted sensitivity W Sopt := W(1+P Copt)1.
When the plant P contains a time delay, and the sensitivity weight W is bi-proper,
the indirect approach outlined above requires internal stabilization of C (which con-
tains infinitely many unstable modes) by F. In Chapter 2, a solution to the two-block
H control problem involving a plant with infinitely many poles in the open right
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r
z
y
+ +
C P
FK
W
Figure 4.1: (i) F = 0: the weighted sensitivity is H optimal; (ii) F = 0: thecontroller K is stable.
half plane is given. Then, the results of that Chapter will be used to derive sufficient
conditions for solvability of the stable H controller design problem considered here
for systems with time delays. Namely, in this Chapter we investigate the indirect
method of obtaining a strongly stabilizing controller for systems with time delays,
subject to a bound on the deviation of the sensitivity from its optimal value. First
we examine the optimal sensitivity problem for stable delay systems and illustrate
that the corresponding optimal controller has the structure of the plant introduced
in Chapter 2.
4.1 Optimal Sensitivity Problem for Delay Systems
Consider the feedback system shown in Figure 4.1, where P(s) = ehsNp(s) and
W(s) = 1+ss+
. We assume that Np,N1
p
H. By using the method developed in [2,
15], we calculate the optimal controller Copt(s) minimizing the weighted sensitivity
W(1 + P C)1 over all stabilizing controllers, as follows (See Appendix A for details).
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The smallest satisfying the phase equation given below is the optimal (smallest
achievable) sensitivity level:
h + tan
1 + tan
1
=
(4.1)
where =
122
22, and < < 1
. Once opt is computed as above, the corre-
sponding optimal controller is
Copt(s) =(1 2opt2) + (2opt 2)s2
opt(+ s)(1 + s)
N1p (s)
1 + opt
s1+s
ehs
. (4.2)
Also, define the optimal sensitivity function as Sopt(s) = (1 + P(s)Copt(s))1, then,
Sopt(j) =1 + opt(j)
1+j ejh
1 +
1jopt(+j)
ejh
. (4.3)
In [47], it was mentioned that H-optimal controllers may have infinitely many right
half plane poles. Here we will give a proof based on elementary Nyquist theory:
if S1opt(j ) encircles the origin infinitely many times, we can say that Copt(s) has
infinitely many right hand poles, because P(s) does not have any right half plane
poles. For s = j as , we have
S1opt(j ) 1
optejh
1 opt
ejh
and |S1opt(j)| opt . Since < opt < 1, we can say that |S1opt(j)| has constant
magnitude between 0 and 1 for sufficiently large . For k =2k
h, as k the
phase of S1opt(jk) tends to
. In other words, S1opt(j ) intersects negative part of
the real axis near k =2k
h, as k . Similarly, S1opt(j) intersects positive part of
the real axis near k =(2k+1)
has k . Thus S1opt(j) encircles the origin infinitely
many times, which means that Copt(s) has infinitely many poles in C+.
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Remark. Let m1(j) =
j+j
ejh , m2(j) =
1j1+j
ejh and g(j ) =
opt
+j1+j
. Then,
W(j)Sopt
(j) = opt g
1(j) + m1(j)
1 + g1(j)m2(j) = opt1 + g(j)m1(j)g(j ) + m2(j ) and hence |W(j )Sopt(j )| = opt as expected.
4.2 Sensitivity Deviation Problem
Recall that the H optimal performance level was defined as
0 := opt = inf C stab. P
W(1 + P C)1
where W(s) = 1+ss+
, with > 0, > 0, < 1, and P(s) = Np(s)Mp(s), with
Np, N1
p H, and Mp is inner and infinite dimensional, e.g. Mp(s) = ehs. We
have obtained the optimal controller for the sensitivity minimization problem in (4.2).
Claim: The optimal H controller is in the form
Copt(s) =N1p (s)Nc(s)
Dc(s)(4.4)
where Dc is inner infinite dimensional and Nc, N1c H.
It is easy to verify this claim by comparing (4.2) with (4.4): we see that
Nc(s) =2optW
2(s)m2(s) m1(s)1 + 1optW(s)m2(s)
,
=1
2opt(+ s)2
(1 2opt2) + (2opt 2)s2
1 + 1opt
1s+s
ehs
(4.5)
Dc(s) =1optW(s) + m1(s)
1 + 1optW(s)m2(s)= 1optW(s)
1 + m1(s)optW1(s)1 + m2(s)
1optW(s)
,
= 1optW(s)(1 + P(s)Copt(s))1,
=
s+s
ehs + 10
1+s+s
1 + 10
1s+s
ehs
(4.6)
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where m1(s) =
s+s
ehs, m2(s) =
1s1+s
ehs. Note that Nc(s) has no right half
poles or zeros (it has only two imaginary axis poles that are cancelled by the zeros
at the same locations). Therefore Nc, N1c
H. Also, it is easy to check that Dc is
inner and infinite dimensional.
