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University of Calgary
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Graduate Studies Legacy Theses
1997
Optimal design of multirate systems
Shu, Huang
Shu, H. (1997). Optimal design of multirate systems (Unpublished doctoral thesis). University of
Calgary, Calgary, AB. doi:10.11575/PRISM/22106
http://hdl.handle.net/1880/26848
doctoral thesis
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THE U ~ R S I T Y OF CALGARY
Optimal Design of Multirate Systems
by
Huang Shu
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDLES
M PARTIAL FULFILL-MEXT OF THE REQb?REMENTS FOR THE
DEGREE OF DOCTOR OF PEILOSOPHY
CALGARY, ALBERTA
APRIL, 1997
@ Huang Shu 1997
Acquisitions and Acquisitions et BiMi~graptrÈ Sewke~ - services bibliographiques
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Canada
ABSTRACT
Multirate -stems are o h used to achieve a cost advantage in implementation.
This thesis is devoted to both theory and application of optimal design of multirate
systerns-
In the theory part, we develop new algorithms for mdtirate optimal design mith
X2 and 31, criteria. In particular, we present state-space solu~ions to multirate X2-
optimal and suboptimal control and a solution to multirate ?dm-suboptimal control in
tems of a bilinear tansformation. Explicit formulas are given in terms of solutions of
Riccati equations and nest matrices are used to handle the causality constraint arising
h m multirate system structures. Compared to existing results in the frequency
domoin, the coctrollers obtained have the advantage of comput ational efficiency and
ease of implementation. Based on the new dgorithms, a software package is developed
in MATLAB for a general multirate 3d. design-
In the application part, we study via 31, optimization two different systems in
control and signal processing: a power system and a hybrid filter bank. For the
power system, we present a thorough study of power system stabilizers with four
types of 31, stabilizers designed and investigated, which include analog design based
asd discrete design based, single input and multi-input, and single-rate and multi-
rate. The multirate stabilizers are designed based on the new algorithm developed
and are superior to the single-rate ones in the sense that they require substantially
lower sampling rates. For d designs, we propose a systematic method of choosing
weight ing functions to meet certain operating requirements. The st abilizers designed
provide an implementation advantage - they have low complexity and require only
slow samphg rates - and outperform the conventional stabilizers on a series nonlinear
dqmamic tests. For the multirate filter bank. we present a direct design method for
the hybrid structure which consists of continuous- and discrete-time systems. Specifi-
c a synt hesis filters are designed to min-e the worst-case energy gain of the error 111
system, suitably weighted, between the hybrÎd filter b d and an ideal system. This
mdtirate hybrid design problem is converted into a single-rate discrete-the one of
optimization, which is then solved by the standard 36, design technique. An
example is discussed in detail to illustrate the design process. The filter bank design
also represents a new application of ?&, contrd theory
1 a m deeply indebted to my supervisor Dr. T. Chen for his tremendous support, en-
couragement, and motivation throughout the course of this research. I wodd dso lilce
to t h a d him for the extremely carefiil reading and correcting of Mnous manuscripts
related to this work and for ail the valuable suggestions.
1 wodd like to thanli Dr. O.P. Malik for his suggestions. .&O a special note of
thanks goes to Jian He and Shen Chen for several interesting discussions during the
course of this work. The test resdts for the power system in this thesis are based on
the power system simulation program developed by Jian He-
I am gratefid to my research coueagues, N. Rafee and A. Saadat-Mehr, for proof-
reading the manuscript of this t hesis.
1 would like to adcnowledge the financial support for this research, provided in part,
by the Natural Sciences and Research Council of Canada, made a d a b l e by Dr. T.
Chen. and by the Depaztment of Electrical and Cornputer Engineering through the
Graduate Research Scholarship.
Finally, I owe special thanks to m y d e Lin and my son Jimmyfor being a constant
source of encouragement, support and optimism throughout rny doctoral work.
To My Parents.
CONTENTS
- 0 APPROVAL PAGE. .... .... .... .. ..... .... . . - . - .-... .....,. ii
A B C . . . . - . . . . . . . . . - - . . . . . . . . . . . . . . . . . . - . . . . . . iü
DEDICATION.. . .. . . . . .. . - . ,.. - . . . . . . .. - . . . . . .. . . . . . . . .. . . M
TABLE OF CONTENTS.. . . . . . . . . . . . . . ., . . . . . . . . . . . .. . . . . - . . vii
LIST OF TABLES. . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES. . . . . . . . . . . , . - - . . . . . . . . . . . . - .. . . - . . . . . . . . x
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . .. . - - . . xiii
CHAPTERS
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . - . . . . . - . . . . . . . . . . . . . 1 1. t Single-Rate Sampled-Data Systems . . . . . . . . . . . . . . . . . . . 1 1.2 Yultirate Sampled-Data Systems . . . . . . . . . - . . . . . . . . . . 3 1.3 Properties of Multirate Controllers . . . . . . . . . . . . . . . . . . . 6 1.4 O u t h e of the Thesis . . - . . . . . . . . . . . . . - . . . . . . . . . . 9 1.5 Xotation . . . . . . . . . . . . . . . . . . . . . . . - . - . . . . . . . . 11
2. CONSTRAINED 3C2 CONTROL: A STATESPACE APPROACH 13 2.1 Causality and Nest Operator . . . . . . . . . . . . - . . . . . . . . . . I I 2.2 The Unconstrained Case . . . . . . . . . - . . . . . - . . . . . . . . - 16 2.3 Main Results: The Constrained Case . . . . . . - . . . . . . . . . . . 19 2.4 Proof of the Main Results . . . . - . . . . . . . . . . . - . . . . . - . 13 2.5 Conclusions . . . . . . . . . . , - - - . . . . . . . - . - . . . . . . . - 16
3. CONSTRAINED 31, CONTROL: AN ALTERNATIVE APPROACH 28
3.1 An EExting Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 A Solution Process via Bilinear Transformation . . . . . . . . . . . . 31 3.3 P r o o f o f I n v e r t i b i l i ~ ~ f D ~ ~ a n d D ~ ~ . . . . . . . . . . . . . . . . . . 33 3.4 A Design Procedure for klultirate 3d. Controllers . . . . . . . . . . . 36 3.5 Conc~usions . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . 41
4 . 31, DESIGN OF DIGITAL POIKER SYSTEM STABILIZERS ... 42 . . . . . . . . . . . . . . . . . . . . . . . 4.1 Motivation and Introduction 4'2
. . . . . . . . . . . . . . . 4.2 Linearized Plant and Weighting Functions 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Linearization 46
. . . . . . . . . . . . . . . . . . . . . . . 1.22 Weighting Functions 48 . . . . . . . . . . . . . . . . . . . 4.3 Design of Robust Digital Stabilizers 51
. . . . . . . . . . . . . . . . 4.3.1 Analog Design and Discretization 54 . . . . . . . . . . . . 1.3.2 DisueteTime Design of SIS0 Stabilizers 60 . . . . . . . . . . . 4.3.3 DisueteTime Design of MIS0 Stabilizers 66
. . . . . . . . . . . . . . 4.3.4 Multirate Design of FNSO Stabilizers 67 -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusions ta
..... 5 . 'fl. DESIGN OF HYBRLD MULTlRATE FILTER BANKS 77 . . . . . . . . . . . . . . . . . . . . . . . 5.1 MotivationandIntroduction 77
. . . . . . . . . . . . . 5.2 Conversion to a Pioblem of 'flm Optimization 81
. . . . . . . . . . . . . 5.2.1 Conversion to a Discrete-The Problem Si . . . . . . . . . . . . . . . . . . 5-22 Conversion to an ?&,, Problem 83
. . . . . . . . . . . . . . . . . . . . . . . 5.2.3 A Design Procedure 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Example 91
. . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Performance Limitation 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions 99
6 . ON CAUSALITY AND ANTICAUSALITY OF CASCADED LIN- ........................... EAR DISCRETETIME SYSTEMS -101
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Motivation 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Special Cases 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 General Case 106
............................... 7 . CONCLUDING REMARKS 110
............ A . P O m R SYSTEM MODEL AND PARAMETERS 121
LIST OF TABLES
. . . . . . . . . . . . . 4.1 Parameters for the lineaxized power system mode1 45
. . . . . . . . . . . . . . . . . . . . 4.2 Parameters for the tmed analog CPSS 52
. . . . . . . . . . . 4.3 Nonlineu simulation tests of the CPSS and the figures 53
. . . . . . . . . . . . . . . . . . . A.1 Parameters for the power system mode1 122
LIST OF FIGURES
1 2 LineanZed power system mode1 . . . . . . . . . . . . . . . . . . . . . . . . 5
. . . . . . . . . . . . . . . . . . . . . . . 1.3 The hybrid multirate filter bank 6
. . . . . . . . . . . . . . . . . . . . . . . 1.4 Singlerate discrete-time system 8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The lifted system 13
. . . . . . . . . . . . . . . . . . . 3.1 Standard multirate sampled-data setup 37
. . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mdtirate discrete-the systern 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The lifted LTI setup 38
. . . . . . . . . . . . . . 3.4 -4 design procedure for multirate & controuer 40
. . . . . . . . . . . . . . . . . . 4.1 A schematic diagram of the power system 43
. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linearized power system model 17
. . . . . . . 4.3 Recodgured power system with PSS and weighting functions 50
. . . . . . . . . . . . . . . . . . . 2.4 The IEEE standard CPSS configuration 52
4.5 Torque disturbance with normal load: CPSS (dot). analog stabilizer (solid) . . . . . . . . . . . . . . . reduced-order analog stabilizer (dash-dot) 54
4.6 Torque disturbance with light load: CPSS (dot). analog stabilizer (solid). . . . . . . . . . . . . . . . reduced-order analog stabilizer (dash-dot) 55
4.7 Torque disturbance with lead load: CPSS (dot). analog stabilizer (solid). . . . . . . . . . . . . . . . reduced-order andog stabilizer (dash-dot ) 55
4.8 Voltage disturbance with normal load: CPSS (dot). analog stabilizer (solid). . . . . . . . . . . . . . . . reduced-order analog stabilizer (dash-dot ) 56
4.9 Voltage disturbance with light load: CPSS (dot). analog stebilizer (solid). . . . . . . . . . . . . . . . reduced-order andog stabilizer (dash-dot ) 56
4.10 Voltage dist urbance with lead load: CPSS (dot). analog stabilizer (solid). . . . . . . . . . . . . . . . reduced-order analog st abilizer (dash-dot ) 57
4.11 Ground fadt test with normal load: CPSS (dot), analog stabilizer (solid), reduced-order analog stabilizer (dash-dot), discretized analog stabilizer
. . . . . . . . . . . . . . . . . . . with sampling period 20 ms (dash)
. . . . . . . . . . . . . . . . . . . . . 4.12 The singlerate digital control setup
. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 The discrete-the setup
4.14 Torque disturbance with normal load: CPSS (dot), digital SISO stabilizer . . . . . . . . . (solid), digital MISO singlerate stabilizer (dash-dot )
4.15 Torque disturbance with Eght load: CPSS (dot), digital SISO stabilizer . . . . . . . . . (solid), digital MISO single-rate stabiher (dash-dot)
4.16 Torque disturbance with lead load: CPSS (dot), digital SISO stabilizer . . . . . . . . . (solid): digital MISO single-rate stabilizer (dash-dot)
4.17 Voltage disturbance tvith n o m d load: CPSS (dot), digital SISO stabilizer . . . . . . . . . (solid), digital MISO singlerate stabilizer (dash-dot )
4.18 Voltage disturbance with light load: CPSS (dot): digital SISO stabilizer . . . . . . . . . (solid), digital MIS0 singlerate stabilizer (dash-dot)
4.19 Voltage disturbance with lead load: CPSS (dot), digital SISO stabilizer . . . . . . . . . (solid), digital MISO single-rate stabilizer (dash-dot)
1.20 Ground fadt test with normal load: CPSS (dot), digital SISO stabilizer . . . . . . . . . (solid) , digital MIS0 singlerate stabilizer (dash-dot )
4.21 Magnitude fiequency responses from Kef to p (solid) and w (dot); p and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w a x e b o t h i n p . ~ .
. . . . . . . . . . . . . . . . . . . . . 4.22 The multirate digital control setup
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.23 The lifted LTI setup
4.24 Torque disturbance with normal load: CPSS (dot), digital MISO multirate stabilizer with sampling period 120 ms for w (solid), and digital MISO multirate stabilizer with sarnphg period 180 ms for w (dash-dot) . .
4.25 Torque disturbance with light load: CPSS (dot), digital MISO multirate stabilizer with sampling period 120 ms for w (solid), and digital MISO multirate stabilizer 6 t h sampling period 180 ms for w (dash-dot) . .
4.26 Torque disturbance with lead load: CPSS (dot), digital MISO multirate stabilizer with sampling period 120 ms for w (solid), and digital MISO multirate stabilizer with sampling period 180 rns for w (dash-dot) . .
xi
4.27 Voltage disturbance with normal load: CPSS (dot), digital MISO multi- rate stabilizer with sarnphg period 120 ms for w (solid), and digital MISO multirate stabiüzer with samphg perÏod 180 ms for w (dash-dot) 73
4.28 Voltage disturbance with Iight load: CPSS (dot), digital MISO d t i r a t e stabilizer with sampling period 120 ms for w (solid), and digital MIS0 multirate stabilizer 6 t h sampling period 180 ms for w (dash-dot) . .
4.29 Voltage disturbance with lead load: CPSS (dot), digital MIS0 multirate stabilizer with samphg period 120 ms for w (solid), and digital MISO mdtirate stabitizer with sampling period 180 ms for u (dash-dot) . .
4.30 Ground fault test with normal load: CPSS (dot), digital MISO multirate stabilizer with sampling period 120 ms for w (solid), and digital MIS0 multirate stabilizer with sampling period 180 ms for w (dash-dot ) . . 71
-- 5.1 The hybrid multirate filter bank. . . . . . . . . . . . . . . . . . . . . . . i i
5.2 The pre-filtered error system. . . . . . . . . . . . . . . . . . . . . . . . . 78
3.3 The equident discrete-time error system. . . . . . . . . . . . . . . . . . 85
5.4 Final equivolent system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 1 ~ 1 (solid), IG~:,I (dot), and /G1l (dash) in dB versus w. . . . . . . . . . . 92
5.7 1 Fol (solid) and I& 1 (dash) in dB versus w / r . . . . . . . . . . . . . . . . 94
3.8 Steady-state error (solid) and desired output (dash) versus k. . . . . . . 95
5.9 Unit step response $ versus k. . . . . . . . . . . . . . . . . . . . . . . . . 96
5-10 Impulse responses verms k for Fo (upper) and FI (lower). . . . . . . . . 97
6.1 The cascade of two systems . . . . . . . . . . . . . . . . . . . . . . . . . 10 1
xii
LIST OF SYMBOLS
superscript L
G(z) -
G ( s ) -
II-II
Hardy space
ditto
Hilbert space over the set of real numbers
Hilbert space over the set of integers
nest space
ditto
class of nest operators fiom y,- to Ur
open unit disk
unit circle
red-rational
orthogonal complement
G( l/z)'
G(-s)'
Euclidean nom
2-norm on &, 12: or ?&
-nom on 3C,
transpose of matrix -4
cornplex-conjugate transpose of matrix A
orthogonal complement of matrix A
trader matrix of system G
sampling operator
hold operator
sampler with period h
zereorder hold with perïod h
maximum singular value . .. Xlll
inf idmilm
SUI? supremum
S direct s u m
J direct subt raction
Lm image
+ Redheffer star product
( ) linear fiactional transformation
xiv
INTRODUCTION
In this chapter, we shall motivate our study of single- and multirate sampled-data
control systerns, present an o u t h e of the thesis and introduce some notation.
1.1 Single-Rate Sampled-Data S ystems
Digit al technology has brought dramatic change in process control instrumentation
and control system strategy. A digital control system, sometimes c d e d a sampled-
data control system, is a feedback system which consists of four basic components: a
plant, a analog-tedigital converter, a data-processing unit or digital controllerl and a
digit al-to-analog converter. The control loop samples and quant izes cont inuous-t ime
signds from the plant (process), processes the digital signals via a control algorithm.
and converts the renilting digital signals into continuous-time ones to regdate the
plant.
The schematic sampled-data control diagram is given in Figure 1.1, where the
four basic components are illustrateci by G (plant), Kd (digital controller), S (A/D
convener) and 31 (DIA converter). This system is hy6Rd in that it involves both
continuous- time and discrete-time signais. G is a continuous-tirne linear t i r n e - ~ ~ ~ a n t
(LTI) plant, which has two analog inputs, the exogenous input w and the control in-
put u, and two analog outputs, the regulated output t and the measured output
y. £Cd is a digital controller, which receives the measured sequence yd and generates
a control sequence ud. S is a sampler: It converts the continuous-the y into the
discrete-time yd by samphg y. 31 is a hold: It converts the discretetime ud into
the continuous-the u by holding ud between each two samples. In this setup? all
the signals involved are possibly vector-valued. To iimprove the clarity: we have used
-7 L
continuous lines to represent continuous signais and dotted Iines to represent disnete
signals. This convention will be kept throughout the thesis.
Figure 1.1. Standard sampled-data setup
A/D and D/A conversions involve sophisticated technologies. The above sampler
and hold are idealized models which ignore quantization errors. The simplest A/D
and D/A scheme is to do alI the conversions at different channels symhronously with
the same period. Such systems are cded singlerate systems. .More precisely. a
single-rate scheme is the hllowing:
0 S = Sh: an ided sampler with a k e d period h, defined via
O 3L = HA, a zereorder hold with the period h: defined via
& is updated every h seconds, synchronized with Sh and Hh.
Due to the hybrid nature, the sampled-data -tem is harder to deal with than ei-
ther a purely continuous-t ime systern or a pure- discrete-time system. The sampled-
:3
data system may be designed based on specifkations given in continuous tirne, discrete
time, or even both.
Thzre are in gened three methods to design digital controllers for sampled-data
systems. The h t is to do an andog design and then a discretization. This is the
simplest method because the analog specifications are natural and familiar. However,
in order to recover the analog pedormance specifications, usually one must use fast
samplers and computw. This introduces a trade-off between performance and hard-
m e cost. The second is to discretize the plant mode1 and then do a discrete-time
design. This is dso a simple method: If the original continuous plant is LTI, the cor-
responding discretized plant is also LTI. Also, since the sampling rate is incorporated
into the design process, one can attempt design with a slow sampling rate, thereby
reducing implementation cost. In Chapter 1, we will make a concrete cornparison
between these two methods for a power system stabilizer design. The limitation of
the second method is that poor intersample ripple may occur because the method
deds only with information at sampling instants. The third approach to design dig-
ital controllers is by a direct design in continuous tirne. This would be a preferred
approach because it solves the problem with no approximation. However, since a
sampled-data system is hybrid and tirne-varyhg, the method is mathematicdy less
tract able.
Single-rate sampled-data systems have b e n widely studied, as reflected in many
books, see, e.g., Kuo [Ml, Astr6m and Wittenmark (61: Franklin et al- [29] y and Chen
and Francis [17].
1.2 Multirate Sampled-Data Systems
In multi-input: multi-output ( M M O ) systems, it is sometimes advantageous to
use different rates at different sampling and hold chonnels. The resultant systems are
referred to as m u k a t e systems.
Muitirate sptems are abundant in indust . Examples indude aerospace control
systems [a], power systems [10], robotic systems [48], chernical processes [77], and
digital Nter b& [72]. One reason to use mnltirate controllen is to achieve a cost
adçantage in implementation. In this thesis, we s h d study two different multirate
systems in control and signal processing.
The first system is a multirate stabüizer for a singlemachine infinite-bus power
system. The power system is in general a nonlineax system and consists of a s p -
chronous generator, a govemor, an exciter, an automatic voltage regulator (-liR)o
and transmission lines. The role of the power system stabilizer (PSS) is to provide ad-
ditional damping to low fiequency disturbances bj- measuring rotor speed and povver
and generating a voltage control signal which is fed back to the AVR and exciter.
By lineoripng the nonlinear model at a certain operating condition, one can obtain
a iomh order model of Figure 1.2, where Ap, Aw, 46, and Au, are the deviations
in power, rotor speed, power angle, and terminal voltage, respectively, AI& is a
reference voltage distur bance: and AT, is a mechanical torque disturbance. Gsing
the machine pasameters (for details, see Chapter 4): it may be found that the tram-
fer function from A L f to 4 p has a wider band of significant frequency components
than the transfer function fiom to Au. Hence, instead of using a single fast
sampling rate, one can sample the power signal fast and the speed signal slowly, hence
reducing the implementation cost.
To convert the powet system in Figure 1.2 to the standard setup in Figure 1.1, we
define (for convenience, we &op d the prefix A)
and r to be a vector of any group of LY., p and 6. Then Figure 1.2 is reconfigured
into Figure 1.1. In mdtirate sampling schemes, the sampler S and the hold 7f have
different rates. In this case, if we take the sampling periods for the power and for
Figure 1.2. Linearized power system model.
the speed to be h and 2ht respectively, and the period for the hold to be h, then the
multirate sampler corresponds to two different rate samplers
and the single-input hold reduces to the single-rate one: H = HA.
The second multirate system to be studied in the thesis is a hybrid filter bask
which find applications in subband coding [72] and fast A/D conversion [53]. Shown
in Figure 1.3 is the 2-channel filter bank, which includes LTI analog analysis filters Go
and Gl, LTI digital synthesis fdters Fo and Fi, two slow samplers (Sah) with period
9h, and two expanders (E) with the factor of 2. The system is typically used to
process bandlimited analog signals, z(t ) , with Nyquist fkequency l/ h. Usually~ Go is
lowpass and G1 highpass. Fo and FI axe designed to achieve certain reconstmction
performance characteristics. Because slow samplers are used, such filter b d sys-
tems can achieve overall bit-rate reduction in tansmitting analog signals [Tl] : or- if
designed properly, can approximate the fast A/D converter with period h 1531.
Figure 1.3- The hybrïd multirate filter bank.
1.3 Properties of Multirate Controllers
In general, multirate controllers are more complex t han single-rate controllers. The
desired multirate controllers should satisfy three proper5es: periodicity, causalit- and
finite dimensionality [19].
Periodicity is a direct generalization of time invariance from single-rate syst erns
to muhirate systems. As an example, consider the multirate power system stabilizer
with the sampling and hold strategy taken in the preceding section. It is easily
seen that the least common period for the two sampiing charnels and the one hold
channe1 îs T = 2h- To keep the overd system to be T-periodic, the two-input and
one-output stabilizer must be T-periodic. This idea can be extended to the general
MIMO controller case: Let T be the le& common period for all sampling and hold
channels; The overd system is T-periodic 8 the controller fi is T-periodic in real
tirne. It can be checked that to keep the periodicity for all sampling and hold channeisl
the skmpling and hold rates must be rationdy related (i.e., the ratio of any two rates
is rational) [19].
Causality and finite dimensionality are requïred for Kd to be implementable in
real time on microprocessors with finite memory. With finite dimensionaüty, Kd can
be described by state-space difierence equations. For the multirate power system
stabilizer, we have the following equations for fi:
(Xote that we have used continuous s isals evaluated at respective sampling instants
to represent the corresponding discrete signais, because two sampling rates CO-exist.)
For causa&- of the mdtirate model, we must have a3 = 0.
Equations in (1.1-1.3) can be used for implementing the mdtirate stabilizer on
computers, the state vector z being updated every T seconds. This controuer &
is two-input, one-output, but it is ad~ontageous to view the model in (1.1-1.3) as a
three-input? t m u t p u t LTI system:
This corresponds to grouping the inputs and outputs properly over every period T.
Then i t follows from (1 .LU) that & has a statespace model given by
This three-input: two-output controller is almost like a standard MIMO. LTI discrete-
time system except for the causality constraint = O, arising from the multirate
structure. The method of obtaining such equident LTI models is formally referred
to as the lifting technique [38,44, 191.
In general, the multirate problem of Figure 1.1 can be recast using the lifting tech-
nique as a discrete-t ime singlerate problem involving only LTI systems in Figure 1.4,
where & is an LTI plant and & is an LTI controller to be designed; mr a and
y are respectively the lifted exogenous input, control input, regdated output. and d
measured output. The lifted sigoals a and usually have higher dimensions than ud
and yd? respectively, and yd is nrbject to a causdity constraint: Its direct feedthrough
Figure 1 -4. Singlerate discret e-t ime system
term is a block Iower triangular matrix [44, 59, 74, 191. Multirate controller design
thus is reduced to constrained LTI controller design.
'ilultirate control problems have received much attention in the last four decades-
Recent studies inchde the parametrization of all stabilizing controllers [a' 591, LQG/
LQR designs [2: 45, 31: X2/31, designs [?5? 54, 74, 19, 551. Meyer [44] and Ravi et
al. [SI presented a controller parametrization method for multirate systems which is
the direct extension of the Youla parametrization (se, e.g.? [27]). This method was
used by [Gy 54: 74, 191 to reduce the multirate R2 and 31, control problems into
the constrained R2 and 3C, model rnatching problems in the fiequencq- domain: In
[4 Meyer noted that the causality constraint is a convex one and hence a numericd
technique based on convex optimization [11] was proposed for the multirate LQG
problem; Voulgôris et al. [74] proposed a projection method for the constrained R2
and 3C, p r o b l v Chen and Qiu [19] developed an effective framework based on
nest operators to handle the causality constraint and gave an explicit solution to the
9
constrained R, model-matching problems using factorizations associated Nith nest
operators; this framework was also used in 1551 to obtain a characterization of aJ1
suboptimal 36. controllers.
1.4 Outline of the Thesis
ki the next two chapters, we study more computationally efficient methods for
both constrained 7i2 and 31, control problems.
In Chapter 2, a statespace approach to the constrained optimal and suboptimal
3t2 control problems is developed and explicit formulas for controllers are given in
terms of solutions of two Riccati equations. The solution process takes advantage of
nest operators to handle the causality constraint [19] and involves a series of modi-
fications of the standard Youla parametrization. The resdts include the solution to
the standard, unconstrained 3CÎ problem as a special case.
In Chapter 3, a different approach to obtaining constained 31, controllers is pre-
sented. More specifically, it is shown that the bilinear transformation can be used to
convert the constrained discrete-time Ra problem into a continuous-t ime R, pîob-
lem. As a resdt , a constrained discretetime 360 solution can be obtained in tenns
of the standard, unconstrained continuous-time & solution and the methodology
proposed by Chen and Qiu [19] to handle the causality constraint. Based on this
solution, we can take advantage of existing sohaxe, e.g., MATLAB, and modify
amilable ut ility functions to solve multirate 'tl, problems.
Chapter 4 includes a case study for Chapter 3 - a multkate 'fl, power system
stabilizer for a single-machine, infinite-bus power system. Since 31, design of power
system stabilizers has received an increased attention in recent years [JI, 5: 14, f 1: 11,
we present a comprehensive study with four ?Lw stabilizers designed and investigated:
(1) an analog SISO stabilizer measuring the speed signal and then irnplemented digi-
tally: (2) a SISO digital stabilizer designed in discrete time: (3) a rnultivariable digital
10
stabilizer measuring both speed and power signds with a single samphg rate: (4)
a multirate digital stabilizer involving two samphg rates and designed using the
method in Chapter 3. Our designs give rise to controllers which have a relativel- low
implementation cost; in partidar, aIl stabilizers designed have low ordeq require
considerably slow samphg rates, and outperform the conventional (analog) power
system stabilizer on a series of nonlineax dynamic performance tests. The impor-
tance of weighting finctions is stressed in the design process.
Chapter -5 studies the multirate filter bank shown in Figure 1.1 using 'H, opti-
mization. Since the system does not f a in the standard sampled-data setup, there
is no method a d a b l e to carry out the design directly. We take the viewpoint that
the analysis filters have already been designed and -thesis filters are then to be de-
signed to minimize the worst-case energy gain of the error system, suitab- weighted.
between the filter bank and an ideal system - a fast A/D converter with some time
d e l - By using the lifting techniques. this hybrid and multirate problem is reduced
to one of 'fl, optimization involving only LTI, discrete-time systems, which is then
solvable. An example is diswsed and studied in detail to illustrate aspects in the
design process. In contrast to digital filter banks, performance limitations exist in
this hybrid filter bank.
Chapter 6 treats a fundamental problem motivated by both the filter bank design
md X2 and N, control: Uihen two finite-dimensional iinear time-invariant (FDLTI)
discrete-the systerns, causal or noncausal, are cascaded, under what conditions is the
cascaded system causal or anticausal? Wë answer this question in general. For the
special cases when one system is causal and the other is anticausal: we give necessary
and sufficient conditions. The results can be used to ver* whether a filter bank is
causal or not if the synthesis filters are chosen to be noncausal.
Finally, Chapter 7 offers some concluding remarks dong Nith some suggestions for
1.5 Notation
The notation used throughout the thesis is quite standard.
In discrete time, the frequency-domain spaces 3L2 and 31, are the Hardy spaces
of mat&-vdued functions analytic outside the unit disk, Le., Ir1 > 1, with noms
defined as, respectively,
11611, = SUP W ~ m a z [ & e i ~ ) l ,
where * means the cornplex-conjugate transpose and cm, the maximum singular
value. Prefkv 12 means real-rational; hence RU2 and RR, are the real-rational
subspaces of Xz and respectively. 3Cz can be regarded as a subspace in the
Lebesgue space L2 dehed on the unit circle Ir1 = i. The orthogonal complement of
3L2 in L2 is denoted R$ with RN& its rd-rational subspace.
Similar notation is used in continuous time for the fkequency-domain spaces. For
example, iH, is the Hardy space defhed on open right-half plane Re(s) > O with the
ZR, is the real-rationd subspace of X,. The context will prevent confusion.
To represent signals, L2 denotes the Hilbert space of squareintegrable continuous-
time signals, perhaps vector-valuedt defined over the time set of a l l real numbers; the
inner product and corresponding nom on L2 are
where ' denotes transpose and the nom on t ( t ) is the Euclidean one. Similady. &
denotes the Hilbert space of square-summable discrete-time signals, perhaps vector-
valued, defined over the time set of aU integers with the b e r product and nom:
For an LTI system G, either in continuous or discrete t h e , its trânsfer matrix is
denoted G. For a state-space realization (A, B, D) of an LTI system, we use the
packed notation
to represent its transfer matrix D + C ( s I - .4)-lB in continuous tirne or D + C(r I -
A)-'B in discrete tirne. For a discrete transfer rnatrix &), ~ ( z ) " is defined to
be G ( I / P ) ' , which corresponds to the adjoint system of G; for a continuous t r a d e r
rnatrix ~ ( s ) , the correspondhg operation G(s)- is G(-s)'.
FinaUy, given an operator K and *O operator matrices
the linear fractionai transformation associated with P and is denoted
and the Redheffer star product [60] of P and Q is
Here: we assume that the domains and co-domains of the operators are compatible
and the inverse exists. With these defmitions, we have
CHAPTER 2
CONSTRAINED 3C2 CONTROL: A STATESPACE APPROACH
As noted in the preceding chapter, a multirate control probiem can be converted
into a singlerate control problem involvïng only LTI discrete-time systems. However,
the LTI controller to be designed is subject to a causality constraint on its direct
feedthrough term. In this chapter, we study design of the constrained controller with
an X2 optimali- criterion'.
With respect to the dismete-tÏme setup in Figure 2.1, the constrôined 3t2 problem
can be stated as follows:
Given G. find a contrder K which stabilizes G, satisfies a causality constraint.
and further minimizes the ?& norm f!rom w to z.
In Section 2.1, we will associate the causa&@ condition to nest operaton.
Figure 2.1. The lifted system
There are several ways for such a constrained 3L2 problem to arise:
'The r d t s in th5 chapter have been published in [6j] -
14
In discrete-time periodic systems, a common technique to convert a periodic
design problem into an LTZ one is lifting [38]. However, causaIity in periodic
control systems puts a condition on the direct feedthrough terms of the lifted
systems: They must be (block) lower-triangular. R2 or LQG periodic control
leads naturally to the constained problem [74].
O In discretetime multirate systems, again one uses lifting to get LTI systems
[@, 591; but the LiRed controllers s a t i e a similm causality constraint: Certain
blocks in the feedthrough terms must be zero. An LQG multirate control design
gives rise to the constrained 3Lz problem [45].
In X2-optimal control of sampled-data multirate systems, by a certain reduction
process [75,64] based on a continuous lifting technique: see, e-g., (81, one arrives
at the same discrete-time constrained X2 problem.