Note that,
Dc = 10 W S0 =
10 W(1 + P Copt)
1 =
10 W
Dc
Dc + MpNc
.
Our goal is to have a stable controller K, by an appropriate selection of F:
K(s) =
Copt(s)
1 + F(s)Copt(s) .
At the same time we would like to have the resulting sensitivity function,
S(s) = (1 + P(s)K(s))1 =
1 + Mp(s)Np(s)
N1p (s)Nc(s)Dc(s)
1 + F(s)N1p (s)Nc(s)Dc(s)
1, (4.7)
to be close to the optimal sensitivity, Sopt = (1 + P Copt)1. By the parameterization
of the set of all stabilizing controllers for Copt [44], F can be written as,
F(s) =X(s) + Dc(s)Q(s)
Y(s) N1p (s)Nc(s)Q(s)
with N1p (s)Nc(s)X(s) + Dc(s)Y(s) = 1 which can be solved as Y = 0 and X =
N1c Np where Q H, Q(s) = 0. Then, in terms of the design parameter Q, the
functions F(s), K(s) and S(s) can be re-written as,
F(s) = N1c (s)Np(s) + Dc(s)Q(s)
N1p (s)Nc(s)Q(s) = (Q1
(s) + C1
opt(s)) (4.8)
K(s) =Copt(s)
1 Copt(s)(Q1(s) + C1opt(s))= Q(s) (4.9)
S(s) = (1 + Mp(s)Np(s)(Q(s)))1. (4.10)
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Also, the sensitivity function S(s) should be stable. We can define the relative devia-
tion of the sensitivity as WS0SS
, then minimizing this deviation over Q H,Q(s) = 0 is equivalent to
1,opt = inf QH
WS0 SS
,
= inf QH
W(MpNc)(1 + DcNpN1c Q)Dc + MpNc
. (4.11)
Note that, |Dc(j) + Mp(j)Nc(j)| = |10 W(j)| as shown before. Then,
1,opt = inf
QH
0Nc(1 + Dc
Q) (4.12)
where Q = NpN1c Q. For stability of the feedback system formed by the resultingcontroller K and the original plant P, we also want the sensitivity function S =
(1 MpNpQ)1 to be stable. Once the optimal Q is determined from (4.12), a
sufficient condition for stability of S (and hence the original feedback system) can be
determined as
|Np(j)| < |Q(j )|1 (4.13)
Note that the problem defined in (4.12) is equivalent to a sensitivity minimization
with an infinite dimensional weight 0Nc for a stable infinite dimensional plant
Dc. For the case where both the plant and the weight are infinite dimensional,
sensitivity minimization problem is difficult to solve. So, we propose to approximate
the weight by a finite dimensional upper bound function: find a stable rational weight
W1 such that |0Nc(j )| |W1(j )|. We suggest an envelope which is in the form,
W1(s) = 0 Ks + 1s + 1
,
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where
K = 1 + 1opt
1 = (opt + )(1 opt)1
1
and 1 is determined in some optimal fashion.
Then, we can solve the one following block problem as in Chapter 2
1,opt 2,opt = inf QH W1(1 + DcQ). (4.14)Note that 2,opt is the smallest value of 2, in the range 0K < 2 < (0K)
11
,
satisfying
= tan1
1
+ h + tan1
10 cos(h) + ( 10 sin(h))(+ 10 cos(h)) +
10 sin(h)
+tan1
1
tan1
10 cos(h) ( 10 sin(h))(+ 10 cos(h)) 10 sin(h)
(4.15)
where =
(oK)22122
21
22(0K)2
. After finding 2,opt, we can write the C2,opt as,
C2,opt(s) = A(s) 11 Dc(s)B(s) (4.16)
where,
A(s) =(20K
221 22,opt21) + (22,opt 20K2)s20K2,opt(1 + s)(1 + s)
B(s) =
2,opt
0K
1 s1 + s
.
In order to calculate Q2,opt(s) corresponding to C2,opt(s), we will use the transforma-tion
Q2,opt(s) = C2,opt(s)1 + P(s)C2,opt(s)
.