Recent studies on the constrained X2 problem are ail in the fiequency domain,
based on the parametrization of d stabilizing controllers [44: 591 to reduce the prob-
lem into a constrained 7i2 model-matching problem [45? 54: 14. 191. In this chapter,
we present the explkit statespace realizations for the optimal and suboptimal 3C2
controllers satisfying the causrtlity constraint.
2.1 Causality and Nest Operator
In this section we s h d characterize the causality condition on by nest operators
as in [NI. Let u = Ky. If the direct feedthrough term in EL is DA-: we have u(0) =
Dr; ~(0). Arising from lifting, u(0) and y (O) are vectors of the form
The elements in the two vectors may occur at different time instants in a real-time
system, sas yi(0) at tirne ti and uj(0) at hj . Then the causality condition means that
15
the output ~ ~ ( 0 ) cm only depend OR inputs occurring up to hj-
Let C be the set of t h e instants ti and hi; Iet n + 1 be the number of elements in
this set (not counting repetitîons); order E increasingly (cr < or+&
c ={O, : r=0,1, ..., n).
For r = 0,1, ...' n, define
y, and Ur correspond to, respectively, the inputs and outputs occurred after and
inciuding time or during the Iarger period associated with the lifting. It follows that
{y,) and {Ur) are nests of subspaces s a t i e n g
and that the causaliq condition on K means that DK is a nest operator.
The set of d such operators is wrïtten N({Y,), (U,)) and abbreiiated N((yr)) if
{Yr) = {U,). In the foflowing lemmas, the spaces involved are ail finite-dimemional.
Lernma 2.2 Let D be an opemtor on Y .
16
(a) There ezist a unitary operator Ul on Y and an operator & in N({Y,)) svch
thut D = UIRl-
(6) There ezrst a unitary opemtor U2 on y and an operator R2 in N({y,)) such
that D = R2&.
Lernma 2.3 Let D be a nonnegatiue Hennitian opemtor on Y. Then there ezist
operators Ri, R2 E N({yr) ) sach that D = R; Ri = R&.
The results in the lemmas f011ow easily fiom matrix theory by noting the relation-
ship between nest operators and (block) lower-triangular matrices: For an operator
D mapping Y to 24, if we decompose the spaces Y and U in the following way
where 3 and @ denote the direct subtraction and direct sum, respectively, then the
associated matrix representation of D is
D E JV({y,)z {U,)) means that this mat* representation is (block) lower triangular,
i-e., DG = O if i > j-
With these, the constrained 7i1 problem is as follows: Given G: design a stabilizing
K sati-g &(oc) E N({Yr), {U,)) to minimize ~~T'llz.
2.2 The Unconstrained Case
The solution to the constrained ?12 problem will relate to that of the standard?
unconstrained X2 problem.
Uk start with B date-space model for G:
The standing assumptions for the unconstrained N2 problem me given as follows:
(i) (-4, Bz) is stabilizable and (Cz, A) is detectable;
(ii) M := Df,Di2 and :Y := are nonsinylar;
(iii) For any r on the unit circle 3Zi: the matrkes
have fdl column and row d, respectivek
These assumptions are standard in 'Hz and 3L, literature [3'7]. Assumption (i) is
n e c e s s q and sufficient for the existence of stabilizing controllers. Based on this
wumption, there evist a state feedback matrLz F and an output injedion matri2 L
such that A + B2 F and A + CCz are stable. The optimal ?L2 solution is obtained
b - suitab- specifyïng F and L (see below). Assumption (ü) means that the control
and sensor weightings must be nonsingular matrices, which implicitly assumes that
the system must have at least as many outputs to be controlled as control inputs and
at l e s t as many =ogenous inputs as measured outputs. This assumption insures
that the 3L2 problem is nonsingular. Findy, 'flroptimal controller c m be obtained
by solving two Riccati equations, whose stabilizing solutions ore guaranteed to be
unique by assump tions (i)- (iii) . Let -4, QI R be real n x n matrices with Q and R symmetric. The solutions of the
algebraic Riccati equation
is known [32] to relate to the generahed eigenspaces of the sympiectic pair
If H has no generalized eigenvalues on ûV and the two subspaces
are complementary, where x(H) denotes the eigenspace of H corresponding to gen-
eralized eigenvalues inside D, then &(H) defmes uniquely a stabilizing solution of
(2.2): denoted Ric(X); see, e.g.? [3Tl 171.
Now the ?&-optimal controuer Kwt can be given as follows. Introduce the sym-
plectic pairs & and Sy and the matrices F, Fo, Lt Lo:
Then h,t is given by the following state-space mode1 [16]
It cm be verified that A+ B2F and A+CC2 are stable and the closed-loop tmnsfer
The optimal solution can be obtained as follows: Solve the following two Riccati
equations for X and Y, respectively:
and then compte Fo7 L, Lo to get K , . Note that &&o) = Lo7 which does not
belong to N({yr) , {Ur)) in general. Hence the optimal controller for the constrained
problem is tqpically different from this Kqt.
2.3 Main Results: The Constrained Case
In this section, we s h d develop complete state-space solutions to the constrained
R2 control problems based on the three assumptions in the preceding section.
Introduce the same matrices: XI Y: F, Fo, L, Lo, as in Section 2.2. Since A + B2F and A + LC2 are stable: the set of intemallq- stabilizing controllers for G can be
characterized by the following Youla parametrization foxmula [ ~ f ] :
To sirnpliS; rnatters to follow2 we modify this parametnzation as follows: With the
mat& Lo defined in Section 2.2, replace the parameter Q by Lo + Q and then absorb
Lo into &; this leads to the following characterization of stabilizing controllers
where
Such controllers May not sati* the causality constraint? which will be considered at
a later stage.
Uiith this controller parametrization, the closed-loop system T' can be given in
terms of Q:
where f is the Redheffer star product G * j [60], whose state-spâce mode1 can be
computed h m the state-space models of G and j:
It follows that
with
and
T' =,+T~QT~~ Q €ER2-
Further examining fi, we obtain &er some computation
Tl = TF + P2fL,
where
(The matrix Fo is defined in Section 2.2.) Hence, our constrained 3Cz control problem
becomes: Find a Q E RN2 wïth
to minimize the 3L2 nom of
Xow we consider some causality issues raised in Section 2.1. Introduce factoriza-
tions for the two positive definite matrices R and S by Lemma 2.3,
s = SIS;,
where Ri and Si are invertible nest matrices satisfying Ri E rV({Ur)) and SI E
,Ve({X}). Xote that the matrix RIL& maps y to U.
Recall that hp({yr}, {Ur)) is a subspace in the space of matrices mapping Y to Li:
the latter can be regarded as a Hilbert space with the 2-nom (compatible with the
3L2 nom for transfer matrices). Let NL({J&), {Ur)) be the orthogonal complement
of n i ( {X) . (Ur)). Decompose
RILoSl = W + WL
FVe are now set up to state the main result of this chapter.
Theorem 2.1 The optimal controller solaing the constrained 3L2 problem is giuen by
Moreover, the minimum cost is giuen by
The proof will be given in the next section. From the theorem, we note that
Km in (2.7) satides the causaity condition: By the facts that RI E N({U.)) and
S1 E M((yr)) and Lemma 2-1, V E M({X) , {U,)) iff RIVSI E N ( { X ) , {U,));
but RIVSl = W E N({Y,), {Ur)). -41~0, in the expression of & in (2.8): the
hs t two terms correspond to the unconstrahed problem; and hence the last tenn,
Il WLll?. is due to the causality constraint. Based on Theorem 2.1, rhe computation of
the constrained R2 solution just reg& a simple modification to the unconstrained
solution in (2.3).
In control system design nith multiple objectives [39], ofken one is interested in
suboptimal controllers. The proof of Theorem 2.1 yields easily a characterization of
all suboptimal 3C2 controllers. For completeness, this result is stated below.
Theorem 2.2 Let 7,t be given as in Theotern 2.1. Assume y > yWt. Then the set of
al1 stabilicing controllers which satisfy the causality constraint and achieve 11f"112 < Y
is giuen by
2.4 Proof of the Main Results
The proof of Theorem 2.1 requires the fouowing lemma.
Lemma 2.4 With the matrices R and S used in Sections 2.2 and 2.3, we have
(a) f ; f F E R3ck and f;P2 = R;
(6) E and = S-
Proof We s h d prove part (a); part (b) follows similady. Define
AF = A+B2F,
Gr = Ci+Dl2F,
BIF = BI + &Foy
D l l ~ = Dll + Dl2F0-
The matrix X satisfies the Kccati equation in (2.5) associated with the s!mplectic
pair Sx , and hence satisfies the equivalent L yapunou equation (after some algebra)
This equation admits a solution in the
00
.Usq it can be vedîed that the following identity holds:
24
To see E R;H:kl write in te- of povser series of r:
Each Ek is an infinite sum; however, using the infinite sum solution (2.10): we can
simpw Ec for k 2 O as follows:
Then it follows h m the definitions of Dllr, BIF and Fo that
and h m (2.11) that Ek = O for k > 1. Hence, pcfF E RRi. Similar ilrgument dl
give TTT2 = R. If
Sow we proceed to prove Theorem 2.1.
Proof of Theorem 2.1 We staxt with T, in (-2.6): namely,
Fm = FF + T;(TL + QT;).
Since FcZF E R3c: [Lemma 2.4 (a)], the two tems on the right are orthogonal in
X2 and hence
the last equality following from the fact that PTT; = Ri Ri. Similady, by Lemma 2.4
(b): we have
II~JI: = [IF& + I I R ~ ~ ~ I I ; + IIR~QS~II;.
To simplî@ further, redefine R ~ Q s ~ as Q. (Note that R ~ Q S ~ E 7E31* iff Q E RR2
because RI and Si are invertible.) This corresponds to modifying the controller
parametrization into
w here
and then
I I~~UI I ; = I I ~ F I I ~ + ~IR~TLI I~ + 1 1 ~ 1 1 2 - (?.la)
Without the causality constra.int, the optimal Q is obviously zero. With the causal-
ity constraint, we must have
This is the sarne as
by Lemma 2.1. For (2.14) to be tme, ~ ( o o ) must completely cancel WL, the corn-
ponent of RILoSt in NL({X), {&)); Le.: Q must be of the folloming orthogonal
decomposit ion
Substitute (2.13) into (2.12) and absorb -WL into .fo to get a parametrization of
controllers sati&ing the causaliw constraint :
Some calculation and the definition of V show
Also, substituting (2.15) into (2.13), we have
since WL and are orthogonal in 7f2.
'Yow the results follow easily: The optimality is achieved when = O: put this
into (2.16) to get kopt as in (2.7)- c
2.5 Conclusions
In this chapter, we have obtained the complete state-space solutions to the optimal
asd suboptimal R2 control problems mbject to a causality constra.int . The solutions
require no additional assumptions: The constrained % problem is solvable iff the
standard: unconstrained 'flz problem is solvable. Compared to the existing solutions
in the fiequency domain [74? 19: 451 the results have the advantage of cornputational
efficiencg and ease of implementation on cornputers. Li fact , the algorit hms developed
in this chapter have been implemented in a MATLAB multirate control software which
is cmently under development [56].
For ease of derivation, we have assumed D22 = O to obtain the constrained X2-
optimal controuer. This results in no loss of generality. In fact, if Dn # O: typically
due to the lifting procedure [44, 191, thus it is an easy matter to veri- that the
optimal contrder l?lnnu,wz for Dn # O is [76]
S i
where h',, is the optimal controk for the case taking Dn = O and is given in (2.7).
'iote that hrnwrwt satisfis the causality constraint by Lemma 2.1:
The results in this chapter are usefid in studying multirate control systems: A case
study was given in [56] for design of multirate 3C2 power system stabilizers; another
application to optimal multirate discretization of analog controllers may be found in
[ s i ] - Fina&, we mention that the results in this chapter can be extended directly to
the case involving a multiple objective 7tz measure in the sense of Pareto optimality
dong the lines of [39] and the techniques in this chapter to hande the causality.
CHAPTER 3
CONSTRAINED 3t, CONTROL: AN ALTERNATIVE APPROACH
Since a bilinear transformation of a system p r e s e m its 36, nomst one ofien con-
verts discrete R, controuer design into a continuous one via the bilinear transforma-
tion. The reason is that continuous %, design requires considerably less computation
(compare [33] for the continuous-time 3C, solution with [37] for the discrete-time 31,
solution). For the Rw design with a causality constraint on the controller feedthrough
terms, solutions have been given numerically via convex optimization by Voulgaris et
al. [71] and explicitly by nest operators by Chen and Qiu [19, 5S]. Hence the follow-
ing question arises: Can we do the constrained disaete-time 31= design also by the
bilineôr transformation? This chapter attacks this probIem if o d y one suboptimd so-
lution is sought. More spe~ifically~ we discuss how to use the bilinear transformation
toget her with the methodology proposed by Chen and Qiu [19] to obtain a constrained
36. solution. The resdts will be used for designing multirate power system stabilizers
- a topic studied in the next chapter.
In viem of Fieme 2.1, the constrained ?fw suboptimal control problem c m be
stated as follows:
Giwn G and 7 > 0, h d ô stabilizing K with ~ ( m ) E ,V({Y,). (24,)) to achieve
llFmll, < 7: if such K exists at d.
In what follows, we take y = 1.
Similar to the constrained 3C2 probIem, the constrained 7& problem may arise
in N, designs of discrete-time periodic systems, discrete-time multirate systems and
sampled-data multirate systems. For example, to design a sampled-data multirate
system via the 3L, approach, one can reduce the design probkm into a single rate LTI
problem of 31, optimizatioc via continuous and discrete lifting techniques [X, 191.
The liftings put the causality constraint on the direct feedthrough terms of the Iïfted
LTI controllers-
3.1 An Existing Solution
Uë first summarize the solution process in [19], which is given in the fiequency
domain. -4 key step there is the parametrization of alI 31, suboptimd controllers in
the unconstrained case. Since such a result is adab le using the state-space methods
[37]. we s h d proceed further from here. The state-space mode1 of the plant is assumed
to be
G ( Z ) = [y-+] - (72 D21
W e have assumed that D22 = O. The extension to the case with Dm # O follows dong
the lines of discussion in Section 2.5.
Let us first drop the causality constraint and find a characterization of a l l stabi-
lizing controllers satisfiing 11 f, 11, < 1, which is given in [3?]:
where
with D~~ and ~~1 being square and invertible. We refei to [37] for details of diecking
the expression of It follows that &oc) depends on ~ ( m ) in an affine way:
In order to get k(m) E Aœ((y,),{Ur)), &oc) must be specified. Since l l ~ l l ~ ~ 2
11 &(m) 11 the equident problem is to find a constant matrix oc) xith 11 Q(W) 11 c i such that k (m) in (3.3) belongs to N({yr), {Ur)).
B y Lemma 2.2? introduce rnatrix factorizations (QR factorizat ions)
where Ri, R2: Ui : & are aU invertible, U;, U2 orthogonal, and R1 E N({U,}), R2 E
,V({Yr}). Substitute the factorizations into (3.3) and pre- and post-multiply by RF'
and RF' respectively to get
we rewrite (3.1) as
P = T - W .
It follows that &cc) E ,V({Yr), {Ur)) iff W E ,V({Y,), {Ur}) (Lemma 2.1) and
1 1 ~ ( c o ) 11 < 1 iff 1 1 Pl[ < 1. Therefore, we arrive at the following problem: Given T'
find W E N ( { X ) ? {Ur)) such that
This problem can be related to a mat& distance problem [21, 191: Given T I find a
matrix W E N ( { X } , {&)) to minimize IIT - WII. Let
The problem in (3.5) is solvable, and hence an 'MW-suboptimd controller satis@ing
the causality constraint exists, if p < 1.
To summarïze? assume p 5 1. Define WW such that
and
3.2 A Solution Process via Bilinear Tkansformation
The bilinear transformation is a bijective function between lzl > 1 in the r-plâne
and Re(s) > O in the s-plane:
Since the bilinear transformation preserves 31, noms for trader matrices (se, e-g..
[76j), one can convert a discrete-time 36, problem into a continuous-the one.
The standard, unconstrained 76, design via bilinear transformation involves the
following steps:
Step 1 Convert the discrete plant G into a continuous one G, via bilinear
transformation.
Step 2 Design a continuous X, controuer Kc for G,.
Step 3 Do inverse bilinear transformation to convert Kc into K: which is a
discrete RO3 controuer.
To carry out Step 2. one can use the standard r e d t s in [33,22] which is implemented
as the function hinfsyn in MATLAB p-halysis and Spthesis SooIbox [ i l -
3 '5
For the constrained R, problern, we need a chamcterization of all 'W, suboptimal
controllers, which is, in te- of the results in 133, 221,
where
~ 5 t h DCl2 and being square and invertible matrices. Hence, the conesponding
characterization of all discrete-time '& controllers is
where
In the above, the bilinear transformation has done the foilowing operations: It con-
and to L~ via (see, e-g., [17])
In the next section, we will prove that D~~ and hl are invertible matrices. These
properties are important in treating the causality constraht.
3 o w we consider the causality issue based on the chasacterization of (3.7) that is,
find ô ~ ( o o ) with 11$(0o)ll < 1 such that %(m) E N({y,) , {Ur)): where
Sote that now &(oc) relates to Q(CG) by a linear fiactional transformation; we can
follow a similar step in [19. 551 to simpli& this relation. In the following, we assume
that is normalized with
and perfonn a transformation
Q = ~ ( ~ 0 1 ) -
Ey [60], this transformation translates 1 1 ~ 1 1 ~ < 1 into 118, 11- < 1- It foUoWs that
with
Now it is an easy matter to verify that iz2(oo) = O? and iiz(m) and L2&) are
nonsingdar since we have assumed that D12 and hi are invertible. In this way, we
arrive at an affine rnapping ~ ~ ( o o ) H ~ ( o o ) :
The problem then becomes: fmd a Q ~ ( C Q ) with ~l~~(rn)ll < 1 such that h;(o=) E
,V({K), {Ur)). This is similar to the problem studied in the preceding section asso-
ciated with (3.3) and solvable with the obvious modifications.
3.3 Proof of Invertibility of and D~~
Ln the preceding sectionl the invertibility of Dl* and Dzl played an important role
in deriving the constrained solution: It guarantees that î12(00) and tzl(m) are
invert ible, and hence the associated QR factorizations have invertible factors. In t his
34
section, we prove these properties in terms of continuou-time ?& solutions. For ease
of reference, we nunmarize the properties as a claim.
Clairn d12 and Dzi ore invertible.
The proof to follow will be based on a simplified version of 3C, theory proposed in
[2]. This treatment captures the essentid features of the general problem [33], but
simplifies the derivation a great ded. The general proof is much more algebraically
involved and is omitted-
The simpEed plant mode1 considered in [22] is
which implicitly assumes that Ddl = O ônd Dc22 = O. Moreover, the following
assump tions are made:
(i) (-4, Bci) is stabilizable and (Cci, A,) is detectable;
(ii) (-Act Bc2) is stabilizable and (Cd, &) is detectable;
Wit h t hese; the cont inuous-the %--subopt imal controllers are given as follows. As-
sume the conditions (i)-(iv) are satisfied. The set of ail stabilizing controllers such
that IIF(G~;,. &) 11, < 1 are chasacterized by
with Xc and Y , sati-g the Riccati equations
respectively? and
zc = I - y,&.
Examing Mc: we see that BCl2 (= 1) and bal (= I ) are invertible. Mso? if we
it can be verified that Ac + BcFc and Ac+ LcC, axe stable (their eigendues are in the
open left-half plane). The claim needs two expressions for A,; whkh are first given
as follows.
Lemma 3.1 A, c m be -tten in two versions:
Proof The first equali- follows easily from (3.12). To see the second one, we need
to apply two Riccati equations (3.13) and (3.14): In view of (3.12)-(3.15),
From here, (3.1 7) then follows.
Uë are now ready to prove the claùn.
Proof of the Claim In view of (3.8) and (XI?),
It foIlows that Dlz is invertible iff 1 - (Ac + LeCC) is. The latter is true because the
eigendues of Ac + LcCc can only be negative. Similady, using (3.16): we can show
t bat
821 = [I - (Ac + BcFc)][I - (4 + BcFc + C&Bcl ) l - ' ,
and hence the çtability of rl, + B,Fc implies that & is invert ible. O
3.4 A Design Procedure for Multirate X, Controllers
The constrained ?& solution can be used to design multirate R- controllers - in
the next chapter we will study multirate R- stabilizers for a power system. A t this
point, it is appropriate to summârize a design procedure based on the results of this
chapter.
Let us start with the standard multirate sampled-data setup of Figure 3.1. Uk
assume that G is LTI with a state-space mode1 given. S and X axe mdtirate sampling
Figure 3.1. Standard multirate sampled-data setup
and hold operaton defined via
where mi and n j are integers and h is a real number referred to as the base pen'od.
The above operations correspond to sampling the i-th channel of y with period n i h
and holding the j-th channel of u with period njh. Based on the analysis in [19]: S
and U can be fixther factored as
where Sh and Hh are fictitious single-rate sampler and hold, respectively? and
are discrete multirate sampler and multirate hold
defined via
respectively, with Srni and H,,
+ = Smi4 +(k) = +(kmi),
t. = Hn, v(knj + r ) = 4(k), r = 0?1, - O - :nj .
The design procedure goes as follows:
Step 1 Disaetize G into Gd. There are in g e n e d two ways to do this. The
first is by the sarnple and hold conversion; this corresponds to the traditional
discretization with sampling period h. The second is by a nom-presening
conversion; see [19]. Either Say, Figure 3.1 is converted into a multirate discrete
Figure 3.2. Multirate discrete-the system
design setup of Figure 3.4.
Step 2 Apply lifting, Le., convert Figure 3.4 into a single-rate discrete LTI
setup of Fiope 33.. The general formulas for the multirate lifting may be found
in, e.g.? [19]. Then we arrÏve at the constrained 3t, controi problem.
F i p e 3.3. The lüted LTI setup
39
Step 3 Design & to stabilize G7 s a t i e the causality constraint, and achieve
This is sohable by the discussion in Section 3.2.
MATLAB provides a nice platform to implement the above procedure. In fact.
we have developed a software package to design rnultirate 31, controllers using the
bilinear transformation method proposed in Section 3.2. To summarize, let us brie&
discuss the routines we have written, as illustrated in Figure 3.4.
a plant: Store a statespace mode1 of the continuous-the plant.
a dmet-p: Discretize the plant by traditional sample and hold conversion. If one
would Like to do a direct sampled-data design, a nom-preserving discret k a t ion
method is atailable ini e.g., [1'7].
0 m l i ' Multirate lifting of the discretized plant. According to [19], we need to
lift each of sub-trader finctions of the discretized plant, which is implemented
in the function Zfprt, and then combine them. The function rnlift is to get a
state-space mode1 of the whole 1if'ted plant, çd, by calling Zfprt.
hisynaldc: Discrete-time 'H, design for the lifted plant. The output is a charac-
terkation of aIl unconstrained R, suboptimal controllers. This function carries
out the t h steps given in Section 3.2: It converts the lifted plant into a contin-
uous one via bilinear tansformation (BL); it c d s a continuous design function
hisyn-al which generates a characterization of all continuous 7.1, suboptimal
controllers; it then converts the controllers into discrete ones via the inverse bi-
linear transformation (BL). The new huiction hisyn-al is a modification of the
MATLAB built-in function hinfsyn, which cornputes only one continuous 31,
solut ion.
Figure 3.4. A design procedure for multirate 310, controller
e lff2l-d: Simplify the linear fractional relation in (3.9) into an ofiine function in
(3.11).
perm: Permute the columns of kli(OO) and &1(00) aad the rom of L~~(oG) and &+û) in (3.11). The reason to do this is that the elements of the input
and output of the iifted controller may not be ordered in the time order due
to the lifting procedure: hence the causiility constraint may not necessarily
41
imply that the direct feedthrough texm of the Mted controller is a block lower
tnangula matrix. The fùnction p e m is buiit to reorder the columns of Lll(oo)
and i21(~) and the rom of Lll(oo) and &cc) according to the t h e order
of the controuer input and output, thus the methodology in Section 3.1 can be
applied to handle the causality constraint.
go-dist: Reduce the &e function in (3.11) into a mat& problem by doing
QR factorizations for itz(ffi) and &O) (reordered) . The M N L AB built-in
function qr can be used for the QR factorization for tI2 (00) - Another function
prl is built for the dual factorkation required for &(m).
disprb: Solve the distance problem in tenns of the solution process given in
pi1 191.
kd-causa[: Finab, calculate the controller which satisfis the causality con-
straiat .
Conclusions
In this chapter, we have shown that the bilinear transformation method: usually
used in the standard, unconstrained discrete 31, design, is also appropriate for the
U- problem Nith a causality constraint on the c o n t r o b feedthrough terms. The
methodology used to handle the causality constraint followed closely that in [19], but
the solution is more computationdy efficient. Based on this, we have developed a
MATLAB software package with severd utility functions and subprograms to design
general discrete-time multirate controllers using 31, optimization.
One limitation of the bilinear transformation method is that it yields only one
constrained 3I, solution. In contrast, the direct discrete-time design approach can
give a,II Xfl, suboptimal solutions [S].
3C, DESIGN OF DIGITAL POWER SYSTEM STABILIZERS
This chapter looks at an application of ?& optimization: Robust digital stabilizers
(controllers) are designed for a singlemachine infinite-bus power s yst em' . These
digital stabilizers. including a multirate stabilizer ushg the technique given in the
preceding chap ter, outperform the conventional anaIog stabilizer and require only
low cost in implementation.
4.1 Motkation and Introduction
The system to be studied is the singlemachine, infinitebus power system whose
schematic diagram is shown in Figure 4.1. In the power system? the spchronous
generator generates power which is transmitted through the transmission lines with an
infinite bus the exciter and automatic voltage regulator ( .N IL ) are used to maintain
the terminal voltage profde; the associated governor monitors the shaft frequency
and controls the mechanical power and speed. Often in operation this nonlinear
system is subject to various disturbances due to changes in, e-g., generation schedulest
transmission-line structures, load condit ions, and network interconnections; in order
to provide damping to oscillations caused by these disturbances, the power system
stabilizer (PSS) is instded via modulation of the generator excitation. Design of
robust digital power system stabilizers against different disturbances is the focus of
this chapter.
Conventional power system stabilizers (CPSS) are widely used in industry for their
simplicity in structure and hence ease in implementation via analog circuitry. These
analog stabilizers have low complexity; for example, a tq-pical cIass of stabilizers
'The tesults in this chapter have ben reported in [68].
1 - 1 Transmission
Figure 4.1. A schematic diagram of the power system.
axe of fourth order, with a ht-order prefdter, a second-order lead-lag compensator.
and a first-order washout factor in cascade connection. The analog CPSS design
is usually based on an LTI model linearïzed about the normal operating conditions
and employs classical lead-lag compensation techniques, typically graphicd in the
frequency domain. Stabilizers designed using this graphical technique offer a certain
degree of robustness against parameter variations and &ovm disturbances; some
improvement can be made using the s o - d e d enhanced technique [47] which provides
extra compensation within a certain range of operating conditions.
The ?&,-based optimization can considerably Ïmprove performance and robust-
ness of power systems, as is evidenced by recent work on analog stabilizer designs
[SI, 5: 14, 7.1, 11. 'H, optimal control design minimizes the worst-case energ'- gain
(R, nom) of a certain dosed-loop trander mat&, suitably weighted. For properly
selected weight ing functions, the optimal controllers designed have good performance
in the face of uncertainties in plant modeling and/or disturbances; moreover, tradeoffs
between performance and robustness can be studied in this framework. One property
of the 31= optimal controller is that its order equds, roughly speaking, the order of
44
the plant plus orders of the weighting functions. This can be a drawback if the plant
and weighting functions are of hi& order; in this c- mode1 reduction techniques
are usually applied after the optimal design: e-g., in [Tl], the power system controUer
designed is initidy of order 20, which is then reduced to 10; in [14], the order of the
stabilizer designed using p synthesis is reduced fiom 35 to 10. In this study. we target
stabilizers with orders comparable to the conventional power system stabilizen.
Our main purpose is to design digital power system stabilizers which can be im-
plemented on microprocessors with low cost. To meet this objective, the digital
stabiIizers designed have low complexity and require only slow A/D and DIA con-
versions. We s h d attempt two approaches to design 36, stabilizers. The fmt is to
design analog îi, controllers and then discretize them for digital implementation.
This approach involves approximation in discretization and normally requires high
sampling rates in order to emulate the analog system - see one of the PSS designs
studied later. The second approach is to fix the sampling rate and discretize the plant.
and then perfonn 'H, control design directly in discrete tirne. This approach has the
advantage that because the samphg rate is incorporated in the design process. one
can attempt design with slow sarnpling rates. However, weighting functions should
be selected properly to avoid overdesigning the system at sampling instants. Sev-
eral of our designs studied later follow this second approach, yielduig power system
stabilizers with high performance and low sampling rates.
Several control schemes and robust stabilizer designs are investigated for digital
implementation. Stabilizers can have access to a single input, the speed signal, or
two inputs, the speed and power signals, and generate a voltage control signal for the
automatic voltage regulator (.4VR.) and exciter; in the two-input case, the sampling
schemes can be single-rate or multirate. Thus four robust digit d designs are studied
in t his chap ter:
43
1. The stabilizer is SISO, meanving the speed signal; a robust analog stabilizer
is designed first using 'H, optimization and is then discretized via bilinear
transformation (Tustin's approximation). To maintain good performancet fast
sarnpling is required; the samphg period in this case is 10 ms.
2. This has a sim3a.r control setup to Case 1 - SISO and single-rate, but the plant
and weighting functions are discretized first and the digital stabilizer is designed
based on discrete-time 31, optimization. To keep comparable performance, we
can reduce the sampling rate by a factor of 4, the sampling period being 10 ms.
3. ùi this case, the control setup has two inputs and one output - MIS0 (multi-
input, singleoutput) setup, and a single sampling rate is used. The design
is based on the discretized model as in Case 2. Because an additional power
signal is meanired, the sampling rate can be further reduced without sacrificing
performance; the sampling period in this case is 80 ms.
4. Similar to Case 3, the stabilizer measures the speed and power signas but
we choose a multirate sampling scheme. Since multirate system is the main
subject in this thesis, we present two sampling schedules to compaze: In the
iirst one, the speed and power signals are sampled with periods 120 ms and
60 ms, respectively, and the control signal (DIA conversion) is updated every
60 ms. In the second schedule, the samphg period at the speed is further
increased to 180 ms. One reason for using multirate controllers is to achieve
a cost advantage in impIementation in cases when the signds involved have
different bandwidths.
AU the digit al power system stabilizers designed compare favorably 6 t h the analog
conventional one, as will be shown by a series of pedormance tests via nonlinear
simulations. Moreover? the digital stabilizers have relatively low orders (46) ; this is
46
important for rd-t ime implementation. This property is obtained by using a low-
order linearized model in design (this is partially justified by the robustness of the
design approach taken) and carefidy selecting simple weighting functions.
The multirate design problem imlved in Case 4 can be solved using the bilinear
transformation method given in Chapter 3 - a 3IATL.A.B softwaze package has been
dedoped t O implement the multirate design approach. The software takes advantage
of the existing MAT LAB fiinctions: in particularo the cont inuous-time 'H, design
function hinfsyn is used as one of the key functions for design - see Section 3.1 for
det ails. The result h g mult irate robust stabilizers have good performance comparable
with single-rate stabilizers, but use lower skmpling rates.
4.2 Linearized Plant and Weighting Functions
A spchronous generator connected to an infinite bus is in generd described by
noniinear difierential equations with combined order seven. As we mentioned eariier.
for low-order stabilizers: it is advantageous to use a low-order hearized model in
?lm optimization. Hence we use the simplified Park's two-axis model [4]: obtained
by ignoring transients in the stator circuit and the effect of the rotor amortisseur -
see the Appendix at the end of the thesis for the equations and the parameters which
are derived from an experimental singlemachine power system at the University of
Calgary.
Linearizing about the normal operating conditions [power P = 0.9 p.u. and power
factor Pj = 0.95 (hg)]: we obtain a linear model [4] in Figure 4.2 for the power
system? where Ap, Au, 46: and 4~ are, respectivelq; the deviations in power, speed,
power angle, and terminal voltage. The reference voltage disturbance is 4 Kej and
the mechanical torque disturbance AT,. For ease of reference, we will drop the prefix
av I
Figure 4.2. Linearized power system model.