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That gives
Q2,opt(s) = A(s)
1
1 Dc(s)B1(s)
=(2
1 2
K
2
1) + (2
K 1)s2
(1 + s)(K(1 + s) Dc(s)(1 s)) (4.17)
where K =2,opt0K
. After finding Q2,opt(s), F(s) can be calculated via (4.8),F(s) = (Q12,opt(s) + C1opt(s))
where Q2,opt(s) and Copt(s) are found in (4.17) and (4.2) respectively.Similarly, the resulting controller K(s) is determined as
K(s) = Q2,opt(s)which is shown in (4.9).
4.3 Summary of Results
An indirect approach to design stable controller achieving a desired H perfor-
mance level for time delay systems is studied. This approach is based on stabilization
ofH controller by another H controller in the feedback loop. Stabilization of the
controller is achieved and the sensitivity deviation is minimized. However, there are
two main drawbacks of this method. First, the solution of sensitivity deviation brings
conservatism because of finite dimensional approximation of the infinite dimensional
weight. Overall system does not achieve the exact performance level, since the op-
timal H controller is perturbed by deviation. Second, and more importantly, the
stability of overall sensitivity function (feedback system stability) is not guaranteed.
It is possible to keep the feedback system stable as long as the resulting coprime
factor perturbation in the controller is small. However, the perturbation needed to
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stabilize the controller may be relatively large. For this reason we tried an alternative
method, which is the subject of the next Chapter.
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CHAPTER 5
ON STABLE H CONTROLLERS FOR TIME-DELAYSYSTEMS
In this Chapter we focus on the strong stabilization problem for infinite dimen-
sional plants such that the stable controller achieves the pre-specified suboptimal H
performance level. When the optimal controller is unstable (with infinitely or finitely
many unstable poles), two methods are given based on a search algorithm to find
a stable suboptimal controller. However, both methods are conservative. In other
words, there may be a stable suboptimal controller achieving a smaller performance
level, but the designed controller satisfies the desired overall H
norm. The stability
of optimal and suboptimal controller is discussed and necessity conditions are given
for stability ofH controllers.
It is known that a H controller for time-delay systems with finitely many un-
stable poles can be designed by the methods in [17, 16, 15, 45]. In general, weighted
sensitivity problem results in an optimal H controller with infinitely unstable modes
[47, 48].
We assume that the plant is single input single output (SISO) and admits the
representation as in [15],
P(s) =mn(s)No(s)
md(s)(5.1)
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where mn(s) = ehsM(s), h > 0, and M(s), md(s) are finite dimensional, inner,
and No(s) is outer, possibly infinite dimensional. The optimal H controller, Copt,
stabilizes the feedback system and achieves the minimum H cost, opt:
opt = W1(1 + P Copt)1W2P Copt(1 + P Copt)1
, (5.2)
= inf C stabilizing P
W1(1 + P C)1W2P C(1 + P C)1
where W1 and W2 are finite dimensional weights for the mixed sensitivity minimization
problem.
In the next section, the structure of optimal and suboptimalH controllers will
be summarized. The optimal controller with infinitely many unstable poles case is
considered in Section 5.2. The conditions and a design method for stable suboptimal
H controller is given in the same section. Similar work is done in Section 5.3 for
the optimal controller with finitely many unstable poles. Examples related for these
design methods are presented in Section 5.4, and concluding remarks can be found in
Section 5.5.
5.1 Structure ofH Controllers
Assume that the problem (5.2) satisfies (W2No), (W2No)1 H, then optimal
H controller can be written as [2] (for controller calculations, see Appendix A),
Copt(s) = Eopt(s)md(s)N1o (s)Fopt(s)L(s)
1 + mn(s)Fopt(s)L(s)
. (5.3)
Similarly, the suboptimal controller achieving the performance level can be de-
fined as
Csubopt(s) = E(s)md(s)N1o (s)F(s)LU(s)
1 + mn(s)F(s)LU(s). (5.4)
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Note that the unstable zeros of Eopt and md are always cancelled by the denomi-
nator in (5.3). Therefore, Copt is stable if and only if the denominator in (5.3) has no
unstable zeros except the unstable zeros of Eopt and md (multiplicities considered).
Same conclusions are valid for the suboptimal case, Csubopt is stable provided that
the denominator in (5.4) has unstable zeros only at the unstable zeros of E and md
(again, multiplicities considered).