Table 4.1. Parameters for the lùieaxized power system model.
Parameter Value Parameter Value '
4 in a.lI variables from now on. Xote also in Figure 4.2 that a simplified model is
given for the AVX and exciter block which is in the form of a first-order system:
Kz Tas + 1'
The parameters in Figure 4.2 can be determined from linearizing the equations in the
Appendix with the given operating conditions; these are shown in Table 4.1.
with this linea,rïzed model in Figure 4.2, the overd order is four. Note that the
power system stabilizer is not included in Figure 4.2; but it measures w or both s. and
p and generates a control signal Au, which is then fed back to the AVR and exciter.
8 LW h', 1.5495
4s
This mode1 has ben used for conventional stabilizer design and robust analog sta-
bilizer design [14, 131 based on 36. optimization. However, the orders of the robust
st abilizers designed are st ill considembly higher t han t hat of the convent ional s t abi-
lizers due to the inclusion of weighting functions in design ( recd that the controLler
order equais rougkdy the order of the plant plus those for the weighting functions). To
have reasonable orders for stabilizers via Ra optimization, one must select weighting
functions for good performance while keeping their orders to a minimum - a topic to
be discussed next.
4.2.2 Weighting Funetions
For good PSS design, let us note two desirable properties for the power system
under control:
1. By standard practice, the stabilizers should provide damping to oscillations
due to only low-frequency disturbaaces. Typicdy these occur in the frequency
range of approximately 0.2 to 2.5 Hz, see, e-g., [42]. Thus stabilizen are to be
designed to attenuate only low-fiequency disturbances.
2. In the case that the stabilizer has input w and output u, i t is desirable that
both w and u be zero at steady state when the disturbances are step signals.
This reflects the requirements that at steady state, the power angle 6 (integral
of w) be constant and the stabilizer be disconnected in effect. The latter prop
erty means that weU-designed stabilizers are brought to action o d y during the
transient process.
From the first property, good designs should be focused on at tenuating low-fiequency
components in the disturbance inputs Tm and K.,. For this purpose, two input
weighting functions (fictitious premters) Wt for Tm and W, for Kef are introduced
49
for design. Wt and W, are chosen to be lowpass to cover the fiequency band of inter-
est: to keep the order increase to a minimum, they are taken to be first-order lowpass
filters with the same time constants:
(This way the order of the genealized plant including the two weighting functions is
increased only by one due to the fact that both Wt and Wv have the same denomina-
tor.) Note that the target frequency range can be determined by ody one parameter
T. The gains kt and kv reflect relative weighting between attenuated low-fiequenc-
disturbances in Tm and Ker: e-g., if kt is mu& larger than +: more emphasis is
placed on disturbance fiom Tm.
So provide damping to the oscillations, one can use as regulated outputs the power
( p ) : speed (w), power angle (6): or any combination of the t h e . However. the
standard R, technique does not parantee the second propem; that w and u at
steady state converge to zero due to step disturbances: If #(oc) # 0, 8 wilI r m p to
infmity; if the stabilizer is overdesigned so that u(m) # O (it provides overdamping),
&(cc) WU be lower than the expected level which is determined by the parameters of
the power system without the stabilizer. The first case could happen if one used ody
;2 a,nd/or p as the regulated outputs; and the second case could happen if u is not
correctly weighted as a regulated output.
For all desipso we will use three signalç, p, 6, and ul as regulated outputs subject
to appropriate weighting. Then the power system in Figure 4.2 can be recodgured
into an equivalent plant Po with three inputs Tm, K e f : and ul and four outputso the
reeplated outputs p, 6, and u: and the measured output y (y can be w or a two-
dimensional vector consisting of (*. and p); the stabilizer is in feedbadc connection
with Po using y as input and u as output, as is shown in Figure 4.3, where h' is the
stabilizer. Io weight the regulated outputso we introduce a constant weighting c; for
Figure 4.3. Reconfigured power system with PSS and weighting functions.
d: For a nonzero c ~ , the ?&, designed closed-Ioop system guarantees that due to step
disturbances &(oc) is weu-defined and finite so that w ( m ) must be zero, because 6
is an integral of u;. Similady, to get u(oo) = O due to step disairbances, we use an
integrator as the weighting function for u:
where E is a s m d positive number whose role is to slightly perturb the weighting
function so that Wu becomes stable (this makes 'fl, problem nonsingular). Findy,
we also use constant weighting for p and normalize it to 1: in this case, experience
shows that c~ shodd be much smaller than 1 to avoid overdesign.
To summarize, we have introduced input weighting functions for disturbances Tm
and Kef as shown in Figure 4.3,
wl and to2 being prefiltered disturbance inputs with no spectral limitations. We have
dso introduced output weighting functions for the original regulated outputs:
a l
Absorb the weighting functions into Po to get the generalized plant P. the shaded
blodc in Figure 4.3: defmed as follows:
This way we arrive at the standard setup with the following equations
The associated analog design problem can be stated as follows:
Design an LTI, causal stabilizer K to stabilize the generalized plant P and
minimize the 31, n o m of the 3 x 2 closed-loop transfer matrix fiom w to r .
Of course, to achieve good performance one needs to adjust the weighting functions
Ui,? WVf Wu, and CJ; more ~pecificdy~ the following five parameters are tuned in
design:
TT kt, 5, CJ, ku-
Xotice that the number of puameters involved is even l e s than that of a standard
convent ional power svstem st abilizer.
Find- we mention that a state-space model can be obtained for P which is of order
six [order of Po (four) plus the order increase (two) due to the weighting functions].
This state-space model is our basis for subsequent analog and digital designs.
4.3 Design of Robust Digital Stabilizers
Robust stabilizer designs here are based on the generalized plant P in (4.2) which
includes the linearized power system model Po and the weighting functions. Because
.j 2
our choices of the linearized model and the weighting functions, we can easily compute
a sixth-order st atespace model for P :
where DZ2 = O for both cases of y by dadation.
Our digital stabilizers designed Iater will be compared against the analog conven-
tiond stabilizer (CPSS) of the form given by the IEEE standard CPSS type PSSlA
shown in Figure 4.4. This CPSS measures the speed signal (u) and output the volt-
Figure 4.4. The IEEE standard CPSS confgguation.
age signal (u) to the -4VR and exciter. In Figure 4.4, u,, and u,, represent the
control Iimit constraint; with our model, u,, = u,in = 0.1 p.u. The parameters
of the CPSS are tuned carefully for good response in 6 (power angle) when a torque
disturbance is applied to the nonlinear power system model under normal operating
conditions. The parameters are given in Table 4.2. Xote there that Ai and .A2 are
both zero, the CPSS is effectively a fourth-order system.
Table
b l T2 0.02 Ts T3 0.1 Ts 0.005 hfs 0.05
4.2. Parameters for the tuned analog CPSS.
Throughout pon-er system stabilizers are tested for three cases:
a Tm J.: a torque disturbance applied with 0.1 p.u. step decrease and removal:
a t: a voltage disturbance with 0.05 pu. step increase and removal:
a Ground fault: three-phase-to-ground short circuit accident and redosure:
and under three load conditions:
a Yormd load conditions: P = 0.9 p-u., P' = 0.85 (lag) (slightiy Lower than
PI = 0.95 for the operating conditions);
a Light load conditions: P = 0.2 p.u., Pf = 0.85 (lag);
a Lead load conditions: P = 0.5 p-u., PI = 0.9 (lead).
Xote that for lag Pf , we have 6 > 0; and for lead Pj: we have b < O. For ground fault
accident, only normal load conditions me tested. In all cases, the power angle b is
monitored. For example, using the malog CPSS, we simulate and show the responses
of b in the seven cases in Figures 4.5411 (dotted cunres) as detailed in Table 4.3.
II Nomal load 1 Light load 1 Lead load 1 I - 1
Tm L II Figure 4.3 1 Figure 4.6 1 Figure 4.7 ] I - , - L - 1 Figure 4.8 1 Figure 4.9 1 Figure 4.10 1
Table 4.3. Nonlinear simulation tests of the CPSS and the figures.
In what foIlows we will design four robust digital stabilizers via 'fl, optimization;
our purpose is to fmd stabilizers whkh outperform the CPSS: have low orders, and
require low sampling rates, thereby providing an advantage in implementation.
4.3.1 Andog Design and Discretization
We will b t design a robust analog stabilizer and then obtai. the digital stabilizer
bq. discretizing the analog one. The stabilizer is SISO, meMuing only the speed
signal.
In view of Figure 4.3: the analog 36. design problem can be stated as follows:
Given the generaked P? design an LTI stabilizer K to provide closed-loop stabiliw
and minimize the 3L, nom of the trander môtrix from w to z, denoted f . ( s ) . Recall
that the 'fl, norm of Fm is the supremd maximum singular d u e on the imaginary
Such an optimization problem can be solved by y-iteration; each iteration involves
solving two Riccati equations. The standard solution [33] has been implemented in
MATLAB as function hinfsyn in the p-Analysis and Synthesis Toolbox [f]. In pneral.
tirne (sec)
Figure 4.5. l'orque disturbance with normal load: CPSS (dot), analog stabilizer (solid) reduced-order analog stabilizer (dash-dot )
0.14' 1 L 1
0 1 2 3 4 5 6 7 0 9 1 0 time (sec)
Figure 4.6. Torque disturbance with light load: CPSS (dot), analog stabilizer (solid): reduced-order analog stabilizer (dash-dot)
Figure 4.7. Torque disturbance with lead load: CPSS (dot), analog stabilizer (solid), reduced-order analog stabilizer (dash-dot)
. - - - 0.48
, Q , 1
0 1 2 3 4 5 6 7 8 9 t O time (sec)
Figure 4.8. Voltage disturbance with normal load: CPSS (dot), analog stabilizer (solid), reduced-order analog stabilizer (dash-dot )
Figure 4.9. Voltage disturbance with light Ioad: CPSS (dot), analog stabilizer (solid), reduced-or der analog s t abilizer (das h-dot )
o'*O 2 6 9 1 1 0 time (sec)
Figure 4.10. Voltage disturbance with lead load: CPSS (dot), analog stabilizer (solid), reduced-order analog s tabilizer ( dash-dot )
Figure 4.11. Ground fault test with normal bad: CPSS (dot)? analog stabilizer (solid), reduced-order analog stabilizer (dash-dot ) , discretized analog stabilizer with sampling period 20 ms (dash)
one inputs a state-space model for P and gets a state-space model for K fiom the
optimization: the order of K equals that of P. Since the order of our P was kept as
low as six, the optimal stabilizer designed will be also of order six, which is comparable
with the standard CPSS given in Figure 4.4.
Before ninning the program hinfsyn, a little massage is necessary to regularize the
problem. The problem with the original data for P is singular because DI2 and D21
in (1.3) are both zero matrices and do not have MI r d . This can be fked by slight-
perturbing the two zero matrices to have full r d . For example, we can introduce a
fictitious input wj and an output zj weighted by a small number c = 0.1. Thus P is
The problem then becomes nonsingular and the design cm be done based on Prim.
The foms of the weighting functions have been chosen in Section 4.2.2: the param-
eters involved are selected by trial and error (and experience) during the design and
simulation process to achieve good dynamic performance for the closed-loop power
system. In thk analog design, we use
These weighting hctions will help achieve the two desirable properties discussed at
the beginning of Section 4.2.2 for the controIled power system.
The optimal analog stabilizer K can be designed now by 36. optimization using
the M.4TLAB function hinfqn:
59
Xote that similar to the washout factor in the CPSS in Figure 4.4, the numerator of
thÏs robust ~ ( s ) contains a derivative factor whose function is to force the output
of K to converge to zero when there axe step disturbances. This justifies our use
of the integrator-Ue weighting function Wu. Nonlinear simulations are conducted to
e d u a t e this analog controller: The seven test results are also contained in Figures 4.5-
4.11 as the solid m e s . W e conchde that this robust analog stabil'zer is considerably
superior to the conventional st abilizer.
For an order cornparison, note that 'fl, based power system stabilizers designed
in the literature have higher orders: in [71, 141 the robust stabilizers both have order
ten, which have alreadq- undergone a model reduction process. Our analog stabilizer
is of order six and we can also consider pedonn model reduction. Using MATLAB
fûnctions hankmr, we compute the fourth-order. optimal Hankel-nom approximation
to the sixth-order stabilizer K , based on a balanced realization of k ( s ) (obtained via
MATLAB function sysba2):
This ha. the sarne order as the CPSS tested. hlthough the derivative factor has
disappeored in &(s), it can be vedied that it still satisfies the second propertt-. Le.:
once connected to the power system, the control input u is still zero at steadÿ state for
step disturbances. The simulation tests are shown in Figures 4.5-4.11 in dot-dashed
cuves: In general, the reduced-order Kr suffers slight peâonnance degradation corn-
pared with the original K; but the test results are still much better than those of the
CPSS. (From now on: we s h d skip model reduction and content with the si-xth-order
s t abilizers generated from R, op t imization.)
Our purpose is to implement stabilizers via microprocessors. Thus we discretize
the andog robust stabilizer using bilinear transformation (Tustin's approximation)
to get a sixth-order digital stabilizer. the sampling period used being 10 ms. The
60
closed-loop responses with this digital stabilizer for the standard tests axe almost
indistinguishable fiom those with the anôlog r;'(s), and are hence omitted. However,
if larger sampling periods are used for the discretization, the performance of the digital
stabilizer is noticeably worse than the d o g performance: For fistration, take the
samphg period to be 20 ?O; the ground fadt test for the resultant digital stabilizer
is showvn in Figure 4.11 (dashed curve), from which we see a much higher overshoot
for the fault recoverq- than those using the andog stabilizer (solid curve) and even the
CPSS (dotted curve). This shows the disadvantage of discretizing an analog design
- it requires relatively fast sampling rates. If slower sampling is desired without
compromising performance, design should be done directly based on discrete-time
4.3.2 Discrete-The Design of SIS0 Stabilizers
The setup in this case is shown in Figure 4.12, where P is again the generaiized
Figure 4.12. The single-rate digit al control set up
power system with a state-space mode1 in (4.3) but the stabilizer is a sarnpled-data
one: The measured signal y (t) is first sampled by the ideal periodic sampler Sh with
sampling period h to get the sampled signal yd(k); then yd(k) is processed by the
61
digital stabilizer Ka to generate the control sequence ud(k), which is then transformed
to a piecewise continuous signal u(t) by the zero-order hold Hh.
In this subsection, we take y to be the speed signal and so the stabilizer is SISO.
The setup invdved is a single-rate control system with only one rate ( I / h ) . For an
implementation advantage, we take a sampling rate which is a factor of four slower
than the one used in the analog design, Le., h = 40 ms. We remark here that the
sampling rate cannot be made arbitrady slow: If low-fiequency oscillations of up to
j Hz are to be attenuated by the digital stabilizer, roughly speaking, the samphg
rate should be at least twice as fast as the maximum oscillation fiequene- (Shannon's
sampling theorem) and so the sampling period has an upper limit: h < 100 ms.
To perform a discrete-time design, the first step is to discretize the generalized
systern P to get Pd [l'il. The discrete-time system Pd and the digital stabilizer fi
f o m a discrete-time feedback system shown in Figure 4.13- -4 state-space model for
Figure 4.13. The discrete-tirne setup
Pd can be cdcdated fiom that for P in (4.3) based on sample and hold equivalence:
Here,
Then the associated disaete-time 3L, design problem is as foI.lows:
Given Pd, design an LTI digital stabilizer Ka to provide closed-bop stability
and mirrimize the 9& nom of the cIosed-loop transfer matrix T~~~ : ZQ rt zd:
an optimization problem c m be solved purely in discrete time aaalogously via
two dgebraic Riccati equations [37, 691. However, we will use the bilinear transfor-
mation to convert it to an equivalent contipuous-time problem. see, e.g.. [17] and
Section 3.2. and then solve by applyuig the M-4TL-4B funaion hinfsyn.
Xote that the weighting functions we used for digital design are still given in con-
tinuous time. However, the parameters are not necessarily the same as in Section 4.3.1
in the d o g case and are slightly re-tuned in the design and simulation process for
good performance. W e have
The corresponding digital robust stabilizer is computed to be of order six:
Xotiote that the numerator of &(z) again contains the derivative factor ( r - 1): which
guaantees that the final value of ua(k) due to step disturbances is zero. But this
factor can be eventually côncelled out by the factor (z - 0.9999) in the denominator.
yielding a f%h-order stabilizer. The test resdts are given in Figures 4.14420 (solid
cuves) . To compare, the test resdts for the CPSS are redrawn as dotted curves. It
is dear that this digital stabilizer is comparable in performance to the analog robust
stabilizer, outperforms the CPSS a great ded, yet requires a sampling period of ody
40 ms.
tifne (sec)
Figure 4.14. Torque disturbance with normal load: CPSS (dot)? digital SISO stabi- lizer (solid), digital M I S 0 single-rate stabilizer (dash-dot)
Figure 4.15. Torque disturbance with light load: CPSS (dot), digital SISO stabilizer (solid): digital MIS0 single-rate stabilizer (dash-dot)
time (sec)
Figure 4.16. Torque disturbance with lead load: CPSS (dot), digital SISO stabilizer (solid) digital MIS0 singlerate stabilizer (dash-dot )
Figure 1.17. Voltage disturbance with normal load: CPSS (dot), digital SISO stabi- lizer (solid), digital MIS0 singlerate stabilizer (dash-dot)
Figure 4.18. Voltage disturbance with light load: CPSS (dot), digital SISO stabilizer (solid), digital MIS0 single-rate stabilizer (dash-dot )
o n ; L I 1 2 3 4 5 6 7 8 9 1 0
tirne (sec)
Figure 4.19. Voltage disturbance with lead load: CPSS (dot), digital SISO stabilizer (solid), digital MlSO singlerate stabilizer (dash-dot)
0.45~ 1 O t 2 3 4 5 6 7 8 9 1 0
izrne (sec)
Figure 4.20. Ground fault test with normal load: CPSS (dot), digital SISO stabilizer (solid). digital MIS0 singlerate stabilizer (dash-dot)
4.3.3 Discrete-Tme Design of MIS0 Stabilizers
The MIS0 control setup is similar to the SISO control setup in Figure 4.12 except
that the signal y now is a twdiruensional vector,
y=[ ; ]
m e a s h g both the speed and power. The digital stabilizer has two inputs and one
output, ail sampled at the same rate. With this additional power signal, we hope to
further reduce the sampling rate with Little expense in performance.
The disaete-the design process is almost the same as in Subsection 4.3.2. But
in this case, it is helpfid to bring in a weighting function at the measured output
s; (Figure 4.12). For 31- design, the weighting function we use for this measured
output is a constant, denoted by cd. Let us take the sampling period h = 80 ms:
which is twice as much a . that used for the SIS0 stabilizer in Subsection 4.3.2. For
the following design, the weighting b c t i o n s are
The digital stabilizer designed again via discrete-time R, optimization has the 1 x 2
transfer rnatrk
with
Again: the factor (z - 1) in the numerators can be cancelIed out by the factor ( z -
0.9938) in the denominatoc the stabilizer is actudy mh-order. The performance
test results are given in Figures 4.14-4.20 (dot-dashed curves). As can be seen, this
'rlISO digital stabilizer still outperforms the analog CPSS a great deal, although the
sampling rate is quite slow.
4.3.4 Multirate Design of MISO Stabilizers
The MISO digital stabilizer studied in Section 4.3.3 in the single-rate setting can
admit a multirate sampling scheme. To see why multirate control is appropriate. we
plot the magnitude fiequency responses of the two transfer functions in Figure 4.2
from, e-g., the voltage input Kef, to the measured signds, w and p (both in p-u.),
as are shown in Figure 4.21. From this figure, the transfer function from to p
has a wider band of significant fiequency components than the transfer function from
to w. Hence it is reasonable to sample p fast and w slowl. Let us consider
two sampling schedules. First, we use a sampling period of h = 60 ms for the power
Figure 4.21. Magnitude frequency responses from to p (solid) and w (dot); p and a are both in p.u.
signal p and a samphg period of 2h = 120 rns for the speed signal sr: the hold (DIA
conversion) also has a period of 60 ms. Second. we inaease the sampling period at
the speed signal further to 3h = 180 ms in order to see the performance degradation.
The associated design setup is shown in Figure 4.22, where the signals w and u
are as before, and
Figure 4.22. The multirate digital control setup
69
(Recd that as in Section 4.3.3, we have weighted m e m e d output w by 6.) Let
us focus on the b t sampling schedule. The multirate sampler S corresponds to two
and the multhte hold 3L reduces to the single-rate one: 'fl = Rh. Let T be the least
common period for all skmpling and hold channels - in this case T = Sh = 120 ms-
Discretizing the s jstem and applq-ing lifting, we amive at as LTI system of Figure 4-23
with a t hree-input , two output controuer:
Figure 4.23. The lifted LTI setup
which corresponds to grouping the inputs and outputs properiy over the period T .
Then it follows that & has a state-space mode1 given by
Xote that D13 = O &ses fiom the causality constraint on the multirate controller.
The 3L, design for this constrained problem is as follows:
70
Design an LTI & of the form in (4.4) satisfying the causality constraint Dl, =
O to provide dosed-loop stability and min;mize the ?& n o m of the transfer
rnatrix fiom a to a in Figure 4.23.
The rnultûate stabilizer can be obtained by running the M.4TLAB programs given
in Section 3.4. The following weighting functions are used for our subsequent design:
The robust multirate stabilizer Kd, in the lifted state-space form of (4.4, is computed
Note that the order of this multirate stabilizer is again six and that the zero entry
in the D-matrix indicates causal i . The test results for this multirate stabilizer are
given in Figures 4.24-4.30 (solid m e s ) in cornparison with those for CPSS. From
these plots, we conclude that dthough the sarnpling periods for w and p are as l q e
as 120 and 60 ms? the rnultirate stabilizer is considerably better than the CPSS.
Finally, let us replace the sampling period of 120 ms for the speed signai with 180
rns and keep the remaining rates unchanged. In this case, T = 32 and the controller
& becomes a fout-input, th.e+~utput one: -
Its state-space mode1 is
wheie, again, the zero entries &se fiom the causality constra.int. Ushg the following
set of weighting funaions
we can design the multirate stabilizer &, whose test results are shown in Figures 4.24-
4.30 (dot-dashed m e s ) . It is seen that the performance of the mdtirate stabilizer
starts to deteriorate; this is particularly reflected in the ground fadt test of the
stabiliw (Figure 4.30): It is worse than the solid curve? but still no worse than
the dotted curve (CPSS). This suggests that if we use the CPSS performance as the
benchmark, we have almost corne to the limit in using slow-rate digital stabilizea via
Figure 4.24. Torque disturbance with normal load: CPSS (dot), digital MIS0 multi- rate stabilizer with sampling period 120 ms for w (solid), and digital MIS0 multirate stabilizer with sampling period 180 ms for w (dash-dot)
Figure 4.25. Torque disturbance with light load: CPSS (dot), digital MISO multirate stabilizer with samphg period 120 ms for w (solid), and digital MIS0 multirate stabilizer with sampling perïod 180 ms for w (dasidot)
tim (sec)
Figure 4.26. Torque disturbance with lead load: CPSS (dot), digital MIS0 multirate stabilizer with sampling period 120 ms for w (solid), and digital MISO multirate stabilizer with sampling period 180 ms for w (dash-dot)
time (sec)
Figure 4.27. Voltage disturbance with normal load: CPSS (dot), digital MISO multi- rate stabilizer with sampling period 120 rns for w (solid), and digital MIS0 mdtirate stabilizer with sampling period 180 ms for w (dash-dot)
Figure 4.28. Voltage disturbance with light load: CPSS (dot), digital MISO multirate stabilizer with sampling period 120 ms for w (solidj, and digitd MISO multirate stabilizer with sampling period 180 ms for w (dash-dot)
Figure 1.29. Voltage disturbance with lead load: CPSS (dot), digital MISO multirate stabiEzer with sarnpling period 120 ms for w (solid), and digital MISO multirate stabilizer with sampling period 180 ms for w (dash-dot )
0.450 , . , 1 1 2 3 4 5 6 7 8 9 1 0
time (sec)
Figure 4.30. Ground fault test with normal load: CPSS (dot), digital MISO multirate stabilizer with sampling period 120 ms for w (solid). and digital MISO multirate stabilizer with sampling period 180 ms for w (dash-dot)
the optimization.
4.4 Conclusions
In this chapter, we have studied stabilizer design for a single-machine, infinite-
bus power system using the robust ?& optimization. Several designs have been
accomplished includïng asdog and digital, single-rate and multiate. .AH stabiIizers
designed have low complexity - they are LTL and of low orders - and outperfom
the analog, conventional power system stabilizer a great deal on a series of nonlinear
d ~ o m i c tests; the digital stabilizers designed also enjoy the additional property t hat
they use considerably iow samphg rates in implementation.
Sorne other concluding remarks are:
In 'H, control design, it is important to use appropriate weighting functions.
Weighting functions are chosen to reflect the operational requKements of the
system and are sometimes based on experience. But it is desirable to use simple
weighting functions, for they introduce minimum order increase in controllers.
Power systems are generdy described by high order nonlinear equations. In
some previous work (se, e.g., [71]). high order linearized models are used to
approximate the nonlinear equations. In our study, we showed that the low
order linearized model, which has been used in design of the conventional power
system stabilizer, is still usehl in designing high performance stabilizers.
a Mdtirate control systems offer a cost advantage in implementation. Design of
multirate control systems involves handling a new causality constraint and it is
feasible to accompli& %-optimal designs based on recent advances in the the-
ory [74, 191. To implement this theory in MATLAB, the bilinear transformation
approach proposed in Chapter 3 offers a computationd admntage.
76
a For reiatively low sampling rates, digital design based on discretized models
works better than discretizing analog designs. Poor intersample rippIes can be
avoided by properly selecting weighting functions in the design process. To
further reduce samphg rate, the direct sampled-data design methods (se,
e.g., [l?]) would be a good choice, because they take into account intersample
information.
-- l I
CHAPTER 5
Na DESIGN OF HYBRID MULTLRATE FILTER BANKS
The focus of the previous chapters was placed on control systems. In th% chapter.
we tuni to the study of a signal processing system, in particular? a hybrïd multirate
filter bank. Our purpose is to design the multirate filter bank using 3L, optimization.
thus providinp a new application of RI, control theory'.
5.1 Motivation and Introduction
The subject of this chapter is the hybrid multirate filter bank shown in Figure 5.LI
where Go and G1 are LTI andog analysis fikers, SZh the slow sampler with period
Figure 5.1. The hybrid multirate filter b d .
2h, E the ezpander by the factor of 2, and Fo and FI LTI digital synthesis filters-
The expander E is a new component whose function is to inaease sampling rate by
inserting zeros between sarnples:
3ote that we have denoted expanders by E instead of the traditional symbol t 2
because the former is more appropriate for equations.
lThe r d t s in this chapter have been published in [67].
78
Such systems are potentidy usefil in digital trammÏssion of andog audio signds.
Suppose the input analog signal r(t) is stnctij- bandlimited. To f a i t M y represent
x ( t ) ? we need to sample at least as fast as its Nyquist fiequency w ~ ; this leads to
the constraint on the sampling period: h 5 Zir/wN. Using the two-channel filter
bank. one codd design Go to be lowpass and G1 highpass, splitting the band into
two subbaads; this way one codd effectively reduce the samphg fiequency in each
channel by a factor of two and at the same time achieve overall bit-rate reduction in
transmission by exploiting fiequency characteristics in coding [72]. For such a system
to work, the two synthesis filters must be designed so that the output + approximates
the desired output 4 obtained by fast sampling the input: t$ = Shx. By standard
practice, sorne time deiay wil l be tolerable and so we should compare tb(k) with
d(k - m): where rn is the timedelay integer.
Let U be the unit time-delay system in discrete t h e with trollder function z-'.
The error system is P S h - Ko as is depicted kom z(t) to the error signal, r ( t ) =
&(k - m) - tb(k), in Figure 5.2. However, this error system presents a difficulty for our
Figure 5.2. The pre-filtered error system.
design (see below) because of the bandlimited assumption of x ( t ) . Aence we introduce
a pre-filter G which is stable, causal, and lowpass with passband compatible with the
bandwidth of x ( t ) . In t 6 i s way we get the ptefiltered error system in Figure 5.2 hom
v ( t ) to ~ ( k ) , namely, (UmSh -K)G. From now on, this WU be called the error system.
We remark that G shodd be thought of as fictitious: introduced ody for design,
mhose role is to capture the fiequency components of x ( t ) of interest; then the input
to G, v ( t ) , is no longer bandlimited. In this sense, G can be regarded as a fiequency-
selective weighting filter. In practice, one could &O think of G as the anti-aliasing
filter before A/D conversion (with period h) .
The performance criterion we w is the hybrid worst-case energy gain of the error
system, namely,
where the supremum is taken over d nonzero v in L2- Recall that for a fixed finite
energy signal v(t) , the energy gain of the error system is ( I E [($/ llv 11;. Hence J relates
to the worst-case energy gain: If J is smd, then the energy gain of the error systern
is unifonnly s m d for all possible inputs.
It is worth mentioning that without the loqass pre-filter G, J becomes infinite
(the error system is unbounded from L2 to 12) no matter what synthesis filters are
used. This is due to the presence of severe aliasing; see, e.g., [E] and [l ï? Section 9.31.
It can be verified that we get perfect reconstruction ( J = O) with no time delay
(m = O) if ideal flters are used. However, if stable and causal flters are to be used.
one often has to tolerate some time delay to get 3 smd; see the design example in
Section 5.3.
In this study, we take the viewpoint that the andysis filters have been designed
already, and the synthesis filters aze now to be designed to minimize J. The precise
design problem is as follows:
Given causal. stable, finite-dimensional analog fiters G: Go, Gi with G lowpass
80
and given the fast sampling period h and a tolerable the-delay integer m.
design causal: stable, digital synthesis filters Fo and Fi to minimize J.
The optimal synthesis fiIteis designed will be IIR in geneal. The optimal perf'ormance
achieved is Jqt(rn), which caa be shown to be a nonincreasing tunction of the time
deIay m.
The optimization problem involved can be written as follows:
11412 JWt(m) = inf J = inf sup -. Fo FI 625 4 0 1127112
This is a minimaz design problem, which cannot be solved directly by known met hods.
because of the hybrid and multirate nature of the error system. The main contribution
of this chapter is therefore the development of a procedure to convert this problem
into an equivalent %-optimal model-matching problem involving only discrete-the
LTI systems, which is then solvable.
ho ther possible application of the problem studied is to achieve fast A/D conver-
sion by slow A/D conversion in the twcxhannel filter bank in Figure 5.1. Properly
desiped filter banks can increase the conversion rate by a factor of two. An>- integer-
factor increase is possible if one uses multiple channels - a motivation to e--end the
results of this paper to the multiple-charnel case. For such an application, practi-
cal issues such as filter complexity and jitter effect need to be addressed; these are
not discussed in this study. Some practicd issues were studied in [53] for fast A/D
conversion using digital filter bank th-
Digit al filter banlrs have been studied extensively in signal processing; Vaidyanathanos
book ['i-] provides a detailed historical survey of the literature up to 1992. Optimal
design using l2 induced n o m was advocated and studied by Shenoy et al. [63' 641
in digital multirate filter design and by Chen and Francis [18] in digital filter bank
design for perfect reconstruction. Though possible applications of hybrid filter banks
81
were mentioned in [72], to our best knowledge, no work was reported on design of such
hybrid filter banks. The work here seems to be the first attempt in this direction.
Hybrid filter b d s may be more appealing than digital filter b d s in transmitting
analog signals. First. digital filter b a h require A/D conversion (prior to processing)
which is twice higher in speed than hybrid filter b a h . Second, practical speech and
video signals are mdog and hybrid filter banks treat them directly. Third, because no
analog signds are strictly bandlimïted in practice, aliasing caused by .4/D conversion
always exists; this aliasing is out of consideration in digitd filter banks but wiU be
under control in hybrid filter b d s since analog filters are incorporated in the design.
5.2 Conversion to a Problem of 3C, Optirnization
The design problem stated in the precedhg section is hybrîd and multirate. to
which no current methods are applicable directly. In this section we s h d reduce it
fUst to a discrete-the rnultirôte problem and then M e r to a discrete-time single-
rate problem of ?lm optimization, which can be solved readily by existing software.