It is clear that the optimal, respectively suboptimal, controllers have infinitely
many unstable poles if and only if there exists o > 0 such that the following inequality
holds
lim
|Fopt(o + j )Lopt(o + j)| > 1, (5.5)
respectively,
lim
|F(o + j)LU(o + j)| > 1. (5.6)
The controller may have infinitely many poles because of the delay term in the de-
nominator. All the other terms are finite dimensional.
Even when the optimal controller has infinitely many unstable poles, a stable
suboptimal controller may be found by proper selection of the free parameter U(s).
In Section 5.2, this case is discussed.
Note that the previous case covers one and two block cases (i.e., W2 = 0 and
W2=0 respectively). When Fopt is strictly proper, then the optimal and suboptimal
controllers may have only finitely many unstable poles. Existence of stable suboptimal
H controllers and their design will be discussed in Section 5.3 for this case.
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5.2 Stable suboptimal H controllers, when the optimal con-troller has infinitely many unstable poles
The following lemma gives the necessary condition for a suboptimal controller to
have finitely many unstable poles.
Lemma 5.2.1. Assume that the optimal controller has infinitely many unstable poles
and U(s) is finite dimensional. Then the suboptimal controller has finitely many
unstable poles if and only if
lim |
F(j)LU(j)
| 1. (5.7)
Proof Assume that the suboptimal controller has infinitely many unstable poles,
then the equation
1 + eh(+j)M( + j)F( + j)LU( + j ) = 0
has infinitely many zeros in the right half plane, i.e. (recall Section 3.1.1) there exists
= o > 0 and for sufficiently large ,
1 + eh(o+j) lim
(F(o + j)LU(o + j )) = 0 (5.8)
will have infinitely many zeros. Since F and LU are finite dimensional,
lim
F(j ) = lim
F( + j )
lim
LU(j ) = lim
LU( + j)
> 0.
By using this fact, we can rewrite (5.8) as,
1 + eh(o+j) lim
(F(j)LU(j )) = 0 (5.9)
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which implies that in order to have infinitely many zeros, the condition in lemma
should be satisfied. Conversely, a similar idea can be used to show that (5.7) implies
finitely many unstable poles.
Note that this lemma is valid not only for only finite dimensional U(s) term, but
also for any U H, U 1 provided that
lim
U(j) = lim
U( + j) = u, > 0. (5.10)
is satisfied where u R. Also, we can find conditions on U which guarantees finitely
many unstable poles by using the lemma.
Assume that U(s) is finite dimensional and bi-proper, and define
f = lim
|F(j)| > 1,
u = lim
U(j ),
k = lim
L2(j)
L1(j).
Lemma 5.2.2. The suboptimal controller has finitely many unstable poles if and only
if the following inequalities hold:
|k| 1f
, |u| 1 f|k|f |k| (5.11)
when (n1 + l) is odd (even) and ku < 0, (ku > 0), and
|k| < 1, f|k| 1f |k| < |u| 0, (ku < 0).
Proof By using Lemma 5.2.1, when (n1 + l) is odd (even) and ku < 0, (ku > 0),
we can re-write (5.7) as
f|k| + |u|
1 + |k||u| 1.
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After algebraic manipulations and using f > 1, we can show that (5.11) satisfies
this condition. Similarly, when (n1 + l) is odd (even) and ku > 0, (ku < 0), (5.7)
is equivalent to
f |k| |u|1 |k||u| 1,
and (5.12) satisfies this condition.
Note that u is a design parameter and the range can be determined, by given f
and k.
Theorem 5.2.1. Assume that the optimal and central suboptimal controller (when
U = 0) has infinitely many unstable poles, if there exists U H, U < 1 suchthat L1U has no C+ zeros and |LU(j)F(j )| 1, [0, ), then the suboptimal
controller is stable.
Proof Assume that there exists U satisfying the conditions of the theorem. By
maximum modulus theorem,
|1 +ehsoM
(s
o)F
(s
o)L
U(s
o)|>
1 eh
|F
(j
)L
U(j
)|>
0,
therefore, there is no unstable zero, so = +j with > 0. Since, all imaginary axis
zeros are cancelled by E, the suboptimal controller has no unstable poles.
The theorem has two disadvantages. First, there is no information for calculation
of an appropriate parameter, U. Second, the inequality brings conservatism and
there may exist stable suboptimal controllers even when the condition is violated. It
is difficult to reveal the first problem, therefore it is better to use first order bi-proper
function for U. For the second problem, define max and max as,
max = max|LU(j)F(j)|=1
,
max = max[0,)
|LU(j)F(j )|.
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It is important to design max and max a