The techniques used relate to both sarnpled-data control and polyphase decomposition
in signal processing.
5.2.1 Conversion ta a Discrete-The Problem
The first step is to convert the hybrid problem into an equivalent discrete-tirne
one. Let the error system frorn v to E in Figure 5.2 be T, namely,
This involves samplers Sh and SZh with diRetent periods. Let S2 (usudy represented
by the symboIJ 2) be the dismete-time dounisampler defined via
It can be i r d e d that is the cascade of Sh and S2:
Substituting this into (5.2). we have
Csing (5.1) and (5.3), it foUows that the cascade connection ESÎ (downsample. then
upsarnple) is &en by
that is, it zeros the input at odd times. Thus ES2 is a 2-periodic system (but S2E = 1,
the identity system). The overall system T is 2h-periodic in continuous time. Define
the continuous- and discretetime systems P, and Ld, respectivelqi as follows:
Thus P, is a l-input, %output system, Ld is Sinput, 1-output, and the error system
the cascade of three systems: the continuous-time system P,, the %input, 3-output
sampler, also denoted by Sh . and the discrete-time system Ld. In our case, P, is stable
and causal with a strict- proper transfer matrix. Thus ShPc is a bounded operator
from L* to e2 [El.
Now we wish to relate the performance meanue, J = llTll, to the nom of a purely
discrete-time system. For this we need two lemmas. The first lemma is from operator
theory [9]:
S 3
Lemma 5.1 Suppose X and Y a n iwo Hilbert spaces and T is a linear bounded
operator from X to y. Then
whete Tn is the adjoint operator mapping Y to X.
The second is from sampled-data control [17: Section 10.51 :
Lemma 5.2 For a stable, causal continuous-time system Pc unth strictly pn
fer rnatrtx
theie e=Çists a stable, causal discrete-time system Pd satisfying
and
where Ad = eu and Bd iS any mat* satisfying
h Bd B; = et" B ~'e'"' dt.
Ipplying the tmTo lemmas to the error system, we get
oper tran
The latter nom is &-induced and is on a purely discrete-time system Ld Pd. Thus we
have reduced the hybrid performance measure to a discrete-time measure.
Suppose P, has a realization
1
GL (s)
Since P,(s) is 3 x 1, the output mat& is partitioned into three rom. From Lemma 5.2,
the state-space model for Pd is given by
This Bd given by (5.4): is not unique: for example, we can take Bd to be a ~ y Choleshc
factor or even the square root of the right-hand mat&. .Also, the r a d of Bd is in
general greater than one, meaning that although P, is singleinput. Pd is in general
multi-input In fact, if (A, B) is controllable, the matrix integrai in (-5.4) is nonsingular
[Hl and hence Bd must be a square matrix.
Partition Pd as below
Then the equivalent error system becomes
which corresponds to the discrete-time system shown in Figure 5.3, and the equivalent
design problem is:
Given discrete-time systems Ho, Ki, IV, ail stable and causal: and given the
integer rn, design stable, causal, discrete-time Fo and FI to minimize 11 Tdll : the
&induced n o m of Td.
' ----. -.*{-q.. *-. . - .{Y[--- ----. {y- - - . {FI. .-- - . --.'
Figure 5.3. The equivalent discrete-the error system.
This is a discrete-time multirate design problem. It looks like the problem one
would get from design of muhirate filter banks for perfect reconstniction [la]; but
in fact it is different: First, the system to be matched is PN, not a pure time
delay; second, the three systems Ho, Hl, Ai are d multi-input. The second fact
leads to a limitation on achievable performance levels in design; see Section 5.4 for a
rigorous discussion. Next, we s h d reduce the multirate problem further to a single-
rate problem of R, optimization.
5.2.2 Conversion to an % Problem
From Figure 5.3, it follows that
At tthis point we need the polyphase matrices for the four LTI systems Hot Hl: Fo,
and FI [72]: For the analysis bank,
and for the spthesis b d ,
The synthesk polyphase mat& R is as usud 2 x 2. However, note that since Ho
and Hi have multiple inputs in general, i ) ( z2 ) is a generalization. -4ssume Ho(-.) and ( z ) are both 1 x n; then the two identity matrices in (5.7) are both n x n and ~ ( i ' )
is '5 x 2n.
With the two polyphase matrices, we hope to convert design of Fô and t to that
of R. Substituting equations (5.7) and (5.8) into (5.6) and using the noble identities
Here, O-: again, corresponds to the time delay z-' and Q and R are the LTI systems
with tiaasfer mat& &(z ) and ~ ( r ) respectively.
To proceed further, we need the following lemma which can be proven readily [BI.
Lemma 5.3 The operator and its right inverse
are both nom-preseming on &, where V-l, the inverse of U* on &: is the unit time
advance.
Thus the nom of Td remains unchanged if we pre- and post-multiply Td by
respectivek This leads to
This system W c m be shown to be LTI. In this way we arrive at the final equivalent
s-stem, shown in Figure 5.4 back in the fiequency domain. The sizes of the three
matrices &(z). ~ ( r ) , and ~ ( z ) are 2 x 273, 2 x 2, and 2 x 272, respectivel. (The
integer n is the dimension of the input in Figure 5.3.) Hence, the P2-induced nom of
Figure 3.4. Final equivalent system.
W - RQ equals the 'fl, n o m of its transfer matrix, 11 w - RQ 11,. Uë condude that the original design problem, minimizing the Cî-to-Q-induced
n o m of the hybrid, multirate system v E in Figure 5.2, is equivalent to the following
Ha optimization problem:
Given stable and causal transfer matrices ~ ( z ) and ~ ( t ) , design stable and
causal R ( t ) to minimize 11 w - 8~11,-
Equidently, we can write:
The latter optimization is a standard ?&,-optimal model-matching problem.
Uë s h d condude this subsection by denving explicit expressions for Q(z) and
~ ( z ) . The key lemma is the following:
Lemma 5.4 Let F 6e a causal, LTI discrete-tirne system m3h a transfer matriz
and Po and its polypbase components:
Then Fo(z) and &(z) are the tramfer matrices of the causal, LTI systems Sz F E and
S2L7-' F E respectively and
Proof The state-space formulas can be easily obtained by noting the power series
expansion of &). To see that S2FE has a transfer matrk Fo(r), we need to use
noble identities and Lemma 5.3. Wnte
S, F (z) E = s2 (F, (zZ) + z-' Pl (z2)) E
= S~E&(,(Z) +s2z-'~Fi(z)
- î, - A O*
In the second equality, we have applied noble identities, and in the last one, we have
applied Lemma 5.3. The daim then follows. Similady, we can prove that S&-'F E
has t rader matrix &). O
R e d from (5.5) the state-space realization of Ho and Hi are given by
Based on this, it is immediate from Lemma 5.4 that &) takes the fomi
Dependhg on whether the the-delay integer rn is even or odd, the mat& W ( Z )
takes on different forms. From (5.5) the state-space realization of N:
.(il = [Y] C O *
Let us first assume rn is even, w, m = 2d for some d 3 O. Then fiom (5.10)
hoking the identity = VdS2 [72], the (1, 1) and (2,2) blocks equd to Lrd(S&-1 .\-E):
this has a transfer matrix
by Lemma 5 . 4 Similady, the (2 , l ) block L 1
is Cld(LINE), which has a transfer matria
and the (1,2) block has a t r d e r matrix
Based on the four transfer matrices, we have
If m is odd, n = 2d + 1 (d 2 O ) ? the expression for &(z) can be derived in the
5.2.3 A Design Procedure
Much of the computation involved is based on statespace data. At this point, let
us summarize the steps into a design procedure:
Input: fast period h, deIay integer m, analysis Hters Go(s) and G&). and
pre-filter G ( s ) ;
Output: optimal performance J&(rn) and synthesis f i tes Fo(z) and Fi(;).
Step 1 Compute a state-space realization for the 3 x 1 matrix:
Step 2 Compute the matrices 4 = eu and Bd satis%ng
h &Bi = e t A ~ ~ Y ' ' d t ,
via. e-g., Choleslcy factorkat ion. The matrix integral involved can be computed
using matrix exponential htnctions [73].
Step 3 Compute the tR.o matrices Q(Z) and ~ ( 3 ) :
Step 4 Solve the following %-optimal model-matching problem
The optimal ~ ( r ) is the 2 x 2 polyphase matnx for the spthesis flters.
Step 5 Obtain & and via
It should be noted that here we do not need to haadle the causality constra.int due
to the particular structure of the filter bank. Thus we can carry out the standard
?& optimization procedure in Step 4 and this can be solved by the existing MAT-
LAB function hinfsyn, for which one needs to convert the discrete-time &, problem
equivaZent1y to a continuous-time one using bilinear transformation.
5.3 Example
In this section we study in detail an example to bring forward some points that we
feel important, e-g., performmce limitation, tradeoff between the time-delay integer
(m) and reconstruction performance, and model reduction by FIR tnuication to the
optimal IIR sqnthesis filters. It is emphasized that the example is chosen to illustrate
the new design procedure in the preceding section; the fdters are not necessarily meant
for some realistic application.
For simplicity, we assume that the hybrid filter bank in Figure 5.1 is used to
process analog signals r (t ) with fiequency spectrum mostly limited to Iw 1 5 1. Hence the corresponding Xyquist sampling period is a seconds; this is taken to be the fast
sampling period h. The design method proposed in this paper allows us to design the
andog ana&& filters &st based on considerations in coding and transmission and
then design the digital synthesis flters for best reconstmction.
In this example, the andysis filter Go should be lowpass with passband Iwl 5 0.5.
t V e t h Go to be a third-order elliptic filter with trader function
The magnitude Bode plot of Go is showm in Figure 5.5 (dotted cuve). The other
Figure 3.5. IGI (solid), jGol (dot), and lGl 1 (dash) in dB versus W.
analysis filter Gi should be bandpass with passband 0.5 5 1 ~ 1 5 1. This is designed
via the Y.&TLAB function Ip26p ushg the prototype (lowpass) third-order elliptic
So we get a si-xth-order bandpass Gl with the transfer fimction
whose magnitude Bode plot is given again in Figure 5.5 (dashed line) .
The role of the fictitious pre-ater G is to retain the desired fiequency components
of input signals and so G is lowpass with passband Iwl 4 1. This is realized by
cascading a second-order Chebyshev type1 filter and a fourth-order elliptic filter:
The Bode plot of G is also shown in Figure 5.5 (solid curve).
With these analog filters, we can proceed to design optimal synthesis filters given
the time-delay integer m. The procedure in the preceding section is implemented in
93
MATLAB. To carry out Step 4? we tmnsform the discrete-time 3L, problem into a
continuous-time 3L, problem via bilinear transformation and then apply the function
hinfsyn. The optimal performance Jm is in general a nonincreasing function of m. In
this Jqt versus rn is given in Figure 5.6, where Jwt drops relativel~ apidy
Figure 3.6. Jet versus m.
for 2 $ m 5 8 but stays at about the same level for m 2 10. This illustrates the per-
formance limitation: B y increasing m, one cannot get arbitrarily good reconstruction.
More discussion on this will be given in the next section.
The optimal synthesis Hters computed are in general IIR with orders that increase
linearly with the integer m and with the orders of G: Go, and Gi. In this example, for
s m d JWt we t&e rn = 15; the correspondhg Jopt is 0.0633 and the optimal synthesis
filtea Fo and FI are computed to be IIR of order 89. The magnitude Bode plots of Fo
and FI are given in Figure 5.7, fkom which we see t hat Fo is lowpass and FI highpass.
To test the two optimal s ~ t h e s i s filters and also to get some sense of what the
performance ( Jmt = 0.0633) means. we apply two analog signals at the input x( t ) of
Figure 5.7. 1 F* 1 (solid) and 1 fil 1 (dash) in dB versus w/a.
Figure -5.1 for the desi@ hybrid bank, simulate the output $(le), and compare with
the desired output ~ ( k - 15) = x[(k - 15)hI. (Recall that the pre-Mter is introduced
O* for design; for testing one uses Figure 5.1.) The first input is bandlimited.
consisting of high- and low-fkequency components (chosen arbitrarily):
The desired output 4(k - 13) and the steady-state reconstruction error [e(k - 15) -
&(k)] are shown in Figure 5.8. It is seen that the maximum error (0.035) in mae~tude
is quite s m d relative to the size of the desired output. The second test input is a
continuous-the unit step, x ( t ) = l(t). The dynamic response of the hybrid f3ter
bank is plotted in Figure 5.9. Although this input is not bandlimited to Iwl 5 the
output of the filter bank tracks the desired output, a discrete-time unit step delayed
by 15 samples, fairly wek The steady-state error is 0.0110. Note that the system is
not particul& designed for step-tracking.
To design high-quality hybrid filter banks, one would tend to use more sophisti-
-21 1 1 I I
O 20 40 60 80 1 O0
Figure -5-8. Steady-state error (solid) and desired output (dash) versus k.
cated analog filters and, perhaps, tolerate relat ively large time delay S. These neces-
sarily result in high-order synthesis filters. In the example, if rn = 15, the method
generates optimal Fo and FI which are IIR and of order 89. However? the proposed
method is still useful in that starting from here one could approximate the optimal Fo
and Fi by FIR ones via tmcation or by lower-order IIR ones using mode1 reduction.
Yow let us truncate the designed optimal Fo and Fi to get FZR approximations. The
impulse responses of Fo and Fi are IIR and are shown in Figure 5.10 for the initial
40 samples. W e see that both responses settle to zero rough-y in 30 samples. Thus
we tnincate the impulse responses to get FIR filters of length 30. The truncated
synthesis filters PoFrR and f iFIR achieve a performance JrIR = 0.0633, accurate to
three significant digits (compare with JW = 0.0633). The Bode plots for Fo - FoFrR
and FI - FiFIR are given in Figure 5.11. The maximum errors occur at fiequencies
around the transition bands of Fo and FI:
12 i 1 ?
-020 I I I
IO 20 30 40 so 60 70 WI 90 rm
Figure 5.9. Cinit step response @ venus k.
Hence the 31, approximation errors are
Finally, we mention that the order issue of the synthesis filters is attacked by
model-reduct ion techniques in [Ml.
5.4 Performance Limitation
In the preceding example, we saw that as m increases, JO@ converges to some
nonzero value, indicating a performance limitation. In contrast, in the 7&, design
of digital multirate filter banks in [BI, it is proven that under some mild regularity
condition, one can always get arbitrarily close to perfect reconstniction (Jqt = O) by
tolerating large enough time delays. In this section, we will give an exphnation using
R, model-matching theory.
The problem at hand is as follows:
Figue 5.10. Impulse responses versus k for Fo (upper) and FI (lower) .
Here, R is 2 x 2 and w and Q are both 2 x 2n, n being the number of inputs in
Figure -5.3. The nonsquareness of this Q implies that the associated 36, problem is
a tweblodr one; whereas in [la], the corresponding matrix is squaze, leading to an
one-blodr problem only.
For sorne rigomus andysis, assume that ~ ( r ) always has full row rank on the unit
circle. Then Q has a CO-inner-outer factorization [;?7]
where the outer Q,, the inverse 0;' and Q~ are ail in Go and Qn, being dl-pass,
satides Q ~ ( ~ ) Q ~ ( z ) = I . It foUows then that the matnx fimction
= @ 11 [ &QX - RQ- W(I - Q ~ Q ~ ~ ) ] 11- (5.12) R
This is a *O-block ?&, problem: The fist block in the matrix in (3.12) now involves
a square Q,: the second block is independent of R. From here? we conclude that J,,
R e c d that ~ ( r ) can be expressed as (m = 2d or rn = 2d + 1) multiplying
some transfer matrix which is independent of d. This delay factor does not affect the
infinity nom in (5.13). This means that increasing the time delay m cannot improve
the lower bound on Jqt and so performance limitation exists.
A condition on perfect reconstmction (JOpt = 0) can be obtained too. For perfect
reconstruction, it is necessary that
In thîs case,
99
which is a one-block problem. Sim.ïlarl,t* as in [18] me can show that this JW converges
to zero as rn tends to idhity. In other words, under the condition in (3.14), aibitrarily
perfect reconstruction is possible if one is w i b g to tolerate d c i e n t l y Iarge time
delay
The above analysis also shows that the lower bound on Jm in (5.13) is asymptot-
i c d y tight: Jm converges to this Iower bound if the time delay rn tends to infini%
In the example in the precedïng section, the limiting value JWt = 0.0633 corresponds
to this lower bound, which can be pre-computed without going through the 'fl, op-
timization, if a factor Q, cm be computed for Q. Some interesthg questions &se:
How to select andysis filters to make the lower bound smd? Wlat is the tradeoff in
terms of filter complexity?
In the case of perfect reconstruction, we have
In view of the factorkation in (5.11), postmultiply by to get
Since QX is anticausal, & and accordingly the synthesis filters Fi and F2 are non-
causal in general (see Step 5 in Subsection 5.2.3). In the next chapteq we s h d give
çome verifiable conditions for RQ to be causal.
5.5 Conclusions
In this chapter, we proposed an 31,-based procedure for minimax design of hybrid
filter banlis involving analog and digital Nters. The work forms an extension of
31, optimization to multirate systems in signal processing. The filters designed are
robust in the sense that uniformly good performance is achieved for the class of all
finite-energy signals. -4 design example was included for illustration.
100
Our study treats only the twcdannel case. Extension of the r d t s to muitipk-
channel case is possible dong the hes in Section 5.2. However, the complexity of the
optimization problem grows with the number of channels and may become an issue
at some point-
In signai processing systems, FIR mers me ofien used to achieve a cost admage.
In the design example, the optimal IIR synthesis flters were tmcated to get FIR
approximations. -4 better approach for design of FIR synthesis filters of a given length
is by optimizing the filter coefficients direct1y. This leads to finitdknensiond convex
optimization which is studied for digital filter banks in [58]. We remark here that the
method used in [58] cm be applied to design of hybrid filter banks with FIR synthesis
a t e n using the fkmework proposed in this chapter.
CHAPTER 6
ON CAUSALITY AND ANTICAUSALITY OF CASCADED LINEAR DISCRETETIME SYSTEMS
This brief chapter treats a fundamental question arising fiom both filter bank
design and UÎ/'fl, control: When two FDLTI systems, causd or noncausal, are
cascaded, what can be said about the causality properties of the overd system?
W e gïve conditions under which the cascaded system is causal or anticausd; these
conditions are related to Sylvester equations obtained fiom state-space data of the
sys t ems. For brevity, ody discrete-time syst ems are considered' .
6.1 Motivation
Consider cascading two FDLTI systems, Gl and G2, together as in Figure 6.1.
where both Gi and G2 can be MBlO. It is weIl-known that G2G1 is causal if both G1
Figure 6.1. The cascade of two systems
and G2 are causd. U l a t about if one or both Gi and G2 is not causai? For example,
suppose Gi and G2 are both single-input, single-output in d i ~ r e t e time with transfer
fimctions
Gl(z) = r-l: &(z) = ri, k,l > O;
it follows that G1 is causal, Gz is not, but GzGl may be either causal or noncausal.
depending on the two integers k and 1. The goal in this chapter is to answer such
lThe results in this chapter have been published in [66].
10'2
questions in g e n d cases, in partidar, we give state-space conditions for testing
causality properties of G2Gi in the MIMO setup. To our best knowledge, such results
were not available in the literature,
This question can be motivated fiom both filter bank design and optimal control
t h e o i In the preceding chapter, the hybrid filter bank design was translated to
mode1 matchhg problem of a discrete-time system and the perfect reconstmction
condition is that a cascade connection equals a causal system
We have shonm that in general the synthesis polyphase rnatrix R (and so synthesiç
filters) must be non-causal to satisfjr this condition. Similar results arise in digitai
filter bank design, where, in some cases, a cascade connection are required to equd an
aaticausal transfer rnatriv [72,66]. The results in this chapter will give some verifiable
conditions for the cascaded system RQ to be causal or anticausal. In (361 a design
scheme was proposed using causal analysis filters and anticausal synthesis filters with
irnplementation issues discussed in [36, 461.
In control, though all practical controilers are causal for implementabiliw non-
causal systems appear in deriving various optimal controllers- For example. in deriva-
tion of Ra optimal controllers, we have cascaded an anticausd system with a causal
system to get an anticausa overall system (see Lemma 2.4). This is a special case of
Proposition 6.1 in this Chapter. -4nother case where noncausd control systems arise
is in discrete-time 31, controi [69] in which it is shown that it is possible to attain
optirnaiity via non-causal controllers.
In order to treat noncausd systems in discrete time, we need to consider signals
and systems dehed on the (time) set of aJ integers, both negative and positive. With
this in mind. a discretettirne LTI, MIMO system G is completely characterized by its
impulse response: a rnatrix sequence,
This is in general a two-sided sequence. To make things precise, we define the fouowing
terms: G is causal if g(k) = O whenever k 5 -1; G is strictly causal if g ( k ) = O
whenever k 0; G is anticBusal ifg(k) = O whenever k 2 1; G is strictiy anticausa1 if
g(k) = O for k > 0; and G is noncausal i f g ( k ) # O for some k -1. It is clear from
the definitions that a noncausal systern G can be decomposed into G = G, + Ga with
G, causal (stnctly causai) and Ga strictly anticausal (anticausd).
m e n tw-O discrete-the systems are cascaded as in Figure 1, is the resultant system
causal or anticausal? Wk s h d Grst look at some specid cases involving a causal
system and an anticausd system and then tadde the generd case. Throughout the
chapter we çhall assume systems in discussion are finite-dimensional.
6.2 Special Cases
In this section, we assume Gl is causal and G2 is anticausal. What c m be said
about causality properties of GzGl and GiGz? The analysis to follow is based on
state-space data.
Since Gl is LTI, finitedimensional and causal, it has a state-space realization (with
input ull output ~ 6 ~ : and state 6):
The transfer matrix is written
(with the subscript c denoting a causal system), or as a power series in z-l :
104
Since Gz is anticausal, there exist matrices BZ, c2 and i)2 such that G2. mith
input v2 and output is describeci by the following kckwutd statespace equations:
The corresponding t rader matrk is
where subscript a means anticausality. In terms of power series: we have
The causality properties of G2Gl will relate to the Sylvester equation
ahich is more general than the Lyapunov equation we saw in (2.9). Assume that
both Al and .a2 are stable, i.e., the eigendues of Al and L& are in D. (6.3) admits
a solution of infinite series:
If t his equation has a unique solution, or equitdentiy [30], no eigenvalue of At is
L ow we c m reciprocal of some eigenvalue of A*: then this solution equals X in (6.4). Y'
state our fist result-
Proposition 6.1 Assume X in (6.4) ezists.
(a) G2G1 is causal if B2L4 + Â 2 ~ ~ ~ 1 = O and is strictly causal if, in addition, B2& + = O. Moreooer, the converses hold i f the pair (&: Â2) iS o bsentable.
(6) 4 G 1 is anticausal if D ~ C ~ + C?XA~ = O and is strictly anticausal if, in addition.
D ~ D ~ + & X B ~ = O. Moreover, the converses hold if(&, B I ) is controllable.
Proof The trader rnatriv for G2Gi is
The coefficient matrices g(k) can be obtained by multiplying (formally) the two series
in (6.1) and (6.2) together and using (6.4) to simpiify the expressions; then we geto
Thus the d c i e n c y proof follows easily from respective definitions. For necessi@
note that causality of G2G1 implies g(k) = O for k E. -1: which implies in tuni
where nt is the dimension of the square matrix Â2. Now observability of (&A2) implies that the observability matrk, appeared on the right-hand side of (6.5): is
injective and hence we get B ~ D ~ + $x& = O. The rest of the necessin- proof
-lote from the proof that if B& + Â2xl?i = 0, the causal system G2Gi has a
st ate-space reahzation
This means that all the anticausal poles introduced by G2 are cancelled by zeros of
Gt for the resuitant system to be causal. Similar interpretations can be given for the
ot her cases.
The dual of Proposition 6.1, which concerns causality properties of Gi G2 (G1 and
Gz as in Proposition 6.l), dl involve a dinerent rnatrix
with the assotiated Sylvester equation
The following r d t s can be proven andogously.
Proposition 6.2 Assume Y in (6.6) &ts.
(a) G1G2 is causai if D ~ C ~ + C~Y-& = O and is strictly causai if, in addition, D ~ & + C~Y& = O . Moreover, the converses hold if the pair (&: $) is controllabk.
(b) G1 Gz is anticattsal if B~ D~ + A ~ Y B~ = O and is strictly anticausa1 if, in addition,
~~b~ + C ~ Y & = O. ~ M o ~ ~ o v ~ T , the converses hold if (Cl: .4i) is obsentable.
6.3 General Case
Consider G2Gi, where Gi and G2 are LTI, finite-dimensional, but general non-
causal systems. Decompose them into causal and strictly anticausa parts:
Bring in state-space matrices:
&te that the direct feedthrough terms in G&) are zero. Then G2G1 is the sum of
four cascaded systems
W e wish to h d the impulse response of each of the four systems.
First? given the state-space realizations of the causal Gi, and G2,, we get a real-
ization of G2,G1, via a formula in: e.g., [27],
107
To simplify this, introduce the Sylvester equation
and assume it has a solution O-. (It has a unique solution in A2 and Al have no
common eigenvalue [30] .) Performing a similarity transformation, with
Hence the impulse response g&) of GacGiC is given by: g&) = O for k 5 -1:
g c m = 9D1: and
Similady, introducing the Sylvester equation
and assuming this equation has a solution V : we can obtain the impulse response of
G2.G1.: g,,(k) = O for k 2 O and
g , , (k)=-~2 .4;k-1~Êl+&~A;kœ1~17 k s -1.
The other t~vo cascaded systems in (6.7), &Glc and G2,GI,, have been studied
in Propositions 6.1 and 6.2; so we define
Again, if such X and Y exist, they must satidy the associated Sylvester equations:
108
Then as in the derivations of Propositions 6.1 and 6.2, the impulse responses of GZa Glc
and G2,Gia are respectively (note that the d w c t feedthrough terms of bl, and
Xow putting things together, fiom (6.7) we get the impulse response g for the
overd system G2G1:
Based on this. the next results follow easily.
Proposition 6.3 Assume U and V are solutions of equations (6.8) and (6.9) respec-
tiüely; and X and Y defined in (6.10) ezist.
(a) G2G1 iS causal if
and is stn'ctly causal if, in addition,
(b) G2G1 is anticausal if
and is strictly anticausal if, in addition, (6.13) holds.
109
Partial necessity conditions can be given too. For example, if (6.11) holds. then
causai^. of G2Gi and controllability of (A1, BI) imply (6.12); if (6.12) holds, then
causality of G2G1 and obseftability of (&A) imply (6.11). Similar staternent can
be made about anticausa3ity and conditions (6.14) and (6.15)-
Finally, we mention that the resdts in this chapter can dso be extended to
continuous-time systems; for details, see [66].
CHAPTER 7
CONCLUDING REMARKS
This thesis has been devoted to both theory and applications of multirate systems.
Xenr dgorithms have been developed for multirate optimal control with X2 and 7&
criteria. C o m p d to the previous solutions [45, 74, 191, the resdts obtained have
the advantage of computational efficiency and ease of implementation on cornputers.
A software package based on the multirate 'H, dgorithms has ben developed and
applied to a power system stabilizer design. The multirate 3tz atgonthms were applied
to case studies in -6,571. .Uso. new MATLAB softwxe for multirate controllers based
on the work in Qiu and Chen [54: 19, 581 and the algorïthms in this thesis is under
development [56]. In addition, we have provided a study for hq-brid multirate filter
banks via Xm optimization.
It is no doubt that multirate systems have many advantages .A lot of research
efforts are required in this area. In what follows, we s h d suggest some future work
relative to the topics in the thesis.
Multirate Optimal Control
In recent study of multirate optimal control, the focus was on LQR, LQG, 3C2, R,
and el criteria (2; 43,20; 74,54,55,19: 651, all with a single optimal criterion. Further
research should tadde more sophisticated specs such as mixed H2/7frn, mixed l1 /'Mm :
ceneralized ;H2 , lfCoc with time-domain constraints, and multiple objective ?&/?Lw. Of b
course. the studies of those subjects depend on the corresponding results for single-
rate problems. For example, for the single-rate mixed ?&/Ra problem? there are
many forms of expressions so far (see, e.g., [IO, 61, 23, 70, 621): and the solutions
to the general problern remain, to a large extent. u n h o m . In spite of this. i t is
111
still worthwhile to look at the multirate setting in terms of the available single-rate
solutions. Chen and Qiu's work (19, 551 provides an effective h e w o r k based on
nest matrices for the problems with the mixed/multiple specs. Future study shouid
focus on how to use the tool to handle causality constaints. Wë &O mention that
the solution to the mdtirate minimum entropy probIem, a specid case of the mixed
X2 /lz/'flo problem [49], has been given in [55].
Another interest ing topic on multirate control is finitehorizon optimization prob-
lems wïth LQG or 36, type of performance meastues [35]- Controllers for finite-
horizon measures are dways timevarying. Bence, it may be possible to look at
time-va.ryZng systems directly. For the multirate LQG problem, we believe that the
idea to do the lifting in [54] may be borrowed to reduce the multirate problem into
a single-rate LQG problem subject to a causality constraint. However, it should be
recognized that finitehorizon problems require solving diffeerential Riccati equations
and the techniques involved to handle causality constrkints will be essentially dinerent
from the infinite-horizon cases.
Power System Stabilizers
Today's power -stems feature mdti-machines, multiple areas and htercomected
complex networks. Design of power system stabilizers in such a saphisticated environ-
ment becomes extremely dïfEcuIt in the sense of controller coordination and parameter
tuning. Hence, it is desirable to use simple weighting hct ions to reduce the amount
of trial and error involved. However, weighting functions must be chosen to reflect
the basic operational requirements of the system, as mentioned in Section 4.2.2. The
method for choosing weighting functions presented in Chapter 4 lays the groundwork
for syn t hesising mult i-machine systems. In addit ion, since mult i-machine systems
involve more sipals with diflerent bandwidths, we beüeve that multirate sampling
schemes provide more freedom of design. But the coordination of controllers may
become more difncult than singIemachine systems.
Hybrid Filter Banks
Compared to digital fdter banks, hybrid filter b d s , though more attractive con-
cept~*, are more diaicult to design due to non-ideal analog filters. At ppresent,
selection of the ânafog filters is stiU based on trial and error. W e feel that a sys-
tematic procedure is needed to help choosi~g the analog filters. For this, one would
need to look into the connedion between the state-space representations of the analog
filt ers and the performknce limit derived in Section 5.4.
S tudy of hybrid filter banks is relatively new in signal processing. More research
efforts are required to analyze hybrid filter banks in the frrquency domain. Some
related questions are: How to define for hybrid filter banks the f d a r concepts in
digitd filter banks such as phase distortion and magnitude distortion? How to relate
these distortions to the 3t, performance masure proposed in Chapter 5?
[1] S. S. .r\hmed, L. Chen and A. Petroimu, "Design of suboptimal X- excitation
controllers", IEEE Trans. on Power Systems, vol. 11, no. 1, 312-328, 1996.
[-21 H. M. -41-Rahmani and G. F- Franklin, "A new optimal mdtirate control of
h e a r periodic and tirneinvariant systems", IEEE Trans. ilutomat. Control, vol.
35,406415, 1990.
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APPENDIX A
P O m R SYSTEM MODEL AND PARAMETERS
First defme the fouowing signak involved in modeling of the singlemachine infinite-
bus pomr system: 6: power angle
machine rotor slip speed mechanical torque electricd torque quadrature-Oms transient voltage generator field voltage armature m e n t , direct component armature m e n t : quadrature component armature voltage, direct component mature voltage: quadrature component bus voltage (us = 1 p-u. for idhite bus) reference (disturbance) voltage terminal voltage control voltage from the stabilizer (if any).
The Park's two-axis representation [24,4] is used in the thesis for the generator.
and for the transmission lines,
1 22
These are obtained by neggecting tansients in the stator cirait and the effect of
the rotor amortisseur. The AVR and exciter are modeled by a simpMed first-order
system:
Taif = -ef + K,(Ker - ut tu).
The parameters involved in the above equations axe given in Table A.1, which are
Table Al. Parameters for the power system model.
based on a Iaboratory setup at the University of Calgayy [13]. In this table, aIl the
time constants are given in seconds, ;J. is in radfsec, and the other quantities are in
p-u-
